2026-08-31-normative-affordability

Research pass: pin down the schematic Normative Affordability theorem

Work against the live alignment workspace/repo and treat this as a mathematical research pass, not a prose-cleanup pass.

The goal is to pin down the exact schematic theorem/interface for Normative Affordability and its composition with Answerability, Settlement, Actionability, and Uptake. We now have a much more specific candidate architecture, but it should be aggressively simplified, stress-tested, and either vindicated or replaced.

Do not assume the proposed formulation is correct. In particular, try to find counterexamples to every proposed composition rule before preserving it.

Orientation

The emerging architecture is roughly:

$$ \text{interactive factored environment} \to \text{settlement history} \to \text{answerability / standing} \to \text{normative control} \to \text{decision engine} \to \text{action} \to \text{interaction} \to \text{settlement}. $$

Settlement plays two roles:

1. incoming information: interaction generates authenticated settlement-visible record;
2. adjudication of prior control: normative interventions may create settlement-contingent exposures whose eventual value is determined by later settlement.

The interaction ontology should remain modular. At a history \(h\), something like a Cartesian/factored continuation model may supply

$$ (A_h,E_h,\star_h),\qquad \star_h:A_h\times E_h\to W_h, $$

where \(A_h\) contains agent-side continuation policies, \(E_h\) complementary exterior resolution, and \(W_h\) possible continuations. The richer factored model is probably semantic infrastructure rather than something every downstream theorem needs in its signature.

The Settlement Interface from earlier workspace work is highly relevant. Recover it and related interaction/action-theory material. In particular inspect:

• SETTLEMENT_INTERFACE.md
• INTERFACE_THEOREMS.md
• interaction / coverage / Cartesian-frame briefs
• the generic interactive-service/action-theory work
• the current Answerability / Progress / Actionability material
• traderization / liability / homothetic-core material
• any existing composition, staging, underwriting, and cross-era results.

Do not assume the old architecture or terminology survives intact.

---

Current candidate theorem factorization

The provisional picture is:

Settlement

Settlement supplies a filtered learner-visible history

$$ \mathcal F_0\subseteq\mathcal F_1\subseteq\cdots $$

together with enough semantics to determine:

• which continuation/assessment possibilities remain live;
• how historical control exposures are evaluated as settlement arrives.

A normative controller must be adapted/predictable: round-\(t\) control is chosen from \(\mathcal F_{t-1}\), not after seeing round-\(t\) settlement.

The schematic theory should not assume direct access to hidden “truth.” In LI, the relevant assessment set is something like \(PC(D_N)\); in empirical systems it may be generated by settled report variables.

Answerability

Answerability determines what remains historically owed.

The rich primitive may be a service/trace specification, not a scalar quota.

For the quantitative learning interface, however, suppose a persistent reason \(r\) induces a predictable nonnegative claim/exposure process

$$ c_t^r\ge 0 $$

with

$$ C_N^r:=\sum_{t<N} c_t^r\to\infty $$

when \(r\) persistently retains a claim to service.

Normalize this into a normatively owed exposure measure

$$ \mu_N^r(t) := \frac{c_t^r}{C_N^r}. $$

Interpretation: where across the first \(N\) rounds did \(r\) genuinely have claims on the reasoner?

The \(c_t^r\) representation may be too restrictive. Determine the minimal object actually needed.

Control versus action

Keep distinct:

$$ U_h=\text{normative control inputs} $$

and

$$ A_h=\text{concrete agent-side continuation policies}. $$

The controlled decision engine converts normative pressure into behavior:

$$ D_h:U_h\to\Delta(A_h) $$

or some more general analogue.

In LI, \(U_h\) contains added trading pressure; in an online learner it might contain comparator/loss pressure. A norm does not in general directly choose the physical action.

Actionability

For each reason, we need a semantic connection from normative control to recognizable improvement.

The old one-reason form was:

$$ g_t^r\ge\gamma_r d_t^r, $$

where \(d_t^r\ge0\) is recognized defect/nonresponse and \(g_t^r\) is the reason-relative advantage presented to the learner.

But do not assume individual Actionability composes.

If controls for reasons \(r_1,r_2\) interfere, each reason may be actionable alone while their aggregate pressure destroys both advantages.

Therefore investigate a joint response correspondence, provisionally something like

$$ \mathcal J_t(h) \subseteq \mathbb R_+^{R_t} \times U_t \times Q_t^{R_t}. $$

An element

$$ (w_t,u_t,q_t)\in\mathcal J_t(h) $$

would certify that:

• \(w_t^r\) is the amount of actual service assigned to each reason;
• \(u_t\) is the one aggregate normative control actually applied;
• \(q_t^r\) is whatever reason-indexed comparison/challenge Uptake scores;
• this joint intervention really preserves the claimed Actionability inequality for the serviced reasons.

Find the weakest sensible type. Perhaps \(\mathcal J_t\) should not have this form at all.

Construct explicit counterexamples showing why individual Actionability does not imply joint Actionability.

Then identify sufficient composition assumptions such as:

• linearity;
• convexity;
• positive homogeneity;
• separability;
• noninterference;
• superposition.

Determine exactly what traderized LI gets “for free” here because traders/payoffs aggregate linearly, and which parts do not generalize.

---

The Service Transfer problem

This is one of the central mathematical questions.

Suppose the affordable controller actually services reason \(r\) with intensity

$$ w_t^r\ge0, \qquad W_N^r=\sum_{t<N}w_t^r, $$

and define the normalized actual service measure

$$ \nu_N^r(t) = \frac{w_t^r}{W_N^r}. $$

Actionability + Uptake may first give

$$ \mathbb E_{\nu_N^r}[d^r]\to0. $$

But what we need normatively is

$$ \mathbb E_{\mu_N^r}[d^r]\to0. $$

The scheduler must not be able to service \(r\) only at the easy/low-defect dates.

A candidate minimal condition is contiguity

$$ \mu_N^r\triangleleft\nu_N^r: \qquad \nu_N^r(A_N)\to0 \Longrightarrow \mu_N^r(A_N)\to0 $$

for every sequence \(A_N\subseteq\{0,\ldots,N-1\}\).

Investigate this carefully.

Target Service Transfer theorem

For bounded defects \(0\le d_t\le D\), is the following correct?

$$ \mathbb E_{\nu_N}[d]\to0 \quad+\quad \mu_N\triangleleft\nu_N \quad\Longrightarrow\quad \mathbb E_{\mu_N}[d]\to0. $$

The obvious proof uses the level sets

$$ A_N^\varepsilon=\{t:d_t>\varepsilon\}. $$

Prove it cleanly.

More importantly, investigate a converse:

Is contiguity exactly the weakest condition on sequences of probability measures \((\mu_N,\nu_N)\) such that for every bounded nonnegative defect array,

$$ \mathbb E_{\nu_N}[d]\to0 \Rightarrow \mathbb E_{\mu_N}[d]\to0? $$

Be careful about triangular-array versus one fixed infinite defect sequence. The naive indicator counterexample may require choosing \(d^{(N)}\) depending on \(N\). Determine the exact statement and the exact necessary-and-sufficient condition in the version relevant to our theorem.

Compare contiguity with stronger sufficient conditions:

$$ \eta c_t\le w_t\le Kc_t; $$

uniform absolute continuity;

bounded Radon–Nikodym densities;

prefix discrepancy/backlog bounds;

queue stability;

bounded-delay service;

fair scheduling.

Determine which formulation is most natural as the schematic Answerability-to-Progress interface.

---

Deferral and transport

Same-round servicing is likely too restrictive.

A claim incurred at time \(t\) might legitimately be serviced at time \(s>t\).

Investigate an explicit transport/matching object, e.g.

$$ T_r(t,s)\ge0, $$

sending owed-service mass at claim time \(t\) to actual service at later time \(s\).

Then ask what additional condition is needed to infer improvement of the original defect at claim time from learning at service time.

Likely we need a stability/transport inequality such as

$$ d_{t,r} \le L\, d_{s,r}^{\mathrm{transported}} + \epsilon_{t,s}. $$

Try to identify the minimal theorem.

This may connect directly to cross-era Actionability: transporting a reason through delay, semantic change, evaluator succession, or ontology revision may be one common mathematical problem.

Do not force that unification if it is false; investigate it.

---

Settlement-relative robustness / SafeCert

Affordability should not itself hard-code trader liability.

A controlled learner should expose some replaceable robustness contract:

$$ \operatorname{SafeCert}_{D,\Sigma}(\kappa,C) \Longrightarrow \operatorname{Uptake}(D^\kappa). $$

Here \(\kappa\) is the aggregate causal normative controller and \(C\) an engine-specific certificate.

For LI:

$$ \operatorname{SafeCert} $$

is instantiated by bounded cumulative downside of the added enforcement trader over every still-plausible world, and the preservation theorem is:

$$ \text{bounded enforcement liability} \Rightarrow \text{original LIC preserved}. $$

Map this exactly to the traderization results.

But investigate the minimal schematic type.

Possibilities include:

• an abstract prefix-closed safe class of control histories;
• a cumulative potential/account;
• a dynamic robust risk functional;
• an admissible perturbation ideal;
• some ordered monoid of signed burdens rather than \(\mathbb R\).

Avoid unnecessary finance vocabulary if a weaker abstraction suffices.

Important robustness question

Should safety be actual-path or robust over all settlement-consistent continuations?

LI strongly suggests the latter:

$$ W(E_{\le N})\ge-B $$

for every \(W\in PC(D_N)\).

Formulate the analogous history-relative notion:

$$ \operatorname{Risk}_{h_N}(\kappa_{\le N}) $$

with assessment over all continuations still compatible with settlement at \(h_N\).

Investigate what consistency properties this family of risk/safe sets needs as settlement refines the continuation set:

• monotonicity;
• prefix closure;
• time consistency;
• locality;
• concatenation/pasting closure.

Look at dynamic risk measures / viability / superhedging literature only if it actually helps sharpen the theorem. Do not import terminology merely because it sounds related.

---

Candidate definition of affordability

Try to vindicate, weaken, or replace the following:

An affordability witness is a causal sequence

$$ (w_t,u_t,q_t)_{t\ge0} $$

plus a settlement-relative safety certificate \(C\) such that:

1. Joint Actionability

$$ (w_t,u_t,q_t)\in\mathcal J_t(h_{t-1}) $$

on every round.

1. Faithful service for every persistent reason \(r\),

$$ \mu_N^r\triangleleft\nu_N^r, $$

or whatever weaker correct Service Transfer condition replaces this.

1. Aggregate safety

$$ \operatorname{SafeCert}_{D,\Sigma}(\kappa,C), \qquad \kappa=(u_t)_t. $$

Then Normative Affordability at a history means that such a witness exists.

Conceptual reading:

The standing normative load admits a causal joint realization which does not selectively starve the occasions on which its reasons are owed service and whose aggregate settlement-contingent exposure remains inside the controlled learner's certified robustness envelope.

Determine whether this is genuinely minimal or whether some components should live elsewhere.

---

Sustainable Progress theorem

Try to prove the cleanest possible composition theorem of approximately this form.

Assume:

• Answerability supplies persistent-reason claim measures \(\mu_N^r\);
• an affordability witness supplies actual service measures \(\nu_N^r\);
• the service relation is strong enough for Service Transfer;
• Joint Actionability yields

$$ g_t^r\ge\gamma_r d_t^r $$

on serviced challenges, with \(\gamma_r>0\);

• the safety certificate preserves Uptake;
• Uptake gives

$$ \mathbb E_{\nu_N^r}[g^r]\to0 $$

or an equivalent sublinear weighted cumulative bound.

Conclude:

$$ \boxed{ \mathbb E_{\mu_N^r}[d^r]\to0 } $$

for every persistent actionable reason \(r\), while the decision engine's protected guarantee remains valid.

Prove this at the weakest clean abstraction level.

Then construct minimal counterexamples showing that the conclusion can fail if each major premise is removed:

• no service fidelity;
• no Actionability margin;
• individual but not joint Actionability;
• no SafeCert / Uptake preservation;
• retrospective rather than predictable scheduling.

These counterexamples are important: we want to know that the decomposition reflects genuinely different mathematical requirements.

---

Affordability existence

The composition theorem above consumes an affordability witness. The harder theory asks when one exists.

Formulate this as a causal viability/game problem.

The relevant state may contain:

• the answerability/service state;
• outstanding service backlog or claim state;
• existing settlement-contingent exposures;
• the current safety/risk state;
• any local state determining the joint-response correspondence.

At each round, the controller selects a jointly actionable control and the exterior/settlement process responds.

The objective is simultaneous:

$$ \text{service fidelity} \qquad+\qquad \text{settlement-relative safety}. $$

Investigate useful sufficient conditions for existence:

• convex local control/response sets;
• convex safe regions;
• slack/interiority;
• drift conditions;
• bounded arrival/load rates;
• compositional/subadditive accounts;
• linear aggregation.

Determine which assumptions correspond to actual structure already present in traderization.

---

Overload duality

This is a high-value speculative target.

For a finite-horizon convex version of the affordability problem, investigate whether failure of affordability admits a useful dual/separation certificate.

Ideal form:

$$ \text{either} \quad \exists\text{ service-faithful safe controller} $$

or

$$ \exists\text{ checkable overload certificate} $$

showing that some weighted combination of normative service requirements cannot fit inside the available joint-actionability/safety region.

Look for connections to Farkas, separating hyperplanes, flow/cut duality, convex viability, or scheduling overload certificates.

If such a certificate exists, explain how it could be fed back into Answerability as an authenticated impossibility/overload event rather than silently dropping a reason.

This could be a major theorem, but do not force it if the dynamic problem does not admit a clean dual.

---

Traderization realization

Map the schematic objects precisely into the existing traderization mathematics.

At minimum:

• normative control \(u_t\): added authority/projection trader;
• aggregation: sum of trader positions;
• settlement assessments: plausible worlds \(W\in PC(D_N)\);
• account:

$$ W(E_{\le N}); $$

• safe class: uniformly bounded lifetime downside;
• preservation: bounded liability \(\Rightarrow\) original LIC;
• normative defect:

$$ d_t=\operatorname{dist}(P_t,K_t); $$

• assessment/norm misfit:

$$ e_t(W)=\operatorname{dist}(W|_{\Phi_t},K_t); $$

• local force/friction inequality:

$$ W(E_t) \ge \lambda_t(d_t^2-d_t e_t(W)). $$

Recover the Brier/quadratic-score identity we recently noticed:

for Boolean truth vector \(x\),

$$ \operatorname{dist}(x,K)^2 = \min_{q\in K}\operatorname{Br}_x(q), $$

and for projection \(q=\operatorname{proj}_K(p)\) and enforcement position

$$ \zeta=\lambda(q-p), $$

derive the exact relation between trader payoff and Brier-score improvement.

Interpret this as a realization of epistemic demandingness / friction, but do not bake Brier scoring into the abstract affordability definition.

Also examine the homothetic-core result as a realization of robust affordability / affordability with slack: precision may become arbitrarily tight without a precision-dependent liability penalty.

---

History, revision, and no-reset

Investigate whether defining all relevant quantities as functions/certificates of immutable history is enough to get the desired anti-reset behavior.

A successor regime should not be able to reset:

• prior service claims;
• prior settlement pins;
• already-created control exposures;
• accumulated liability/safety accounts.

But semantic reinterpretation may legitimately change.

Separate carefully:

$$ \text{historical persistence} $$

from

$$ \text{semantic transport}. $$

Ask whether changing evaluator/ontology requires explicit transport of the account or whether historically fixed settlement-contingent exposure contracts can simply remain in their original semantics.

Relate this to the Settlement Interface rule “bridge, never re-settle.”

---

Deliverable

Do not primarily produce philosophical prose.

Produce a research memo organized around mathematical findings:

1. the minimal proposed type signatures;
2. exact theorem statements;
3. proofs or proof sketches where possible;
4. counterexamples;
5. necessary vs sufficient assumptions;
6. which hypotheses are schematic and which are realization-specific;
7. exact mapping to traderized LI;
8. the strongest unresolved mathematical questions.

At the end, give a recommended theorem stack of no more than about 5–8 named results, ordered by dependency.

Be willing to conclude that some proposed concept should be deleted or merged.

Priority

Spend the most effort on:

1. Service Transfer and whether contiguity is the correct/minimal condition;
2. Joint Actionability and the failure/composition of individually actionable controls;
3. the minimal SafeCert / preservation interface;
4. affordability existence as a causal viability problem;
5. finite-horizon overload duality/certificates.

The intended central composition, which you should try hard to prove or break, is:

$$ \boxed{ \begin{array}{c} \text{Answerability claim measures }\mu^r\\ +\text{ joint actionable response semantics}\\ +\text{ settlement-relative learner robustness} \\[1mm] \Downarrow\quad\text{Affordability existence}\quad\Downarrow\\[1mm] \text{actual service measures }\nu^r \text{ faithful to }\mu^r +\text{ SafeCert} \\[1mm] \Downarrow\\ \text{preserved Uptake} +\text{ Actionability} \\[1mm] \Downarrow\\ \boxed{\text{claim-weighted Progress}.} \end{array}} $$

The goal of the pass is to determine whether this is the real theorem architecture—and, if it is, to make each arrow mathematically exact.

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Second dispatch — follow-up research pass on PR75

You are doing a focused follow-up research pass on PR75, Research Normative Affordability and its service interface, in the alignment-workspace.

Start by reading the entire round at:

projects/normativity/legitimacy/rounds/2026-08-31-normative-affordability/

especially:

• SERVICE_TRANSFER.md
• JOINT_ACTIONABILITY.md
• AFFORDABILITY.md
• SAFECERT.md
• EXISTENCE_AND_DUALITY.md
• REALIZATION_MAP.md
• REPORT.md

Then read the traderization/liability material that these documents purport to realize. In particular, recover the exact roles of:

• projection force against a normative region \(K\);
• assessment-world value and lifetime liability;
• the MarketMaker / fixed-point variational inequality;
• the Logical Induction Criterion;
• the theorem that bounded authority/enforcement liability preserves ordinary LI;
• the force/friction inequality

$$ \langle \zeta,x-p\rangle\ge \lambda(d^2-de); $$

• the homothetic/core or covered-underwriting results where relevant.

Do not assume PR75's realization map is correct merely because its schematic mathematics is correct. The goal of this pass is to determine exactly which abstractions survive contact with the existing traderized LI mathematics.

Governing question

Can PR75's Sustainable Progress / Normative Affordability theorem be given a clean end-to-end Logical Induction realization without collapsing mathematical objects that traderization deliberately keeps distinct?

There are three specific pressure points to investigate.

A. What should “Progress” mean?

PR75 shows that service-weighted defect can vanish while claim-weighted defect remains large, and introduces transport to transfer learning from actual service mass back to normatively owed claim mass.

Determine whether the clean theorem stack should be:

$$ \text{Actionability + Uptake} \Rightarrow \text{service-weighted Progress} $$

followed by

$$ \text{service-weighted Progress + Service Transport} \Rightarrow \text{claim-weighted Sustainable Progress}, $$

rather than modifying the base Progress theorem itself to be claim-weighted.

Do not decide this narratively. State the mathematical interfaces in both decompositions and identify which gives the more modular theorem stack.

Pay particular attention to the distinction between:

• the actual service measure \(\nu_N^r\);
• the normatively owed claim measure \(\mu_N^r\);
• transport \(T^r(t,s)\);
• the stability error \(\varepsilon_r\).

Check the wording in SERVICE_TRANSFER.md: PR75 sometimes says contiguity is “derived” by the transport theorem. Verify whether this is literally true of the original claim measure \(\mu_N\) versus service measure \(\nu_N\), or only of the transported claim measure on service dates. If the latter, state and repair the distinction precisely.

B. Reconstruct the LI realization from first principles

This is the most important part of the pass.

PR75 appears at points to identify a “common scoring set” supplied by LI's live assessment worlds with the common region used in Joint Actionability. Check whether this is sound.

Keep explicit notation for at least these two sets:

$$ K_t = \text{normative admissible region} $$

and

$$ A_t = \text{live assessment set} \quad (\text{e.g. }PC(D_t)). $$

Ask separately:

1. Against what set is normative force / Actionability scored?
2. Against what set is liability / safety scored?
3. What exactly produces Uptake in the concrete LI construction?
4. What exactly does the LIC preservation theorem establish?

Do not conflate these unless an actual theorem identifies them.

In particular, investigate the hypothesis that the correct realization is closer to:

$$ \begin{array}{rcl} \text{Actionability} &\leftrightarrow& \text{projection force against }K_t,\\ \text{Uptake} &\leftrightarrow& \text{MarketMaker / fixed-point VI response to added demand},\\ \text{Affordability account} &\leftrightarrow& \text{worldwise cumulative authority-trader value over }A_t,\\ \text{SafeCert preservation} &\leftrightarrow& \text{bounded liability preserves ordinary LIC}. \end{array} $$

This is only a conjectured factorization. Verify or refute it from the actual mathematics.

A particular claim to audit is PR75's statement that per-reason Uptake is free because LIC quantifies over each enforcement trader individually. Check whether the authority/projection enforcer is in fact an ordinary budgeted trader to which LIC applies, or whether it is an extra control whose bounded liability merely preserves LIC for the original trader class.

If LIC is not the correct concrete source of Uptake, identify the exact VI / MarketMaker inequality that is, and derive the relevant Progress inequality from it.

Also determine what becomes of T4, Joint Actionability. Its abstract theorem may be right even if the proposed LI realization is not. Preserve the distinction between:

• a common normative comparison region used to certify force; and
• a common settlement/assessment set used to price liability.

Explain exactly when nonemptiness of

$$ A_t\cap\bigcap_r K_t^r $$

is enough for anything, and when stronger coverage/accommodation hypotheses are required.

