2026-08-16-traderized-enforcement

PR38 proof-closing pass: make the generalized-LI / traderized-deduction spine trustworthy

Repository: A-M-Berns/alignment-workspace Existing PR: #38 — Normativity: traderized enforcement — the mechanism, and what it costs Existing branch:

traderized-enforcement

This is a CONTINUATION OF PR #38.

Do not open a new PR.

The scientific purpose of this pass is unusually narrow:

Take every load-bearing mathematical statement in the proposed generalized-LI / traderized-enforcement / traderized-deduction arc and either raise it to a level where a skeptical maintainer should have ~99% confidence in the exact stated theorem, or explicitly weaken, refute, or quarantine it.

Do NOT optimize for preserving the exciting story.

Optimize for making the surviving story extremely hard to break.

If the grand arc fails, that is a successful result.

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0. Branch and provenance discipline

Before doing anything:

1. Fetch the latest remote state of PR #38.
2. Confirm its current head SHA.
3. Read the PR body in full.
4. Read AGENTS.md, relevant governance, DECISIONS.md, PRIORITIES.md, and the current traderized-enforcement round.
5. Work ONLY on the existing traderized-enforcement branch.
6. Do NOT open a new PR.
7. Do NOT merge PR #38.
8. Do NOT merge or rebase main into the branch merely to make GitHub say mergeable unless doing so is genuinely required for the research and the repository rules allow it.
9. Record starting and ending SHAs.

The current branch has been through many correction passes. Treat old prose, old theorem numbers, and old interpretations as potentially stale. Re-derive the important claims from the live tree.

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1. The target architecture to prosecute

The candidate architecture is:

semantic source | v C_t ⊆ Δ(Ω) / \ / \ v v L_t K_t = π_t(C_t) | | | v | finite row presentation | | v v Budgeter^L E^K \ / \ / v v TradingFirm^L + E^K | v MarketMaker | v LIC_L + finite-time K_t-conformance

Here:

L_t = { ω : ∃ μ ∈ C_t, μ(ω) > 0 }

is the assessment/live-world process.

The claimed deductive specialization is:

C_t^D = Δ(PC(D_t)) L_t^D = PC(D_t) K_t^D = π_t(C_t^D) = conv(PC(D_t)|_{Φ_t})

and the claimed traderized-deduction result is roughly:

MarketMaker(TF^D + E^{K^D})

still satisfies the ORIGINAL Logical Induction Criterion relative to D, while also satisfying explicit finite-time approximate coherence bounds.

This pass must determine exactly how much of that diagram is theorem and how much is still suggestive.

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2. Success criterion: no load-bearing ~85% statements

At the end, produce a table of EVERY load-bearing claim with one of:

• SOURCE-EXACT
• LEAN-PROVED
• PROVED
• DERIVED-FROM-NAMED-THEOREMS
• EXHAUSTIVE-FINITE
• WITNESS
• CONJECTURE
• FALSE
• BLOCKED

For each surviving headline theorem, state:

1. exact hypotheses;
2. exact conclusion;
3. exact source dependencies;
4. whether it is formalized;
5. whether the executable construction actually inhabits the theorem's types;
6. whether the theorem composes into the next theorem without a hidden assumption.

The desired outcome is NOT “assign 99% confidence.”

The desired outcome is:

remove the reasons one would have for assigning substantially less than 99%.

If a claim cannot reach that standard, narrow the claim.

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3. First task: independently reconstruct the original LI dependency graph

Do not trust the current SOURCE_AUDIT merely because it exists.

Read directly:

• the pinned Logical Induction paper source;
• the pinned Formalized-Agent-Foundations Logical Induction implementation;
• the exact definitions and proofs of:
  • exploitation / LIC;
  • Budgeter;
  • the three Budgeter properties;
  • TradingFirm;
  • trading-firm dominance;
  • MarketMaker;
  • the LIA capstone/existence theorem.

Re-establish or correct the claim:

PC(D_t) is consumed by the exploitation criterion and by Budgeter/TradingFirm, but not by MarketMaker itself.

Produce a dependency graph at theorem/definition granularity.

For each occurrence of DeductiveProcess or PC(D_t) in the construction, classify whether it is:

• semantically load-bearing;
• only used for computability;
• only used for nesting;
• definitional packaging;
• irrelevant to the proof.

Look especially for hidden uses that the previous audit could have missed.

This is the foundation of everything else.

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4. PRIMARY TARGET — formalize the live-world Budgeter/TradingFirm lift

This is the most important task.

The current candidate theorem is:

Let L be an assessment process over the fixed LI world space satisfying:

(L1) temporal nesting: L_{t+1} ⊆ L_t

(L2) effective finite restriction: for every t and finite sentence support S, restrict(t,S) = { W|_S : W ∈ L_t } is computably obtainable as a finite set

(L3) nonemptiness: L_t ≠ ∅ on every queried support.

