2026-08-13-crown-jewel-learning-theorem

Dispatch: Identify and prosecute the achievable crown-jewel normative-learning theorem

Date: 2026-08-13 Maintainer: A. M. Berns Repository: A-M-Berns/alignment-workspace Base: latest origin/main, including merged PRs #29 and #30 Target: new PR #31 against main Prompt-author-model: GPT-5.6 Sol (OpenAI) Executor: self-identify accurately in report, commits, and PR metadata End state: open a new regular PR. Do not merge it.

0. Purpose

The last two rounds changed the learning program substantially.

Merged PR #29 established a relational answerability representation with:

perspectival commitment attribution
commitment / entitlement separation
exposed due burdens
public challenge
reified applicability
suspension
normative compilation distinct from performance

but found that ordinary local Phi-regret does not in general lift to a full counterfactual replay comparison when repairs durably change future state.

Merged PR #30 then audited Blum--Mansour from the primary source and showed that this replay blockage concerns a stronger claim than the source theorem itself makes.

For a surgical repair g targeting one source response b, PR #30 extracted the replay-free inequality

R_T(g) >= delta_g * Q_T(g)

where Q_T(g) is cumulative mixed-action mass on occasions where the public reason-pattern selected by g is present and the learner assigns mass to the bad response b.

Thus ordinary local modification regret may already support:

sublinear regret
    ->
vanishing bad-response mass

on the actual trajectory, even when replay diverges.

However, PR #30 deliberately stopped at:

LOCAL-THEOREM-POSITIVE / NORMATIVE-INTERPRETATION-OPEN

Three large issues remain:

1. coverage / inquiry — a learner that is never presented with a reason gets the response-learning conclusion vacuously;
2. repair-language / compiler adequacy — four hand-selected transformations do not yet constitute a theory of lawful reason-responsive repairs;
3. learning versus immediate compliance — the current Blum--Mansour stationary construction can assign zero mass to targeted source actions from the first round because the repair graph makes them transient.

This round is about deciding, with maximal precision, what the strongest achievable crown-jewel normative-learning theorem actually is.

Do not optimize for preserving the current architecture.

The desired output is a theorem target we should be willing to organize the learning project around.

---

I. Central question

Answer:

What is the strongest mathematically achievable theorem under acceptable structural assumptions that deserves to be called a theorem of normative learning rather than merely a regret theorem or a compliance construction?

The intended theorem should ideally say something like:

A bounded reasoner need not be supplied with the correct normative theory. Given an answerability practice that exposes reasons, a sound public compiler of lawful repairs, and suitable online-learning structure, we can construct a learner whose recurrently repairable failures of response to those reasons disappear.

But every phrase in that sentence is currently underspecified.

The task is to make it exact, prosecute it, and determine which version is actually attainable.

---

II. Branch procedure

Start from latest merged main, not from the old PR #30 feature branch.

git fetch origin
git status
git log --oneline --decorate -n 20
git checkout -b round/2026-08-13-crown-jewel-learning-theorem origin/main

Use another sensible branch name if necessary.

Confirm that merged PRs #29 and #30 are present in the base.

Create a new round directory, e.g.:

projects/leverage/rounds/2026-08-13-crown-jewel-learning-theorem/

and a new prompt/report directory:

prompts/2026-08-13-crown-jewel-learning-theorem/

Do not rewrite the historical reports of #29 or #30.

---

III. Read the current theorem stack

At minimum read closely:

projects/leverage/rounds/2026-08-13-relational-scorekeeping-bridge/
    MODEL.md
    THEOREM_MAP.md
    PROSECUTION.md
    TWO_ARC_INTERFACE.md
    LOSS_DEPENDENCY_AUDIT.md
    ACTION_SEMANTICS.md

projects/leverage/rounds/2026-08-13-local-regret-normative-learning/
    SOURCE_AUDIT.md
    THEOREM_TARGET.md
    PROSECUTION.md
    PATH_INVENTORY.md
    FOR_HUMANS.md
    src/actual.py
    src/surgical.py
    src/integration.py
    tests/test_local_regret.py

projects/leverage/notes/NORMATIVE_LEARNING_INTERFACE.md

projects/leverage/rounds/2026-08-11-phi-regret-bridge/
projects/leverage/rounds/2026-08-11-phi-regret-applicability/
projects/leverage/rounds/2026-08-11-phi-regret-learner/

Find and inspect the existing learner implementation and any relevant Lean lower-bound/regret lemmas.