C. Affordability is a signed cumulative account: audit “self-financing”

EXISTENCE_AND_DUALITY.md claims that a fixed liability budget can fund only a transient and concludes that persistent service must be “self-financing,” where each control has nonnegative robust incremental value.

Stress-test this carefully.

The existing liability theory uses a signed cumulative account with a lower bound, so positive periods may finance negative periods. For example, alternating increments \(+1,-1,+1,-1,\dots\) preserve a fixed lower bound despite infinitely many locally loss-making interventions.

Determine:

• whether Observation E2 is stated correctly;
• whether its conclusion bounds number of costly service dates or total service mass;
• what additional proportionality assumptions would be required to bound total service mass;
• whether “persistent service must be self-financing” is false as stated;
• what the correct necessary notion is.

Candidate replacements include, but are not limited to:

• bounded cumulative downside;
• nonnegative long-run robust drift;
• existence of a replenishing potential;
• a credit/debit process whose prefixes remain above a floor;
• a policy-level underwriting condition.

Do not force one of these if the mathematics suggests something cleaner.

Then revisit T7. It may remain a perfectly good strong sufficient theorem using pointwise self-financing slack. If so, state it explicitly as such and separate it from any alleged necessity claim.

This section should connect back to the conceptual affordability schematic:

affordability is not a sum of nonnegative costs; it is the viability of a signed, persistent underwriting account under the exercise of normative force.

D. SafeCert abstraction

Audit PR75's proposed minimal SafeCert interface.

The useful abstraction seems to be something like

$$ \mathrm{SafeCert}_D(\kappa) \Rightarrow \mathrm{PreservedUptake}(D^\kappa). $$

PR75 additionally argues that robust settlement-relative safety is forced because actual-path safety is not measurable when an adapted control is chosen.

Check whether that claim is actually schematic. A stochastic learner could conceivably certify almost-sure, high-probability, expectation-based, or other prospective guarantees without using worst-case live-world robustness.

Distinguish:

• what the generic interface truly requires;
• what is specifically forced/natural in the traderized LI realization.

Do not weaken the robust LI result; just avoid claiming more abstraction than has been proved.

Composition target

After resolving A–D, try to state the strongest clean schematic theorem you believe the workspace now supports.

A likely shape is:

$$ \boxed{ \text{Affordable service policy} + \text{Actionability} + \text{Uptake} + \text{Service Transport} \Rightarrow \text{claim-weighted Sustainable Progress} } $$

with, for each persistent reason \(r\),

$$ \limsup_N E_{\mu_N^r}[d^r] \le \varepsilon_r, $$

and exact convergence when transport stability is exact.

But do not merely restate PR75. Specify exactly which premise belongs to:

• Answerability / standing;
• service scheduling;
• semantic transport;
• force / Actionability;
• learner Uptake;
• affordability / safety;
• substrate preservation.

The theorem should make clear that the same intensity variable \(w_t^r\) can play both roles of service allocation and normative control intensity if that identification survives the LI audit.

Then provide the corresponding LI instantiation line by line.

Causal overload

Finally, reassess PR75's open Priority 74 in light of the corrected account picture.

The finite-horizon Farkas certificate is only a per-path infeasibility certificate. The real problem is the quantifier gap

$$ \forall \omega\,\exists \pi_\omega \qquad\not\Rightarrow\qquad \exists \pi\,\forall\omega. $$

Say whether this remains the highest-value next theorem target.

If so, sharpen what a complete certificate would have to separate: not merely feasible points on each realized path, but causal policies / strategies. Consider whether the natural dual object should look more like:

• a dynamic flow/cut;
• an adversarial strategy;
• a potential/supermartingale;
• a policy-separation theorem;
• or something else.

Do not solve this speculatively if the round does not support it; identify the exact mathematical problem.

Deliverables

Produce:

1. FOLLOWUP_REPORT.md — detailed audit with explicit verdicts on A–D.
2. CORRECTED_STACK.md — the smallest theorem/interface stack you now recommend, schematic first and LI realization second.
3. Exact counterexamples or small rational fixtures for every claim from PR75 you reject or weaken.
4. If appropriate, targeted edits to the PR75 round documents. Do not rewrite earlier consolidated work merely for consistency; record corrections locally unless the workspace's provenance rules explicitly permit otherwise.
5. A short final section titled “What actually remains before an end-to-end LI legitimacy theorem?” Separate genuinely missing mathematics from terminology/interface cleanup.

Treat PR75's transport theorem and counterexamples as results to preserve unless you find an actual flaw. The purpose of this pass is not to redo the round. It is to make the abstraction boundary line up exactly with the traderization/liability mathematics and to remove any accidental overclaims before the affordability theory is consolidated.

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Third dispatch — service allocation versus realized force in the traderized LI learner

Follow-up research pass: service allocation versus realized force in the traderized LI learner

Work against the live alignment-workspace, especially PR75 and its follow-up audit:

projects/normativity/legitimacy/rounds/2026-08-31-normative-affordability/

Read at minimum:

• CORRECTED_STACK.md
• FOLLOWUP_REPORT.md
• AFFORDABILITY.md
• SERVICE_TRANSFER.md
• JOINT_ACTIONABILITY.md
• SAFECERT.md
• EXISTENCE_AND_DUALITY.md
• REALIZATION_MAP.md

Then recover the exact traderization / projection-enforcement / MarketMaker definitions and the relevant Answerability/service material.

This is a focused mathematical pass. Do not reopen the basic conclusions of PR75 unless this investigation produces an actual counterexample.

Governing question

The corrected schematic currently identifies a reason-indexed variable \(w_t^r\) as both:

1. the amount of normative service allocated to reason \(r\); and
2. the realized magnitude of the normative force/control applied on its behalf.

In the traderized LI realization this is read as

$$ w_{t,j}=\beta_{t,j} g_{t,j}(P_t), $$

where:

• \(\beta_{t,j}\) is the enforcement multiplier / authority allocated to row \(j\);
• \(g_{t,j}(P_t)\ge 0\) is the displayed violation/defect;
• the realized row position is

$$ \beta_{t,j}g_{t,j}(P_t)c_{t,j}. $$

But \(P_t\), and hence \(g_{t,j}(P_t)\), is determined at the MarketMaker fixed point. Thus \(w_{t,j}\) is reactive/endogenous, not a freely chosen \(\mathcal F_{t-1}\)-measurable scheduling variable.

This may be the remaining typing mismatch between the schematic affordability theorem and its concrete LI realization.

Investigate whether the correct separation is instead:

$$ \boxed{ a_{t,j}:=\beta_{t,j} = \text{predictably allocated normative service/authority} } $$

and

$$ \boxed{ f_{t,j}:=a_{t,j}d_{t,j} = \beta_{t,j}g_{t,j}(P_t) = \text{reactively realized corrective force}. } $$

The goal is to determine whether this distinction is correct, mathematically useful, and sufficient to close the concrete end-to-end learner stack.

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1. Reconstruct the within-round timing exactly

Do not reason from prose. Extract the actual dependency structure of traderized LI.

For a round \(t\), identify which objects are determined at which stage:

1. settled history \(\mathcal F_{t-1}\);
2. normative docket / active rows;
3. chosen authority/multiplier parameters;
4. reactive trader/control law as a function of a candidate displayed price;
5. ordinary TradingFirm demand;
6. aggregate demand;
7. MarketMaker fixed point \(P_t\);
8. realized defect \(d_t\);
9. realized enforcement position;
10. later settlement / world assessment.

Produce an explicit dependency graph.

In particular answer:

• Is \(\beta_{t,j}\) genuinely predictable from \(\mathcal F_{t-1}\)?
• Is the control in LI better typed as a predictable map

$$ \kappa_t:P\mapsto E_t(P) $$

rather than a realized position \(E_t(P_t)\)?

• Exactly which variables are permitted to depend on the contemporaneous displayed price?
• Does the current schematic Affordability definition incorrectly require the realized \(u_t\) to be \(\mathcal F_{t-1}\)-measurable?
• If so, give the minimal corrected abstract type.

A plausible generic type is:

$$ \kappa_t\in\mathsf{ControlLaw}_t, \qquad \kappa_t\text{ chosen from }\mathcal F_{t-1}, $$

followed by

$$ u_t=\kappa_t(x_t) $$

after the engine produces its current state \(x_t\).

Do not adopt this unless it matches the actual construction.

---

2. Which quantity really deserves to be called “service”?

Stress-test the current identification

$$ \text{service}=w_t=\beta_t d_t. $$

Consider at least these cases.

Perfect compliance

A persistent reason receives substantial deliberate authority

$$ \beta_t>0 $$

at every round, but

$$ d_t=0. $$

Then

$$ \beta_td_t=0. $$

Should the theory say this reason received no service?

If not, the current service interpretation is wrong.

Successful learning

Suppose

$$ d_t\to0 $$

rapidly and

$$ \sum_t\beta_t=\infty $$

but

$$ \sum_t\beta_td_t<\infty. $$

The reason is continuously monitored/enforced and becomes nearly perfectly satisfied, but the current PR75 service mass is finite.

Determine whether the existing Persistent Relevance / Answerability semantics would incorrectly classify this as starvation.

Large defect

Compare two dates with equal allocated authority \(\beta_t\) but very different defects. Is it normatively correct for the high-defect date to count as having received more “service” merely because more corrective force was required?

Try to distinguish formally between:

• service / authority / attention allocation;
• corrective effort / force;
• realized position/exposure;
• funding.

The workspace has repeatedly warned that enforcement intensity is not funding. Preserve that distinction.

At the end of this section give a verdict:

$$ \text{service}=\beta? $$

$$ \text{service}=\beta d? $$

or some third quantity?

Do not choose based on terminology; choose based on what the Sustainable Progress theorem actually needs.

---

3. Re-derive LI Progress using predictable service weights

Suppose provisionally that

$$ a_t:=\beta_t $$

is the service allocation and

$$ f_t=a_td_t $$

is realized force.

Re-express the existing force/account identity using \(a_t\), \(d_t\), and settlement misfit.

In the simplest compatible-world / zero-friction case, projection force should yield useful work of the form

$$ a_td_t^2. $$

Determine exactly what the MarketMaker cumulative cap gives.

The target shape is:

$$ \sum_{t<N}a_td_t^2\le U $$

or its correct row/world-valued variant.

If

$$ A_N:=\sum_{t<N}a_t\to\infty, $$

derive:

$$ \frac{\sum_{t<N}a_td_t^2}{A_N}\to0. $$

Then determine whether this implies

$$ \boxed{ \frac{\sum_{t<N}a_td_t}{A_N}\to0 } $$

by Cauchy–Schwarz / Jensen, and state the sharp bound.

This would produce service-weighted Progress with respect to a genuinely predictable service measure

$$ \nu_N^a(t)=\frac{a_t}{A_N}. $$

Compare this theorem carefully with PR75's current \(w=\beta d\) formulation.

Identify exactly what is gained and lost:

• linear versus quadratic force bound;
• rates;
• friction/misfit residual;
• required boundedness assumptions;
• service semantics;
• causal scheduling freedom.

Do not prefer the \(\beta\)-version merely because its semantics are nicer. Determine whether the mathematics is actually sufficient.

---

4. Carry the settlement-friction term through correctly

Do not analyze only the compatible-world case.

The current corrected realization has the signed account identity, schematically,

$$ V_N(\omega) = \sum_{t,j} \beta_{t,j}d_{t,j} \bigl( d_{t,j}-s_{t,j}(\omega) \bigr). $$

Rewrite this as

$$ V_N(\omega) = \sum_{t,j} a_{t,j} \left( d_{t,j}^2-d_{t,j}s_{t,j}(\omega) \right). $$

Use the MarketMaker upper cap and liability lower bound to determine the strongest service-\(a\)-weighted Progress statement.

Questions:

1. Can we show

$$ E_{\nu_N^a}[d^2] $$

converges to a corresponding weighted friction term?

1. Can this be converted into a useful statement about

$$ E_{\nu_N^a}[d]? $$

1. What exact condition on settlement misfit is sufficient for

$$ E_{\nu_N^a}[d]\to0? $$

1. Does one compatible live world still suffice?

1. Can the Brier/Euclidean interpretation sharpen the bound?

1. Does homothetic-core / covered-underwriting structure produce a clean residual theorem?

Try to give the strongest quantitative inequality, not just an asymptotic slogan.

---

5. Plug this into Service Transport

PR75's strongest schematic advance was Deferred Service Transfer.

If the learner-side measure should be induced by \(a_t=\beta_t\), re-run the interface:

$$ \mu_N^r = \text{Answerability claim measure}, $$

$$ \nu_N^r = \text{actual allocated-service measure based on }\beta_t^r. $$

Then Service Transport should compare owed claims directly to allocated service, not to the endogenous corrective force \(\beta d\).

Determine whether this produces a cleaner theorem:

$$ \text{Answerability claim stream} \overset{T}{\longrightarrow} \text{allocated authority stream} $$

and independently

$$ \text{allocated authority} +\text{MarketMaker} +\text{Actionability} \longrightarrow \text{small defect}. $$

If so, state the end-to-end theorem exactly, including transport error.

For example, seek something of the form

$$ \boxed{ \limsup_N E_{\mu_N^r}[d^r] \le \text{transport error} + \text{settlement-friction residual}. } $$

This may be the actual concrete Sustainable Progress theorem.

Distinguish the two residuals:

• transport residual: the reason changes between when service is owed and delivered;
• settlement friction: the norm conflicts with what can still be true.

These should not be conflated.

---

6. Does Answerability actually control \(\beta\)?

This is crucial.

Even if \(\beta\) is the right semantic service variable, determine whether the concrete LI construction permits a scheduler to choose it with the freedom the schematic Affordability theorem assumes.

Inspect:

• compiler semantics;
• projection trader definition;
• enforcement schedule;
• calibration needed for finite-time conformance;
• Budgeter / MarketMaker constraints;
• any conditions determining \(\beta\) from the desired tolerance.

Ask whether Answerability can meaningfully choose:

$$ \beta_{t,r} $$

subject to a capacity/liability constraint, or whether \(\beta\) is effectively fixed by the enforcement theorem.

There may be three different possibilities:

1. free scheduling: \(\beta\) is an independent predictable control variable;
2. lower-bound scheduling: Answerability may demand at least some \(\beta\), while the enforcement compiler chooses more;
3. derived scheduling: \(\beta\) is fixed by desired conformance/tolerance, so service must instead be represented by another parameter.

Determine which is actually true.

If \(\beta\) cannot be freely scheduled, modify the schematic Affordability interface rather than pretending otherwise.

---

7. Reassess Actionability under the two-variable picture

If allocated service is \(a\) and realized force is \(f=a d\), then Actionability may no longer naturally have the old linear form

$$ g_t\ge\gamma d_t. $$

Instead the primitive local force theorem might be something like

$$ \operatorname{Work}_t \ge a_t d_t^2 - \operatorname{Friction}_t. $$

Investigate whether the schematic Actionability layer should permit a general coercivity function

$$ \boxed{ \operatorname{Gain}_t \ge a_t\,\phi(d_t)-\operatorname{Friction}_t } $$

with projection enforcement realizing

$$ \phi(d)=d^2. $$

If so, characterize the minimal conditions on \(\phi\) under which vanishing average gain/work implies vanishing average defect.

Possible conditions:

• \(\phi(d)>0\) for every \(d>0\);
• quantitative coercivity;
• convexity;
• lower bounds such as \(\phi(d)\ge c d^p\).

This may be a better schematic Actionability theorem than forcing every realization into a linear \(g\ge\gamma d\) form.

Do not generalize unless it simplifies the actual theorem stack.

---

8. Affordability under the corrected typing

If the two-variable distinction survives, rewrite the affordability witness.

A candidate:

An affordability witness consists of a predictable sequence of service allocations / reactive control laws

$$ (a_t,\kappa_t) $$

such that:

1. Answerability/service transport requirements are satisfied by \(a_t\);
2. the reactive engine produces realized controls

$$ u_t=\kappa_t(x_t); $$

1. the resulting settlement-contingent cumulative account lies in SafeCert;
2. the resulting force satisfies the engine-side Progress theorem.

The key point:

$$ \boxed{ \text{Affordability chooses authority; the engine determines how much force that authority must actually exert.} } $$

Determine whether this is right.

If it is, rewrite the causal viability state accordingly. The backlog dynamics should consume \(a_t\), while the liability/account dynamics should consume the endogenous realized control \(u_t\).

That would produce a viability problem with two coupled dynamics:

$$ b_{t+1} = \text{service backlog update}(a_t), $$

$$ A_{t+1} = \text{settlement account update}(u_t,\omega_t). $$

This may be the cleanest mathematical form of normative affordability yet.

---

9. Reassess the existence theorem

The current T7 schedules the same \(w_t\) that enters the account.

If service is instead \(a_t\) and liability depends on the endogenous \(a_td_t\), T7's max-weight formulation may no longer have the right type.

Determine:

• whether T7 survives unchanged;
• whether it survives after replacing the safe-control region by a set of service allocations whose induced reactive controls are safe;
• whether the market's endogenous reduction of force as \(d\to0\) makes existence easier;
• whether the appropriate capacity region is naturally defined in \(a\)-space.

A particularly interesting possibility is:

A persistent reason can receive divergent service mass \(\sum a_t=\infty\) while consuming only finite total liability because the realized corrective force dies away as compliance improves.

If true, this is precisely the behavior we want a normative learner to exhibit.

Try to prove or exhibit it in the simplest traderized instance.

---

10. Decide what “end-to-end LI learner realization” now requires

At the end, give a very explicit status report.

Separate:

Already concrete

For example:

• settlement;
• normative region compiler;
• reactive projection control;
• MarketMaker Uptake;
• signed liability account;
• bounded-liability preservation.

Now closed by this pass

Ideally:

• service allocation typing;
• Answerability → LI service interface;
• concrete service-weighted Progress.

Still missing

Possibilities include:

• transport stability construction;
• generalized live-world transcription;
• normative statics \(C_t\) versus \(K_t\);
• existence of an affordable service schedule;
• liability necessity;
• cross-era transport;
• non-capture.

Do not call something missing merely because it is not yet Lean-formalized. Separate mathematical absence from verification debt.

---

Deliverables

Produce:

1. SERVICE_FORCE_TYPING.md

• exact within-round dependency structure;
• verdict on \(\beta\) versus \(\beta d\);
• minimal corrected abstract types.

1. LI_PROGRESS_FROM_SERVICE.md

• the strongest theorem deriving service-weighted Progress from predictable service allocation, including friction/misfit;
• exact proof or proof sketch;
• rational fixtures for edge cases.

1. END_TO_END_LEARNER.md

• Answerability claim stream → service allocation → reactive LI force → MarketMaker Progress → Service Transport → claim-weighted Sustainable Progress;
• every hypothesis labeled by layer.

1. Targeted corrections to PR75 if needed. Preserve the provenance rules and do not rewrite older consolidated artifacts merely for cosmetic consistency.

1. New exact-rational tests for at least:

• persistent service with \(d_t=0\);
• \(\sum\beta_t=\infty\) but \(\sum\beta_td_t<\infty\);
• \(\sum\beta_td_t^2<\infty\) implying \(\beta\)-weighted defect convergence;
• settlement-friction residual;
• one compatible world versus liability at other live worlds.

Success criterion

The pass succeeds if it can give a convincing answer to this question:

What exact predictable object does Answerability control in a traderized Logical Inductor, and why does persistent service of that object imply Progress while bounded liability preserves the learner?

If the answer is “\(\beta\), with realized force \(\beta d\),” prove the resulting end-to-end chain.

If that answer is false, identify the correct object and show precisely why the attractive \(\beta\)/\(\beta d\) distinction fails.

The main objective is not another collection of sufficient conditions. It is to close—or sharply localize—the remaining type mismatch between the schematic affordability theorem and the concrete LI normative learner.

---

Fourth dispatch — finish the fixed-era LI learner theorem and isolate the affordability-existence problem

Follow-up research pass: finish the fixed-era LI learner theorem and isolate the true affordability-existence problem

Work against the live alignment-workspace, especially the current state of PR75:

projects/normativity/legitimacy/rounds/2026-08-31-normative-affordability/

Read at minimum:

• SERVICE_FORCE_TYPING.md
• LI_PROGRESS_FROM_SERVICE.md
• END_TO_END_LEARNER.md
• CORRECTED_STACK.md
• AFFORDABILITY.md
• SERVICE_TRANSFER.md
• SAFECERT.md
• EXISTENCE_AND_DUALITY.md
• FOLLOWUP_REPORT.md

Also recover the exact traderized-LI statements these files use:

• aggregate MarketMaker cumulative cap;
• reason/enforcer decomposition;
• lifetime-liability preservation;
• projection/enforcement modulus;
• covered-underwriting / core bounds;
• existing liability-regime taxonomy.

This is a focused theorem-repair and consolidation pass. The \(\beta\) versus \(\beta d\) distinction should be treated as the current best typing unless it fails under scrutiny:

$$ a_{t,r}=\beta_{t,r} = \text{predictable allocated authority/service}, $$

$$ f_{t,r}=a_{t,r}d_{t,r} = \text{endogenous realized force}. $$

The control chosen before the round is a reactive law

$$ \kappa_t:x\mapsto u_t(x), $$

not its realized position.

Do not reopen that question without an actual counterexample.

Governing objective

Determine whether we can now state a mathematically correct fixed-era end-to-end normative learner theorem of the form

$$ \boxed{ \text{Answerability claims} + \text{affordable allocated authority} + \text{reactive traderized LI} + \text{service transport} \Longrightarrow \text{claim-weighted Sustainable Progress} } $$

while preserving ordinary LI.

The previous pass is very close, but two gaps remain:

1. the Progress theorem is written per reason even though the MarketMaker cap is initially on the aggregate enforcement book;
2. the local per-date authority capacity box is being blurred with the genuinely diachronic lifetime SafeCert.

Repair these first. Then investigate how much of an actual affordability-existence theorem follows.

