2026-10-22-normative-inductor-integration

Prompt: Normative Induction — End-to-End Integration and Theorem Consolidation

You are working in A-M-Berns/alignment-workspace, on the existing branch for PR #129. Do not merge the PR. Your task is to bring the Normative Induction construction into a mathematically coherent, substantially integrated, pre-merge state.

This is a research and formalization round, not a documentation cleanup. Exercise mathematical judgment. Challenge existing formulations when necessary, repair genuine interface gaps, and distinguish results established by the formalism from assumptions imported through theorem signatures.

The goal is not to solve normative learning in full. It is to reach a point where the maintainer can confidently say:

We have one well-defined Normative Inductor, one substantive theorem characterizing its guarantees, an informative end-to-end example, and a precise account of what remains conditional or open.

0. Orientation and constraints

Before making changes, read:

• AGENTS.md, DECISIONS.md, PRIORITIES.md, the normativity claims registry, and applicable contribution rules.
• PR #129 in full, especially WarrantEntitlement.lean, SeedStatics.lean, ConvergenceMeasure.lean, EndToEndWitness.lean, and the two new rounds' reports.
• NormativeInductionInterface.lean, NormativeInductorComposition.lean, NormativeInductor.lean, EnforcementStrategy.lean, and the relevant preservation and effective-representation modules.
• The canonical wiki pages Normative-Inductor.md, Normative-Induction.md, Seed-Theory.md, Legitimacy.md, and Theorem-Spine.md.
• The existing practical-response/adequacy-gate and decision-adapter work, wherever it now lives.
• The relevant traderization affordability results and their stated limits.

Follow the repository's proof-layer/specification-layer permissions. Do not silently modify protected specifications or settled maintainer decisions. If a necessary change requires maintainer authority, formulate the exact proposed change and continue as far as permitted.

Preserve the established theorem substrate and claim/provenance discipline. All new Lean proofs must be sorry-free, with explicit axiom audits. No strengthening of theorem status merely because an existing conditional result has been instantiated with similarly conditional hypotheses.

1. Target: one mathematical Normative Inductor

Work backward from the following intended construction:

\[ \text{Accounted normative history} \longrightarrow \text{Licensed, revisable warrants} \longrightarrow \text{Canonical joint constraint region }K_t \]

\[ \longrightarrow \text{Affordable enforcement trader} \longrightarrow \text{Augmented Logical Inductor }P \longrightarrow \text{Practical response }\Pi(P) \longrightarrow \text{Progress}. \]

Revision by defeat, settlement, and rebuttal affects the same warrants and constraints throughout. The public record carries the corresponding provenance and authorization information.

There should not be one operative interpretation of a warrant for its forced interval, another for its certificate, and a third for the market's enforcement position.

The target theorem should have three connected conclusions, referring to the same actual market and normative state.

A. Preservation of Logical Induction

The augmented market remains a logical inductor relative to the underlying assessment process:

\[ \operatorname{LI}_L(P). \]

The trader's cumulative assessed liability must be bounded by an explicit \(B\). Where possible, produce that bound using the warrant and revision structure, rather than assuming it independently.

The new odds-bankroll and reopening results in PR #129 are central inputs.

B. Finite-time normative responsiveness

The actual market's prices approximately respect the legitimately compiled constraints, at a quantitatively declared tolerance.

Ideally, for a suitable public defect,

\[ d_t(P_t,K_t)\leq\delta_t. \]

Use the force layer's actual guarantee. If it provides rowwise violations rather than distance to the region, do not silently identify the two. Either establish an appropriate quantitative conversion or use a well-justified row-defect interface.

The theorem should apply to the actual augmented market history, not merely to arbitrary stipulated prices satisfying the constraints.

C. Practical Progress

A concrete or parametrically specified decision adapter consumes this market and satisfies the appropriate practical-response certificates, yielding the existing Progress bound:

\[ \operatorname{Progress}(P) \leq \Gamma \sqrt{\frac{\sum_s\rho_s}{\sum_s\lambda_s}} +\operatorname{Error} +D\operatorname{Residual}. \]

Critically, do not merely assume PracticalUptake in the final integration theorem if the relevant practical/uptake conditions can be derived from the construction and existing adequacy-gate results.

A useful conditional theorem is acceptable. A conjunction of previously conditional theorems that leaves all their interesting premises untouched is not the desired outcome.

2. First task: audit the actual handoffs

Before implementation, build a concise, mathematically precise map of the current interfaces.

For every arrow in the proposed pipeline, identify:

1. The actual Lean definitions on each side.
2. Whether the two sides refer to the same objects and histories.
3. The theorem establishing the handoff, if any.
4. Any assumptions the handoff currently leaves undischarged.
5. Whether the gap is a routine integration problem, a substantive mathematical problem, or a genuinely missing semantic input.

In particular, investigate the following.

Accounted obligation state versus warrant grounds. The existing ObligationState/accounted-export architecture and PR #129's Grounds are distinct objects. Clarify how a legitimately incurred obligation or reason yields an admitted warrant occurrence. Do not conflate authorization of a verdict-selection event with authorization to add a constraint to the market.