Then replacing PC(D_t) by L_t throughout Budgeter and TradingFirm yields well-defined computable objects and analogues of the source's:

Budgeter.1 Budgeter.2 Budgeter.3 TradingFirm dominance.

And therefore:

MarketMaker(TradingFirm^L)

satisfies the generalized Logical Induction Criterion LIC_L.

4a. Do not prove a toy analogue

A toy finite model is not sufficient.

The target must be connected to the ACTUAL LI formalization/types strongly enough that the theorem really establishes the claimed generalization.

Preferred outcomes, strongest first:

A. Generalize the actual FAF definitions locally in the workspace and prove the lifted theorems against them.

B. Factor the existing definitions through a new AssessmentProcess interface and prove that the old deductive construction is an instance.

C. If library architecture makes A/B impossible without modifying FAF itself, isolate the minimal upstream refactor exactly and prove every downstream theorem that can be proved without pretending a surrogate is the real thing.

Do NOT write a structurally similar finite theorem and call the lift formalized.

4b. Attack L1–L3

For each L1–L3:

• identify the exact source proof step using it;
• try to remove it;
• if it is necessary, construct a failure witness when feasible;
• check whether some hidden fourth hypothesis is actually needed.

In particular prosecute:

• resurrection of previously dead worlds;
• inconsistent finite restriction or non-functorial restriction;
• non-effective restriction;
• empty live sets;
• support growth;
• arbitrary changes outside the queried support;
• closedness/topological assumptions;
• whether literal global nesting is stronger than required;
• whether nesting only on queried finite restrictions suffices.

The best result may be a theorem with weaker hypotheses than L1–L3. Do not preserve those names by inertia.

4c. Deductive specialization

Prove, ideally definitionally where possible:

AssessmentProcess.ofDeductiveProcess(D)

has

L_t = PC(D_t),

and therefore:

Budgeter^L = Budgeter^D TradingFirm^L = TradingFirm^D LIC_L = LIC_D.

I want a real specialization theorem, not merely tests showing equal outputs on small examples.

This is what earns the phrase “generalization of Logical Induction.”

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5. Audit which original LI asymptotic theorems actually generalize to LIC_L

Do not say “all LI properties generalize” without checking.

Separate two questions.

A. Properties of an arbitrary market satisfying the original LIC

For each major theorem family in the LI paper, determine whether its proof uses only:

no efficient trader exploits relative to the assessment process

or whether it also uses substantive facts specific to deduction / the theory.

At minimum classify:

• convergence;
• limit coherence;
• affine coherence / inductive coherence;
• learning of logical patterns;
• calibration / unbiasedness families;
• self-trust / introspection families;
• conservatism;
• statistical learning;
• any other major theorem family the paper presents as a property of logical inductors.

For each family classify:

GENERALIZES VERBATIM TO LIC_L GENERALIZES AFTER REPLACING A SEMANTIC CONCLUSION REQUIRES EXTRA STRUCTURE ON L SPECIFIC TO DEDUCTION / Γ NOT AUDITED

For example, do not conflate:

"prices converge"

with

"the limit is a probability measure over completions of Γ."

Those need not have the same status under arbitrary L.

This section should let a reader know exactly what “generalized LI” means.

B. Traderized deduction

Separately verify:

if the modified deductive market still satisfies the ORIGINAL LIC_D, then every theorem in the paper stated for arbitrary logical inductors over D applies unchanged.

List any exceptions that are statements about the PARTICULAR construction rather than the criterion.

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6. SECOND PRIMARY TARGET — inhabit the actual LI Strategy type

The existing round argues that the violation-proportional trader is expressible in the LI feature grammar.

Close this.

For a finite rational row presentation

K_t = { p : <c_j,p> >= r_j },

define

g_j(P) = max(0, r_j - <c_j,P>)

and

ζ_E(P) = Σ_j β_j g_j(P)c_j.

Actually construct the corresponding source-side Strategy n term, or the closest theorem-bearing object in the pinned Lean formalization.

Prove/check:

1. it has finite support;
2. all features have legal rank;
3. coefficients are continuous in the current price;
4. the strategy is computable from the prior history and current-price input;
5. row data and β are available before MarketMaker selects P_t;
6. no illicit dependence on the realized violation is used to set β;
7. the strategy can be added to the MarketMaker aggregate without changing the fixed-point theorem's hypotheses.

If a rational approximation or representation issue exists, quantify it rather than silently treating reals as rationals.

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7. THIRD PRIMARY TARGET — prove computability of the modified algorithm

The theorem map currently treats something like

effectively presented regions -> modified algorithm is a computable belief sequence

as less settled than it should be.

Close it or find the obstruction.

Given:

• computable assessment process L;
• computable finite row presentation of K_t;
• computable tolerance schedule δ_t;
• computable ordinary aggregate volume bound M_t;
• the generalized TradingFirm if §4 succeeds;

prove that

P_t = MarketMaker_t(TF^L_t + E_t)

is computable.