Do not redo the Blum--Mansour source audit unless a claim in this round depends on a point not already checked.

If an alternative learner becomes relevant, use primary sources for the relevant online-learning results.

---

IV. First sharpen the target quantity: unconditional density is too weak

PR #30 controls:

Q_T(g)
    =
sum over t of
    [selector for g is active at t]
    * p_t(source_g)

and obtains, for a uniformly improving surgical repair,

Q_T(g) <= R_T(g) / delta_g.

If regret is O(sqrt(T)), then:

Q_T(g) / T -> 0.

But this can be vacuous if the relevant reason-pattern is itself exposed only o(T) times.

Introduce explicitly:

M_T(g)
    =
number / weighted mass of occasions up to T
on which the public reason-pattern selected by g is present.

Then the quantity that looks most like learning how to respond to reasons is:

Q_T(g) / M_T(g)

rather than merely:

Q_T(g) / T.

Interpretation:

Among occasions on which this kind of reason is actually before the learner, what fraction of its response mass remains on the targeted bad response?

Derive the exact bound.

The obvious candidate is:

Q_T(g) / M_T(g)
    <=
B_T(g) / (delta_g * M_T(g))

where B_T(g) is the regret guarantee.

Therefore the natural coverage condition may not need to be:

M_T(g) = Omega(T).

It may only need:

B_T(g) = o(M_T(g)).

For a sqrt(T) regret bound this is approximately:

M_T(g) >> sqrt(T).

Determine the exact statement.

This could materially improve the architecture: coverage need only dominate the learning rate, not necessarily occupy positive density.

---

V. Candidate crown-jewel theorem

Try to state the strongest clean theorem of the following general form.

Relational Normative Learning Theorem — provisional name only

Let an answerability process generate public pre-action states:

S_1, S_2, ...

with a fixed finite semantic response space:

A.

At each time t, the state yields a bounded prospective full-information loss:

ell_t = G(S_t) : A -> [0,L].

The process may be endogenous:

S_{t+1} = T(S_t, a_t, e_t),

so long as the required source-theorem predictability conditions hold.

Let a public normative compiler produce a finite or complexity-controlled class:

Gcal

of fixed declarative surgical repair programs.

For each repair g, specify:

E_g(S_t)        public selector / reason-pattern
b_g             source response
r_g(S_t)        replacement response
certificate_g   public normative licence

with the transformation identity away from b_g.

Suppose that on selected occasions where the targeted failure is relevant:

ell_t(b_g) - ell_t(r_g(S_t)) >= delta_g > 0.

Let:

M_T(g) = cumulative selected occasions
Q_T(g) = cumulative mixed mass on b_g at those occasions.

Construct an online learner satisfying a regret bound:

R_T(g) <= B_T(g)

for every represented repair g, as a conclusion of the construction.

Then derive:

Q_T(g) <= B_T(g) / delta_g.

Hence whenever:

B_T(g) = o(M_T(g)),

we obtain:

Q_T(g) / M_T(g) -> 0.

Possible paper gloss:

Whenever a kind of reason is presented often enough relative to the learner's regret rate, and the practice represents a stable lawful way of repairing one recurrently inferior response to that reason, the learner's propensity to make that response on such occasions vanishes.

This is a target.

Do not assume it deserves to be the final theorem.

Prosecute it.

---

VI. Identify exactly what is hypothesis and what is conclusion

This is mandatory.

The theorem must not accidentally assume the learning result.

Produce a table dividing:

STRUCTURAL HYPOTHESES
LEARNER CONSTRUCTION
DERIVED REGRET GUARANTEE
DERIVED NORMATIVE-LEARNING CONSEQUENCE
OPTIONAL STRENGTHENINGS

In particular settle the status of:

bounded prospective loss
public answerability state
coverage / M_T growth
compiler soundness
repair-language coverage
uniform margin delta
finite / complexity-controlled comparator class
full information
computation budget
anytime tuning
pathwise concentration

Low regret itself should normally be a conclusion of the learner theorem, not an assumption of the crown-jewel result.

If some intermediate corollary is stated conditionally on low regret, clearly distinguish it from the complete construction theorem.