Canonical constraint semantics. Grounds.region, Grounds.settledRegion, forcedRegion, and the region consumed by the actual enforcer must be reconciled.

Licence validity under revision. The current Supported/Licensed chain is defined using the port-only bundle. Determine how a certificate remains valid or becomes invalid under the actual settlement-sensitive survivor rule.

Market conformance. Determine whether the warrant-generated schedule is actually connected to the force layer's quantitative conformance theorem at the realized market.

Practical response. Determine exactly which existing adapter lemmas can convert enforced price constraints into adequate behavior, and which semantic assumptions they require.

Computability. Investigate ComputableMarket for the augmented warrant-enforcement construction. If its proof requires a substantial separate development, leave it as a clearly named and isolated assumption. Do not disguise it through an opaque wrapper.

Use this audit to prioritize implementation. Do not allow it to become an end in itself.

3. Consolidate the normative state and revision semantics

The first major implementation objective is one canonical operative warrant state.

A plausible design includes grounds, provenance, a docket, settlement, and authorization/admission data, with a single derived set of active occurrences.

The precise representation is your choice; prefer reusing existing abstractions over introducing redundant ones.

The active-occurrence rule must properly handle:

• Direct docket defeat.
• Dependency on another occurrence's applicability.
• Occurrence-level defeater conditions.
• Rebuttal of a conditional form when its consequent is settled false.
• Vacuity when its antecedent is settled false.
• Retirement of refuted premises.
• Appropriate propagation of retirement or defeat through dependency chains.

One concrete problem: in the current fixture, occurrence 3 cites applicability occurrence 2 and carries {2} as a defeater, yet occurrence 3 survives when occurrence 2 is undercut. Determine whether this is consistent with the intended semantics. If not, repair it.

Do not rely on the subsequent refutation of \(Y\) to hide an earlier defect in dependency propagation.

Similarly, identify exactly which settlement-closure conditions the construction needs. Either compute the required closure or make a clear, checked closure hypothesis. The rules should not work only because the fixture manually supplies convenient extra pins.

Establish a canonical compilation path such that the active warrants generate:

• The joint price-space region.
• The cited bundle for certificate checking.
• The inputs to forced-interval calculation.
• The rows held by the enforcement trader.
• The public record of operative commitments.

Distinct views are acceptable; distinct incompatible semantics are not.

Investigate duplicate occurrence IDs and the provenance consequences. Establish an appropriate well-formedness discipline if necessary.

Preserve the distinction between logical/structural constraints and defeasible substantive constraints.

Acceptance criterion: given a normative state, there is a single authoritative answer to which warrant rows are currently enforced, and all the principal consumers agree.

4. Connect warrant enforcement to preservation and quantitative conformance

Use the actual force and market interfaces, not a new toy market invented for this round.

The new odds theorem gives the original \(3\rho\) liability bound with a suitable live slack world. The reopening theorem gives \(3\rho(2+\rho)\) for the relevant one-reopening case.

Preserve these results and investigate their generalization only insofar as necessary for the end-to-end construction.

In particular:

• Separate the general affordability interface from its specific warrant-schedule discharge.
• Connect the canonical survivor schedule to the actual enforcement trader.
• Establish the conformance guarantee for the augmented market under the real market-maker contract.
• Make the budget/intensity/tolerance dependence explicit.
• Preserve joint feasibility: simultaneously serviced constraints must form one joint region, not competing independently enforced obligations.
• Handle empty or conflicting regions accountably; never derive effective conformance from an inconsistent constraint set.

There is a potential architectural mismatch between the older projection-based Normative Inductor and the rowwise violation-proportional trader used in PR #129. Investigate it carefully.

Prefer one clear canonical enforcement architecture, with a mathematically explicit bridge if the older projection and newer rowwise interfaces are both retained.

Do not attempt to prove a fully general repeated-reopening affordability theorem unless it falls naturally out of this work. One general conditional interface plus a rigorously discharged nontrivial instance is enough for this round.

5. Actually connect the practical-response layer

This is a central objective, not an optional extension.

The current NormativeInductionInterface has a strong Progress theorem, but EndToEndWitness.Run.witness assumes the complete PracticalUptake certificate.

Investigate the existing adequacy-set, soft-gate, Lipschitz adapter, and practical-certification results. Reuse them to build an actual connection:

\[ \text{Constraint region} \to \text{Market defect} \to \text{Response adequacy} \to \text{PracticalCert} \to \text{Progress}. \]

A suitable simple finite response problem is encouraged.

The adapter should be nontrivial: it should genuinely depend on the market's prices and use those prices to make a meaningful choice among responses.

Do not satisfy the requirement through an arbitrary constant adapter, identically zero loss, zero serviced mass, or an unconstrained PracticalUptake assumption repackaged under another name.

The important semantic bridge should remain visible: a price lying near an admissible region does not by itself prove that an action is adequate. The region must be connected to a declared adequacy interpretation, and the adapter must have the requisite stability or margin property.