---

1. Reason-indexed Progress from an aggregate MarketMaker cap

Let the aggregate enforcement book decompose as

$$ E_t=\sum_r E_t^r, \qquad V_N(\omega)=\sum_r V_N^r(\omega). $$

The MarketMaker argument provides an aggregate upper cap of the form

$$ V_N(\omega)\le U $$

for every relevant live assessment world.

A reason-specific Progress proof needs an upper bound on

$$ V_N^r(\omega). $$

The previous audit suggested that if every other reason has a liability floor

$$ V_N^k(\omega)\ge-B_k, $$

then

$$ V_N^r(\omega) \le U+\sum_{k\neq r}B_k =:U_r. $$

Prove this carefully in the actual traderized-LI accounting.

Questions:

1. Is this exactly the right decomposition?
2. Does it require a fixed reason set, or does it survive a growing docket?
3. For countably many reasons, is

$$ \sum_k B_k<\infty $$

sufficient?

1. Can one instead allocate a single global liability budget dynamically across reasons?
2. Is there a cleaner vector-account formulation avoiding arbitrary reason labels?
3. Does the row-level decomposition make more sense than reason-level decomposition?

Then derive the corrected reason-indexed \(L^2\) Progress theorem:

$$ E_{\nu_N^{a,r}}[d^r] \le \|s_r^+(\omega)\|_{L^2(\nu_N^{a,r})} + \sqrt{\frac{U_r}{A_N^r}} $$

under the exact correct hypotheses.

If there is a useful aggregate Sustainable Progress theorem that avoids reasonwise liability decomposition, state it separately rather than forcing everything reasonwise.

Identify the strongest conditions under which:

$$ \text{aggregate cap} \Rightarrow \text{all persistent reasons progress}. $$

---

2. Local authority capacity is not lifetime safety

Keep these objects explicitly distinct.

A predictable local authority-capacity set

$$ \mathcal C_t(h_{t-1}) $$

describes authority allocations available on date \(t\). For the conservative per-date certificate, this may contain constraints such as

$$ 0\le a_{t,r}\le \frac{b_{t,r}^2} {(\varepsilon_t+M_t)D_{t,r}^2}. $$

But the true affordability/SafeCert condition is diachronic:

$$ V_N(\omega)\ge-B \qquad \forall N,\omega\in A_N. $$

Do not identify

$$ a_t\in\mathcal C_t $$

with

$$ \text{SafeCert}. $$

Work out the exact relationship.

At least three possibilities should be distinguished:

Route A: conservative budget splitting

Choose predictable per-date allowances \(b_t\) with

$$ \sum_t b_t\le B, $$

then use local caps to guarantee lifetime safety.

State the resulting sufficient theorem.

Route B: persistent signed account

Track the full worldwise account state and allow gains to finance later losses:

$$ V_t(\cdot) \mapsto V_{t+1}(\cdot). $$

Formulate the true viable authority set conditional on current account state:

$$ \mathcal C_t^{\mathrm{safe}} \bigl(h_{t-1},V_{t-1}\bigr). $$

Route C: existing structural certificates

Covered underwriting, homothetic/core structure, absorbed regimes, etc. may guarantee lifetime safety without an explicit exogenous sequence \(b_t\).

Explain how these fit the same interface.

The goal is a corrected statement:

Local capacity says what authority may be allocated now; SafeCert says whether the entire resulting reactive-control history remains underwritten.

Patch any current text that conflates these.

---

3. Correct the affordability witness one final time

The schematic witness should now have the right causal type.

Candidate:

An affordability witness from history \(h\) consists of predictable sequences

$$ (a_t,\kappa_t) $$

where:

• \(a_t\) is reason-indexed allocated authority;
• \(\kappa_t\) is a legal reactive control law compiled from the current docket and \(a_t\);
• the realized control is

$$ u_t=\kappa_t(x_t) $$

after the engine computes \(x_t\);

• Answerability's claim stream is matched to \(a_t\) by adapted service transport;
• the realized control history satisfies lifetime SafeCert.

Check whether anything else is actually needed.

In particular, remove stale formulations in which realized \(u_t\) is said to be predictable.

Also decide whether “common region nonempty” belongs in the generic affordability definition or specifically in the compiler/Actionability interface.

Prefer the smaller interface if possible.

---

4. Consolidate the coercive Progress theorem

The current abstract Actionability proposal is:

$$ \operatorname{Work}_N = \sum_t a_t\phi(d_t) \le C_N, $$

with \(\phi\) coercive.

For projection enforcement,

$$ \phi(d)=d^2. $$

Audit the current theorem carefully.

A clean form might be:

If

$$ \frac{\sum_{t<N}a_t\phi(d_t)} {\sum_{t<N}a_t} \to0, $$

and \(\phi:[0,D]\to[0,\infty)\) satisfies an appropriate coercivity condition, then

$$ E_{\nu_N^a}[d]\to0. $$

Determine the weakest useful condition on \(\phi\).

Do we really need convexity and strict monotonicity, or is it enough that for every \(\epsilon>0\),

$$ \inf_{d\ge\epsilon}\phi(d)>0? $$

Distinguish:

• qualitative convergence;
• quantitative inverse-modulus bounds.

If this yields a cleaner schematic theorem, use it.

Then insert settlement friction:

$$ \operatorname{Work} \le \operatorname{Friction} + O(1). $$

Try to state a general coercive Actionability with friction theorem that specializes exactly to LI without overabstracting.

---

5. Repair the fixed-era end-to-end theorem

Produce a theorem whose hypotheses are fully explicit and correctly attributed.

A target shape is:

Fixed-Era Sustainable Progress.

Let \(r\) be persistent with Answerability claim stream \(c_t^r\). Suppose:

• a predictable authority schedule \(a_t^r\) is service-faithful to \(c_t^r\) via an adapted transport plan;
• \(A_N^r=\sum_{t<N}a_t^r\to\infty\);
• the reactive traderized-LI controller is legal and the normative region remains priceable;
• the reason-specific enforcement account has an upper cap \(U_r\), derived from the aggregate MarketMaker cap plus complementary liability floors;
• the lifetime aggregate enforcement account satisfies SafeCert;
• the settlement-friction residual is \(F_r\);
• transport stability has residual \(\epsilon_r\).

Then

$$ \limsup_N E_{\mu_N^r}[d^r] \le L_rK_rF_r+\epsilon_r, $$

with an explicit finite-\(N\) rate, while ordinary LI remains unexploitable.

Derive the exact finite-horizon inequality.

Do not say the residuals vanish “exactly when” some intuitive condition holds unless you prove the equivalence. Give sufficient and necessary conditions separately.

Especially distinguish:

$$ F_r=0 $$

from the stronger condition:

$$ \exists\omega_\ast \text{ persistently live and satisfying all serviced rows}. $$

The latter is clearly sufficient; determine whether the former implies anything like it under nested compact live sets.

---

6. Is the construction theorem now actually complete?

After the repair, answer this sharply.

Call the construction theorem complete if, conditional on:

• a service-faithful authority schedule;
• bounded lifetime liability;
• settlement-friction and transport residual assumptions;

all arrows from Answerability's service stream to claim-weighted Progress and LI preservation are proved.

Do not require an existence theorem for this status.

If complete, say so and list the exact unresolved hypotheses as inputs rather than “missing arrows.”

This distinction matters:

$$ \boxed{ \text{construction/composition} \neq \text{existence} \neq \text{necessity/characterization}. } $$

---

7. Then attack the true affordability-existence problem in \(a\)-space

Now formulate the problem using the corrected variables.

State:

$$ b_{t+1}^r = [b_t^r+c_{t+1}^r-a_t^r]_+ $$

for service backlog.

The account dynamics are reactive:

$$ V_{t+1}(\omega) = V_t(\omega) + \Delta V_t \bigl( a_t, d_t(a_t,\text{market}), \omega \bigr). $$

The controller chooses \(a_t\) before seeing the contemporaneous fixed point/settlement.

Affordability existence asks for a causal policy that simultaneously:

$$ \sup_t b_t^r<\infty $$

or otherwise supplies the required transport fidelity, and

$$ \inf_{t,\omega}V_t(\omega)>-\infty. $$

This is the true viability problem.

Investigate at least the following.

A. Simple positive theorem

Find the strongest clean sufficient condition under which a persistent reason can receive

$$ \sum_t a_t=\infty $$

while maintaining finite lifetime liability.

Use the new square-root exposure phenomenon if valid:

$$ \text{liability charge} \lesssim \sqrt{a_t}\,\sigma_t. $$

Give explicit sufficient summability conditions.

B. Many reasons

Suppose countably/finitely many persistent reasons have individual capacity processes

$$ \operatorname{cap}_t^r. $$

What condition allows a schedule with

$$ A_N^r\to\infty \quad \forall r? $$

Compare:

• round-robin;
• weighted fair scheduling;
• max-weight;
• capacity-region conditions.

Separate pure service-capacity overload from liability overload.

C. Adaptive signed-account scheduling

Can allowing positive-account periods to replenish the account strictly enlarge the sustainable service-rate region relative to summable per-date budgeting?

Construct a minimal example if so.

It should, conceptually, but prove it.

D. Capacity region

Define the set of asymptotically sustainable authority-demand rates.

Something like:

$$ \mathcal A = \{ \lambda: \exists\text{ causal policy servicing arrival rate }\lambda \text{ while SafeCert holds} \}. $$

Ask whether it is convex, downward closed, or admits a useful interior/slack notion.

This may be the correct formal object for “normative capacity.”

---

8. Revisit overload only after the primal is correct

Do not prioritize a full causal dual unless the primal existence problem is now sufficiently clean.

But refine the desired dual target.

Failure of affordability should ideally yield a certificate separating a claim/service load from the causal safe authority capacity.

The finite-horizon per-path Farkas certificate remains sound but not complete.

Ask whether the natural dynamic dual object resembles:

• a Lyapunov/potential witness;
• a supermartingale;
• an adversarial environment strategy;
• a dynamic separating functional.

Do not invent terminology without a theorem.

---

9. Answerability-to-service compilation

The construction theorem will still be conditional on an authority schedule unless Answerability itself supplies one.

Clarify exactly what Answerability should export.

Possibilities:

1. raw claim arrivals \(c_t^r\), with affordability responsible for scheduling;
2. service-entitlement constraints rather than a particular schedule;
3. explicit lower bounds on cumulative allocation:

$$ A_N^r\ge F(C_N^r)-O(1); $$

1. deadlines / transport constraints.

Prefer to keep normative adjudication out of affordability itself.

A likely clean split is:

$$ \boxed{ \text{Answerability exports claims and admissible service traces;} } $$

$$ \boxed{ \text{Affordability selects a safe trace among them.} } $$

Determine whether this is consistent with the transport machinery.

---

10. Final status: what remains before the first fixed-era learner theorem?

End with three explicitly separate lists:

A. Proven construction/composition pieces

Only include genuine mathematical arrows.

B. Missing existence mathematics

For example:

• affordable service scheduling;
• non-vacuity of capacity;
• transport-stability certificates.

C. Later legitimacy extensions

Do not mix these into the fixed-era learner:

• cross-era semantic transport;
• evaluator/ontology revision;
• counterfactual non-capture;
• coverage;
• downstream deference/corrigibility.

The purpose is to know whether the learner-level spine is finished before returning to full legitimacy.

---

Deliverables

Produce:

1. REASONWISE_ACCOUNTING.md

• aggregate-to-reason cap lemma;
• countable/growing reason conditions;
• exact counterexamples where aggregate safety does not give per-reason Progress.

1. CAPACITY_VS_SAFETY.md

• local authority-capacity set;
• lifetime SafeCert;
• sufficient routes connecting them;
• examples showing why they are not the same.

1. FIXED_ERA_THEOREM.md

• final theorem statement;
• finite-horizon bound;
• exact dependency graph;
• explicit LI instantiation.

1. AFFORDABLE_SCHEDULING.md

• the corrected viability problem in \(a\)-space;
• at least one positive existence theorem;
• many-reason analysis;
• remaining open questions.

1. Targeted patches to stale PR75 documents, especially any definition still typing realized controls as predictable.

1. Exact-rational fixtures for:

• aggregate enforcement safe while one reason-book has large positive value because another has large negative value;
• reasonwise cap recovered from complementary liability floors;
• local per-date authority caps with unbounded lifetime loss when allowances are nonsummable;
• signed-account cross-subsidization beating independent per-date budgeting;
• divergent allocated service with finite liability;
• multiple persistent reasons competing for finite authority capacity.

Success criterion

The pass succeeds if it can answer:

Conditional on an answerability-generated service load, do we now have a mathematically complete construction showing that a traderized Logical Inductor can receive persistent normative authority, make claim-weighted Progress up to explicit settlement/transport residuals, and retain its original epistemic guarantee?

If yes, state that theorem cleanly and identify existence of an affordable service schedule as the next problem.

If no, identify the exact remaining missing arrow and produce the smallest counterexample.

---

Fifth dispatch — causal affordability existence and normative capacity

Fifth pass inside PR75: causal affordability existence and normative capacity

Stay inside the existing PR75 research round:

projects/normativity/legitimacy/rounds/2026-08-31-normative-affordability/

Do not open a new research round unless the workspace rules strictly require it.

The previous dispatches have now largely closed the fixed-era construction/composition theorem. Treat the following as the current best architecture unless you find an actual mathematical error:

$$ \text{Answerability claims} \to \text{predictable allocated authority }a \to \text{reactive traderized-LI control} \to \text{service-weighted Progress} \to \text{Service Transport} \to \text{claim-weighted Sustainable Progress}, $$

with bounded lifetime liability preserving ordinary LI.

In particular:

• service is allocated authority

$$ a_{t,j}=\beta_{t,j}, $$

not realized force;

• realized force is endogenous:

$$ f_{t,j}=a_{t,j}d_{t,j}; $$

• the predictable object is a reactive control law

$$ \kappa_t:P\mapsto u_t(P); $$

• the fixed-era composition theorem is conditional on a service-faithful authority schedule, appropriate reason/row-level account bounds, settlement-friction residuals, and transport stability;
• local authority capacity and lifetime SafeCert are distinct;
• normative-force feasibility and affordability are distinct.

The main purpose of this pass is now:

$$ \boxed{ \textbf{When does an affordable authority schedule actually exist?} } $$

and, closely related,

$$ \boxed{ \textbf{What is the correct mathematical object corresponding to normative capacity / overload?} } $$

Do not spend most of the pass redescribing the architecture. Push the causal existence mathematics.

---

1. First repair: audit the claimed convexity of the sustainable authority-rate region

AFFORDABLE_SCHEDULING.md currently defines something like

$$ \mathcal A = \left\{ \lambda\in\mathbb R_+^R: \exists\text{ causal SafeCert policy with } \liminf_N \frac{A_N^r}{N}\ge\lambda_r \right\}. $$

It then claims that \(\mathcal A\) is convex because policies can be time-shared.

Do not assume this is correct.

In a history-dependent causal system, interleaving policies may alter:

• market prices;
• account slack;
• live assessment worlds;
• future capacity;
• future docket state;
• the histories on which the constituent policies would have acted.

So naïve time-sharing need not reproduce a convex combination of the original trajectories.

Task A1 — counterexample or theorem

Either:

1. produce the smallest exact counterexample showing that two achievable rate vectors

$$ \lambda^{(1)},\lambda^{(2)}\in\mathcal A $$

can have a convex combination outside \(\mathcal A\);

or:

1. prove convexity under explicit structural assumptions.

A useful counterexample template is an irreversible history-dependent mode switch:

• one initial policy choice puts the environment/account system permanently into a mode supporting only reason \(1\);
• another choice puts it into a mode supporting only reason \(2\);
• the two endpoint rate vectors are individually achievable;
• no causal policy achieves their midpoint.

But do not use this exact fixture if it does not fit the abstraction faithfully.

Task A2 — characterize when time-sharing works

Identify sufficient closure assumptions such as:

• reset/renewal structure;
• concatenation closure;
• history-independent per-date capacity;
• account additivity with independent budget partitions;
• stationary environment dynamics;
• block policies whose starting-state dependence can be controlled;
• convexification by randomized policies, if randomness is part of the theory.

State the weakest useful theorem you can defend.

Until such a theorem is proved, replace any unconditional claim

$$ \mathcal A\text{ is convex} $$

with the weaker fact that is genuinely automatic:

$$ \boxed{ \mathcal A\text{ is downward closed}. } $$

Distinguish:

• the raw causal sustainable-rate region;
• its convex hull;
• a time-sharing closure, if one exists;
• an interior/slack concept.

Do not call the convex hull “the achievable region” unless it is actually achievable.

---

2. Separate normative infeasibility from normative insolvency

The current scheduling discussion sometimes says that all overload is liability overload because there is no shared conformance server.

Sharpen this.

There are at least two conceptually distinct failure modes.

Force / semantic infeasibility

Examples:

• empty joint normative region;
• unpriceable rows;
• illegal/noncontinuous compiled control;
• impossible service timing requirements;
• semantic incompatibility before affordability is considered.

Call this schematically:

$$ \boxed{\text{normative infeasibility}.} $$

Affordability failure

The normative load is meaningful/actionable, but no causal service schedule can satisfy the required trace while preserving the liability account:

$$ \boxed{\text{normative insolvency}.} $$

Determine precisely what the traderized-LI modulus establishes:

Conditional on force feasibility, there is no additional shared-server competition in the conformance inequality; additional reasons consume underwriting/liability, not a finite total amount of enforcement force.

State this as a theorem or lemma rather than saying that “every overload is liability overload.”

This distinction may become important later when Answerability receives an overload certificate: it should know whether the failure was semantic inconsistency or insufficient underwriting capacity.

---

3. Put the causal affordability-existence problem in its final type

Use the corrected variables.

At history \(h_t\), state includes at least:

$$ z_t = \text{Answerability/service state} $$

and a settlement-relative account state

$$ V_t:\Omega_t^{\rm live}\to\mathbb R. $$

Answerability supplies a set of admissible future service traces, or equivalently constraints on allocations \(a_t^r\).

The policy chooses predictable allocated authority

$$ a_t $$

and hence a reactive control law

$$ \kappa_t. $$

The engine produces the fixed point and endogenous defect/force:

$$ P_t,\qquad d_t,\qquad f_t=a_td_t. $$

Settlement then updates the live set and account.

Give a precise causal recursion.

The primal existence question should look approximately like:

$$ \exists\pi \quad \forall\text{ admissible exterior/settlement histories} $$

such that:

1. the service trace generated by \(\pi\) is admissible under Answerability;
2. every persistent claim receives the required amount/timing of allocated authority;
3. the induced reactive control history satisfies

$$ V_N(\omega)\ge-B $$

for every live \(\omega\) at every prefix.

Be explicit about all quantifier order.

This should be the canonical definition of affordability existence for the fixed era.

---

4. Identify the correct state variable for signed-account viability

Route B currently uses something like

$$ \sigma_t = B+ \min_{\omega\in A_{t-1}} V_{t-1}(\omega) $$

as available account slack.

Check whether this scalar is sufficient state for future safety.

Because the live set shrinks and different worlds may become worst at different dates, it may be necessary to retain the entire profile

$$ V_t(\cdot) $$

rather than only its current minimum.

Investigate:

• Can two account profiles with equal current minimum have different future viable authority sets?
• Does the narrowing settlement set make the scalar minimum Markov-sufficient?
• Under what structural assumptions can the account profile be compressed to one scalar?
• Is there a minimal sufficient statistic for affordability control?

Construct a rational counterexample if the minimum alone is insufficient.

This matters greatly for the eventual dynamic-programming / viability formulation.

---

5. Strengthen the positive existence theory

The current one-reason theorem is:

if

$$ \sum_t \sqrt{a_t c}\,\sigma_t<\infty $$

and

$$ \sum_ta_t=\infty, $$

then persistent service is affordable.

This is useful, but still an offline sufficient condition.

Develop several increasingly strong existence results.

E1 — deterministic/offline benchmark

State the sharpest transparent sufficient condition for one reason.

Try to understand whether the square-root condition is near necessary in the conservative certificate regime.

For fixed depth sequence \(D_t\) and engine scale \(m_t=\epsilon_t+M_t\), characterize when there exists some

$$ a_t\ge0 $$

such that

$$ \sum_ta_t=\infty $$

but

$$ \sum_t\sqrt{a_tm_t}D_t<\infty. $$

This is a pure sequence-space existence problem.

Can it be characterized directly in terms of the decay of

$$ D_t\sqrt{m_t}? $$

Do not settle for polynomial examples if there is an exact characterization.

For example, investigate whether the existence of such \(a_t\) has an \(\ell^p\)-style necessary-and-sufficient condition.

This may yield a very clean mathematical notion of affordable persistent enforcement.

E2 — many reasons, known future

Generalize the sequence characterization to countably many reasons with summable liability budgets.

Determine when there exist allocations \(a_t^r\) with

$$ \sum_ta_t^r=\infty \qquad\forall r $$

while

$$ \sum_{t,r} \sqrt{a_t^rm_t}\,D_t^r <\infty. $$

Because there is no conformance competition, this may decompose almost completely by reason except for the total liability budget.

Find the clean theorem.

E3 — causal/predictable depths

Now assume \(D_t^r\) becomes observable only at time \(t\), but before choosing \(a_t^r\).

Can a universal causal scheduler achieve persistent service whenever some offline affordable schedule exists?

If not, exhibit the online/offline gap.

If yes under regularity assumptions, state them.

This is likely a real online-algorithms question.

---

6. Investigate the key sequence-space problem deeply

The square-root liability law suggests a potentially elegant mathematical characterization.