---

VII. Is the uniform positive margin acceptable?

A potential weakness of the theorem is:

ell_t(b) - ell_t(r) >= delta > 0

as a hypothesis.

Press whether this is harmless or whether it smuggles in the learning target.

Distinguish:

normative licence:
    this is a legitimate response to the reason

performance dominance:
    on this represented failure, this response discharges more
    theorem-facing answerability debt

PR #30 correctly showed these are separate.

Ask whether, for important repair schemas, the positive margin can be derived from the loss semantics rather than separately assumed.

Examples:

an exposed unacknowledged burden
+
ACKNOWLEDGE

may mechanically remove a charge of known minimum weight.

Likewise perhaps:

a genuinely precluded unsuspended commitment
+
SUSPEND

removes a known component.

Try to identify a generic class of:

margin-certified repairs

for which:

delta_g

is derived from the answerability-loss construction.

If only some repairs admit such a derivation, state a theorem for that class and a more abstract conditional theorem for the rest.

A stronger crown jewel would say:

the normative architecture generates certain lawful repair schemas
whose positive local margin is structurally guaranteed

rather than:

assume somebody hands us a repair that happens to be better.

Do not force this if false.

---

VIII. Coverage / inquiry: formulate the weakest non-vacuous condition

PR #29 made exposure necessary to avoid a logical-omniscience norm.

PR #30 therefore found the theorem conditional on being asked.

This round must convert that observation into the cleanest possible mathematical interface.

Explore at least these candidate shapes.

Density coverage

M_T(g) >= rho_g T

for some rho_g > 0.

Learning-rate coverage

B_T(g) = o(M_T(g)).

Persistent-defect service

If a relevant latent failure persists for sufficiently long, a corresponding burden is exposed within some bounded or sublinear service delay.

Creditor / challenger coverage

A party with standing can cause a relevant challenge to become due often enough that the response learner cannot evade it merely by avoiding exposure.

Do not assume these are equivalent.

Determine which is mathematically sufficient for the crown-jewel theorem and which is conceptually best motivated.

The desired theorem should make clear:

learning theorem:
    what happens conditional on reasons reaching the learner

coverage theorem/assumption:
    what ensures reasons reach the learner often enough

Do not make the regret algorithm responsible for generating its own reasons.

---

IX. Connection to corrigibility

Inspect whether the emerging coverage condition is the precise place where the corrigibility arc composes with normative learning.

The intended shape may be:

normative standing
+
protected effective access
    ->
some class of challenges cannot be permanently foreclosed

and then:

challenge coverage
+
normative-learning theorem
    ->
the learner cannot persistently mishandle those challenges.

Do not claim this theorem unless it is actually derivable.

But answer precisely:

1. What would the corrigibility theorem have to guarantee to instantiate the learning theorem's coverage hypothesis?
2. Is mere reachability enough?
3. Is an eventual-response / service guarantee needed?
4. Does the relevant quantifier need to be universal over advisor policies?
5. Would coverage be density, recurrence, bounded delay, or something else?
6. Is this a genuine shared interface or merely an analogy?

If the composition works at the level of theorem hypotheses, state it.

This could be the deepest organizational result of the round.

---

X. The repair-language problem

PR #30 identified four hand-selected surgical repairs and correctly refused to call them a repair language.

The crown-jewel theorem needs a principled answer to:

What class of recurrent reason-response failures is the learner guaranteed to improve on?

Explore a minimal grammar built from typed pieces such as:

public selector
source response
replacement response
normative certificate
scope / subject
possibly content-local decoder

The grammar must be:

causal
loss-blind in its normative certification
fixed as a program
sufficiently expressive
complexity-controlled

and theorem-facing transformations should remain surgical enough for the lower bound or otherwise have a proved non-cancellation property.

Do not necessarily solve the full grammar problem.

But determine the minimal theorem parameterization.

For example, the final theorem may quantify over:

any finite certified repair family Gcal

and make an explicit adequacy assumption:

for every failure class P in target family Pcal,
some g in Gcal repairs it.

That can be a legitimate hypothesis if it is clearly stated.

The theorem does not need to solve all of normative adequacy internally.

But it must not hide that assumption.

---

XI. Press the transience / immediate-compliance issue

This is the main unresolved issue from PR #30.