It is acceptable to supply that adequacy semantics as a typed assumption and prove the decision-theoretic consequences. It is not acceptable to assume the conclusion that the actual response has small loss.

Use the existing distinction between competence, permission, and adequacy. Do not unnecessarily fold authority or corrigibility constraints into the enforcement trader.

The existing Progress theorem must apply to the same market history constructed by the warrant-enforcement stage.

6. Strengthen the end-to-end witness

Replace or extend EndToEndWitness.Run.witness so that it is genuinely informative about Normative Induction.

Retain the useful structure of the current fixture: a principal warrant, an independent warrant on \(Z\), settlement, undercut, rebuttal, and a surviving commitment.

But improve it substantially.

Coherence. The current fixture includes \(X\land A\leq X\) but omits other obvious logical relations, such as \(X\land A\land Y\leq Y\). Consequently the initial warrant constrains the conjunction without necessarily constraining \(Y\) itself. Add appropriate structural coherence constraints or explain rigorously why an intentionally incomplete fragment is necessary.

Genuine normative influence. Show that the warrant meaningfully restricts some relevant market prices and that the resulting conformance guarantee affects the adapter's response quality.

Revision. Show an undercut and rebuttal changing the operative constraints through the canonical rule. The independent warrant must continue to function.

Market behavior. The current clearing schedules are independently stipulated and satisfy the older port-only region. Distinguish these from the actual market's output. The improved witness should establish nontrivial properties of the actual constructed market, even if its exact prices cannot be computed.

Response behavior. Include at least one concrete finite decision problem with a nonconstant market-consuming adapter and a derived practical certificate.

Nonvacuity. Exhibit positive enforcement intensity, an active substantive warrant, genuinely restrictive constraints, positive service mass, and meaningful evaluation loss. The example should not satisfy its properties merely because nothing substantive happens.

Prefer a small example proving a meaningful behavioral theorem over a large conjunction of shallow nonvacuity properties.

For theorem packaging, distinguish:

• Construction of an object.
• A conditional result about that object.
• A realized witness discharging the conditions.

Do not label the second as the third.

7. Produce the consolidated theorem

Create a canonical Lean theorem, in an appropriately named module, that gives the closest rigorous formulation of the intended Normative Inductor.

It should have a readable signature whose assumptions correspond to independently meaningful properties:

• Legitimate admission/authorization and well-formedness.
• Correct compilation of active warranted constraints.
• Joint feasibility.
• A computable augmented market, where required.
• Affordable enforcement.
• An appropriate quantitative conformance interface.
• Adequacy semantics and a sound/stable decision adapter.
• Evaluation coverage and amplification conditions.

Avoid including premises equivalent to the entire desired conclusion.

As far as practicable, the theorem should discharge the subordinate conditions from constructions developed in the repository rather than taking them as primitive assumptions.

Its conclusions should include:

1. Preservation of ordinary Logical Induction.
2. Quantitative conformance to the operative normative constraints.
3. The canonical practical Progress bound.

I expect some hypotheses to remain semantic inputs, and possibly computability inputs. That is acceptable if they are sharply isolated and shown nonvacuous in a meaningful example.

If full consolidation fails, produce the strongest nonvacuous theorem that is actually proved and state the exact missing lemma or incompatible interface preventing the stronger version.

Do not introduce a theorem whose primary accomplishment is syntactically conjoining the three desired conclusions under assumptions that already state those conclusions.

8. Clarify the learning claim

Write down exactly what is being learned.

Distinguish:

1. Learning future normative evaluations or verdicts.
2. Learning and revising the strengths or applicability of warrants.
3. Learning to make practically adequate responses under changing warranted constraints.

The present substrate most directly supports the third, conditional on adequate handling of the first two.

Do not claim that bounded-liability enforcement makes normative judgments true, automatically discovers good warrants, or guarantees convergence to moral truth.

At the same time, do not undersell the real result: a reasoner whose beliefs and responses can be made quantitatively responsive to revisable warranted constraints, without destroying Logical Induction, is a substantive construction.

State a clean conditional asymptotic corollary if it follows straightforwardly from the Progress bound as the average work budget, error, and residual mass vanish. Identify the coverage and stability conditions it needs.

Do not embark on a general theory of convergence between multiple reasoners, full legitimate inquiry, or LI+BRIA decision theory in this round.

9. Paper-facing theorem story

Produce a concise, serious mathematical exposition alongside the Lean.

Suggested organization:

A. Problem. How can a bounded logical reasoner acquire and respond to revisable normative commitments without either ignoring them or allowing their enforcement to destroy its underlying rationality guarantees?

B. Normative state. Grounds, warrants, licence, revision, and the canonical active constraint region.

C. Enforcement. The compiled trader, quantitative responsiveness, and financial liability.

D. Main theorem. The exact hypotheses and three connected conclusions: Logical Induction, conformance, and practical Progress.

E. Example. A warrant introduced, acted on, challenged, revised, and tracked through the resulting market and response behavior.