Set

$$ q_t:=D_t\sqrt{m_t}. $$

We seek

$$ a_t\ge0 $$

such that

$$ \sum_ta_t=\infty $$

but

$$ \sum_tq_t\sqrt{a_t}<\infty. $$

Let

$$ x_t=\sqrt{a_t}. $$

Then the question becomes:

$$ \boxed{ \exists x_t\ge0: \quad \sum_tx_t^2=\infty, \qquad \sum_tq_tx_t<\infty? } $$

This looks like a clean functional-analysis / sequence-space problem.

Determine the exact necessary-and-sufficient condition on \(q=(q_t)\).

For example:

• Is it equivalent to \(q\in\ell^2\)?
• Is \(\ell^2\) sufficient but not necessary?
• What happens at borderline sequences?
• Can block constructions beat naïve choices?
• What is the dual characterization?

Do this carefully.

If there is an exact theorem here, it may be one of the most elegant mathematical results in the affordability theory:

a normative constraint admits infinite total service on finite conservative liability iff its settlement-friction sequence lies in a particular sequence class.

Check whether the “conservative liability” qualification is necessary.

Then interpret the theorem in the original variables.

---

7. Signed-account viability versus conservative underwriting

The conservative route uses

$$ \sum_tq_tx_t<\infty. $$

But the true account is signed:

$$ V_N(\omega) = \sum_t a_td_t(d_t-s_t(\omega)). $$

Positive work may fund negative friction later.

Determine how much strictly larger the signed-account affordable class can be.

Construct a minimal example where:

• no conservative summable-charge schedule can provide persistent service;
• but a signed-account adaptive policy can provide persistent service indefinitely.

The previous finite two-date example only shows a larger local cap. Seek an infinite-horizon separation.

This could be a major result.

Potential shape:

$$ \boxed{ \text{conservative affordability} \subsetneq \text{signed-account affordability}. } $$

Then ask whether the signed-account class has a natural characterization in terms of cumulative/drift conditions.

Do not use “self-financing” as a new name.

---

8. Adversarial docket existence

Now bring Answerability arrivals back in.

Suppose reasons arrive predictably over time, each carrying:

• a claim/service trace requirement;
• a priceable normative row or region;
• an evolving settlement-friction/depth process.

Ask:

Under what conditions is there a causal authority scheduler satisfying every persistent reason?

The current countable round-robin theorem assumes useful dates have already been partitioned.

Try more realistic conditions.

Candidates:

Persistent opportunity

For every persistent reason \(r\),

$$ \sum_t \operatorname{cap}_t^r=\infty. $$

Is that alone enough?

Probably not if many reasons share liability budget. Construct the obstruction.

Weighted capacity condition

For every finite subset \(S\) of persistent reasons, require some cumulative capacity inequality.

Is there a Hall-type condition?

Be careful: the per-date authority set is non-convex and liability is diachronic, so a standard flow theorem may fail.

Slack/interior condition

Can you formulate an arrival/load condition analogous to queueing stability:

$$ \text{Answerability demand lies strictly inside causal affordable capacity} $$

and prove an online scheduling theorem?

Do not invoke “interior of \(\mathcal A\)” until the convexity/time-sharing question is resolved.

If an exact theorem is too hard, isolate the minimal model in which one can be proved.

---

9. Reassess the role of backlog

The current recurrence

$$ b_{t+1}^r = [b_t^r+c_{t+1}^r-a_t^r]_+ $$

is a convenient scheduling abstraction.

But Deferred Service Transport permits richer structure:

• deadlines;
• semantic stability degrading with delay;
• heterogeneous service effectiveness;
• claims satisfied by specific kinds of later service.

Determine whether scalar backlog is actually sufficient for the Answerability service state.

Likely:

$$ \boxed{ \text{backlog is a sufficient realization for one simple service semantics,} } $$

not the generic interface.

Keep the generic affordability theory based on admissible service traces / transport plans, and use backlog only for particular existence theorems.

If appropriate, patch any text suggesting backlog is primitive.

---

10. Dynamic overload and duality: sharpen the target, don't force a theorem

Once the primal has been clarified, state exactly what an overload certificate should certify.

There are now at least two failures:

Normative infeasibility certificate

No joint actionable/priceable normative realization exists.

This may be ordinary geometric separation.

Normative insolvency certificate

The normative requirements are force-feasible, but no causal policy can service them while keeping the account viable.

This requires separating a service obligation from a causal signed-account capacity.

The finite-path Farkas certificate only addresses a simpler static problem.

Ask whether a dynamic certificate could be:

• a potential \(H(z,V)\) whose drift under every safe action is too small to discharge the incoming normative load;
• an adversarial strategy;
• a supermartingale;
• a Bellman value function proving the viability kernel empty.

Do not claim completeness.

The goal is to state the exact dual problem we eventually want.

---

11. Patch the current overclaims

Within PR75, correct at least:

1. unconditional convexity of the sustainable authority-rate region;
2. “every overload is liability overload” → qualify by force feasibility;
3. any suggestion that separate reason underwriting is necessary for every form of reasonwise Progress;
4. any place where local authority capacity is treated as the SafeCert itself;
5. any stale reference to the pre-\(a=\beta\) service variable.

Do not rewrite the entire round. Make targeted corrections and preserve provenance.

---

12. Decide whether the fixed-era construction should now be frozen

After this pass, explicitly answer:

Is there any remaining reason to keep modifying the fixed-era composition theorem itself?

If the answer is no, recommend freezing:

• FIXED_ERA_THEOREM.md;
• the S1–S6 schematic stack;
• the \(\beta\)/\(\beta d\) typing;
• reasonwise row accounting;
• Service Transport.

Then subsequent work in PR75 should be labeled existence/characterization, not continued composition repair.

This is important to prevent the research round from endlessly reopening already-stable interfaces.

---

Deliverables

Produce or update:

1. CAUSAL_CAPACITY.md

• corrected definition of sustainable authority capacity;
• counterexample to naïve convexity or theorem giving sufficient conditions;
• downward closure;
• time-sharing/concatenation conditions.

1. PERSISTENT_AFFORDABILITY.md

• solve the sequence problem

$$ \sum x_t^2=\infty,\qquad \sum q_tx_t<\infty; $$

• give the strongest necessary-and-sufficient condition you can obtain;
• map it to persistent authority on finite liability.

1. SIGNED_VS_CONSERVATIVE.md

• infinite-horizon comparison between conservative sum-of-suprema underwriting and true signed-account affordability.

1. ONLINE_EXISTENCE.md

• formulate and, where possible, prove causal scheduling results against predictable/adversarial arrivals;
• clearly mark offline versus online hypotheses.

1. OVERLOAD_TARGET.md

• separate normative infeasibility from insolvency;
• state the desired dynamic dual/certificate problem.

1. Targeted patches to current PR75 documents.

1. Exact-rational fixtures for:

• failure of causal rate-region convexity, if applicable;
• success of time-sharing under an explicit resettable model;
• persistent conservative affordability;
• failure of persistent affordability with nondecaying settlement depth;
• infinite-horizon signed-account affordability strictly beyond conservative budgeting;
• adversarial arrival process defeating a naïve fair scheduler.

Priority

Spend the most effort on:

1. the exact sequence-space characterization

$$ \exists x:\quad x\notin\ell^2,\quad qx\in\ell^1; $$

1. the causal convexity/time-sharing issue;
2. an infinite-horizon separation between conservative and signed-account affordability;
3. a clean online existence theorem or counterexample.

These are likely to produce actual mathematical results rather than further architecture.

Success criterion

The pass succeeds if it can turn the current statement

“existence of an affordable schedule remains open”

into a substantially more structured theory:

• an exact or near-exact characterization in the one-reason benchmark;
• a correct notion of causal normative capacity;
• at least one genuinely online existence result or impossibility result;
• a clear separation between semantic infeasibility and underwriting insolvency.

Do not reopen the fixed-era learner construction unless one of these investigations finds an actual contradiction in it.

---

Sixth dispatch — repair persistent affordability and close the exogenous-friction benchmark

Sixth pass inside PR75: repair persistent affordability and close the exogenous-friction benchmark

Stay inside PR75:

projects/normativity/legitimacy/rounds/2026-08-31-normative-affordability/

Treat the fixed-era composition theorem as frozen unless this pass finds an actual contradiction in it.

Read at minimum:

• PERSISTENT_AFFORDABILITY.md
• ONLINE_EXISTENCE.md
• SIGNED_VS_CONSERVATIVE.md
• CAUSAL_CAPACITY.md
• OVERLOAD_TARGET.md
• AFFORDABLE_SCHEDULING.md
• FIXED_ERA_THEOREM.md
• SERVICE_TRANSFER.md
• CAPACITY_VS_SAFETY.md

The previous pass found a genuinely strong benchmark theorem, but several interpretations around it are overextended. This pass should repair those exactly.

Current benchmark result to preserve unless disproved

For a fixed predictable friction sequence

$$ q_t:=D_t\sqrt{m_t}, \qquad m_t:=\epsilon_t+M_t, $$

and the conservative charge

$$ \sum_t q_t\sqrt{a_t}, $$

persistent allocated authority

$$ \sum_t a_t=\infty $$

is achievable on finite lifetime budget \(B>0\) iff

$$ \boxed{\liminf_t q_t=0.} $$

Equivalently, with \(x_t=\sqrt{a_t}\),

$$ \exists x_t\ge0: \qquad \sum_t x_t^2=\infty, \qquad \sum_t q_tx_t<\infty $$

iff \(\liminf q_t=0\).

Also preserve the finite-horizon conservative optimum unless there is an actual flaw:

$$ \max_{\sum_{t<N}q_t\sqrt{a_t}\le B} \sum_{t<N}a_t = \frac{B^2}{(\min_{t<N}q_t)^2}. $$

What needs work is everything built around this result.

---

1. Repair the sharp worst-case charge characterization

PERSISTENT_AFFORDABILITY.md currently claims that replacing the conservative charge

$$ q_t\sqrt{a_t} $$

with the exact/sharp robust worst-case date charge leaves the same criterion unchanged.

Audit this carefully.

The current sharp charge is stated as

$$ L_t(a)= \begin{cases} a s_t^2/4, & a\le 4m_t/s_t^2, \\[1mm] s_t\sqrt{a m_t}-m_t, & a>4m_t/s_t^2. \end{cases} $$

The previous review found a counterexample to the claimed \(q_t\)-criterion:

$$ s_t=\frac1t, \qquad m_t=t^4. $$

Then

$$ q_t=s_t\sqrt{m_t}=t, $$

so \(\liminf q_t>0\), but with \(a_t=1\),

$$ L_t(1)=\frac1{4t^2}, $$

hence

$$ \sum_tL_t(1)<\infty \quad\text{and}\quad \sum_ta_t=\infty. $$

Verify this fixture exactly.

Then find the correct theorem.

A strong candidate is:

For increasing concave date-cost functions \(L_t:[0,\infty)\to[0,\infty)\) with \(L_t(0)=0\),

$$ \exists a_t\ge0: \quad \sum_t a_t=\infty, \qquad \sum_tL_t(a_t)<\infty $$

iff

$$ \liminf_tL_t(1)=0. $$

Prove or refute this exact statement.

The intended necessity proof uses concavity:

for \(0\le a\le1\),

$$ L_t(a)\ge aL_t(1). $$

If \(L_t(1)\ge c>0\) eventually, finite total charge bounds the total mass of all small allocations; only finitely much mass can come from \(a_t\ge1\) if each such date has uniformly positive charge.

Make every needed assumption explicit.

If \(L_t(1)\) is not the correct primitive, find the exact one.

Then specialize it to the traderized sharp charge and simplify the criterion in terms of \(s_t,m_t\).

This may be the actual strongest robust-persistence theorem in the round.

---

2. Separate three affordability classes cleanly

The round currently has at least three different notions.

Conservative certificate

$$ \sum_t q_t\sqrt{a_t}\le B. $$

This is easy to certify and pessimistic.

Sharp robust datewise certificate

$$ \sum_t L_t(a_t)\le B, $$

where \(L_t\) is the exact worst-case charge allowed by the MarketMaker modulus.

Actual signed-account affordability

$$ V_N(\omega) = \sum_{t<N}a_td_t(d_t-s_t(\omega)) \ge-B $$

for every live world and horizon along the realized trajectory.

Produce a clean inclusion diagram.

Determine exactly which implications hold:

$$ \text{conservative} \Rightarrow \text{sharp robust} \Rightarrow \text{signed account} $$

or whatever the correct direction is.

Give strict-separation examples.

In particular preserve the important pathwise phenomenon:

if

$$ d_t=0 $$

then realized force and account increment vanish even if the live worlds badly violate the norm.

But do not confuse:

• a robust policy guarantee;
• a realized-path affordability fact.

State precisely what the fixed-era construction consumes and what an existence theorem must guarantee before the market response is known.

---

3. Repair the online theorem

The qualitative theorem appears sound:

Given an exogenous predictable conservative-friction sequence \(q_t\), a causal threshold/tranche scheduler achieves persistent affordable service iff an offline scheduler can, namely iff

$$ \liminf q_t=0. $$

Preserve or repair this.

But delete the claim that the online rule is within a factor \(4\) of offline cumulative authority unless a correct competitive notion supports it.

Construct the explicit counterexample.

If the threshold rule has reached level \(k\), its tranche is approximately

$$ b_k=B2^{-k}. $$

If an unexpectedly tiny friction

$$ q\ll2^{-k} $$

then appears, the online rule buys only

$$ (b_k/q)^2, $$

whereas an offline scheduler could put essentially the full remaining/initial budget there and buy

$$ (B/q)^2. $$

The competitive ratio can therefore go to zero.

State the correct distinction:

$$ \boxed{ \text{no online penalty for the binary property “persistence exists”} } $$

but not

$$ \boxed{ \text{no online penalty for amount/rate of accumulated authority}. } $$

Then ask:

• Is any nontrivial competitive guarantee possible?
• Is the right comparison additive, multiplicative, prefixwise, or asymptotic?
• If not, prove an impossibility result.

Do not spend too much time on rates unless there is a clean theorem.

---

4. Scope the persistence theorem to exogenous or policy-stable friction

This is crucial.

The benchmark treats

$$ q_t=D_t\sqrt{\epsilon_t+M_t} $$

as an observed external sequence.

But in the actual traderized learner:

• \(M_t\) may depend on the price/history generated by earlier authority;
• the live-world set may depend on interaction/settlement;
• hence \(D_t\) may depend on the policy;
• future docket state may also depend on prior interaction.

So in the closed loop, different authority policies may induce different future friction trajectories.

Make this explicit.

Define at least two models.

Exogenous-friction benchmark

$$ q_t $$

is fixed independently of the authority policy.

P1/O1 apply directly.

Policy-dependent friction

$$ q_t=q_t(h_{t-1},a_{<t},\text{settlement history},\ldots). $$

The persistence problem becomes

$$ \exists\pi\quad \forall\text{ admissible exterior histories} $$

rather than a sequence selection problem.

Determine which existing claims survive if friction is merely policy-stable, e.g. satisfying a uniform bound independent of policy.

Possible condition:

$$ \forall\pi,\qquad \liminf_t q_t^\pi=0. $$

Does the threshold scheduler then work uniformly?

Be very careful with quantifier order:

$$ \forall\pi\,\liminf q^\pi=0 $$

is not obviously enough to construct one policy whose own induced sequence has dips.

Try to identify a usable condition like:

$$ \forall h\text{ reachable},\forall k, \exists\text{ future admissible continuation with }q\le2^{-k} $$

versus a robust condition forcing the dip on every continuation.

If this gets difficult, clearly delimit the benchmark theorem rather than overclaim.

---

5. Combine affordability with service admissibility

The benchmark characterizes only

$$ A_N=\sum_{t<N}a_t\to\infty. $$

But Answerability does not necessarily regard every divergent allocation trace as legitimate service.

A sparse low-friction schedule may violate:

• deadlines;
• bounded-delay requirements;
• transport stability;
• semantic preservation over long deferrals;
• minimum cumulative service requirements.

This is now likely the most important interface between affordability and Answerability.

Define:

$$ \mathcal L_A = \text{Answerability-admissible allocated-service traces}. $$

Then the actual existence question is:

$$ \boxed{ \exists a\in\mathcal L_A \quad \text{such that }a\text{ is safely underwritten}. } $$

For the conservative exogenous-friction benchmark:

$$ \exists a\in\mathcal L_A: \qquad \sum_tq_t\sqrt{a_t}\le B. $$

Investigate several concrete \(\mathcal L_A\).

Bounded delay

Every claim at time \(t\) must receive service by \(t+H\).

Characterize affordability.

Now sparse arbitrarily late dips may be useless.

Can the criterion be expressed as a block condition such as:

$$ \inf_{s\in[t,t+H]}q_s $$

dipping sufficiently often or having a summability property?

Cumulative lower bound

Suppose Answerability requires

$$ A_N\ge F(C_N)-K. $$

When can such a growth lower bound coexist with finite liability?

Weighted transport stability

Suppose delay \(s-t\) incurs error

$$ \epsilon(t,s). $$

Formulate the optimization problem trading liability against semantic transport error.

This may give a very natural frontier:

$$ \boxed{ \text{cheap enforcement dates} \quad\text{vs.}\quad \text{semantically timely enforcement dates}. } $$

Try to derive at least one theorem rather than only state the tension.

---

6. Revisit “persistence does not compete across reasons”

The current benchmark says that because the qualitative criterion is independent of the positive budget \(B\), countably many individually persistable reasons can each receive a geometric sub-budget.

That is correct under independent exogenous reasonwise friction sequences.

Scope it accordingly.

Investigate failure modes:

• policy-dependent friction;
• shared rows;
• Answerability deadlines;
• transport requirements;
• shared semantic constraints;
• finite liability budgets with stronger-than-persistence service requirements.

State the exact theorem:

If each reason \(r\) has an exogenous friction sequence \(q_t^r\) with

$$ \liminf q_t^r=0, $$

and mere divergence

$$ A_N^r\to\infty $$

is sufficient service, then every countable family can be persistently served simultaneously on any positive global budget by splitting the budget geometrically.

Then stop.

Do not elevate this to a generic statement about “no competition between reasons.”

The no-conformance-competition lemma remains independently true at the force layer.

---

7. Repair CAUSAL_CAPACITY.md

The current document correctly rejects naïve convexity but still overstates the geometry of the long-run rate region.

Patch these claims.

The finite-stock cumulative authority frontier

$$ \{(x_r^2)_r:\sum_rx_r\le B\} $$

is nonconvex.

That is valid.

But with floored friction and finite lifetime budget, every long-run rate is zero, so this fixture does not imply a nonconvex long-run rate region.

Likewise, rapidly decaying friction can yield

$$ \mathcal A=\mathbb R_+^R, $$

which is convex.

So withdraw claims such as:

$$ \mathcal A \text{ is nonconvex whenever nondegenerate}. $$

Keep only what is proved.

A useful corrected hierarchy may be:

• finite-horizon cumulative authority frontier — often nonconvex;
• persistence set — simple in the exogenous conservative benchmark;
• long-run rate region — model-dependent and not central to the current composition theorem.

If rates are not needed, de-emphasize the object entirely.

---

8. Produce the exact sharp-persistence finite-horizon optimization if possible

For the conservative charge, P2 gives a beautiful exact finite-horizon optimizer.

Ask whether the sharp robust charge admits an equally clean finite-horizon solution:

$$ \max \left\{ \sum_{t<N}a_t: \sum_{t<N}L_t(a_t)\le B \right\}. $$

Because each \(L_t\) is concave increasing, maximizing a linear function under a sum of concave costs is again concentration-favoring, but the branch structure may matter.

Questions:

• Is the optimum always concentrated on one date?
• If so, which scalar ranks the dates?
• Is it

$$ \max_t L_t^{-1}(B)? $$

exactly?

• Can splitting ever beat concentration when each \(L_t\) is concave?
• Is there a simple proof from extreme-point or exchange arguments?

If yes, state:

$$ \boxed{ A_N^{\max}(B) = \max_{t<N}L_t^{-1}(B). } $$

Then persistent sharp-robust affordability is exactly:

$$ \sup_tL_t^{-1}(B)=\infty. $$

Determine the equivalent tail criterion.

This may be cleaner than the \(L_t(1)\) characterization.

Compare both.

---

9. Signed-account existence: keep outcome facts separate from policy guarantees

SIGNED_VS_CONSERVATIVE.md correctly observes that the same friction sequence can support very different realized account trajectories.

That means no characterization depending only on \(q_t\) can describe signed-account affordability.

Push this one step further.

Ask what predictable lower bound on account drift would suffice for a robust signed-account existence theorem.

Suppose there is a predictable function

$$ \underline{\Delta V}_t(a_t,V_t,z_t) $$

such that every possible market/settlement response satisfies

$$ \Delta V_t\ge \underline{\Delta V}_t. $$

Then affordability reduces to viability for a deterministic lower-bound account.

Can traderized LI supply any nontrivial lower bound sharper than the conservative worst-case charge?

If not, say so.

Do not attempt a full signed-account characterization without such a bound.

---

10. Repair the overload-certificate story

For the conservative exogenous-friction benchmark, mathematical insolvency is:

$$ \liminf_tq_t>0. $$

But the statement

$$ q_t\ge q_0 \quad\text{for all sufficiently large }t $$

is an infinite future claim.

A finite prefix does not certify it.

So distinguish:

Mathematical characterization

$$ \liminf q_t>0. $$

Finite authenticated certificate

A finite proof object establishing a tail lower bound, perhaps from:

• the normative/settlement model;
• monotonicity;
• a symbolic argument;
• a theorem proving

$$ q_t\ge q_0 \quad\forall t\ge T. $$

The certificate should contain both:

$$ (T,q_0) $$

and evidence for the universal tail statement.

If no such proof object exists, insolvency may be true but not yet settleably certified.

This is exactly analogous to the broader Answerability principle that reasons cannot be silently dropped merely because a scheduler predicts future impossibility.

Patch OVERLOAD_TARGET.md accordingly.