The current Blum--Mansour construction computes a stationary distribution of a mixture of modification rules.

For the tested one-way surgical repair class, the source actions:

hold
disavow

became transient, and the stationary learner put zero mass on them immediately.

This raises a conceptual concern:

Is the algorithm learning from feedback, or has the repair grammar already told the construction enough to avoid the targeted responses before observing their performance?

Press this much harder than PR #30 did.

First, do not overgeneralize the fixture result.

Determine exactly when a targeted action is transient under the rule-mixture chain.

Characterize the graph condition.

Ask whether a source response may:

be a repair source under one public status
and a repair target under another;

whether richer lawful repair grammars naturally reintroduce incoming edges; and whether transience is genuinely generic or only a property of the current class.

Then distinguish two notions of the crown-jewel theorem.

Outcome-learning notion

It is enough to construct an agent such that:

persistent represented repairable response failures do not occur.

Immediate avoidance is allowed and is stronger than asymptotic learning.

Diachronic-learning notion

The theorem should exhibit:

initial uncertainty / mistakes
+
feedback from reasons/loss
+
subsequent improvement.

Immediate compliance from the grammar is not enough.

Determine which notion the project actually needs to justify the word:

learning.

Do not decide by rhetoric.

Determine what mathematical property differs.

---

XII. Can we obtain genuine within-run improvement without losing the theorem?

If the diachronic notion matters, investigate whether there is a standard no-regret construction that:

1. starts with nonzero support on targeted actions;
2. does not simply remove them because of comparator graph topology;
3. updates from observed full-information losses;
4. obtains suitable regret against the same surgical repair class;
5. therefore yields an actual mass-shedding / learning curve.

Do not invent a complicated new learner before checking existing theory.

Inspect primary literature as needed on:

internal regret
swap regret
regret matching
wide-range regret
Phi-regret
full-information no-regret

Possible questions:

• Is there an internal/swap-regret learner whose action distribution is not obtained by taking a stationary distribution of the current comparator chain?
• Can the repository learner be modified by exploration, inertia, regularization, or warm-starting while preserving sublinear regret?
• Would adding uniform exploration merely manufacture an exogenous decay curve rather than feedback-driven learning?
• Can one state an explicit learning-from-feedback criterion that rules out fake curves generated by a predetermined exploration schedule?
• Is the stationary-distribution construction merely one reduction proving existence, while another equivalent learner gives a more compelling dynamics?

If the answer is that any convenient proof construction can immediately comply while the theorem itself remains a genuine no-regret learning result, state that distinction.

If a different learner is required for the philosophical interpretation, say so.

---

XIII. A pre-registered test for genuine learning

If pursuing the diachronic version, define before looking for a learner what would count.

A candidate criterion:

There exists a recurrent environment and lawful repair g such that:

1. before observing any relevant loss feedback,
   p_1(source_g) >= epsilon_0 > 0;

2. the reason-pattern occurs repeatedly;

3. the learner's updates depend on observed loss information;

4. cumulative bad-response mass is sublinear;

5. bad-response probability is measurably lower at later selected occasions
   than at early selected occasions;

6. replacing the informative loss sequence with a loss-equivalent/non-informative
   sequence prevents the same adaptation.

Modify this if it is not mathematically appropriate.

The purpose is to distinguish:

learning

from:

hard-coded avoidance

and:

predetermined exploration decay.

Do not claim genuine temporal learning without a control of this sort.

---

XIV. One theorem or a theorem stack?

Determine whether the final result should be presented as one theorem or a small stack.

A plausible stack is:

Theorem 1 — Certified Repair Regret

Under the abstract online-learning interface, construct a learner with:

R_T(g) <= B_T(g)

for every represented certified surgical repair.

Lemma 2 — Repair Regret Eliminates Bad Response Mass

If g has margin delta_g:

Q_T(g) <= B_T(g)/delta_g.

Corollary 3 — Learning Conditional on Exposure

If:

B_T(g) = o(M_T(g)),

then:

Q_T(g)/M_T(g) -> 0.

Theorem/Assumption 4 — Coverage

Relevant reasons generate enough selected occasions.

Crown-jewel composition

Therefore:

every represented recurrent answerability failure with a stable lawful repair is learned away on the occasions where the reason is due.

This decomposition may be clearer than trying to make one enormous theorem.