F. Limits. What remains conditional: warrant soundness, legitimacy, settlement semantics, market computability, arbitrary repeated revision, semantic adequacy, and coverage.

The exposition should be understandable without working through the entire repository or its internal priority numbering.

Be especially careful to distinguish:

• Semantic entitlement from operative market force.
• Credal constraints from constraints on actual behavior.
• Verdict licensing from warrant admission.
• A bounded liability guarantee from a guarantee of market computability.
• A genuine end-to-end construction from a conditional composition.
• Normative responsiveness from normative correctness.

10. Priorities and stop rules

Prioritize in this order:

P0 — Canonical semantics. One revision-aware warrant compiler consumed consistently throughout the construction. This should be completed.

P1 — Integrated theorem. One meaningful general theorem connecting affordable warrant enforcement, LI preservation, quantitative conformance, and practical response.

P2 — Strengthened witness. One nontrivial instance testing the actual integration, including the adapter.

P3 — Exposition and consolidation. A paper-facing theorem story and accurate status documentation.

Do not sacrifice P0–P2 to make the documentation comprehensive.

Do not undertake unrelated theorem development just because it is interesting. File open problems rather than continuing indefinitely. In particular, an arbitrary number of reopening events and a full theory of substantive normative correctness are beyond the required scope.

If any objective encounters a hard mathematical obstruction, produce a minimal counterexample or no-go statement where possible, identify the precise interface responsible, and continue with the strongest defensible partial construction.

A discovered obstruction is a legitimate research result. An artificial theorem that conceals the obstruction is not.

11. Verification and reporting

Run the full applicable Lean build, axiom audit, Python tests, and repository checkers. Follow claim registration and provenance rules.

Produce a report with:

1. Final architecture: one concrete diagram naming the actual Lean objects and theorems at every arrow.
2. Main theorem: its exact signature, a plain-language interpretation, and an explicit inventory of assumptions discharged versus still imported.
3. Witness: what is genuinely constructed, what happens to the warrants, what the market guarantees, and what the adapter does.
4. Repairs: inconsistencies corrected from PR #129, especially survivor semantics, licensing, records, and force interfaces.
5. Open problems: ranked by whether they prevent the conditional Normative Inductor, a general repeated-revision construction, or the eventual stronger normative-learning agenda.
6. Verification: exact test/build results and claims whose status changed.
7. Merge recommendation: ready, ready with specified reservations, or not ready, with mathematical reasons.

Do not declare the PR complete merely because all tests pass.

12. Final instruction

Approach this as someone trying to identify the smallest mathematically serious Normative Inductor worth presenting to another researcher.

The existing repository contains considerable relevant work. Prefer discovering and completing the right composition over inventing another parallel formalism.

But be willing to simplify or revise an existing abstraction if the current architecture makes a meaningful theorem impossible.

The decisive question is:

Can we now point to one actual, revision-sensitive, normatively responsive logical reasoner; explain what constraints it accepts and why; prove that its enforcement is affordable and preserves Logical Induction; and derive a quantitative guarantee about its practical behavior?

Push the PR as close as possible to a genuine affirmative answer, with the remaining qualifications mathematically explicit.

The target is not superficial completeness. It is one coherent construction whose theorem means what we want it to mean.

Report — Normative Induction: end-to-end integration and theorem consolidation

Dispatch: prompts/2026-10-22-normative-inductor-integration/PROMPT.md, on the branch of PR #129 at b1c17d30. Prompt author: the maintainer. Executor: Claude Fable 5.1 (Anthropic). Date: 2026-10-22.

Lean	OddsBankroll.lean (22 declarations), WarrantForce.lean (67), WarrantInductor.lean (57), WarrantInductorWitness.lean (111), new; WarrantEntitlement.lean (315, revised), EndToEndWitness.lean (127, revised)
Exact rationals	src/inductor.py, 22 checks under tests/
Documents	HANDOFFS.md (the audit of the arrows), THEOREM_STORY.md (the paper-facing account, §§A–F with the learning claim)
Registered	inductor.grounded-survivors (112), inductor.conformance (39), inductor.certificate-transport (124), inductor.warrant-inductor and inductor.witness (35), all lean-proved
Items	124–128 filed; 35, 39, 112, 115, 116, 118, 121 narrowed

Corrections to the prompt (AGENTS.md §8)