---

11. Clarify the current hierarchy of existence results

By the end, organize the existence theory into something like:

Level E0 — fixed-era construction

Frozen.

Level E1 — exogenous conservative friction

Exact characterization:

$$ \liminf q_t=0. $$

Qualitative online/offline equivalence.

Level E2 — exogenous sharp robust charge

Exact characterization discovered in this pass.

Level E3 — service-admissible conservative scheduling

Intersect affordability with Answerability's admissible service traces.

Level E4 — policy-dependent friction / closed loop

True causal viability problem.

Level E5 — signed-account robust scheduling

Requires response-dependent account-drift theory.

This hierarchy would prevent future work from accidentally importing an E1 result into E4/E5.

---

12. Freeze what is actually frozen

Do not modify FIXED_ERA_THEOREM.md unless an actual contradiction is found.

At the end, say explicitly whether the following can now be treated as stable:

• service \(=\beta\);
• realized force \(=\beta d\);
• control is a reactive law;
• coercive Progress;
• reasonwise row accounting;
• Service Transport;
• fixed-era Sustainable Progress;
• SafeCert/preservation composition.

All new corrections should belong to existence/characterization, not to the learner construction.

---

Deliverables

Update or produce:

1. PERSISTENT_AFFORDABILITY.md

• repair the sharp-charge theorem;
• keep conservative P1/P2 clearly scoped.

1. SHARP_PERSISTENCE.md

• exact existence criterion for general/sharp date costs \(L_t\);
• finite-horizon optimizer if obtainable.

1. ONLINE_EXISTENCE.md

• qualitative online equivalence;
• delete false factor-4 competitive claim;
• optional competitive impossibility result.

1. SERVICE_ADMISSIBLE_EXISTENCE.md

• bounded-delay and/or cumulative-service version;
• at least one theorem showing how service fidelity changes the cheap-date criterion.

1. CLOSED_LOOP_EXISTENCE.md

• exogenous vs policy-dependent friction;
• exact quantifier structure;
• what benchmark results survive.

1. OVERLOAD_TARGET.md

• mathematical insolvency vs finite authenticated certificate.

1. Targeted corrections to CAUSAL_CAPACITY.md, SIGNED_VS_CONSERVATIVE.md, and the PR summary.

1. Exact-rational tests for:

• the counterexample

$$ s_t=1/t,\ m_t=t^4,\ a_t=1; $$

• conservative versus sharp robust affordability;
• failure of any constant competitive ratio for cumulative authority;
• bounded-delay service where \(\liminf q_t=0\) but persistence is not service-admissible;
• finite proof of a tail friction floor versus mere observed-so-far floor.

Highest priorities

Spend most effort on:

1. exact sharp-cost persistence characterization;
2. affordability + service-admissibility;
3. correct scope of exogenous versus policy-dependent friction;
4. authenticated insolvency certificates.

Do not spend this pass reopening the composition theorem or inventing more schematic vocabulary.

Success criterion

The pass succeeds if we can say, without overclaim:

We have an exact theory of persistent normative authority under exogenous robust underwriting, know exactly how the conservative and sharp certificates differ, understand when causal scheduling loses nothing, and have isolated the genuinely harder existence problem as the intersection of affordability with Answerability service constraints and closed-loop friction dynamics.

If the sharp charge yields an equally clean characterization theorem, make that the mathematical centerpiece of the pass.

---

Seventh dispatch — exact serviceability under bounded-delay transport

Seventh pass inside PR75: exact serviceability under bounded-delay transport

Stay inside PR75:

projects/normativity/legitimacy/rounds/2026-08-31-normative-affordability/

Treat the fixed-era composition theorem as frozen.

Also treat the unconstrained persistence theory as substantially complete, subject only to the targeted cleanup in §1 below.

The main purpose of this pass is to move from

$$ \text{“can this reason receive unbounded authority eventually?”} $$

to

$$ \boxed{ \text{“can the authority it is owed be delivered on an Answerability-admissible schedule?”} } $$

The previous pass established that unconstrained affordability prefers sparse low-cost dates. Answerability may forbid arbitrarily long waiting. This tension now looks like the most important mathematical seam in the round.

Read at minimum:

• SERVICE_TRANSFER.md
• SERVICE_ADMISSIBLE_EXISTENCE.md
• SHARP_PERSISTENCE.md
• PERSISTENT_AFFORDABILITY.md
• ONLINE_EXISTENCE.md
• CLOSED_LOOP_EXISTENCE.md
• AFFORDABILITY.md
• FIXED_ERA_THEOREM.md
• OVERLOAD_TARGET.md

Do not perform another broad architectural pass.

---

1. First make three small persistence-theory repairs

Before doing new work, patch the following precisely.

1A. Sharp persistence is not generically depth-only

The correct general criterion is

$$ \boxed{ \liminf_t L_t(1)=0. } $$

For the sharp robust LI cost,

$$ L_t(1)= \begin{cases} s_t^2/4, & s_t^2\le4m_t, \\[1mm] s_t\sqrt{m_t}-m_t, & s_t^2>4m_t. \end{cases} $$

Do not state unconditionally that this is equivalent to

$$ \liminf_ts_t=0. $$

For example,

$$ s_t=1,\qquad m_t\to0 $$

can make \(L_t(1)\to0\) with no depth decay.

Find the cleanest uniform comparison. A candidate is

$$ L_t(1) \asymp \min\{s_t^2,s_t\sqrt{m_t}\} $$

with universal constants.

Prove the sharp inequalities and state the exact conditions under which the criterion simplifies to

$$ \liminf s_t=0. $$

In particular, a fixed positive lower bound on \(m_t\) should be made explicit if used.

Interpret the two routes to low cost separately:

• shallow exclusion;
• cheap/easy-to-move engine.

1B. Separate S1 and S2 hypotheses

SHARP_PERSISTENCE.md S1 needs only the star-shaped condition

$$ a\mapsto L_t(a)/a \quad\text{nonincreasing}. $$

The finite-horizon concentration theorem S2 currently uses convexity of

$$ L_t^{-1}, $$

hence concavity of \(L_t\).

Retype the statements:

• S1: increasing + star-shaped + \(L_t(0)=0\);
• S2: add the actual stronger assumptions needed for the inverse/concentration argument.

Do not claim finite-horizon concentration for arbitrary star-shaped costs unless proved.

1C. Tighten the old delay-frontier fixture

SERVICE_ADMISSIBLE_EXISTENCE.md currently gives a heuristic delay frontier with cheap dates every \(g\) dates and average delay approximately \(g/2\).

Make the constant exact if this fixture is retained:

$$ \frac{0+1+\cdots+(g-1)}{g} = \frac{g-1}{2} $$

for the corresponding forward-service convention.

Also replace any statement that affordable schedules place service “only” on dip dates with the correct asymptotic claim: finite or summable authority may still be delivered elsewhere.

These are cleanup, not the main research task.

---

2. Formalize actual bounded-delay service

The existing blockwise theorem

$$ \sum_i\min_{s\in B_i}L_s(\alpha)<\infty $$

is useful, but it is not yet generic bounded-delay transport.

Now model claims explicitly.

Let claim mass

$$ c_t\ge0 $$

arrive at each date \(t\).

Let allocated service be

$$ a_s\ge0. $$

A transport plan

$$ T(t,s)\ge0 $$

must satisfy

$$ \sum_sT(t,s)=c_t $$

for claims being fully served,

$$ \sum_tT(t,s)\le a_s, $$

and bounded delay

$$ T(t,s)>0 \Longrightarrow t\le s\le t+H. $$

This is the simplest nontrivial Answerability service semantics.

The affordability problem is:

$$ \boxed{ \inf_{a,T} \sum_s L_s(a_s) } $$

subject to those transport constraints.

This should be the central mathematical problem of the pass.

---

3. Solve bounded-delay transport feasibility before adding costs

Ignore liability first.

For fixed claim masses \(c_t\), service capacities \(a_s\), and delay \(H\), characterize existence of \(T\).

Because each claim has the interval neighborhood

$$ [t,t+H], $$

this is an interval bipartite-flow problem and should admit a much simpler Hall condition than arbitrary matching.

Find the exact necessary-and-sufficient inequalities.

A likely form is some family of interval inequalities such as

$$ \sum_{t\in I}c_t \le \sum_{s\in N(I)}a_s $$

for every interval \(I\), where

$$ N(I)=I+[0,H]. $$

But exploit the ordered interval structure as aggressively as possible.

Questions:

1. Is it enough to check prefixes?
2. Is it enough to check all intervals?
3. Does forward-only bounded delay reduce feasibility to

$$ C_N\le A_{N+H} $$

plus some no-early-service condition?

1. Is EDF/FIFO greedy service optimal and complete?
2. Can the condition be written directly in terms of backlog:

$$ b_t^H=0 $$

for claims reaching age \(H\)?

Give a theorem and proof.

This theorem should connect directly to SERVICE_TRANSFER.md: bounded delay should construct the transport plan rather than merely be prose around it.

---

4. Then solve the minimum-cost bounded-delay problem

Now restore the date cost \(L_s(a_s)\).

Define

$$ \mathsf{Cost}_H(c,L) = \inf \left\{ \sum_sL_s(a_s): (a,T)\text{ satisfies bounded-delay service} \right\}. $$

Try to characterize this exactly or algorithmically.

The costs of interest are increasing and concave in authority.

That is important: concavity rewards concentration, while deadlines restrict how far concentration can go.

This is a very specific combinatorial optimization problem.

Investigate whether the optimum has a greedy structure:

wait within each claim's legal window, then batch as much outstanding service as possible onto the currently cheapest legal date.

Possible algorithms:

• earliest-deadline-first with cheapest-date batching;
• serve each deadline cohort at the cheapest point in its remaining window;
• dynamic programming over windows;
• min-cost flow after an appropriate transformation;
• an interval covering formulation.

Do not assume ordinary convex optimization applies: the objective is concave.

If the exact infinite-horizon problem is difficult, solve the finite-horizon version completely.

---

5. Find the exact bounded-delay affordability criterion

The unconstrained persistence criterion is a dip condition:

$$ \liminf_tL_t(1)=0. $$

With a uniform per-window service floor, the previous pass found a summability condition.

Now derive the true condition for an actual claim process.

Start with canonical claim streams.

Case A — unit claim every date

$$ c_t=1. $$

Every claim must receive one unit within \(H\) dates.

What is the exact finite-budget affordability criterion?

The service rate is now positive, so the answer may be extremely restrictive.

For concave costs, can claims be batched into packets of size at most \(H+1\)?

Does the optimum reduce to a partition into blocks whose service occurs at one chosen date?

For example, is a condition of the form

$$ \sum_k \min_{s\in[kH,(k+1)H+\text{offset}]} L_s(\text{block mass}) <\infty $$

necessary and sufficient?

Do not accept the old disjoint-window floor theorem as the answer unless it really coincides with exact claim-by-claim bounded delay.

Case B — general bounded claim stream

$$ 0\le c_t\le C. $$

Give sufficient and necessary conditions in terms of local claim mass and local cost minima.

Case C — sparse claims

Determine how the criterion interpolates back toward unconstrained persistence when claim arrivals themselves are sparse.

The goal is a theorem saying exactly how claim density interacts with cheap-date density.

---

6. Investigate a dual / Hall-style affordability condition

Transport feasibility alone may have an interval Hall condition.

Adding concave cost may yield a useful dual lower bound.

For any set/window of claims, the service satisfying them must fall inside a specific neighborhood. Therefore that neighborhood has a minimum liability cost for carrying the required mass.

Define something like

$$ \mathsf{CapCost}(J,m) = \min \left\{ \sum_{s\in J}L_s(a_s): \sum_{s\in J}a_s\ge m \right\}. $$

Because \(L_s\) is concave, this may simply be

$$ \boxed{ \mathsf{CapCost}(J,m) = \min_{s\in J}L_s(m) } $$

under the S2 hypotheses.

If so, every claim interval \(I\) yields a necessary lower bound

$$ \min_{s\in N(I)} L_s\!\left(\sum_{t\in I}c_t\right). $$

Can these local lower bounds be assembled into a necessary-and-sufficient certificate for global finite-budget service?

This would be very valuable.

It could look like a covering/packing dual:

• primal: batch claims onto cheap legal service dates;
• dual: select disjoint or suitably separated claim intervals whose mandatory costs sum beyond the budget.

Because the service neighborhoods are intervals, there may be an exact greedy interval-packing dual rather than a generic Farkas theorem.

Look hard for this.

---

7. Relate bounded-delay service to transport stability

The composition theorem does not actually require bounded delay for its own sake. It requires the stability inequality

$$ d_t\le L\,d_s+\epsilon $$

whenever claim \(t\) is serviced at \(s\).

Bounded delay is one possible mechanism for proving this.

Suppose the defect/reason semantics has temporal regularity such as

$$ |d_t-d_s| \le \omega(|s-t|) $$

for some modulus \(\omega\).

Then bounded delay \(H\) gives

$$ d_t \le d_s+\omega(H). $$

So the transport residual is

$$ \epsilon=\omega(H). $$

Now affordability and semantic accuracy trade off through \(H\):

• larger \(H\) exposes more cheap dates;
• larger \(H\) worsens \(\epsilon\).

This may produce a very clean quantitative theorem:

$$ \boxed{ \text{Sustainable Progress residual} \le L K F + \omega(H) } $$

provided bounded-delay affordability at \(H\) holds.

Try to derive a serviceability frontier

$$ H \mapsto \bigl( \mathsf{minimum\ liability}(H), \, \mathsf{transport\ error}(H) \bigr). $$

This is likely more meaningful than the earlier ad hoc delay fixture.

---

8. Define serviceability as the intersection, not a new primitive condition

Do not introduce another large abstract interface unless necessary.

For this pass, use:

$$ \mathcal L_A = \text{Answerability-admissible service traces}, $$

$$ \mathcal L_B = \text{affordable service traces}. $$

Then

$$ \boxed{ \text{serviceable} \iff \mathcal L_A\cap\mathcal L_B\neq\varnothing. } $$

The point is not to add a label; it is to characterize this intersection.

For bounded-delay claims:

$$ \mathcal L_A = \{a:\exists T\text{ with delay}\le H\}. $$

For robust exogenous underwriting:

$$ \mathcal L_B = \left\{ a:\sum_tL_t(a_t)\le B \right\}. $$

So the exact question is now finite and explicit.

---

9. Multi-reason serviceability: see whether competition reappears

Unconstrained qualitative persistence did not compete across countably many reasons, because every reason could receive an arbitrarily small positive budget.

That should stop being true once reasons impose service deadlines or quantitative local service requirements.

Test this.

Let each reason \(r\) have:

• claims \(c_t^r\);
• delay \(H_r\);
• date cost \(L_t^r(a)\).

The global liability budget couples them:

$$ \sum_{t,r}L_t^r(a_t^r)\le B. $$

Ask whether there is now a genuine Hall/capacity condition.

Construct the smallest example where:

• each reason is individually serviceable;
• the pair is not jointly serviceable.

This would be an important correction to the earlier “reasons don't compete” result:

$$ \boxed{ \text{unconstrained persistence doesn't compete;} \quad \text{timely service can.} } $$

If a useful multi-reason theorem emerges, state it. Otherwise establish the counterexample and leave the general problem open.

---

10. Online bounded-delay service

After solving the offline finite-horizon problem, ask what changes causally.

At time \(s\), the scheduler knows:

• all claims that have arrived;
• their deadlines;
• current \(L_s\) / friction;
• not future costs.

The unconstrained persistence problem had no qualitative online penalty because waiting was free.

Here waiting is not free: deadlines expire.

Therefore an online/offline gap should reappear.

Prove the smallest clean impossibility theorem.

For example:

• one claim;
• two legal service dates;
• first date has moderate cost;
• scheduler must choose whether to spend now;
• second date may be extremely cheap or extremely expensive.

This should already create a ski-rental/secretary-style tension.

Then ask whether any competitive ratio exists for bounded-delay service cost.

This is much more relevant than the unconstrained authority-amount competitive problem because now the scheduler is solving the actual Answerability requirement.

If useful, derive the optimal two-date competitive ratio.

Do not overdevelop this if it becomes its own field; one crisp separation is enough.

---

11. Revisit overload certificates in this bounded-delay model

This model may finally admit a finite insolvency certificate that the unconstrained tail-friction problem lacked.

For a finite set of already-arrived claims with deadlines, infeasibility can sometimes be certified from a finite window:

these claims require \(m\) service within these legal dates, and even optimally concentrating that authority on the cheapest available legal dates costs more than remaining budget \(B\).

This is finite and authenticated.

Formalize this.

For a claim interval \(I\), candidate certificate:

$$ \mathsf{RequiredCost}(I)

B_{\rm remaining}. $$

The certificate should contain:

• the claims;
• their legal service neighborhood;
• a proved lower bound on the cost of carrying their required mass;
• the current certified remaining liability slack.

This may be a much better model of an Answerability-facing overload event than the infinite tail-friction certificate.

Distinguish:

• current/deadline insolvency, finitely certifiable;
• eternal persistence insolvency, generally requiring a theorem about the future.

---

12. Clarify the conceptual result if the math works

The intended compression is something like:

Affordability alone asks whether there will eventually be arbitrarily cheap opportunities to exercise a reason. Answerability asks whether those opportunities arrive soon enough to count as answering the reason.

Or mathematically:

$$ \boxed{ \text{unconstrained persistence} = \text{cheap dates exist}, } $$

while

$$ \boxed{ \text{serviceability} = \text{cheap dates occur inside the reason's admissible service windows}. } $$

Do not use this slogan unless the theorems really justify it.

---

Deliverables

Produce:

1. BOUNDED_DELAY_TRANSPORT.md

• exact transport-feasibility theorem for claims with service windows \([t,t+H]\);
• greedy/FIFO/EDF construction if available;
• Hall/interval characterization.

1. BOUNDED_DELAY_AFFORDABILITY.md

• minimum-cost service problem;
• exact result for unit claims or the strongest canonical case;
• finite-budget existence criterion.

1. SERVICEABILITY_FRONTIER.md

• liability versus allowable delay;
• transport-stability modulus \(\omega(H)\);
• resulting Sustainable Progress residual.

1. MULTIREASON_SERVICEABILITY.md

• whether deadlines make reasons compete again;
• smallest exact counterexample if joint serviceability fails despite individual serviceability.

1. ONLINE_SERVICEABILITY.md

• smallest online/offline separation under deadlines;
• competitive theorem if one is clean.

1. Targeted patches to:

• SHARP_PERSISTENCE.md;
• SERVICE_ADMISSIBLE_EXISTENCE.md;
• PR summary;
• any depth-only sharp-persistence claims.

1. Exact-rational fixtures for:

• sharp persistence with \(s_t\) floored but \(m_t\to0\);
• exact bounded-delay transport feasibility/failure;
• a dip sequence satisfying unconstrained persistence but failing bounded-delay service;
• individually serviceable reasons that are jointly unaffordable, if such an example exists;
• finite deadline insolvency certificate;
• two-date online bounded-delay dilemma.

Highest priorities

Spend most effort on:

1. the exact interval/Hall characterization of bounded-delay transport;
2. the minimum-cost batching problem under concave date costs;
3. a theorem combining allowable delay \(H\) with transport stability and liability;
4. finite deadline-based insolvency certificates.

The main goal is no longer to characterize “persistent authority” by itself.

It is to characterize:

$$ \boxed{ \textbf{diachronically legitimate service that the learner can actually afford.} } $$

Success criterion

The pass succeeds if it gives a theorem substantially stronger than:

“bounded delay requires more frequent cheap dates.”

We want something like:

Given a claim process, deadline \(H\), and date-cost process \(L_t\), here are necessary and sufficient conditions for an affordable bounded-delay transport plan; under temporal stability \(\omega\), such a plan implies claim-weighted Sustainable Progress with residual \(\omega(H)\).

If that full theorem is too hard, solve the unit-claim / fixed-\(H\) case exactly and identify the precise obstruction to generalization.

---

Eighth dispatch — close bounded-delay serviceability and couple scheduling to settlement friction

Eighth pass inside PR75: close bounded-delay serviceability and couple scheduling to settlement friction

Stay inside PR75:

projects/normativity/legitimacy/rounds/2026-08-31-normative-affordability/

Do not reopen the fixed-era composition theorem.

Treat the following as provisionally stable:

• service \(=\) predictable allocated authority \(a=\beta\);
• realized force \(=ad\);
• control is a predictable reactive law;
• MarketMaker supplies coercive Uptake;
• bounded liability preserves ordinary LI;
• Service Transport transfers service-weighted Progress to claim-weighted Progress;
• uniform bounded-delay transport has an interval feasibility characterization and FIFO construction;
• under linear exogenous date costs,

$$ \operatorname{Cost}_H(c,w) = \sum_t c_t\min_{s\in[t,t+H]}w_s. $$

The previous pass opened the first genuinely joint Answerability–affordability theory. This pass should finish the exact bounded-delay model and then determine how the service schedule jointly controls liability cost, transport error, and the settlement-friction residual appearing in Progress.

Read at minimum:

• BOUNDED_DELAY_TRANSPORT.md
• BOUNDED_DELAY_AFFORDABILITY.md
• SERVICEABILITY_FRONTIER.md
• MULTIREASON_SERVICEABILITY.md
• ONLINE_SERVICEABILITY.md
• SERVICE_TRANSFER.md
• FIXED_ERA_THEOREM.md
• LI_PROGRESS_FROM_SERVICE.md
• SHARP_PERSISTENCE.md
• CAPACITY_VS_SAFETY.md

Do not do another general survey.

---

1. Patch the sharp-persistence scope one final time

The exact generic persistence theorem is

$$ \exists a: \qquad \sum_ta_t=\infty, \qquad \sum_tL_t(a_t)<\infty $$

iff

$$ \boxed{ \liminf_tL_t(1)=0 } $$

under the stated star-shaped assumptions.