Do not hide exponential runtime. We only need computability unless a stronger complexity claim is explicitly made.

Distinguish:

computable efficiently computable.

The privileged enforcement trader is allowed to fail the second.

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8. Re-prove the force spine from the exact MarketMaker contract

Do not assume the current force algebra is correct because parts are already Lean.

Audit and, where useful, strengthen the formal statements.

The core claims include:

F1. Extremal pinning identity

For a realized share vector ζ,

max_W <ζ,W-P> = Σ_φ [ζ_φ^+(1-P_φ) + ζ_φ^- P_φ].

Confirm against the actual LI cash convention.

F2. Enforcement inequality

For

ζ_E(P)=Σ_j β_j g_j(P)c_j

and every x satisfying all rows,

<ζ_E(P),x-P>

= Σ_j β_j g_j(P)^2.

This is already Lean-proved. Recheck the statement matches the executable compiler exactly.

F3. Actual-contract conformance

If MarketMaker sees ordinary aggregate τ with

||τ||_1 <= M_t

and has slack ε_t,

Σ_j β_j g_j(P_t)^2 <= ε_t + M_t.

Already Lean-proved. Audit every inequality and type assumption.

F4. Per-row finite-time tolerance

For desired δ_t, with

β_j >= (ε_t + M_t)/δ_t^2,

obtain

g_j(P_t) <= δ_t

for every row at every date.

Formalize this corollary if it is not already formalized.

F5. From row violation to geometric distance / incoherence

This is where overclaim is easy.

Separate:

• rowwise violation bound;
• distance to K under a norm;
• logical-coherence functional;
• finite support-function approximations.

State the exact Hoffman/net/presentation constants needed.

Do NOT say “within δ of K” when only each displayed row is violated by at most δ.

For deduction, explicitly characterize what row presentation is sufficient for the intended finite-time coherence claim.

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9. Re-prove the safety spine

This pass should make the safety theorem publication-grade.

Use notation that cannot collide with semantic C_t.

IMPORTANT: The current corpus uses C_t both for a semantic credal set in some places and for an ordinary aggregate/volume bound in others.

Fix this conceptual notation collision in the round's theorem-facing prose and, where appropriate, code.

Use e.g.

C_t semantic credal set M_t ordinary aggregate / TradingFirm volume bound

unless repository conventions dictate a better name.

Do not allow

(ε_t + C_t) ...

to be ambiguous between those two types.

S1. Liability identity

Audit/formalize the exact worldwise identity.

S2. Deficit inequality

For exclusion depths

d_j(W)=max(0,r_j-<c_j,W>),

the existing Lean theorem gives the lower bound

value_W(E_t)

= Σ_j β_j g_j^2

• Σ_j β_j g_j d_j(W).

Make sure the sign convention and cumulative interpretation are exact.

S3. World-inclusive zero-liability theorem

If every assessed world W ∈ L_t satisfies every row:

E_t(W) >= 0.

Under temporal nesting, cumulative enforcement liability is therefore B=0.

This is crucial for traderized deduction.

S4. General sufficient safety condition

From the finite-time conformance declaration derive the best justified pointwise bound, currently of the shape

loss_t(W) <= (ε_t + M_t) ||d_t(W)||_1 / δ_t.

Then prove clearly that a summability hypothesis such as

Σ_t (ε_t + M_t)||d_t(W)||_1/δ_t < ∞

over the relevant assessed-world histories yields a uniform cumulative enforcement-liability bound.

Be obsessive about the quantifiers because L_t is shrinking:

• Is the sum for every W that remains live forever?
• Every pair (n,W ∈ L_n)?
• Must prior losses be evaluated at W because nesting gives W ∈ L_i for i≤n?
• What is the exact uniformity required to feed the LIC definition?

Write the theorem at the exact type needed by enforcement preservation.

S5. Support-capacity bridge

Audit separately:

θ_t(W)=sup/max{ μ(W) : μ∈C_t }.

Prove the elementary expectation-to-worldwise bridge with every upper-bound hypothesis explicit.

Then determine whether max_gain is exactly the correct U in the actual LI position semantics.

Keep this as an ALTERNATIVE sufficient route to bounded liability, not a hidden requirement.

S6. Necessity

Do NOT try to force a converse.

The current general converse

unbounded enforcement liability -> efficient exploitation

is open.

Prosecute it enough to state why it is genuinely open and where the one-way proof loses information.

Keep the persistent one-sided exploitation example as a witness, not a theorem of necessity.

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10. FOURTH PRIMARY TARGET — formalize/prove enforcement preservation

The core composition should be written as an exact theorem.

Suppose:

P_n = MarketMaker_n(TF^L_n + E_n)

and

inf_{n,W∈L_n} W(Σ_{t≤n} E_t) >= -B.