1. §3 states that occurrence 3 "cites applicability occurrence 2 and carries {2} as a defeater, yet occurrence 3 survives when occurrence 2 is undercut." Confirmed, and it was a defect of the rule, not of the fixture: Grounds.survivors consumed the occurrence's own port and settleOutcome, retired one step along cites, and read defeaters not at all. The fixture's later refutation of Y hid it exactly as the dispatch suspected. Repaired by the grounded rule (§P0 below); the two earlier rounds' documents carry dated notes where they describe the strength occurrence as surviving until n_E.
2. §4 anticipates "a potential architectural mismatch between the older projection-based Normative Inductor and the rowwise violation-proportional trader." The mismatch is real but lies elsewhere than the trader's shape: the projection enforcer and every computability theorem run on the deductive market (DeductiveEnforcement.history DP), while PR #129's rowwise trader and its bankroll ran on the assessment market (EnforcementPreservation.history L), and no theorem relates the two firms. This round runs the rowwise trader on the deductive market; the assessment-market bankroll becomes an instance of the same abstract account. The two enforcer shapes are not bridged (HANDOFFS.md, the market arrow; item 125).
3. §5 asks for the practical certificate to be derived from "the existing adequacy-set, soft-gate, Lipschitz adapter" results. The soft-gate theorems are stated at a region point (Region u A τ, Margin u A τ δ, Within b u d), which rowwise conformance does not supply. The derivation goes through a displayed-margin route instead (certified bounds transported to the displayed price, then softGate_massOff_le_displayed with a synthetic region point); Within is never established at the real market. Not a deviation from the ask, but the route the dispatch names is not the one that works.
4. §1 writes the target conformance as d_t(P_t, K_t) ≤ δ_t. What is proved is rowwise: every row's excess at the displayed price is at most tol_t. The conversion to a distance is not established and is filed (item 125); the certificate transport is the conversion that is established, to the bounds a licence certifies and an adapter reads.

P0 — the canonical semantics

The grounded rule (WarrantEntitlement.lean). An occurrence is undercut when its own port or a port of its defeat set is defeated; live given an identity set S when not undercut, kept by settleOutcome, and citing into S; the active identities are the least fixed point of activeStep from ∅, reached in G.length steps (activeIds_fixed, by the chain of iterates and a cardinality argument); the survivors are the occurrences live given the active identities. mem_survivors is the characterization: in the grounds, not undercut, kept, citing a survivor. Antitone in the settlement (survivors_anti, as before) and in the docket (survivors_anti_docket, new). The one-step retirement, citesRetired and retiredIds are gone; propagation along citation chains of any length needs no closure on the settlement (item 121 answered). A dangling or cyclic citation is inactive.

defeaters is read as ports whose defeat also undercuts the occurrence — the reading the 2026-10-20 ruling describes ("a warrant's forms share one defeat condition") and the fixture used. The alternative reading, undercutting occurrences that defeat when themselves active, needs the grounded extension with attacks and is filed (item 127).

One bundle, four views (WarrantInductor.lean, NormState). The normative state is the record state with the structural rows. Its bundle is forcedBundle of the seed whose items are the survivors', with the live structural rows and the pins. region_iff reads it by occurrence; bundle_eq_settledBundle says it is the settled bundle when there are no structural rows; interval is forcedInterval on the same seed; the enforcer's rows are its rows through toForceRow; Supported and Licensed are SupportedOn this bundle. The record's Licensed (WarrantEntitlement) is the same predicate on the settled bundle, and Admissible, Supported, the record's position (trace) and positionOf_eq_zero_of_region all now read the settled bundle rather than the port-only one.

Acceptance criterion. Given a state, NormState.bundle is the one answer to which rows are enforced; warrant_inductor's third conclusion is that every enforced row is a survivor's row under its citation, a live structural row, or a pin (mem_compileAt).

Well-formedness and admission. Grounds.WellFormed: identities unique, citations resolve; then a survivor's citation names exactly one survivor (cited_unique). Admission O G: every occurrence realizes an exposed obligation occurrence — a typed link to the accounted state, distinct from the record's authorized verdict-selection events.

P1 — the integrated theorem

The affordability interface (OddsBankroll.lean). Account: a trader playing enforcementStrategy (pres n) n against a history with rational quotes, assessed at live worlds, capped there. The odds bankroll −cap·ρ and the reopened bankroll −cap·ρ(2+ρ) are proved once; the assessment market (WarrantEntitlement.assessmentAccount) and the deductive market (WarrantForce.WarrantSchedule.account) are its instances at cap 3, the second from the source firm's own floor tradingFirmTrader_netWorth_floor.

The enforcer on the deductive market (WarrantForce.lean). WarrantSchedule: fragment map, compiled bundle per date, tolerance. enforcer : EffectiveEnforcer holds the bundle's rows at the calibrated intensity (ε_n + A_n)/tol_n²; market is the augmented deductive market; isLogicalInductor from a liability bound and EffectiveEnforcerComputation, the latter proved for finite-phase schedules (computation: the trade map is a fixed syntax tree in the date and the intensity, tradesOf_primrec, tradesOf_stepBundle_primrec).

Conformance at the realized market. rowViolation_le_tol: with a rational cube point in the day's region, every row's violation at the displayed price is at most tol_n. The slack and volume terms of weighted_square_le_pair are discharged by marketMaker_day_value_le_cube and Strategy.abs_value_le; the calibrated intensity closes with rowViolation_le_of_intensity_ge. excess_le_tol_of_admitted extends it to every row a certificate may cite (bundle rows and cube faces).

Transport. FunctionalCert (multipliers on rows summing to a functional), value, mass; le_of_excess: ℓ·p ≤ value + mass·δ where every positively weighted row's excess is at most δ. BoundCert.transport, bracket_at_market (a supported verdict's bracket at the displayed price, widened by mass·tol on each side), functional_at_market.