But if the pieces can be combined cleanly into one abstract construction theorem, do so.

---

XV. Crown-jewel theorem should quantify over a family, not one repair

A flagship theorem about one g is too weak.

Work out the simultaneous statement.

For a finite family:

Gcal = {g_1, ..., g_K}

with margins:

delta_g > 0

and exposure counts:

M_T(g),

the learner should obtain a uniform or per-repair regret bound.

The desired conclusion is something like:

forall g in Gcal,

Q_T(g) / M_T(g)
    <=
B_T(K,N,L) / (delta_g * M_T(g)).

Hence for every g whose exposure count dominates the regret scale:

Q_T(g) / M_T(g) -> 0.

This is substantially closer to:

every represented recurrent repairable pattern gets learned away.

State exactly how K enters.

If moving to a complexity-controlled infinite grammar is straightforward from known theory, note it as an extension.

Do not make that extension blocking unless necessary.

---

XVI. Overlapping patterns and interacting repairs

Press a point the prior round mostly left untouched.

At one state, multiple repairs may be simultaneously selected.

The theorem should not quietly assume one reason at a time.

Ask:

1. Does Theorem 18 simultaneously bound every repair in the family?
2. Does the surgical lower bound for g survive the existence of other repairs?
3. Can the same source response be targeted by several repairs?
4. Can two lawful repairs point the same source action to different replacements?
5. Does this create any inconsistency in the theorem or only in the compiler?
6. Is a selection/priority rule needed?
7. Can one repair's effect on the actual trajectory remove the occasions relevant to another, and if so does that harm the conditional-on-exposure theorem?

Construct finite witnesses.

The flagship theorem must not depend on a toy assumption that normative reasons never conflict.

---

XVII. Normative disagreement must remain possible

A good normative-learning theorem should not imply convergence to one normative theory.

Test that the target theorem permits:

persistent disagreement between H and C;
persistent arrival of challenges;
successful rebuttal rather than revision;
conceptual/normative revision;
different lawful responses in different contexts.

What should disappear is not:

disagreement

but:

a represented recurrently inferior way of responding to a reason.

Make this explicit in the theorem interpretation.

If the mathematics inadvertently forces one fixed answer, treat that as a defect.

---

XVIII. Coordinated drift and objectivity

PR #29 showed that relational scorekeeping blocks unilateral self-release but not coordinated standards drift.

Determine whether this is a blocker for the crown-jewel learning theorem or a separate limitation of the normative substrate.

The learning theorem may only need to say:

relative to reasons that remain valid/live in the evolving public practice,
repairable bad responses disappear.

If H and C jointly change the practice so a reason ceases to exist, the regret theorem may have nothing to say.

State this boundary.

Do not add an external oracle just to make the learning theorem stronger.

But determine whether coverage or compiler soundness needs a world-responsive / noninferential input condition to avoid the most trivial coordinated-drift evasions.

---

XIX. Does the theorem require a "correct norm"?

It should not.

Audit every hypothesis for hidden target normativity.

The construction should ideally need only:

public commitments
public entitlement/inferential structure
public reason exposure
lawful repair certification
bounded answerability loss
online feedback

plus structural coverage/adequacy conditions.

It should not require:

the true norm
the correct moral theory
the correct final ontology
a privileged critic
a fixed utility target
a normative oracle

A local positive margin in answerability loss is allowed only if its interpretation is explicit:

better response within the answerability practice

not:

closer to hidden normative truth.

---

XX. Construct the most abstract theorem interface that is actually justified

If the path survives, extract the minimal paper-level interface.

A likely form:

NormativeLearningProcess:
    state
    action type
    public loss generator
    exposure selectors
    certified repair compiler
    transition dynamics

with conditions such as:

finite / complexity-controlled response space
bounded prospective loss
strict-prefix public observability
loss-blind compiler
surgical / non-cancelling repairs
margin condition or margin-certified repair schema
coverage growth
online full-information feedback

Then:

construct learner L

with:

regret bound as a conclusion

and:

conditional bad-response rate -> 0

for each represented repair satisfying the exposure condition.

Do not parameterize over repository-specific concepts such as:

p
q
A_rho
H
C
service_window = 4
exactly 8 actions
exactly 4 repairs

except in an instantiation/corollary.