Use the actual MarketMaker cumulative error bound to show:

sup_{n,W∈L_n} W(Σ_{t≤n} TF^L_t) <= market_maker_bound + B.

Then use generalized TradingFirm dominance to conclude:

no efficiently computable trader exploits P relative to L.

This should be proved at the strongest available level.

If the generalized TF lift is Lean-formalized, compose them in Lean if feasible.

At minimum, the theorem statement must expose every dependency rather than saying “LI still works.”

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11. Full general-C_t theorem

Once §§4–10 are done, state the strongest theorem actually earned.

Candidate shape:

Let C_t be a computable semantic credal process from which we obtain

L_t = Live(C_t) K_t = π_t(C_t).

Assume:

• L satisfies the assessment-process hypotheses;
• K_t has an effective finite rational presentation at each date;
• the resulting enforcement strategy is computable;
• the declared enforcement schedule produces a uniform finite cumulative liability bound B on L;
• whatever additional hypotheses the proof-closing pass actually discovers.

Then the market

P = MarketMaker(TF^L + E^K)

satisfies:

1. LIC_L;
2. explicit per-date row-conformance bounds for K_t.

Do NOT put into this theorem:

• legitimacy of C_t;
• derivation of C_t from answerability;
• necessity of the safety condition;
• exact enforcement;
• efficient presentation;
• semantic recovery from K_t.

Those are separate.

If this theorem cannot be closed, identify the smallest failed arrow.

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12. Deduction special case — prove the bonus theorem end to end

This is scientifically independent of the generalized-L lift and should be treated separately.

Start with the ORIGINAL LI construction over D.

Define

C_t^D = Δ(PC(D_t)) L_t^D = PC(D_t) K_t^D = π_t(C_t^D) = conv(PC(D_t)|_{Φ_t}).

D1. Semantic recovery

Prove:

Live(C_t^D) = PC(D_t).

This should be essentially definition-level.

D2. Force recovery

Prove:

K_t^D

is exactly the finite coherence polytope.

Clarify whether the relevant π_t is over all priced sentences / finite support and state the precise fragment.

D3. Zero liability

Because every W ∈ PC(D_t) is a vertex/member of K_t^D, apply the world-inclusive theorem to obtain:

E_t^D(W) >= 0

for every deductively plausible W.

Hence cumulative B=0.

D4. Original LIC preservation

Using the ORIGINAL source TradingFirm dominance, not the generalized theorem, show:

MarketMaker(TF^D + E^D) satisfies LIC_D.

This is one of the most important results in the whole round.

Formalize it if reasonably possible.

D5. Original LI theorem inheritance

Audit the paper and state precisely:

Since the modified market satisfies the original definition of a logical inductor over D, every theorem stated for arbitrary logical inductors over D applies to it.

List construction-specific claims, if any, that do not follow merely from LIC_D.

D6. Finite-time deductive coherence

State the strongest justified result.

Do NOT merely say “finite-time coherence.”

Specify whether the theorem is:

every presented coherence row has violation <= δ_t

or

distance(P_t,K_t^D) <= H_t δ_t

or

an incoherence functional <= δ'_t.

Give the exact presentation/constant/computability hypotheses.

D7. Computational price

Explicitly retain the fact that presenting the full coherence polytope may be exponential or worse in the finite fragment.

The theorem may be computable without being efficient.

The correct result is NOT “efficient finite-time logical omniscience.”

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13. Projection theorem / necessity of C_t

Re-audit the claim that neither L_t nor K_t alone captures the full semantic state.

At minimum preserve/prove:

C ⊆ π^{-1}(π(C))

with equality iff fibre-saturated, plus the minimal two-sentence witness:

C = Δ({00,11})

versus a distinct credal set with the same price projection but different live worlds/support capacities.

This is what prevents the paper from collapsing C_t into K_t.

Also exhibit directly that support information L_t cannot encode quantitative credal constraints such as

μ(φ) >= 1/2 versus μ(φ) >= 3/4

when both worlds remain live.

These are simple claims, but they are architecturally load-bearing.

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14. Constrained-market-maker comparison

Retain only what is actually proved.

Verify the displayed finite counterexample showing that requiring MarketMaker itself to:

satisfy the ordinary market-maker contract AND output P ∈ K

can make the feasible set empty.

This earns:

adding an enforcement strategy preserves the total MarketMaker interface, whereas constraining MarketMaker directly requires a new existence theorem and can fail to have a solution.

Do not inflate the witness into a universal impossibility theorem.

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15. Exactness is secondary

Do not spend most of the pass on exact enforcement.

Audit enough to ensure the paper does not depend on a false exactness story.

Keep separate:

• exact enforcement at zero slack/no opposition;
• exactness of alternative gauge/interior traders in tested dimensions;
• one-dimensional impossibility;
• cube-face settlement witness;
• coherence-segment obstruction;
• face-solidity conjecture.