The adapter (WarrantInductor.Problem). Scores are affine functionals of the fragment's prices; the gate is GatedChoice.softGate on them; the public defect defectOf is the displayed inadequate excess; Excludes and Margins are certificate conditions on the bundle. scoreR_le, scoreR_ge at the market; massOff_le: inadequate mass at most κ·defectOf; defectOf_le: at most massCap·tol; practicalCert. AdapterProtocol.uptake derives PracticalUptake.

The theorem. warrant_inductor, as stated in WarrantInductor.lean; the plain-language reading is THEOREM_STORY.md §D, and warrant_inductor_of_slack is the one-slack-world form.

theorem warrant_inductor {Occ Req : Type*} {S : Protocol Occ Req} {anchor : Occ → Req}
    {O : ObligationState S anchor} {Q : Type} [Fintype Q] [DecidableEq Q]
    (N : ℕ → NormState d) (F : FragmentMap d) (wit : ℕ → Fin d → ℚ) (tol : ℕ → ℚ)
    (htol : ∀ n, 0 < tol n)
    (adm : ∀ n, Admission O (N n).grounds) (hwf : ∀ n, (N n).grounds.WellFormed)
    (DP : DeductiveProcess) (process : DeductiveProcessComputation DP)
    (hE : EffectiveEnforcerComputation (schedule N F wit tol htol).enforcer)
    (B : ℝ)
    (hliab : ∀ n (v : PCWorld), v.ConsistentWith (DP.D n) →
      -B ≤ (realizedEnforcer DP ((schedule N F wit tol htol).enforcer.adaptive DP)).netWorth
        ((schedule N F wit tol htol).market DP) v n)
    (A : AdapterProtocol O d Q)
    (hcheck : ∀ s ∈ A.services, checkCompiled (N s).bundle (wit s) = true)
    (hex : ∀ s ∈ A.services, (A.prob s).Excludes (N s).bundle)
    (hmar : ∀ s ∈ A.services, (A.prob s).Margins (N s).bundle (tol s))
    (hεad : ∀ s ∈ A.services, 0 ≤ A.εad s) (hD : 0 ≤ A.D)
    (hadequate : ∀ r, ∀ s ∈ A.services, ∀ q ∈ (A.prob s).A, A.lam r s q ≤ A.εad s)
    (hbound : ∀ r s q, A.lam r s q ≤ A.D)
    (hlam : ∀ s ∈ A.services, 0 ≤ A.lamS s) (hlampos : 0 < ∑ s ∈ A.services, A.lamS s)
    (hbudget : ∀ s ∈ A.services, A.lamS s * (((A.prob s).massCap * tol s : ℚ) : ℝ) ^ 2 ≤ A.ρ s)
    (hΓ : 0 ≤ A.Γ)
    (hamp : ∀ s ∈ A.services, ∑ e ∈ O.boundary.exposed, A.T e s * (A.D * (A.prob s).κ) ≤
      A.Γ * (A.lamS s / ∑ t ∈ A.services, A.lamS t)) :
    IsLogicalInductor ((schedule N F wit tol htol).market DP) DP ∧
    (∀ n, checkCompiled (N n).bundle (wit n) = true →
      ∀ c ∈ (N n).bundle, excess c.row ((schedule N F wit tol htol).priceVec DP n) ≤ tol n) ∧
    (∀ n, ∀ c ∈ (schedule N F wit tol htol).bundle n,
      (∃ o ∈ (N n).survivors, c.cites = Cite.sub o.id ∧ (adm n).origin o ∈ O.boundary.exposed ∧
        ∀ j, o.cites = .occurrence j → ∃! o', o' ∈ (N n).survivors ∧ o'.id = j) ∨
      (∃ it ∈ (N n).seed.liveStr (N n).docket, c.cites = Cite.str it.id) ∨
      ∃ i, c.cites = Cite.settled i) ∧
    (∀ s ∈ A.services,
      massOff ((A.prob s).gate fun j => (schedule N F wit tol htol).market DP s (F.ι j))
          (A.prob s).A ≤
        (A.prob s).κ * (A.evaluation F.ι).defect s ((schedule N F wit tol htol).market DP) ∧
      (A.evaluation F.ι).defect s ((schedule N F wit tol htol).market DP) ≤
        (((A.prob s).massCap * tol s : ℚ) : ℝ)) ∧
    (A.evaluation F.ι).progress ((schedule N F wit tol htol).market DP) ≤
      (A.evaluation F.ι).modulus + (A.evaluation F.ι).error +
        A.D * (A.evaluation F.ι).residual

Its assumptions, sorted:

kind	hypothesis	status
semantic	Admission, the adequacy losses lam, εad, D	imported, typed
normative state	WellFormed, checked witnesses, Excludes, Margins	checkable on the state; discharged at the fixture
protocol	lamS, ρ (budgets), Γ (amplification), μ, T	declared, as in deductive_normative_inductor
affordability	−B ≤ netWorth	discharged by liability / liability_reopened
computability	DeductiveProcessComputation DP; EffectiveEnforcerComputation	the first is the process's; the second discharged by computation

Conclusions: Logical Induction over the deductive process; rowwise conformance at every checked date and the provenance of every enforced row; the adapter's inadequate-mass bound and the defect's bound; the Progress bound. Compared with PR #129's warrant_normative_inductor and inductor_run: ComputableMarket and PracticalUptake were hypotheses there; both are derived here, and the enforcer, the licence and the elimination read one bundle.