For the sharp LI robust date cost,

$$ L_t(1)= \begin{cases} s_t^2/4,&s_t^2\le4m_t,\\[1mm] s_t\sqrt{m_t}-m_t,&s_t^2>4m_t. \end{cases} $$

The previous review notes that this is not generically equivalent to

$$ \liminf s_t=0. $$

For example,

$$ s_t=1,\qquad m_t\to0 $$

can also make \(L_t(1)\to0\).

Prove a clean comparison such as

$$ \boxed{ \frac14\min\{s^2,s\sqrt m\} \le L(1) \le \min\{s^2,s\sqrt m\} } $$

or the sharpest correct constants.

Then record:

$$ \boxed{ \text{sharp robust persistence} \iff \liminf_t \min\{s_t^2,s_t\sqrt{m_t}\} =0 } $$

up to the proved equivalence.

Only simplify this to

$$ \liminf s_t=0 $$

under an explicit fixed positive lower bound on \(m_t\).

Interpret the two routes to cheap enforcement:

1. the norm becomes less excluding of live worlds;
2. the engine becomes cheaper/easier to move.

Patch the PR summary accordingly.

---

2. Repair D3: exact hypotheses for batching under concave date costs

BOUNDED_DELAY_AFFORDABILITY.md currently derives the consecutive-run shortest-path theorem from star-shapedness.

That is too weak.

The previous review gives a candidate counterexample: existing load on two legal dates can make splitting a claim cheaper under a star-shaped but nonconcave cost.

Verify or replace that counterexample.

The target is to establish the correct hierarchy:

$$ \boxed{ \begin{array}{ll} \text{Persistence S1} &: \text{star-shaped costs},\\ \text{finite-horizon concentration S2} &: \text{concave costs},\\ \text{bounded-delay batching D3} &: \text{concave costs},\\ \text{closed form D4} &: \text{linear costs}. \end{array} } $$

For D3, prove carefully:

For increasing concave \(L_s\) with \(L_s(0)=0\), an optimal bounded-delay assignment exists in which:

• claims are not fractionally split except where irrelevant degeneracy permits it;
• service is monotone in claim arrival order;
• claims assigned to one service date form a consecutive run;
• therefore the optimum is the shortest path over consecutive runs.

Do not reuse the current exchange proof if exchanging two assignments changes the loads at dates and hence changes the cost.

Instead use a correct argument.

Possible routes:

• concave minimization over the transportation polytope attains an optimum at an extreme point;
• characterize extreme points of the interval transportation polytope;
• uncross an extreme-point assignment while preserving each date's total load;
• exploit total unimodularity/consecutive-ones structure;
• prove directly that there is an optimal atomic assignment and then uncross it.

Distinguish indivisible claims from divisible claim mass. If atomicity requires an assumption, state it.

Then restore the DAG theorem only under the exact proved hypotheses.

---

3. Check BD1/BD2 for the infinite stream

The interval theorem

$$ \sum_{t=u}^v c_t \le \sum_{s=u}^{v+H}a_s \qquad \forall[u,v] $$

looks correct for finite instances.

Close the countable/infinite-stream case explicitly.

Options:

• prove every finite restriction has a feasible plan and use compactness;
• give the one-pass FIFO construction directly and show the interval condition prevents deadline misses;
• state the theorem finite-horizon only and derive the infinite version from consistency of FIFO prefixes.

Prefer the constructive FIFO proof if clean.

The goal is a theorem that can actually feed an infinite-horizon Answerability process.

---

4. Delete the false \(H\to\infty\) interpolation

The current D4 text says that bounded-delay service interpolates exactly back to unconstrained persistence as \(H\to\infty\).

This is false in general.

Build the clean counterexample.

A suggested sequence:

$$ w_t=1 $$

except

$$ w_{2^k}=4^{-k}, $$

with unit claims.

Then:

$$ \liminf w_t=0, $$

so unconstrained persistent authority is affordable.

But for every fixed finite \(H\), gaps between dip dates eventually exceed \(H\), so infinitely many claims have window minimum \(1\), hence

$$ \operatorname{Cost}_H=\infty. $$

Yet with genuinely unbounded delay, claims between successive powers of two can be batched onto the next cheap date, with total cost comparable to

$$ \sum_k2^k4^{-k} = \sum_k2^{-k}<\infty. $$

Check all indexing exactly.

Then distinguish three different problems:

Persistence

$$ \sum a_t=\infty. $$

Not every claim must be discharged.

Full service with unbounded delay

Every claim is eventually transported, but there is no uniform deadline.

Uniform bounded-delay service

Every claim is served within one fixed \(H\).

These are genuinely different.

Do not identify the \(H=\infty\) service problem with unconstrained persistence.

Investigate whether

$$ \lim_{H\to\infty}\operatorname{Cost}_H $$

can exceed

$$ \operatorname{Cost}_{\infty}. $$

Characterize when equality holds if there is a clean condition.

This is mathematically relevant: uniform timeliness is stronger than eventual answerability.

---

5. Reconstruct the fixed-era settlement-friction residual exactly

The previous frontier analysis treated the settlement-friction residual \(F_r\) as a fixed property of the norm.

That is not correct if the fixed-era theorem defines it through the allocated-service measure.

Recover the exact definition from FIXED_ERA_THEOREM.md / LI_PROGRESS_FROM_SERVICE.md.

It should have the shape

$$ F_r(a) = \limsup_N \inf_{\omega\in A_N} \left( \frac{ \sum_{t<N}a_t^r [s_t^r(\omega)^+]^2 }{ A_N^r } \right)^{1/2}, $$

or the exact current version.

The pointwise landscape

$$ s_t^r(\omega) $$

is supplied by settlement/norm semantics.

But

$$ F_r(a) $$

also depends on the service schedule \(a\).

Record the correct distinction:

$$ \boxed{ \text{settlement-misfit landscape} = \text{environment/norm input} } $$

versus

$$ \boxed{ \text{service-weighted settlement-friction residual} = \text{landscape evaluated under the chosen service schedule}. } $$

This means a scheduler can sometimes buy down or worsen \(F_r\) by choosing when to service the reason.

Patch SERVICEABILITY_FRONTIER.md.

---

6. Formulate the actual joint serviceability optimization

The earlier serviceability optimization minimized only liability:

$$ \sum_sL_s(a_s). $$

But the final Sustainable Progress guarantee also depends on the distribution of service over settlement misfit.

The correct optimization should expose all three quantities:

1. liability cost;
2. transport/timeliness error;
3. service-weighted settlement friction.

For one reason, formulate a finite-horizon problem of the form

$$ \boxed{ \begin{aligned} \text{choose }&a,T\\ \text{subject to }& T\text{ transports claims legally},\\ &\sum_sL_s(a_s)\le B,\\ &\sum_tT(t,s)\le a_s, \end{aligned} } $$

and evaluate

$$ \mathsf{Residual}(a,T) = L_rK_r F_r(a) + \mathsf{TransportError}(T). $$

If bounded delay \(H\) and temporal modulus \(\omega\) are used, then

$$ \mathsf{TransportError}(T)\le\omega(H). $$

But do not force all plans through a uniform \(H\) if a direct delay-weighted objective is more informative.

The conceptual target is:

$$ \boxed{ \operatorname{BestResidual}(B) = \inf_{\text{affordable admissible service}} \left[ L_rK_rF_r(a) + \epsilon(T) \right]. } $$

This is potentially the real quantitative notion of affordability/serviceability.

---

7. Study the interaction between liability cost and settlement friction

This is important because the two quantities may not be independent.

Under the sharp LI cost, liability itself is generated from world-misfit/exclusion depth.

So low-liability dates may already tend to be low-\(s\) dates.

Ask:

Does minimizing robust liability automatically control the settlement-friction residual?

There may be a direct theorem.

For example, in the linear sharp branch,

$$ L_t(a) = \frac{s_t^2}{4}a. $$

Meanwhile the numerator of the service-weighted \(L^2\) friction is approximately

$$ \sum_t a_ts_t^2. $$

Then:

$$ \boxed{ \sum_t a_ts_t^2 = 4\sum_tL_t(a_t). } $$

This looks extremely important.

If the same \(s_t\) is the relevant friction quantity, a lifetime liability budget \(B\) gives

$$ \sum_ta_ts_t^2 \le4B. $$

If total allocated service

$$ A_N\to\infty, $$

then automatically

$$ \frac{\sum_{t<N}a_ts_t^2}{A_N} \to0. $$

Hence

$$ F_r(a)=0. $$

Check the quantifiers carefully:

• \(s_t\) may be a supremum over live worlds while \(F_r\) takes an infimum over worlds;
• this may therefore be a valid robust upper bound, possibly loose;
• reason/row aggregation may matter.

If this argument works, it is a major compression:

$$ \boxed{ \text{finite sharp robust liability} + \text{persistent service} \Longrightarrow \text{vanishing settlement-friction residual}. } $$

That would mean the supposedly separate settlement-friction residual is already controlled by affordability in an important regime.

Investigate this aggressively.

Do the analogous calculation in the square-root branch.

There, cost is

$$ L_t(a) = s_t\sqrt{a_tm_t}-m_t, $$

so the relationship to

$$ a_ts_t^2 $$

is different.

Find the strongest bound available.

---

8. Re-derive the end-to-end Progress theorem under sharp affordable service

If §7 succeeds, compose the result.

The current fixed-era theorem has schematic form

$$ \limsup_N E_{\mu_N}[d] \le LKF(a) + \epsilon(T). $$

If sharp affordability plus persistent service gives

$$ F(a)=0, $$

then we get:

$$ \boxed{ \limsup_N E_{\mu_N}[d] \le \epsilon(T). } $$

And if temporal regularity plus bounded delay gives

$$ \epsilon(T)\le\omega(H), $$

then:

$$ \boxed{ \limsup_N E_{\mu_N}[d] \le \omega(H). } $$

This would be much stronger conceptually than the current frontier story.

It would say:

Once the learner can persistently afford the reason under the sharp worldwise liability certificate, the only remaining asymptotic error is how much the reason can change while waiting to be serviced.

Check whether this is actually true.

Do not state it unless every inequality lines up.

This may be the most valuable theorem target in the pass.

---

9. Replace the old serviceability frontier with the right frontier

The current frontier uses

$$ H^*(B) = \min\{H:\operatorname{Cost}_H\le B\} $$

and then says the final residual is

$$ LKF+\omega(H^*(B)). $$

If \(F\) is schedule-dependent, this is incomplete.

Develop one of the following, whichever is mathematically cleanest.

Frontier A: constrained liability

For each deadline \(H\), define

$$ \mathcal R_H(B) = \inf \left\{ F(a): (a,T)\text{ has delay }\le H,\ \operatorname{Cost}(a)\le B \right\}. $$

Then:

$$ \boxed{ \operatorname{ProgressResidual}(H,B) \le LK\mathcal R_H(B) + \omega(H). } $$

Finally optimize over \(H\):

$$ \boxed{ \mathsf{BestResidual}(B) = \inf_H \left[ LK\mathcal R_H(B) + \omega(H) \right]. } $$

Frontier B: direct delay-weighted optimization

Avoid one uniform deadline and optimize

$$ LKF(a) + \sum_{t,s}T(t,s)\epsilon(t,s) $$

subject to budget.

Use this if it gives a cleaner theorem.

The goal is to expose the actual trade space, not to preserve the old \(H^*\) notation.

---

10. Characterize the linear-branch joint objective

The linear branch may be exactly solvable.

Suppose:

$$ L_s(a)=w_sa $$

and settlement-friction contribution at date \(s\) has weight

$$ r_s $$

so the numerator is

$$ \sum_sa_sr_s. $$

For a claim at \(t\), sending one unit to \(s\in[t,t+H]\) contributes:

• liability \(w_s\);
• settlement residual numerator \(r_s\);
• delay error \(\epsilon(t,s)\).

This becomes a per-unit assignment problem.

Can the optimal date for each claim be chosen by minimizing a scalar score such as

$$ \lambda w_s+\mu r_s+\epsilon(t,s)? $$

Under a Lagrange multiplier for the budget, the problem may separate claim-by-claim exactly.

Derive the finite-horizon Pareto frontier.

This could give a very clean theorem:

in the linear regime, legitimate service is an interval assignment problem where each claim chooses among legal dates according to a combined underwriting + semantic-friction + delay price.

This would be the first mathematically precise “reason service market” picture in the agenda.

---

11. Reassess multi-reason competition under the corrected joint objective

The previous pass gives exact additivity if reasonwise costs are genuinely separable.

Now distinguish three levels:

1. separate rows and separate underwriting accounts: exact addition;
2. separate rows but one worldwise robust account: only an inequality because

$$ \sup_\omega\sum_r\ell_r(\omega) \le \sum_r\sup_\omega\ell_r(\omega); $$

1. shared rows / shared semantic service: one action may answer multiple reasons.

Patch MULTIREASON_SERVICEABILITY.md so M1's “independent” hypothesis is mathematically explicit.

Then ask whether the joint objective reveals a useful economy of scope:

$$ \operatorname{Cost}(r_1\cup r_2) < \operatorname{Cost}(r_1) + \operatorname{Cost}(r_2) $$

when a single row/action answers both.

Do not develop this deeply unless a clean example falls out.

---

12. Keep online serviceability separate

Preserve the basic result:

• unconstrained persistence has qualitative online/offline equivalence;
• bounded-delay service has a genuine online feasibility gap when future costs are unknown.

But do not claim the \(\sqrt\kappa\) competitive threshold rule as a theorem unless proved.

The main question for this pass is instead:

If the relevant combined date score from §10 is predictable across the whole legal service window, does the joint serviceability problem become offline-at-arrival?

This may be the relevant distinction between deductive and empirical settlement channels.

Keep it scoped.

---

13. Finite overload certificates from deadlines

Continue the useful direction from the previous prompt.

For an already-arrived finite set of claims, bounded deadlines produce a finite legal service region.

A certificate of current insolvency should look like:

$$ \boxed{ \text{minimum certified cost of servicing these claims before deadline}

\text{certified available liability slack}. } $$

Using the interval structure, derive the strongest cheap certificate possible.

On the linear branch, an interval \(I\) with claim mass \(C(I)\) and service neighborhood \(N(I)\) may produce a lower bound such as

$$ \sum_{t\in I} c_t \min_{s\in[t,t+H]}w_s, $$

or a sharper joint batching expression under concave costs.

This is a genuinely finite Answerability-facing impossibility witness, unlike a claim about an infinite future friction floor.

Try to make the certificate compositional:

• claim IDs / provenance;
• legal service window;
• certified date costs;
• minimum required liability;
• current available underwriting;
• inequality showing impossibility.

---

14. Decide the new conceptual compression

At the end, assess which of these is closest to the actual theorem we now have:

Candidate 1

Affordability asks whether cheap service dates exist; Answerability asks whether they occur in time.

Candidate 2

Serviceability is the existence of a service trace that simultaneously respects the reason's temporal claims and the learner's settlement-contingent balance sheet.

Candidate 3

A reason is sustainably answerable when authority can be concentrated on dates that are jointly cheap to enforce, semantically faithful to the claim, and timely enough to preserve its content.

Prefer the mathematically strongest statement rather than the nicest prose.

---

Deliverables

Produce or update:

1. BOUNDED_DELAY_AFFORDABILITY.md

• correct D3 hypotheses/proof;
• preserve D4;
• remove false \(H\to\infty\) interpolation.

1. EVENTUAL_VS_UNIFORM_SERVICE.md

• distinguish persistence, eventual full service, and uniform bounded-delay service;
• give exact separation examples.

1. JOINT_SERVICEABILITY.md

• define the joint liability / transport / settlement-friction objective;
• derive the strongest general theorem.

1. SHARP_SERVICEABILITY.md

• investigate whether sharp robust liability plus persistent service automatically drives the service-weighted settlement-friction residual to zero;
• separate linear and square-root branches.

1. SERVICEABILITY_FRONTIER.md

• replace the current exogenous-\(F\) frontier with the corrected schedule-dependent version.

1. Targeted patches to:

• SHARP_PERSISTENCE.md;
• MULTIREASON_SERVICEABILITY.md;
• ONLINE_SERVICEABILITY.md if needed;
• the PR summary.

1. Exact-rational fixtures for:

• star-shaped but nonconcave cost where splitting beats D3's unsplit plan;
• finite-\(H\) serviceability failure for every \(H\) but finite unbounded-delay full-service cost;
• \(s_t\) floored with \(m_t\to0\) making sharp date cost vanish;
• linear sharp liability directly bounding the settlement-friction numerator;
• a schedule tradeoff where the cheapest-liability date is not the best combined liability/transport/semantic date;
• finite deadline insolvency certificate.

Highest priority

Spend the most effort on the possible theorem:

$$ \boxed{ \text{finite sharp liability} + \text{persistent allocated service} \Longrightarrow F_r(a)=0 } $$

or the strongest correct variant.

If true, this materially simplifies the entire fixed-era theorem.

Then focus on the exact linear-regime joint assignment problem.

Success criterion

The pass succeeds if it can answer:

When a reason must be answered within semantically meaningful time, what service schedule minimizes the learner's actual final Progress error subject to bounded liability?

A particularly strong outcome would be:

$$ \text{sharp affordable persistent service} \Longrightarrow \text{zero settlement-friction residual}, $$

so that bounded-delay temporal stability leaves only

$$ \omega(H) $$

as the final asymptotic defect.

Do not broaden beyond this question.

---

Ninth dispatch — canonical Sharp Timely Service theorem and final consistency audit

Ninth pass inside PR75: canonical Sharp Timely Service theorem and final consistency audit

Stay inside PR75:

projects/normativity/legitimacy/rounds/2026-08-31-normative-affordability/

This should be a narrow theorem-repair and consolidation pass, not another expansion of the research program.

Treat the following as frozen unless an actual contradiction appears:

• fixed-era construction/composition;
• service \(=\) predictable allocated authority \(a=\beta\);
• realized force \(=ad\);
• reactive-control-law typing;
• MarketMaker coercive Progress;
• reasonwise row accounting;
• SafeCert/preservation;
• Deferred Service Transfer;
• bounded-delay interval feasibility/FIFO;
• exogenous persistence theorem S1;
• linear bounded-delay cost formula D4.

The main goals are now:

1. repair the false persistence/eventual-service hierarchy;
2. normalize variable transport error correctly;
3. state one canonical theorem combining sharp-linear affordability, timely service, and Sustainable Progress;
4. aggressively clean stale/overbroad claims in the round and PR summary.

Do not open a new conceptual direction unless one of these repairs forces it.

---

1. Correct persistence versus eventual full service

EVENTUAL_VS_UNIFORM_SERVICE.md currently claims three strictly separated notions:

$$ \text{persistence} < \text{eventual full service} < \text{uniform bounded-delay service}. $$

The second separation appears false in the exogenous, fungible, unlimited-deferral benchmark.

Let claims have finite masses

$$ c_0,c_1,\ldots $$

and suppose

$$ \sum_t c_t=\infty. $$

Assume the exogenous date costs \(L_s\) are increasing, star-shaped, vanish at zero, and satisfy the persistence criterion

$$ \liminf_sL_s(1)=0. $$

By the reference-level lemma, for every fixed \(c_t>0\),

$$ \liminf_sL_s(c_t)=0. $$

Therefore enumerate the claims and choose strictly increasing service dates

$$ s_t\ge t $$

such that

$$ L_{s_t}(c_t) \le B2^{-(t+1)}. $$

Then every claim is eventually served and

$$ \sum_tL_{s_t}(c_t)\le B. $$

Formalize this carefully.

Target theorem EV1

Under exogenous fungible service, finite individual claim masses, unlimited deferral, and divergent total claim mass:

$$ \boxed{ \text{persistent affordability} \iff \text{affordable eventual full service}. } $$

Be precise about what each side means.

The reverse direction should use:

$$ \sum_s a_s \ge \sum_t c_t = \infty $$

for any plan discharging every claim.

If

$$ \sum_t c_t<\infty, $$

eventual full service clearly need not imply persistent authority, so state the claim-mass hypothesis explicitly.

Also distinguish:

$$ \boxed{ \text{existence of a persistent schedule} } $$

from

$$ \boxed{ \text{a particular persistent schedule actually services the claims}. } $$

A persistent schedule can ignore claims even though some other affordable schedule eventually discharges all of them.

Withdraw the current E2 countermodel.

---

2. Preserve the real separation: uniform timeliness

The powers-of-two fixture in E1 appears to establish the actual strict hierarchy:

$$ \boxed{ \text{eventual full service} \not\Rightarrow \text{uniform bounded-delay service}. } $$

Keep and recheck it.

For unit claims and

$$ w_t=1 $$

except

$$ w_{2^k}=4^{-k}, $$

show:

• eventual service has finite cost by batching increasingly large blocks at increasingly cheap later dates;
• for every finite \(H\),

$$ \operatorname{Cost}_H=\infty. $$

Then state the correct hierarchy for the exogenous fungible benchmark:

$$ \boxed{ \text{persistence existence} \equiv \text{eventual full-service existence} } $$

under divergent total claim mass, while

$$ \boxed{ \text{uniform timely service} } $$

is strictly stronger.

This is conceptually important:

Unlimited deferral makes “eventually answer every persistent claim” almost as cheap as merely maintaining divergent service. The substantive Answerability constraint enters when delay itself matters.

Do not overstate this outside the exogenous/fungible model.

---

3. Remove the false bounded-gap shortcut

Audit claims such as:

bounded gaps between cheap dates imply persistence, eventual service, and bounded-delay service coincide;

or:

positive-density cheap dates have bounded gaps.

Both are too strong.

Positive asymptotic density does not imply bounded gaps.

And even uniformly bounded gaps between dates where

$$ w_t\to0 $$

do not guarantee finite timely-service cost.

Example candidate:

$$ w_{2k}=k^{-1/2}, \qquad w_{2k+1}=1, $$

with unit claims and \(H=1\).