The generalized-LI / finite-time-conformance paper should not depend on the face-solidity conjecture.

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16. Required adversarial tests / countermodels

In addition to formal proof, add or preserve regression tests for:

1. resurrection of a world breaks the naive Budgeter induction;
2. finite restrictions changing incompatibly across supports are rejected;
3. empty assessment sets produce vacuity;
4. same K, different C and different L;
5. same L, different quantitative C;
6. generalized Budgeter genuinely differs when L differs;
7. deductive specialization exactly matches old Budgeter outputs;
8. enforcement Strategy inhabits the intended grammar;
9. β cannot depend on the realized post-MarketMaker violation;
10. positive MarketMaker slack permits residual off-K price;
11. ordinary opposing volume permits residual off-K price;
12. world-inclusive K gives zero liability even with violations;
13. non-world-inclusive K can be safe under summable deficits;
14. fixed-depth persistent exclusion can be unsafe;
15. nonnegative expectation under C does not imply nonnegative value at each live world;
16. support-capacity bridge degrades as θ -> 0;
17. deductive K is world-inclusive;
18. traderized deduction has B=0;
19. traderized deduction changes finite prices somewhere;
20. traderized deduction still satisfies the original criterion assumptions;
21. cheap affine logical equalities fail to cut out full coherence;
22. a complete enough presentation gives the claimed finite-time coherence metric;
23. constrained-MarketMaker infeasibility witness;
24. no theorem depends on any withdrawn intensity-free liability claim.

Use exact arithmetic wherever feasible.

Tests are not proofs of general statements; keep their evidence class honest.

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17. Theorem ledger cleanup

The current theorem history contains retractions and superseded readings. That is useful research history but dangerous theorem-facing surface.

Update THEOREM_MAP.md so a new reader can answer immediately:

Proven / essentially settled

• source dependency factorization;
• projection information loss;
• enforcement inequality;
• finite-time row-conformance modulus;
• liability inequality;
• world-inclusive zero liability;
• whatever this pass closes.

Conditional

• only statements whose exact remaining premise is visible.

Open

• safety necessity/converse;
• normative-record -> C_t;
• general exactness characterization;
• any semantic legitimacy result.

False / withdrawn

Keep counterexamples, but do not let withdrawn claims contaminate the current spine.

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18. Paper-level confidence audit

Write a new artifact, e.g.

PROOF_CLOSURE.md

or the most repository-consistent equivalent.

It should present the final mathematical arc in this order:

I. Original LI decomposition

D -> PC(D) -> criterion -> Budgeter -> TradingFirm -> unchanged MarketMaker.

II. Assessment-process generalization

PC(D) -> L

with exact theorem status.

III. Why L is insufficient semantic state

support does not encode quantitative credal restrictions.

IV. Credal semantics

C -> (L,K)

and projection loss.

V. Traderized force

K -> E

with finite-time conformance theorem.

VI. Safety / preservation

bounded cumulative E-liability on L -> LIC_L preserved.

VII. Generalized construction

MarketMaker(TF^L + E^K).

VIII. Deductive specialization

C^D = Δ(PC(D)) -> L^D = PC(D) -> K^D = coherence polytope

and therefore:

original LIC_D + finite-time deductive coherence bounds.

For every arrow, include evidence status.

No philosophical gloss is needed beyond explaining what the mathematical objects mean.

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19. Confidence-kill questions

Before concluding, actively try to kill the entire story with these questions.

K1

Does generalized Budgeter.3 actually survive arbitrary L1–L3, or does it silently use proof-theoretic facts about D?

K2

Does TradingFirm dominance use more structure of D than the current source audit noticed?

K3

Is restrict(t,S) sufficient for every computation, or does some source term need a global witness/world?

K4

Can MarketMaker really consume TF^L+E without a hidden complexity/rank failure?

K5

Does the compiled trader's representation preserve continuity exactly?

K6

Is the claimed M_t volume bound available BEFORE choosing β_t, with no circular dependence on E_t?

This is crucial: if the ordinary aggregate bound changes because the generalized TradingFirm or the enforcement trader changes the belief history, show that β_t is nevertheless computable from the past at the correct information time.

K7

Does the finite-time conformance proof accidentally use a point x∈K whose existence/effective witness is unavailable?

Distinguish theorem existence from algorithmic computability.

K8

Does bounded enforcement liability really imply the exact upper bound on TF needed by tfdom, with the right assessment dates and nesting?

K9

In deduction, is B=0 truly cumulative under the source's time-indexed PC(D_n) definition, or is there an assessment-time mismatch?

K10

Does satisfying LIC_D truly imply every advertised “ordinary LI guarantee,” or are some claims properties of the particular LIA construction rather than the criterion?

K11

Does finite rowwise coherence imply the advertised finite-time coherence metric, or only presentation-relative conformance?