P2 — the witness

WarrantInductorWitness.inductor: warrant_inductor at a five-sentence fixture with no computability and no liability hypothesis. What is constructed, what happens to the warrants, what the market guarantees and what the adapter does: THEOREM_STORY.md §E. The behavioral statements: gate_bounds (1/25, 1/100), gate_nonconstant, exclusion_lapses, bracket, liability (−72), intensity_pos.

Coherence: three structural rows X∧A ≤ X, X∧A∧Y ≤ X∧A, X∧A∧Y ≤ Y (the dispatch's missing relation included) as declared StrItems of the state. clearing in EndToEndWitness stays what it was — a stipulated price path, now documented as such — and nothing in this round's witness is a stipulated price: the gate and the bracket are read at W.market DP, whose values are not computed.

Repairs to PR #129

• Survivor semantics. The one-step rule replaced by the grounded rule; Fixture.survivors_eq, Fixture.rowsRebutted, EndToEndWitness.Run.survivors_eq, firstWarrantAt, rows5, regionRAt_iff recomputed: the strength occurrence leaves at n₀. Fixture.liability_rebutted (registered rebuttal.liability-rebutted) keeps its statement with the recomputed schedule; interval_XA, interval_XAY, hullMass_run are unchanged in value.
• Licensing. Admissible, Supported, Licensed on the settled bundle (SupportedOn a bundle, generally); the countermodels re-proved; EndToEndWitness.Run.licensed_at_entry, undercut_licences on the settled bundle.
• Records. trace carries the position on the settled bundle.
• Force. The bankroll theorems are instances of OddsBankroll.Account; the warrant round's constant-intensity enforcer and assessment market are kept as that instance, and the deductive market carries the calibrated enforcer.
• Workspace. PRIORITIES.md had two ## Workspace friction headings, the first introduced by the rebuttal round when it filed items 121–123; the duplicate is removed and the items sit before the friction section.

What this round does not establish

• Distance-to-region conformance for the rowwise enforcer (item 125); the public defect is the displayed excess and the enforcer's guarantee is per row.
• Market computability beyond finite-phase bundle schedules and computable deductive processes; DeductiveProcessComputation remains an input of the general theorem.
• The bankroll for more than one reopening (item 126), or the sharp constant across one (item 123, unchanged).
• Legitimacy of admission: Admission is a typed map (item 128); the authorized events are the record's data.
• Soundness of any warrant, or adequacy of any declared adequate set; Excludes and Margins are sufficient conditions, and their completeness (whether every region-excluded response has a certificate) is item 118's LP-duality question.
• The undercutter-occurrence reading of defeaters (item 127).
• The two enforcer architectures (projection, rowwise) are not bridged; both are retained, on the same deductive market.

Open problems, ranked

Preventing the conditional Normative Inductor as stated: none; the theorem is proved and run.

Preventing a general repeated-revision construction: item 126 (repeated reopenings); item 123 (the sharp one-reopening constant and the cushion exploit against the pinned maker); item 125 (distance conformance, needed before the rowwise and projection enforcers can share one public defect).

Preventing the stronger normative-learning agenda: item 128 (admission from the account); item 118 (certificate completeness, which decides when the adapter's conditions are checkable rather than supplied); item 127 (defeaters as occurrences); item 117 (strength selection), which this round does not touch — strengths are declared and never learned.

Verification

• lake build of the whole library: Build completed successfully (4094 jobs).
• python3 tests/audit_axioms.py: AXIOM AUDIT: 3320 results across 107 files, all within ['Classical.choice', 'Quot.sound', 'propext'].
• python3 projects/normativity/rounds/2026-10-22-normative-inductor-integration/tests/run.py: Ran 22 tests … OK.
• python3 tests/run.py at the root: ALL GREEN (79 project(s)); python3 -m checkers.run: 71 entries, 1 adjudicated here for the normativity registry, both registries ok; python3 -m checkers.workspace_state --check: WORKSPACE STATE: valid — 6 projects, 176 modern claims, … 124 rounds.

Claims whose status changed: rebuttal.liability-rebutted keeps class and declaration with a recomputed force schedule (the strength row leaves at n₀); warrant.licence-chain, warrant.supported-admissible, warrant.admissible-not-supported, warrant.supported-not-licensed keep class and declaration with Supported now on the settled bundle. Five entries added, all lean-proved.