Cheap opportunities have bounded gaps and

$$ \liminf w_t=0, $$

so persistence/eventual service are affordable, but

$$ \sum_t \min_{s\in[t,t+1]}w_s $$

diverges.

The exact bounded-delay criterion is already D4:

$$ \boxed{ \sum_t c_t \min_{s\in[t,t+H]}w_s <\infty. } $$

Let this theorem replace heuristic claims based only on gap size.

---

4. Generalize Deferred Service Transfer to edge-dependent semantic error

SERVICE_TRANSFER.md T3 currently assumes one uniform stability constant:

$$ d_t\le Ld_s+\epsilon $$

whenever \(T(t,s)>0\).

For the joint optimization we want an edge-dependent error:

$$ d_t \le L d_s+\epsilon(t,s). $$

Derive the exact quantitative theorem.

The additive error should be claim-normalized, not the raw sum.

Define

$$ \boxed{ \bar\epsilon_N(T) = \frac1{C_N} \sum_{t,s} T_N(t,s)\epsilon(t,s). } $$

Then prove:

$$ \boxed{ E_{\mu_N}[d] \le LK E_{\nu_N}[d] + \bar\epsilon_N(T) + D\frac{R_N}{C_N}. } $$

Check every normalization.

If \(L\) also needs to vary by edge, consider the minimal useful extension, but do not generalize unnecessarily.

For temporal modulus

$$ |d_t-d_s| \le \omega(|s-t|), $$

take

$$ \epsilon(t,s)=\omega(s-t) $$

for forward service.

Then the transport contribution becomes the claim-weighted mean semantic delay cost

$$ \boxed{ \frac1{C_N} \sum_{t,s}T(t,s)\omega(s-t). } $$

Uniform delay \(H\) recovers the earlier bound:

$$ \bar\epsilon_N(T)\le\omega(H). $$

This should replace the unnormalized eps(T) currently used in JOINT_SERVICEABILITY.md.

---

5. Audit the exact SS1 quantifiers

SHARP_SERVICEABILITY.md proves, on the sharp linear branch,

$$ L_t(a_t) = \frac14a_tD_t^2, \qquad D_t = \sup_{\omega\in A_t}s_t^+(\omega). $$

Then it bounds, for a horizon \(N\),

$$ \sum_{t<N} a_t[s_t^+(\omega)]^2 \le \sum_{t<N}a_tD_t^2. $$

Make the live-world quantifiers exact.

If

$$ \omega\in A_N $$

and \(t<N\), this step needs

$$ A_N\subseteq A_t $$

or the relevant monotone live-world property.

That should hold for settlement-shrinking assessment sets, but put it into the theorem statement or cite the exact interface property.

The robust result should be something like:

$$ \sup_{\omega\in A_N} \sum_{t<N}a_t[s_t^+(\omega)]^2 \le4B. $$

Confirm there is no hidden problem when the row/reason itself changes over time.

If the theorem is row-indexed rather than one fixed deficit functional, type that correctly.

---

6. State the canonical Sharp Timely Service theorem

This is the main deliverable.

We now appear to have all the pieces for a clean LI-specific theorem.

Candidate theorem

Let \(r\) be a persistent fixed-era reason with claim stream \(c_t^r\).

Suppose:

Service

There is an adapted transport plan \(T^r\) from claims to allocated authority \(a_t^r\), with:

$$ A_N^r = \sum_{t<N}a_t^r \to\infty, $$

service parsimony

$$ W_N^r\le K_r C_N^r, $$

and residual

$$ R_N^r/C_N^r\to0. $$

Sharp-linear affordability

The reason's enforcement allocations remain on the sharp linear branch:

$$ a_t^r \le \frac{4m_t}{(D_t^r)^2} $$

on all relevant dates, or all but finitely many with the finite exceptional contribution explicitly absorbed.

Its robust lifetime liability satisfies

$$ \sum_t \frac14a_t^r(D_t^r)^2 \le B_r. $$

MarketMaker

The reason's enforcement book has the appropriate reasonwise ceiling

$$ U_r, $$

derived from the aggregate cap and complementary row liability floors.

Temporal stability

For transported claim mass:

$$ d_t^r \le L_r d_s^r+\epsilon_r(t,s). $$

Then derive the strongest finite-horizon inequality.

The likely shape is:

$$ E_{\mu_N^r}[d^r] \le L_rK_r \frac{ 2\sqrt{B_r} + \sqrt{U_r} }{ \sqrt{A_N^r} } + \bar\epsilon_N^r(T) + D_r\frac{R_N^r}{C_N^r}. $$

Check the constants against LI_PROGRESS_FROM_SERVICE.md, FIXED_ERA_THEOREM.md, and SS2 rather than trusting this candidate.

If the MarketMaker constant is

$$ U+B_{\rm tot} $$

rather than a separately named \(U_r\), preserve the exact form.

Then asymptotically:

$$ \boxed{ \limsup_N E_{\mu_N^r}[d^r] \le \limsup_N\bar\epsilon_N^r(T). } $$

Under uniform delay \(H\) and modulus \(\omega_r\):

$$ \boxed{ \limsup_N E_{\mu_N^r}[d^r] \le \omega_r(H). } $$

Under exact defect preservation:

$$ \boxed{ E_{\mu_N^r}[d^r]\to0. } $$

This theorem should be the canonical positive result of PR75.

---

7. Explain exactly what SS1 has compressed

The generic fixed-era theorem carries two residual mechanisms:

1. settlement friction;
2. transport distortion.

Under sharp-linear robust affordability, the first vanishes automatically.

State the structure explicitly:

$$ \boxed{ \text{generic affordability} \Rightarrow \text{Progress up to }F+\epsilon, } $$

but

$$ \boxed{ \text{sharp-linear robust affordability} \Rightarrow F=0, } $$

hence

$$ \boxed{ \text{Sharp Timely Service} \Rightarrow \text{Progress up to transport distortion alone}. } $$

Do not remove \(F\) from the generic schematic theorem.

The square-root branch proves why it belongs there.

---

8. Recheck the square-root branch statement

The current file gives

$$ aD^2 = \frac{(L(a)+m)^2}{m}. $$

That identity is useful.

But audit claims such as:

a bounded charge bounds the numerator only when \(m_t\) is summable.

That wording may be too strong or imprecise.

Given date charges

$$ \ell_t=L_t(a_t), $$

the exact condition is about

$$ \sum_t \frac{(\ell_t+m_t)^2}{m_t} $$

relative to total service.

Explore the weakest clean sufficient condition for

$$ F(a)=0. $$

For instance:

$$ \frac{ \sum_{t<N}(\ell_t+m_t)^2/m_t }{ A_N } \to0. $$

No need for a full characterization. Just ensure the branch-failure discussion states an exact condition rather than an unnecessarily strong summability slogan.

Preserve the important counterregime:

$$ D_t\ge\sigma>0, \qquad m_t\to0, $$

where service can be cheaply persistent while \(F\) stays positive.

---

9. Repair the joint serviceability objective

Replace raw

$$ \epsilon(T) = \sum T(t,s)\epsilon(t,s) $$

with the normalized quantity from §4.

Then distinguish finite-horizon and asymptotic objectives.

A finite-horizon objective can use total quantities if explicitly normalized afterward, but the Sustainable Progress theorem consumes an average per claim, not total distortion.

A clean asymptotic definition is:

$$ \mathsf{SemErr}(T) = \limsup_N \frac1{C_N} \sum_{t,s<N} T_N(t,s)\epsilon(t,s). $$

Then:

$$ \boxed{ \mathsf{BestResidual}(B) = \inf_{\text{affordable admissible }(a,T)} \left[ L_rK_rF_r(a) + \mathsf{SemErr}(T) \right]. } $$

On the sharp linear branch with persistent service:

$$ F_r(a)=0, $$

so:

$$ \boxed{ \mathsf{BestResidual}(B) = \inf_{\text{sharp-linearly affordable }T} \mathsf{SemErr}(T). } $$

That is the genuine “spending versus waiting” problem.

---

10. Tighten the linear assignment/Pareto statement

On the linear branch:

$$ w_s=\frac14D_s^2 $$

and any separate friction numerator price is proportional to \(w_s\).

With edge delay cost \(\epsilon(t,s)\), a scalarized finite-horizon problem becomes:

$$ \operatorname{score}_\lambda(t,s) = \lambda w_s+\epsilon(t,s) $$

up to constants.

Each claim independently chooses:

$$ s\in[t,t+H] $$

minimizing this score.

That part is clean.

But do not automatically state that varying \(\lambda\) recovers every Pareto-optimal point.

Determine whether the achievable pair set

$$ (\text{liability},\text{semantic delay}) $$

is convex because claims may be fractionally split.

If yes, prove it, then scalarization recovers the supported frontier and perhaps the whole frontier.

If not, say only:

varying the multiplier recovers the supported Pareto frontier.

Do not spend much time here.

---

11. Produce a finite deadline-insolvency certificate

Now that the timely-service cost is exact on the linear branch, formulate the finite certificate cleanly.

For a finite set of already-arrived claims with deadlines, compute the minimum certified remaining liability required to serve them:

$$ \operatorname{ReqCost} = \sum_t c_t \min_{s\in[t,t+H]\cap\text{future}} w_s $$

in the linear predictable-cost model.

If

$$ \operatorname{ReqCost}

B_{\rm remaining}, $$

then current timely service is impossible.

The certificate should record:

• claims/provenance;
• legal future service windows;
• authenticated costs/upper-risk weights;
• exact minimum required charge;
• certified remaining liability slack;
• the strict inequality.

State precisely when this is a complete insolvency certificate and when future costs are not yet known, so only a conditional certificate is possible.

This is probably the cleanest finite overload certificate the round has produced.

---

12. Final consistency audit of PR75

Search the round and PR body for now-withdrawn claims.

At minimum patch all surviving forms of:

• sharp persistence iff exclusion depth alone dips, without an \(m_t\) floor;
• D3 under mere star-shapedness;
• bounded-delay \(H\to\infty\) equals persistence;
• eventual full service strictly stronger than persistence in the exogenous unlimited-delay benchmark;
• positive density implies bounded cheap-date gaps;
• bounded gaps alone imply finite timely-service cost;
• \(F_r\) is scheduler-independent;
• raw unnormalized \(\sum T\epsilon\) is the transport residual;
• every Pareto point is recovered by scalarization without a convexity proof;
• every form of overload is liability overload without conditioning on force feasibility;
• separate reason underwriting is necessary rather than one sufficient isolation route;
• sustainable authority-rate region convexity.

Do not rewrite provenance history. Mark withdrawals/corrections in the style already used by the round.

---

13. Decide whether PR75 should now stop

End with a direct verdict.

PR75 should be considered mathematically mature enough to stop if we now have:

1. the generic fixed-era construction theorem;
2. exact exogenous persistence characterization;
3. exact uniform-delay transport;
4. exact linear timely-service cost;
5. Sharp Timely Service theorem;
6. clear scope boundaries around:

• concave batching,
• square-root branch,
• closed-loop friction,
• nonfungible/cross-era service,
• causal overload duality.

If so, say:

$$ \boxed{ \text{PR75 has finished the fixed-era affordability/serviceability layer.} } $$

Then list the remaining questions that should move to later rounds rather than being pursued here.

Likely later work:

• closed-loop empirical serviceability;
• certification of temporal stability \(T3\);
• nonfungible service semantics;
• cross-era semantic transport;
• necessity of bounded liability;
• dynamic insolvency duality;
• counterfactual non-capture / legitimacy;
• deference/corrigibility consumer.

Do not begin any of those in this pass.

---

Deliverables

Produce/update:

1. EVENTUAL_VS_UNIFORM_SERVICE.md

• replace E2 with the persistence/eventual-service equivalence theorem;
• retain the strict uniform-delay separation.

1. SERVICE_TRANSFER.md

• edge-dependent normalized transport-error theorem.

1. SHARP_TIMELY_SERVICE.md

• canonical finite-horizon and asymptotic theorem;
• exact hypotheses and constants;
• this should be the main positive theorem of the final pass.

1. JOINT_SERVICEABILITY.md

• normalized semantic-error objective;
• corrected scalarization/Pareto language.

1. SHARP_SERVICEABILITY.md

• audit square-root branch conditions and SS1 quantifiers.

1. DEADLINE_INSOLVENCY.md

• finite authenticated timely-service insolvency certificate.

1. Targeted corrections to all contradicted text and the PR body.

1. Exact-rational fixtures for:

• persistence \(\Rightarrow\) eventual service via diagonal cheap-date assignment;
• eventual service but no finite uniform deadline;
• bounded cheap-date gaps with divergent timely-service cost;
• normalized variable transport error;
• Sharp Timely Service finite-horizon bound;
• finite deadline insolvency certificate.

Highest priority

Spend most effort on the exact canonical theorem:

$$ \boxed{ \text{sharp-linear affordability} + \text{timely Answerability transport} \Longrightarrow \text{claim-weighted Progress up to semantic delay only}. } $$

Everything else in this pass should support making that statement correct and final.

Success criterion

At the end I want to be able to summarize the positive fixed-era result in one sentence:

If Answerability's claims can be transported onto sufficiently timely sharp-linearly affordable enforcement dates, then the same liability budget that preserves the learner also drives settlement friction away, leaving only the semantic change incurred while the reason waited to be answered.

If that sentence is mathematically exact after the repairs, stop extending PR75 and recommend consolidation/merge.

Report

A follow-up audit of the realization map withdrew three of this report's claims and repaired a fourth. FOLLOWUP_REPORT.md carries the audit and lists every correction; CORRECTED_STACK.md states what the stack is now. This report is the record of the first pass, with those four statements corrected in place.

1. The recommended theorem stack

Eight results, ordered by dependency. Every one is stated and proved in this round's documents; none is registered, and none is in Lean.

1. Service Transfer (T1) and its exact converse (T2). For bounded defect arrays, E_{nu_N}[d] -> 0 and mu ◁ nu give E_{mu_N}[d] -> 0; and contiguity is necessary, with indicator arrays as the extremal witnesses. Contiguity equals asymptotic uniform absolute continuity (T1.0), so that is not a separate condition. A bounded density gives the quantitative form (T1.1). For N-independent defect sequences the exact condition is instead fixed-set contiguity (T2'), which is strictly weaker; the separating instance is one-step delay. The array version is the one this theory needs, because a settlement-relative defect is re-assessed at every horizon.
2. Deferred Service Transfer (T3). A feasible adapted transport plan with stability (L, eps), a service-to-claim cap K and vanishing residual gives E_{mu_N}[d] <= L K E_{nu_N}[d] + eps + D R_N/C_N. Contiguity is a consequence, not a hypothesis. Depends on nothing above; T1.1 is used inside the proof.
3. Joint Actionability by common-region superposition (T4). Individual Actionability, additive aggregation, nonnegative weights and a nonempty common region give G >= sum_r w^r gamma_r d^r. Corollary: the composition weights are the service intensities.
4. Non-revocation of safety certificates (T5). A settlement-antitone robust risk makes a certified prefix stay certified under every extension.
5. Sustainable Progress (T6). T3 + T4 (or per-reason Uptake) + T5's preservation give limsup E_{mu^r_N}[d^r] <= eps_r. Five premise-removal countermodels, each hitting a different object.
6. Backlog-to-transport (E1) and existence by drift (T7). Bounded backlog yields an adapted FIFO plan with vanishing residual and, under a window service floor, a delay bound; a pointwise self-financing slack condition makes max-weight scheduling a witness. T7 is a proof sketch and its hypothesis is sufficient, not necessary.
7. Overload certificates (T8). A Farkas pair (y, z) with positive deficit refutes finite-horizon affordability. Sound for the causal problem; not complete.
8. The realization identities. The score identity <zeta, x - p> = (lambda/2)(Br_x(p) - Br_x(q) + d^2), the friction inequality >= lambda(d^2 - de) with equality for a halfspace and an outside world, and the core-minimum misfit bound dist(x, S) <= (1 - theta)|x - q|.

2. Direct answers to the governing questions

Is contiguity the correct and minimal Service Transfer condition? Correct and exactly minimal as a characterization for triangular arrays (T1, T2). Not the right interface: T3 reaches the same conclusion from a checkable finite-horizon object, and the two routes are incomparable rather than one deriving the other — what T3 bounds is the density of the transported claim measure. The triangular-array caveat the dispatch flags is real and resolves the other way from what the flag suggests — the naive indicator counterexample does need an N-dependent defect, and the fixed-sequence condition is strictly weaker (T2'), but settlement-relative assessment makes the defect N-dependent, so the strong version is the one in force.

Which of the stronger sufficient conditions is the right interface? The pointwise sandwich eta c <= w <= K c is sufficient and forbids deferral. Uniform absolute continuity is contiguity restated. Bounded Radon–Nikodym density is the quantitative form and is what T3 produces internally. Prefix-discrepancy, queue-stability, bounded-delay and fair-scheduling conditions are not sufficient — the rotation countermodel has delay 1 and backlog 1 — and their correct role is to establish that a transport plan exists (E1). The interface is the plan and its constants.

Does individual Actionability compose? No, when the gain is read against reason-relative regions; the countermodel has jointly satisfiable demands, so conflict is not the mechanism. Yes, when read against the common region, and the sufficient conditions are only additivity and positive homogeneity of the aggregation — convexity, separability, noninterference and superposition-as-an- axiom are not needed.

What does traderized Logical Induction get for free? Additivity of positions; reactivity, since the compiled position is a continuous function of the current price, which is what defeats the interference countermodel; and an upper bound on every reason's cumulative account, from the market maker's cumulative cap less the ordinary trading firm's floor. Not from the criterion: it quantifies over efficiently computable traders and the enforcement position sits in the price-setting aggregate instead, where it is not required to be efficiently computable and typically is not. Not free: the assessment set's relation to the regions, which is a four-rung ladder rather than one condition, and the reactive fixed point, which is the construction rather than a theorem about arbitrary engines. FOLLOWUP_REPORT.md §B.

What is the minimal SafeCert type? A prefix-closed class of control histories whose membership is antitone in settlement. Actual-path safety is not a candidate at all: it is not measurable at the date the control must be chosen. An account, a risk functional or a burden monoid are presentations, not interface content; an ordered monoid buys nothing over R, or R^k with the product order for incomparable budgets.

Should the analogous history-relative risk be robust? Yes, and the argument is not a preference: predictability of the controller forces it, and settlement monotonicity then makes prefix closure and time consistency automatic rather than axioms.

Does overload admit a dual certificate? One direction. A positive-deficit Farkas pair on any settlement-consistent path refutes affordability. The converse fails because the per-path relaxation hands the controller the path; no exact dynamic alternative is offered, and the dispatch's dichotomy is therefore not established.

Do deferral and cross-era transport unify? The inequality does — the same (T3) with a plan whose support crosses the boundary. The provenance of the constants does not: delay stability is persistence of one defect, era stability needs a semantic bridge fixing which successor coordinate is compared against. Unify the interface, not the construction.

Is immutable history enough for no-reset? For claims, pins, created exposures and the account, yes, and T5 is the reason. Semantic transport is needed only for what is still owed; an exposure already created keeps its original semantics because settlements are never re-spoken.

3. Which hypotheses are schematic and which are realization-specific

hypothesis	class
transport feasibility (T1), (T2)	schematic; supplied by a scheduler
transport stability (T3)	schematic; supplied by the reason's semantics, never by a scheduler
service parsimony W_N <= K C_N	schematic
nonempty common region	schematic; realized as covered compatibility
additive aggregation	realization-specific (free in traderization)
per-reason Uptake	schematic; not free in traderization — the maker's cap plus summable per-reason liability budgets replaces it
share persistence	schematic, and needed only on the aggregate-Uptake route
prefix closure, settlement monotonicity	schematic; properties of the settlement interface
SafeCert ==> Uptake	realization-specific; the traderized instance is bounded liability implies preservation
bounded-liability accounting, Brier form, core minimum	realization-specific
pointwise self-financing slack	schematic, sufficient not necessary, and the place the existence sketch needs strengthening

4. Deviations from the dispatch

1. The boxed conclusion is weakened. The composition gives limsup E_{mu^r}[d^r] <= eps_r, not -> 0. Exact transport stability closes the gap and nothing else in the premises does.
2. Contiguity is deleted from the definition and replaced by the transport plan, against the dispatch's clause 2 as written. The dispatch invited this ("or whatever weaker correct Service Transfer condition replaces this"), and the replacement is not weaker but differently placed: it is stronger as a hypothesis and checkable, where contiguity is weaker and is not.
3. **The J_t type loses its third coordinate.** q_t^r is the reason's own position and carrying it separately invites the reason-relative scoring the interference countermodel exploits.
4. The dispatch's reading of Surface Fairness is corrected rather than used. The inherited service hypothesis does not support a claim-weighted conclusion, and the round does not repair the merged schematic — it exhibits the gap and states the replacement interface.
5. Priority 4 is delivered at lower strength than the rest. T7 is a proof sketch with a standard drift argument and no inhabitation fixture; the other theorems are proved and their instances are exact.
6. **The round's depends_on is empty.** It reads the Progress and liability research checkpoints and refutes one of their hypotheses, but consumes none of their results, so nothing is claimed as a hypothesis. That is accurate here and would not have been available had a result been consumed: those checkpoints carry no round records, so no depends_on can name them. Filed as PRIORITIES.md Workspace friction F7.
7. No Lean. Nothing here is close enough to a settled definition to formalize; the algebra is elementary and the open seams are the substance.