K12

Is full coherence effectively presentable at every finite date under the actual language/fragment assumptions?

K13

Does K_t = π_t(C_t) contain all the information the force compiler needs when C_t constraints are not polyhedral?

If not, state an explicit finite-presentation hypothesis or approximation interface.

K14

Is C_{t+1} ⊆ C_t actually required, or only nesting of L_t? Do not impose unnecessary monotonicity on credal constraints.

K15

Could the source alter C_t while keeping L_t fixed in a way that invalidates safety even though the generalized LIC side is unchanged?

This should be handled by the separate K_t/liability channel, not silently lost.

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20. Formalization priorities

If time is limited, allocate effort in this exact order:

1. actual live-world Budgeter / TradingFirm lift
2. actual enforcement Strategy term
3. computability of MarketMaker(TF^L + E)
4. formal enforcement-preservation composition
5. deductive end-to-end specialization
6. finite-time coherence metric/presentation theorem
7. stronger formalization of general safety summability
8. everything else

Do not spend the pass proving the exactness conjecture before 1–5 are closed.

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21. Tests and verification

At the end run all relevant checks, including at minimum:

python3 projects/normativity/rounds/2026-08-16-traderized-enforcement/tests/run.py python3 tests/run.py

and the full Lean build / axiom audit required by repository policy for any Lean changes.

Run python3 -m checkers.run if any structured claim/registry surface is touched.

Do not register claims merely because they became stronger unless repository policy and maintainer authority explicitly permit it.

Report exact test counts and Lean declarations added.

Every new Lean theorem must be sorry-free and pass the repository's axiom policy.

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22. PR38 body

Update the existing PR38 body to reflect the new proof status.

If the lift succeeds, the body should no longer say:

paper available conditional on formalizing the live-world lift

because that conditional is closed.

If it fails, make the failure the headline.

If it succeeds only under stronger hypotheses, list the new hypotheses prominently.

Do not change “Not for merge” to a merge recommendation unless the branch actually satisfies the repository's merge bar and the research spine is now stable enough to warrant it.

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23. Final maintainer-facing verdict

At the end answer these questions plainly.

1. Is the PC(D_t) -> L_t replacement a genuine theorem now?
2. What is the weakest proved interface on L?
3. Does MarketMaker(TF^L) satisfy LIC_L?
4. Which original LI theorem families generalize to arbitrary L?
5. Which require deduction/theory-specific semantics?
6. Is the enforcement trader an actual legal LI Strategy, not just algebra?
7. Is MarketMaker(TF^L+E) a computable market?
8. What exactly is the finite-time conformance theorem?
9. What exactly is the sufficient safety theorem?
10. Is the safety condition necessary? If not, say NO.
11. Does bounded enforcement liability imply LIC_L preservation?
12. In the deductive special case, is enforcement liability exactly zero?
13. Does traderized deduction satisfy the ORIGINAL LIC_D?
14. Which original LI theorems therefore apply unchanged?
15. What is the precise finite-time deductive-coherence guarantee?
16. What is its computational cost?
17. What remains conditional in the full C_t architecture?
18. Is C_t -> (L_t,K_t) genuinely necessary, or can one object be eliminated?
19. What statements from earlier PR38 passes are now withdrawn or narrowed?
20. What is the strongest paper-level theorem you would sign your name to?

Then give a final theorem spine with NO confidence adjectives:

THEOREM CONDITIONAL THEOREM OPEN FALSE

The aim is for the maintainer not to need phrases like “85% confident” anymore.

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24. Scientific standard

The hypothesis under prosecution is that PR38 has uncovered two distinct mathematical advances:

Advance A — assessment-process Logical Induction

PC(D_t) -> L_t

abstracts the semantic/risk-assessment role of deduction while leaving MarketMaker generic.

Advance B — privileged finite-time force

K_t -> E_t

allows selected effectively presented constraints to receive explicit finite-time conformance guarantees while preserving the relevant LIC whenever the enforcement trader's assessed cumulative liability is bounded.

And deduction may be the especially strong calibration case:

C_t^D = Δ(PC(D_t)) L_t^D = PC(D_t) K_t^D = coherence polytope

for which:

enforcement liability = 0,

so:

traderized deduction = original LIC_D + finite-time approximate deductive coherence.

This is an exciting picture.

Your job is not to confirm it.

Your job is to make every arrow withstand hostile mathematical review, and to delete any arrow that does not.

Traderized enforcement

Whether an admissibility constraint on a bounded reasoner can be given operative force by a distinguished trader that trades against violations of it, what that costs, and whether it can take the role a specially constrained market maker was meant to take.

Verdict: force is purchased out of a finite account, and what it costs is the product of ordinary aggregate pressure, normative exclusion depth and tolerated error — no one of the three has to decay for indefinite force to stay affordable.

The semantic state is a credal set, the price-visible object is its projection, and that projection loses information. Traderized force acts on the projection and is installed as a living interface; the semantics is not.