Merge recommendation

Ready, with two reservations stated as mathematics. The three conclusions are proved on one market for one normative state process, and the run discharges every hypothesis that a theorem in the repository can discharge. The reservations: (i) conformance is rowwise, and the reading "prices within δ of the region" that the wiki's Normative-Inductor page and the projection theorems use is not what this construction proves — a reader comparing the two should be told which guarantee is meant (item 125); (ii) the maintainer's open rulings from the two earlier rounds (M1, M2, M5, the reopened bankroll) are unchanged by this round and the grounded rule is agent-decided; if the undercutter-occurrence reading of defeaters is the intended one, undercut changes and the survivors' fixed point becomes the grounded extension (item 127).

Items filed

124 (certified bounds at the displayed price — filed answered), 125 (distance conformance for the rowwise enforcer), 126 (the bankroll across repeated reopenings), 127 (defeaters as undercutting occurrences), 128 (admission from the accounted state). Narrowed: 35 (second instance; computability and liability discharged), 39 (conformance registered), 112 (answered by the grounded rule), 115 (the licence on the canonical bundle; transport across record transitions remains), 116 (part (c) closed on 2026-10-21), 118 (completeness also for the adapter's certificates), 121 (answered).

Registrations

inductor.grounded-survivors (WarrantEntitlement.mem_survivors, item 112); inductor.conformance (WarrantForce.WarrantSchedule.excess_le_tol, item 39); inductor.certificate-transport (WarrantForce.WarrantSchedule.bracket_at_market, item 124); inductor.warrant-inductor (WarrantInductor.warrant_inductor, item 35); inductor.witness (WarrantInductorWitness.inductor, item 35). Listed and not registered: OddsBankroll.Account.netWorth_ge_neg_odds_reopened, WarrantSchedule.computation, WarrantSchedule.isLogicalInductor, and every other declaration of the four new files.

Names, provisional

grounded rule, active identities, undercut, live given; normative state, canonical bundle; withholding compile; enforcement account, capital; warrant schedule, calibrated warrant enforcer; functional certificate, mass, transport; excludes, margins, public defect (the displayed inadequate excess); adapter protocol; warrant Normative Inductor.

Outstanding maintainer actions

1. **Rule on the reading of defeaters** — shared defeat ports (landed, agent-decided) or undercutting occurrences (item 127): DECISIONS.md, the 2026-10-22 entry's first rejected alternative. The fixture is unchanged either way.
2. The wiki's Normative-Inductor and Normative-Induction pages describe conformance as distance to the region and PracticalUptake as a typed input; this round's status note on Normative-Inductor.md is one paragraph. The synthesis (which construction the page presents, and in what terms) is the maintainer's.
3. The actions of the two earlier rounds stand: M1, M2, M5, the reopened bankroll (DECISIONS.md, Awaiting the author); close PRs #102 and #103; merge PR #129, which now carries three rounds.

Normative Induction — end-to-end integration and theorem consolidation

Verdict: ONE-WARRANT-NORMATIVE-INDUCTOR-ON-THE-DEDUCTIVE-MARKET-WITH-THE-GROUNDED-SURVIVOR-RULE-ROWWISE-CONFORMANCE-AT-THE-REALIZED-MARKET-CERTIFICATE-TRANSPORT-AND-A-DERIVED-PRACTICAL-CERTIFICATE-RUN-ONCE-WITH-COMPUTABILITY-AND-LIABILITY-DISCHARGED — the survivors of the grounds are the least fixed point of the grounded step (not undercut, kept, citing a survivor), under which the fixture's strength occurrence leaves at the undercut and not at the later refutation; one canonical bundle is read as the joint region, the licence, the elimination's input and the enforcer's rows; the rowwise warrant enforcer runs as effective data on the deductive market at the calibrated intensity, where its market is a logical inductor from a liability bound and its effectiveness, every enforced row conforms within the tolerance at the displayed price by the real market-maker contract, and a certificate's multiplier mass transports that conformance to the bounds a licence certifies and an adapter reads; the soft gate on certified scores yields PracticalCert and the Progress bound with PracticalUptake derived; the consolidated theorem warrant_inductor has three connected conclusions on one market, and its run at a five-sentence fixture carries no computability and no liability hypothesis, with the gate's inadequate mass at most 1/25 before the undercut and 1/100 on the surviving warrant after the rebuttal.

Read REPORT.md first. HANDOFFS.md is the audit of the pipeline's arrows; THEOREM_STORY.md is the paper-facing account (problem, state, enforcement, theorem, example, limits). Lean: lean/Workspace/Normativity/Contrib/OddsBankroll.lean (the affordability interface), WarrantForce.lean (the enforcer on the deductive market, conformance, transport, effectiveness), WarrantInductor.lean (the canonical state, admission, the compile, the adapter, the consolidated theorem), WarrantInductorWitness.lean (the run); WarrantEntitlement.lean and EndToEndWitness.lean revised. Exact rationals under src/, run by tests/run.py.

Dispatch: prompts/2026-10-22-normative-inductor-integration/PROMPT.md. Shipped in pull request #129 together with the two earlier rounds.