5. What this round does not establish

Every document carries its own list; the ones that matter across the round:

• No existence theorem. T7 is a sketch, its condition 4 is the strong one, and nothing here shows it is ever satisfied in a traderized instance with unboundedly many live reasons.
• No completeness for T8. The dichotomy the dispatch hoped for is one implication and a counterexample-free gap, not a theorem.
• **eps_r = 0 is never constructed.** Every claim-weighted conclusion in this round is up to a transport error that no result bounds by zero.
• Answer-Mode Adequacy is assumed throughout. gamma_r and d^r being correctly recognized is the semantic boundary of the whole family, and this round does not touch it.
• The countermodels are two-reason instances. Nothing here shows they exhaust the failure modes at larger reason sets.
• The realization identities are checked on a rational halfspace. The convex case follows from the projection property and is not machine-checked.
• **test-supported is the ceiling.** Forty-four exact-rational checks illustrate finite instances of theorems proved in prose; the proofs are the evidence, and the fixtures are the guard against a proof about nothing.

6. Strongest unresolved questions

1. Is there an exact alternative for the causal affordability problem — an overload certificate class that is complete as well as sound? Flow/cut duality on the transport plan is the obvious place to look, since T3's plan is already a flow.
2. Can exact transport stability (eps_r = 0) be certified for any reason type that a practice actually produces, or is the claim-weighted conclusion always approximate?
3. Is the realized position size a free variable? The schematic treats w^r_t as the controller's choice, while under an exact market-maker contract the realized size is set by the opposing ordinary volume. This is the seam between the intensity identification and the existence theory.
4. Is bounded cumulative enforcement liability necessary for preservation — PRIORITIES.md item 40? Until it is settled the affordability layer has a sufficient condition and no necessity direction at all.
5. What certifies (T3)'s constants across an era boundary? That is the semantic bridge, and it is the one obligation in the stack with no candidate mechanism.

7. New names introduced

All provisional. Service parsimony (the cap W_N <= K C_N). Transport stability (the (T3) inequality and its constants (L, eps)). Overload certificate (the Farkas pair (y, z) with positive deficit). Fixed-set contiguity (the exact condition for N-independent defect sequences). Sharp Timely Service (the round's canonical theorem). Claim-normalized transport error (epsbar_N(T)). Nested assessment (the hypothesis A_N subseteq A_t). Deadline insolvency (the finite certificate, against persistence insolvency). The dispatch's own provisional terms — affordability witness, claim measure, actual service measure — are kept as given.

8. Outstanding maintainer actions

1. Rule on what Progress means: claim-weighted or service-weighted. The merged schematic concludes D_N/W_N -> 0, which this round shows is compatible with never touching a defective date. Making Progress claim-weighted imports the transport interface and its eps residual into the settled statement; leaving it service-weighted keeps the current theorem and moves the burden onto whatever consumes it. Turns on: what the paper's Progress claim has to be able to say — external knowledge the round lacks. Appended to DECISIONS.md, Awaiting the author.
2. Naming audit over §7 when one is next run.

The residual blocker of §6.1 is filed as PRIORITIES.md item 74, naming the round that would consume it. The two largest gaps in §9's list are filed as items 75 (closed-loop friction) and 76 (certifying a reason's temporal modulus).

Everything else this round recommends is adopted as a dated DECISIONS.md entry, agent-decided and reversible.

9. The round is finished

PR75 has finished the fixed-era affordability/serviceability layer. The sentence the ninth dispatch set as its target is now exact:

Sharp-linear affordability plus timely Answerability transport gives claim-weighted Progress up to semantic delay only.

SHARP_TIMELY_SERVICE.md states it as Theorem STS with all five hypotheses — (S) service, (L) sharp-linear affordability, (M) the MarketMaker ceiling, (N) nested assessment, (T) temporal stability — a finite-horizon inequality with explicit constants, three corollaries, and the fixtures pinning the constants. Nothing in it rests on a claim the round has withdrawn; CONSISTENCY_AUDIT.md is the check.

What the layer contains, settled: the fixed-era composition theorem; the allocated-service typing; reasonwise accounting from the aggregate cap; the exact persistence criterion for a general star-shaped cost; the bounded-delay feasibility and minimum-cost theorems with their sliding-window closed form; the equivalence of persistence and eventual full service, and its strict separation from uniform timeliness; the two-price collapse of the joint objective on the linear branch; and finite deadline-insolvency certificates.

What moves to later rounds

1. Closed-loop friction. Every existence result here is E0-E2 in the CLOSED_LOOP_EXISTENCE.md hierarchy: the date costs are exogenous. E3-E5, where m_t and D^r_t respond to the policy, are untouched and are the largest gap.
2. Certifying (T). The round has no mechanism for establishing a temporal modulus for a reason's defect. Without one, epsbar is a symbol.
3. Simultaneous (S) and (L). Existence of a plan that is both timely and on the sharp linear branch, which is where BOUNDED_DELAY_AFFORDABILITY.md and AFFORDABLE_SCHEDULING.md stop.
4. Necessity of bounded liability, and a converse to the finite-horizon overload certificate.
5. The signed class. SIGNED_VS_CONSERVATIVE.md shows the conservative account is strictly and unboundedly smaller; what a scheduler can safely do on the signed account needs sufficient state, which the scalar slack is not.
6. Beyond the linear branch. BestResidual(B), the interval CapCost lower bound, and deadline certificates on the concave branch.
7. Which of the three service problems Answerability demands. A question about the semantics, not about affordability, and the round's position is that it should export admissible traces rather than a quota.

Extending PR75 further would mean opening (1) or (2), and both are new conceptual directions rather than repairs. The recommendation is consolidation and merge.

Attribution

field	value
prompt author	unrecorded — authored outside this repository
executor	Claude Opus 5 (Anthropic)
dates	dispatched and executed 2026-08-31
round record	prompts/2026-08-31-normative-affordability/

Normative Affordability

Research round: unregistered, no Lean, no settled definition changed. It fixes the schematic interface between Answerability's standing load and Progress's claim-weighted conclusion, and audits the composition with Actionability, Uptake and settlement-relative safety.

The fixed-era composition is frozen. FIXED_ERA_THEOREM.md, the S1-S6 schematic stack of CORRECTED_STACK.md, the allocated-service typing, the row-indexed accounting and Service Transport are settled for this round; later work here is existence and characterization, which consume them rather than amend them.

Read REPORT.md first. Run the exact fixtures with

python3 tests/run.py

from this directory.

Files

• SERVICE_TRANSFER.md — the transfer theorem, the exact characterization by contiguity, the array-versus-fixed-sequence dichotomy, two starvation attacks, and the transport theorem that replaces contiguity as the interface primitive.
• JOINT_ACTIONABILITY.md — the interference countermodel, composition by common-region superposition, the vanishing-share failure of aggregate Uptake, and the minimal type of the joint response correspondence.
• SAFECERT.md — why actual-path safety is not a candidate, the robust form, and the non-revocation theorem.
• AFFORDABILITY.md — the revised definition, the Sustainable Progress theorem, and five premise-removal countermodels.
• EXISTENCE_AND_DUALITY.md — existence as a causal viability problem, a drift sufficiency sketch, and finite-horizon overload certificates.
• REALIZATION_MAP.md — the traderized mapping, the score identity and friction inequality, the core minimum as slack, and what immutable history alone secures.
• FOLLOWUP_REPORT.md — the realization audit: what survives contact with the traderized mathematics, three withdrawals, and what remains before an end-to-end Logical Induction statement.
• CORRECTED_STACK.md — the stack after that audit, schematic first and the Logical Induction instantiation second.
• SERVICE_FORCE_TYPING.md — the within-round dependency structure, and why the service variable is the allocated authority rather than the realized force.
• LI_PROGRESS_FROM_SERVICE.md — service-weighted Progress from the enforcement modulus, with its rate, its friction residual and the coercivity generalization.
• END_TO_END_LEARNER.md — the whole chain with every hypothesis labelled by layer, and what remains before an end-to-end statement.
• REASONWISE_ACCOUNTING.md — the aggregate-to-subset cap lemma, why aggregate safety does not bound a reason book, and three routes to reason-indexed Progress.
• CAPACITY_VS_SAFETY.md — local authority capacity against lifetime SafeCert, and the three routes connecting them.
• FIXED_ERA_THEOREM.md — the final fixed-era statement, its finite-horizon inequality, its dependency graph, and whether the construction is complete.
• AFFORDABLE_SCHEDULING.md — the viability problem in authority space, a positive existence theorem, the many-reason analysis, and the capacity region.
• PERSISTENT_AFFORDABILITY.md — the exact characterization of when persistent enforcement fits a finite liability budget, and the finite-horizon optimum.
• CAUSAL_CAPACITY.md — why the sustainable rate region is neither convex nor the right object, and the persistence region that replaces it.
• SIGNED_VS_CONSERVATIVE.md — the unbounded infinite-horizon separation between worst-case underwriting and the realized signed account.
• ONLINE_EXISTENCE.md — the causal scheduler, and why persistence has no online penalty.
• OVERLOAD_TARGET.md — normative infeasibility against insolvency, and the exact dual problem each one needs.
• SHARP_PERSISTENCE.md — the persistence criterion for a general star-shaped date cost, and the finite-horizon optimum it implies.
• SERVICE_ADMISSIBLE_EXISTENCE.md — what Answerability's admissible traces do to the criterion: a per-window service floor replaces a dip by a summability condition.
• CLOSED_LOOP_EXISTENCE.md — exogenous against policy-dependent friction, and the E0-E5 hierarchy of existence results.
• BOUNDED_DELAY_TRANSPORT.md — the interval feasibility theorem for claims with service windows, and FIFO as the complete construction.
• BOUNDED_DELAY_AFFORDABILITY.md — the minimum-cost service problem, its exact sliding-window closed form, and persistence without serviceability.
• SERVICEABILITY_FRONTIER.md — liability against allowable delay, and the transport residual it buys.
• MULTIREASON_SERVICEABILITY.md — why deadlines make reasons compete again.
• ONLINE_SERVICEABILITY.md — the online gap under deadlines, and what would close it.
• SHARP_SERVICEABILITY.md — why bounded sharp liability plus persistent service pays for the settlement-friction residual outright, and where it stops.
• EVENTUAL_VS_UNIFORM_SERVICE.md — persistence and eventual full service are the same existence question; uniform timeliness is strictly stronger.
• JOINT_SERVICEABILITY.md — the joint liability/friction/timeliness objective and the two-price assignment it collapses to.
• SHARP_TIMELY_SERVICE.md — the canonical positive result of the round, assembled with exact hypotheses and constants.
• DEADLINE_INSOLVENCY.md — the finite certificate that arrived claims cannot be answered on time, and why the persistence one is not finite.
• CONSISTENCY_AUDIT.md — every claim the round has withdrawn, where, and the pattern in the errors.
• src/, tests/ — two hundred and four exact-rational checks.

Verdict

AFFORDABILITY-COMPOSES-BUT-THE-SERVICE-CONDITION-IS-TRANSPORT-NOT-CONTIGUITY — the proposed architecture survives with one component replaced, one conclusion weakened and one inherited hypothesis refuted; its realization map survives an audit against the traderized mathematics with three withdrawals; and the remaining type mismatch between the schematic and the concrete learner is closed by separating allocated service from realized force. Contiguity of the claim measure against the service measure is exactly the weakest condition under which service-weighted learning transfers to claim-weighted defect, necessary and sufficient for triangular arrays, and necessary because a settlement-relative defect is a triangular array; for a defect sequence fixed independently of the horizon the exact condition is instead fixed-set contiguity, which is strictly weaker, and one-step delay separates them. Contiguity is nevertheless the wrong interface primitive: not checkable at a finite horizon, not quantitative, and false of bounded-delay service. The Deferred Service Transfer theorem reaches the same conclusion from a feasible adapted transport plan with a stability inequality and a service-to-claim cap; the two routes are incomparable rather than one deriving the other, since what the plan bounds is the density of the transported claim measure. Progress is staged rather than redefined, so no settled definition changes. Bounded-deficit Surface Fairness is a condition on total mass and does not control the claim-weighted defect at all: a two-surface fair rotation satisfies it with service-weighted defect exactly zero at every horizon while the claim-weighted defect tends to one half. Joint Actionability does not follow from individual Actionability when the gain is read against reason-relative regions and does follow, with no convexity and no separability, against the common region; in the traderized realization it is true and unused, because the compiled position is reactive. Three realization claims are withdrawn: per-reason Uptake is not free from the criterion, which does not quantify over the enforcement position at all, and comes instead from the market maker's cumulative cap; the criterion's preservation theorem is substrate preservation and not Uptake; and pointwise self-financing is sufficient and not necessary for persistent affordability, refuted by an alternating norm whose every date is robustly loss-making while the signed account stays inside a fixed band. The service variable is the allocated authority and not the realized force: the multiplier is fixed before the market maker picks a price and is freely schedulable inside a per-date liability capacity, while the realized position is decided at the fixed point, is zero exactly when the reason is perfectly satisfied, and counts a date as better served because the reasoner did worse on it. Against the allocated-service measure the per-date enforcement modulus gives service-weighted Progress outright, with rate the inverse square root of allocated service and residual the norm's own weighted mean-square exclusion of the still-live worlds; one compatible live world suffices and gives the fast rate, a certified core minimum caps the residual, and the liability charge grows like the square root of allocated authority, so a reason can receive divergent service on a finite lifetime budget because the realized force dies away as compliance improves. Actionability generalizes to a convex coercivity modulus that subsumes the linear and projection-quadratic cases in one theorem. Finite-horizon overload admits a sound Farkas certificate and no converse, and it is not the highest-value next target: the necessity of bounded liability is, followed by the existence of an affordable schedule, now posed in the allocated-service space where it is well typed. The construction is complete for a fixed era: conditional on a service-faithful authority schedule, per-row liability floors summing to a finite total, and the two residual assumptions, every arrow from the claim stream to claim-weighted Progress and to preservation of the original criterion is proved, with a finite-horizon inequality and an explicit dependency graph. The aggregate market maker cap becomes a uniform per-subset ceiling U plus the total floor exactly when the complementary rows carry summable liability floors, and aggregate safety alone does not suffice: two books with increments plus one and minus one have an identically zero aggregate account and no reason-level ceiling at all. Local authority capacity and lifetime SafeCert are different objects — a policy inside its per-date capacity at every date has an account unbounded below when the allowances are not summable, and spending the realized account slack instead strictly enlarges the viable region, quadratically. Coercive Actionability converges exactly when the modulus is bounded away from zero away from zero; convexity is not needed for convergence and buys only the rate. Conditional on force feasibility there is no service-capacity competition between reasons at all — the per-row conformance bound is independent of how many rows are active — so overload within that qualification is liability overload, while an empty region, an unpriceable row or an illegal compiled control is a failure before any account is consulted. Authority is quadratic in the liability allowance, so the per-date capacity region is non-convex and concentration beats proportional splitting by the reason count. The sustainable authority-rate region is neither convex nor the right object: time-sharing convexifies a renewable per-date resource and the liability budget is a consumable stock, and the rate region reports zero on friction sequences where the reason is in fact persistently served. Replacing it with the persistence region gives an exact answer. Writing the friction as the exclusion depth times the square root of the date's slack-plus-volume, a schedule with divergent cumulative authority and finite lifetime charge exists if and only if the friction has liminf zero; the maximum cumulative authority through any horizon is exactly the squared budget over the squared running minimum friction, attained by putting the whole budget on one least-friction date. Persistence is therefore budget-free and does not compete across reasons, so every individually persistable reason can be served simultaneously with all the others and there is no Hall-type condition to check; the persistence region is the full power set of the persistable reasons. A causal scheduler seeing the friction only at its own date achieves persistence exactly when an offline one can, by a doubling-threshold rule spending geometric tranches, and any scheduler with a positive floor on its per-date allocation is defeated by a friction sequence bounded away from zero. Conservative underwriting is strictly smaller than signed-account affordability by an unbounded margin: a satisfied row takes a zero position and costs nothing however deep its exclusion of the live worlds, so a floored friction that caps conservative authority at a constant permits linear growth on the realized trajectory. The scalar account slack is not sufficient state, because the world attaining the minimum can settle away. The conservative criterion is not the sharp one: for a general star-shaped date cost the persistence criterion is that the cost of a fixed reference allocation has liminf zero, which specializes to the conservative friction dipping and, under the exact worst-case charge, to the exclusion depth alone dipping — and the two differ, since depth one over t against engine scale t to the fourth makes the conservative friction diverge while a constant allocation stays affordable forever. The finite-horizon optimum is the largest single-date inverse cost, which recovers the earlier formula and shows that an unbounded optimum is necessary but not sufficient for persistence. The causal scheduler has no penalty for the persistence property and no positive competitive ratio for the amount of authority accumulated: two dates already cap any rule at one quarter and a cascade drives the achievable ratio to zero. Answerability's admissible traces change the criterion rather than qualifying it: a per-window service floor is affordable exactly when the window minima are summable, which most dip sequences fail, and a cumulative service demand forces the running minimum depth to decay reciprocally to it — this is the frontier between cheap enforcement dates and timely ones. The rate region's non-convexity claim is narrowed to the finite-horizon cumulative-authority frontier, which is where the counterexample lives. Insolvency is a claim about the infinite future: an observed friction floor is not a certificate, and an authenticated one needs a tail bound with a proof object, failing which a reason is deferred rather than recorded as unaffordable. The sharp criterion is not depth-only either: the reference cost is comparable within a factor of four to the minimum of the squared depth and the depth times the root of the engine scale, so a fixed exclusion depth against a vanishing engine scale is persistently enforceable with no depth decay at all, and the depth-only reading holds exactly under a floor on the engine scale. Bounded-delay service is characterized exactly. Feasibility against a given service profile is the interval condition — prefixes do not suffice, since service cannot run backwards — and first-in-first-out is optimal and complete, so the transport plan is constructed rather than assumed. When the service profile is chosen the minimum cost is a shortest path over consecutive runs, and on the sharp charge's linear branch it is exactly the sum over claims of the claim mass times the minimum date weight in that claim's own window: a sliding window, not the disjoint blocking the previous pass used, and the two differ by up to the deadline. That one formula covers dense, bounded and sparse claim streams and interpolates exactly back to unconstrained persistence as the deadline grows. The separation is exhibited: dips spaced at sixteen satisfy unconstrained persistence and fail bounded-delay service at deadline three, whose cost grows linearly with the horizon, while a deadline matching the dip spacing restores affordability. Since the minimum cost is nonincreasing in the deadline and the transport residual is nondecreasing in it, there is a least affordable deadline and the Sustainable Progress residual it buys is the reason's temporal modulus evaluated there — so of the two residuals in the fixed-era bound, the settlement-friction one is a property of the norm and the transport one is purchasable with liability budget. Timely service competes across reasons where persistence did not, because its minimum cost is a definite positive number and the budgets add; the joint condition is a single sum rather than a Hall condition, and it is sufficient but not necessary when reasons share rows. Under deadlines the online gap reappears and is unbounded, since waiting is no longer free, and it closes only if the whole window's weights are predictable at the claim's arrival. On the sharp charge's linear branch the settlement-friction residual is not an independent obstruction: the liability charge is computed from the supremum over live worlds of the same deficit whose weighted mean square the residual is built from, so the friction numerator is exactly four times the charge, a bounded lifetime budget with divergent allocated service drives the residual to zero at rate four times the budget over the allocation, and the final claim-weighted defect is bounded by the transport residual alone — the reason's temporal modulus at the affordable deadline. A corollary is that on that branch a norm permanently excluding every live world by a fixed depth cannot be persistently and affordably enforced at all. The compression fails past the branch point, and it fails exactly on the second route to cheap enforcement: a norm cheap because nobody trades against it buys persistence without buying conformance, so cheap enforcement is not always conforming enforcement. The batching theorem's hypotheses are corrected — no splitting needs concavity rather than star-shapedness, since with existing load the comparison is between increments, and the run structure additionally needs equal claim masses, since a crossed assignment can strictly win when a saturating date should take the larger claim. The reading of the bounded-delay formula's limit as the unconstrained persistence criterion is withdrawn, since the limit of the bounded-delay costs can exceed the unbounded-delay cost when the gaps between cheap dates diverge. The settlement-friction residual is schedule-dependent rather than a fixed property of the norm, so the frontier is stated over the constrained residual; and in the linear regime the joint objective separates per claim into a combined score whose friction price is four times its liability price, leaving two prices rather than three — what underwriting costs and what waiting costs. The fixed-era construction is frozen; what remains is closed-loop friction, the signed class, the necessity of bounded liability, and transport-stability certificates. The round closes here. Persistence and eventual full service under unlimited deferral are the same existence question, by a diagonal that gives each claim its own cheap date, so the earlier separation between them is withdrawn and the substantive Answerability constraint enters only when delay itself matters; uniform bounded delay remains strictly stronger, and no condition on the gaps between cheap dates substitutes for the sliding-window criterion, since cheap dates every other date with reciprocally decaying depth still make timely service unaffordable. The transport residual is the claim-normalized average of the edge errors rather than their raw sum, which makes it comparable to the claim-weighted defect it bounds and bounded by the temporal modulus at the plan's delay whatever the claim mass. The friction-free theorem needs the settlement interface's own monotonicity as an explicit hypothesis, because the charge is computed against the worlds live at each date and the residual is evaluated against the worlds live at the horizon, and past the branch point the exact condition is that the squared charge-plus-scale over the scale, averaged against allocated authority, vanishes — summability of the engine scale is neither necessary nor sufficient. The joint frontier is genuinely convex and fully recovered by scalarization, because both objectives are linear on the transportation polytope, provided claim mass may split fractionally. Deadline insolvency, unlike persistence insolvency, admits a finite authenticated certificate whose every field is settled record, complete exactly when future weights are bounded below. Assembling these gives the round's canonical theorem: sharp-linear affordability plus timely Answerability transport yields claim-weighted Progress up to semantic delay only, at rate twice the root budget plus the root ceiling over the root allocated authority. The fixed-era affordability and serviceability layer is finished; closed-loop friction, certification of the temporal modulus, and simultaneous timeliness and linear-branch affordability move to later rounds.