Force-story verdict: yes. Compiling a row presentation into one trading strategy gives a validated region operative force, with a certified conformance tolerance derived from quantities available before the price is set. The alternative implementation of the same contract — constraining the market maker itself — is not known to be a total function, and there is a displayed date where it demonstrably is not. That is a proof-level asymmetry, not a preference.

Constrained-market-maker verdict: retired for the enforcement column, and nothing else. The settlement interface's reports, timing, persistence, grounding, feasibility, breach stack and answerability are untouched and stay upstream. What traderization replaces is the mechanism that was to make prices respect a region.

**Safety verdict: bounded cumulative enforcement liability, with bound 1 + B.** That abstract theorem is the round's most robust result. The sufficient conditions for bounding B are weaker than a previous pass claimed: the per-date bound is (ε_t + M_t)·‖d_t(W)‖₁/δ_t, the intensity does not cancel, and conformance and liability are traded against each other. A region excluding a live world at every date can still be enforced forever and safely.

Architectural verdict: two objects, not one. The semantic state is a credal set C_t ⊆ Δ(Ω_t); the live worlds are read off it by support; the price-visible object is its projection K_t = π_t(C_t), and that projection loses information — Δ({00,11}) and its fibre saturation share a projection and have different live worlds. So semantics and force are two channels for a reason about information, not about mechanism convenience. Force consumes K_t; it does not determine C_t.

**Normativity application verdict: force-compilable, tolerance-enforceable, safety conditionally discharged.** The motivating statics split into settlement rows, which carry zero liability always, and core rows, whose worst deficit is max(0, r − m_c) — independent of the declared core minimum. Settlement monotonicity makes that non-increasing for a fixed endorsement under irreversible settlement; it does not make it summable, and it is not necessary either. One clause is needed, and the corpus already adopted it for the same failure in another context: a limit on cumulative net outflow. NORMATIVE_SAFETY.md.

Paper verdict: the core generalized-LI spine is kernel-checked. The live-world Budgeter/TradingFirm lift is lean-proved against the pinned dependency's own Budgeter, TradingFirm, Strategy n, Trader and MarketMaker, under hypotheses weaker than the three the round first proposed — support-local nesting and finite sound-and-complete restrictions, with nonemptiness and effectiveness unused. Two links in the arc remain uncheckable by the kernel and are named rather than glossed: the first-order erasure that makes the modified market a program, and the exactness half of the distance duality (DistanceComplete). PROOF_CLOSURE.md is the arrow-by-arrow account and THEOREM_MAP.md the ledger.

Construction verdict: the generalized construction is not the ordinary one. The world process feeds the Budgeter, not only the criterion, so TF^live + E and TF^D + E are different functions of the same belief history — scaling 1/5 against two worlds, 1 against one, on a displayed fixture. They coincide exactly when Ω^live = PC(D_t), which is the deductive case.

python3 tests/run.py     # 200 tests, exact rationals

Five claims from earlier passes are withdrawn, each with its counterexample kept as a regression in tests/test_regressions.py or tests/test_budgeter.py: an intensity-free liability ceiling; an exactness impossibility stated for every empty-interior region; the reading of a live world as one whose own price vector is admissible, with the laundering conclusion that depended on it; and the reading of semantic admissibility off a price region by preimage, with the deductive recovery that depended on that; and the claim that the generalized construction is the ordinary one under a different criterion, which the Budgeter's own dependence on its world process refutes.

file	what it is
SOURCE_AUDIT.md	where D enters Logical Induction, the market maker's contract, what breaks and where, and why constraining the maker costs its totality
MODEL.md	the objects, and which assumptions on the region the proofs consume
ENFORCEMENT.md	Theorems 1–9: the contract resolved, the enforcement inequality, the modulus, and the exactness case split
FORCE_INTERFACE.md	the mechanism-neutral force contract, the two implementations, and the responsibility table
CORE_CONDITION.md	P1's depth condition compiled to a trader, worked end to end
FUNDING_AND_SAFETY.md	the liability identity, the withdrawn ceiling and what replaced it, and the safety condition
DEDUCTION_SPECIAL_CASE.md	deduction as the calibration case; presentation cost; the four equivalence relations
INTEGRATION_MAP.md	objects touched and untouched, the four vocabulary collisions, Legitimacy and Deference
THEOREM_MAP.md	every result with its evidence class, sorted settled / conditional / open / refuted
PROOF_CLOSURE.md	the arc arrow by arrow with what each rests on; which Logical Induction property families generalize; the kill questions answered
NORMATIVE_SAFETY.md	whether the motivating statics discharge bounded enforcement liability: the outflow account, the three-factor cost product, the withdrawn depth-only theorem, and presentation dependence
SEMANTIC_PROJECTION.md	C_t, π_t, K_t, fibre saturation, and the support loss under projection
PAPER_RECONCILIATION.md	the two constructions, the live-world lift with its three hypotheses, deductive recovery, and the paper spine
PROSECUTION.md	forty-one attacks, with a current verdict: what landed, what was withdrawn, what is open

The safety verdict

Safety is implemented at the emission path, and its cost is a product of three factors. compile_safe_force computes the live-world deficit from the very region it is about to enforce, charges the account, and only then constructs the position — so neither an unaffordable request nor a certificate about a different request can produce a safety-certified object. The low-level compile_force promises conformance only and returns a different type.