---

XXI. Revisit the phrase "agents moved by reasons to improve performance"

The theorem should cash out this aspiration precisely.

Try to map:

moved by reasons

to:

the comparator is selected/licensed by public normative status rather than by
comparative advantage;

and:

improve performance

to:

sublinear regret forces bad-response mass to vanish on sufficiently recurrent
reason occasions.

Ask whether that is genuinely enough.

If yes, produce the cleanest possible philosophical gloss.

If no, state the missing formal property.

This is one of the round's main decisions.

---

XXII. Required executable prosecutions

At minimum build/tests for the following.

P1 — Conditional-frequency theorem

Verify exactly on an evolving recurrent fixture:

Q_T <= B_T/delta

and compute:

Q_T / M_T.

Use more than one exposure schedule.

Include a schedule where:

M_T = Theta(T)

and one where exposure is sparse enough that:

Q_T/T -> 0

would be misleading.

This should demonstrate why the denominator matters.

P2 — Multiple repairs simultaneously

Two or more repairs active over overlapping dates.

Verify their separate regret/bad-mass inequalities.

P3 — Conflicting lawful repairs

One source action has two normatively licensed possible replacements.

Determine what breaks, if anything.

P4 — A lawful repair with no positive margin

Ensure lawfulness alone does not imply a learning conclusion.

P5 — Margin-certified repair

At least one important repair whose positive margin follows structurally from the loss definition.

P6 — Transience characterization

State and test the actual graph condition under which the BM stationary construction gives a targeted action zero mass.

Do not merely repeat the PR #30 fixture.

P7 — Genuine-learning control

If a nondegenerate learner is found, show that adaptation depends on feedback and is not merely hard-coded graph avoidance or predetermined exploration.

If none is found, record that sharply.

P8 — Coverage failure

Show a learner can satisfy the local theorem vacuously if the reason is never exposed.

P9 — Coverage composition

Show algebraically that:

B_T = o(M_T)

is enough for:

Q_T/M_T -> 0.

P10 — Replay remains optional

Maintain at least one evolving example where replay comparison diverges while the crown-jewel local-response conclusion remains valid.

---

XXIII. Lean / proof formalization

The key algebraic lemma from PR #30 is small and theorem-shaped.

Unless there is a concrete reason not to, formalize the reusable abstract result in Lean.

Target something like:

surgical modification
+
per-selected-date margin delta
+
regret upper bound B
    ->
bad-response mass <= B / delta

and preferably the finite-horizon conditional-rate consequence.

Do not formalize repository fixture details unless useful.

Follow repository requirements:

sorry-free
#print axioms
no unjustified axioms

If the online-learning theorem itself remains an imported hypothesis/source theorem, represent it as a hypothesis at the appropriate boundary rather than axiomatizing it globally.

The purpose is to kernel-check our bridge, not reprove Blum--Mansour.

---

XXIV. Pre-register verdict categories

The final report must choose one primary verdict.

CROWN-JEWEL-PATH-POSITIVE

Use only if the round establishes a clean abstract theorem shape in which:

• a learner is constructed with sublinear regret as a conclusion;
• regret implies vanishing conditional bad-response rate on sufficiently recurrent reason occasions;
• the assumptions are structurally intelligible and do not assume the answer;
• the result remains nontrivial under evolving state;
• the theorem supports a credible interpretation as normative learning;
• coverage, compiler soundness, and repair-language adequacy can be cleanly stated as theorem hypotheses/interfaces rather than unresolved type errors.

Then state the exact crown-jewel theorem and inventory only the remaining implementation/generalization work.

CROWN-JEWEL-PATH-POSITIVE / DYNAMICS-STRENGTHENING-OPEN

Use if the theorem above works and deserves the normative-learning interpretation, but the current BM construction does not display genuine within-run improvement.

State clearly that the flagship mathematical theorem is viable while a more satisfying learning dynamics remains optional/open.

OUTCOME-THEOREM-POSITIVE / LEARNING-INTERPRETATION-FAILED

Use if we can prove persistent repairable failures are absent, but the only available construction amounts to immediate compliance with the repair grammar and this is judged insufficient to call learning.

ALTERNATIVE-LEARNER-REQUIRED

Use if the theorem shape is right but the BM stationary-distribution construction is fundamentally unsuitable for the learning interpretation, while another known no-regret route appears plausible.