The binding matters because the gap was exploitable twice over: a verified certificate for p ≥ 0 — honestly zero, since nothing can violate it — funded enforcement of p ≥ ½ for nothing; and, with the region held fixed, a certificate from a narrower assessment state funded a wider one for nothing. NORMATIVE_SAFETY.md §7a.

And its boundary. Safety is certified relative to the live worlds the caller supplies. That those are the live worlds of the intended semantic process is upstream and unproved — PRIORITIES.md item 39. The force layer closes certificate substitution within itself; it does not authenticate the semantics handed to it. §7b. Force is purchased out of a finite global account at (ε_t + M_t)·D_t/δ_t per date — a charge computable before the trade is emitted — and the account's discipline implies the bounded-liability hypothesis rather than assuming it. Inverting the charge gives the affordability relation δ_t ≥ (ε_t + M_t)·D_t/b_t: the remaining account determines how tightly the reasoner may be forced.

Per-endorsement finite caps and finite gating are both insufficient — a source obeying both, with one row live per date, drives the aggregate to infinity by admitting fresh endorsements.

The condition is Σ_t (ε_t + M_t)·D_t/δ_t < ∞, and three routes keep it finite: the exclusion depth decays, the ordinary aggregate pressure decays, or the tolerance loosens. A previous version of this round asserted that the first was required — that persistent positive depth alone exhausted any account. That is false, and NORMATIVE_SAFETY.md §9 carries the counterexample: depth fixed at 1/2 and tolerance fixed at 1 forever, against pressure 2^-t, sums to under 1. The corrected limitative statement needs floors on two of the three factors and a ceiling on the third.

Safety does not demand that normative disagreement be deductively resolved, and it does not demand that the disagreement shrink either.

The other verdicts

Exact enforcement: case-split by geometry, and the split is not interior versus empty interior. Available for regions with an interior, and also for regions sitting on a cube face — a settlement pinning a sentence to probability one is enforced exactly by a constant trader. Unavailable for a region strictly inside the open cube with empty interior, which is where a coherence relation lands. The proved theorem is one-dimensional; the general condition (face-solidity: nonempty interior relative to the smallest cube face containing the region) is a conjecture delimited by witnesses on both sides.

And exactness costs the safety property. The compiler that achieves it does not vanish on the region, so it holds positions where there is no violation, and it loses in a plausible world at a price inside a world-inclusive region. The two compilers are not ordered: one is safe and approximate, the other exact and unsafe. Nothing here produces one that is both.

Tolerance: sufficient. The settlement interface's T1 is built to consume a declared tolerance schedule, and the round's modulus converts to the incoherence functional that clause measures — verified against the interface's own displayed instance, where a coefficient net at denominator three recovers 4/15 exactly and coarser nets see none of it.

Constitution boundary: construction of K_t, legitimacy, priceability, effective presentation, nonemptiness, persistence, and the depth of any exclusion stay upstream. FORCE_INTERFACE.md §3 assigns each one. Bounded liability is shared — the source sets the exclusion depth, the mechanism sets the promised tolerance, and the surviving bound carries both. Which worlds the bound is assessed over is the open question.

Traderized deduction: an addition, not a replacement — unchanged from the first pass, and still false under relations R1 and R4 of SOURCE_AUDIT.md §4.

Provisional names

enforcement trader, enforcement intensity, enforcement liability, violation-proportional position, interior-anchored position, exclusion depth, world-inclusive region, support-function presentation, constraint-to-trade compiler, market-maker contract, enforcement inequality, force contract, force declaration, priceable endorsement. None is identified with an existing workspace term; the four live collisions are listed in INTEGRATION_MAP.md §3.

Status

lean-proved and unregistered across nine files under lean/Workspace/Normativity/Contrib/: the assessment-process lift and its deductive specialization, the enforcement strategy and the force inequalities, preservation under bounded assessed liability, traderized deduction's zero liability and its preservation of the original criterion, and the coherence modulus. derived plus exhaustive-finite for the exact dual-distance presentation. test-supported for the fixtures. blocked for market computability. open for safety necessity and for the normative-record bridge. THEOREM_MAP.md sorts all of it; PROOF_CLOSURE.md argues it.

Nothing is registered in CLAIMS.md, and the round adds no living specification note — FORCE_INTERFACE.md and PAPER_RECONCILIATION.md are proposals for one, not one.