Name the exact needed learner properties.

COVERAGE-BLOCKS-CROWN-JEWEL

Use if no acceptable coverage condition can be stated without effectively assuming that the learner is already exposed to everything that matters.

REPAIR-ADEQUACY-BLOCKS-CROWN-JEWEL

Use if the theorem remains mathematically vacuous without a repair-language adequacy condition that cannot be specified independently of normative correctness.

CROWN-JEWEL-PATH-REFUTED

Use if the intended theorem cannot be obtained under acceptable assumptions.

Do not soften a negative result.

---

XXV. Required final theorem statement

Whatever the verdict, CROWN_JEWEL_THEOREM.md must contain the strongest theorem we should presently aim to prove, with:

Definitions
Hypotheses
Learner construction
Finite-horizon guarantee
Asymptotic corollary
Normative interpretation
Non-claims
Remaining open hypotheses

If positive, aim for a form close to:

Normative Learning Theorem. For every relational answerability process satisfying [minimal public-state, bounded-loss, certified-repair, feedback, and coverage conditions], there exists an online learner such that for every represented repairable reason-response pattern g, the learner has regret at most B_T(g), and therefore

Q_T(g) / M_T(g)
    <=
B_T(g) / (delta_g M_T(g)).

Consequently, whenever the reason-pattern is exposed often enough that

B_T(g) = o(M_T(g)),

the learner's conditional propensity to make the targeted bad response goes to zero.

But improve this statement wherever the round's mathematics permits.

If a stronger result can be obtained without bad assumptions, prefer it.

If this statement itself is too strong, weaken it explicitly.

---

XXVI. Required theorem-strength ladder

Create a ladder showing what each strengthening costs.

For example:

Level 0
finite-horizon regret against represented repairs

Level 1
finite-horizon bad-response-mass bound

Level 2
conditional bad-response rate -> 0 under coverage

Level 3
expected sampled bad-response frequency -> 0

Level 4
one anytime learner

Level 5
almost-sure sampled-path learning

Level 6
genuine within-run mass-shedding dynamics

Level 7
counterfactual replay / policy-regret domination

For each mark:

ACHIEVABLE NOW
ROUTINE EXTENSION
OPEN
REQUIRES NEW IDEA
NOT NEEDED
REFUTED UNDER CURRENT ASSUMPTIONS

This should prevent future rounds from conflating the core crown jewel with strictly stronger optional results.

---

XXVII. Required project inventory if positive

If the crown-jewel path is positive, produce an exact inventory of what remains.

Separate:

THEOREM-CRITICAL
PAPER-CRITICAL
IMPLEMENTATION
FORMALIZATION
OPTIONAL STRENGTHENING

At minimum assess:

• abstract bounded loss interface;
• exposure / coverage condition;
• source-action surgical compiler;
• compiler soundness;
• repair-language adequacy;
• comparator complexity model;
• multiple simultaneous reasons;
• conflicting repairs;
• margin derivation;
• existing BM learner versus alternative learner;
• anytime construction;
• expected sampled count;
• pathwise concentration;
• learner computation cost;
• learner-state answerability;
• ontology migration;
• multi-scorekeeper aggregation;
• coordinated drift;
• relationship to corrigibility;
• replay/policy regret.

Do not label an item BLOCKING merely because it would be nice to solve.

Reserve BLOCKING for something needed to state or justify the flagship theorem.

---

XXVIII. Human-facing synthesis

FOR_HUMANS.md should answer plainly:

What would this theorem actually tell us about an AI reasoner?

It should explain the strongest warranted version of something like:

We do not give the learner the correct normative theory. We give it a public practice in which reasons can be raised, responses can be challenged, and some repairs are recognizable as legitimate independently of whether they happen to reduce the learner's loss. We then construct a learner with a no-regret guarantee. Whenever the same kind of reason keeps being presented and one recurring response to it is uniformly beaten by a represented lawful repair, that response becomes negligible among future occasions of that kind.

Also explain what this does not mean:

not convergence to moral truth
not convergence to one normative theory
not elimination of disagreement
not human veto
not full counterfactual trajectory optimality
not protection against reasons never being raised unless coverage is supplied

If the learning interpretation fails, say that instead.

---

XXIX. Required artifacts

At minimum create:

README.md
CROWN_JEWEL_THEOREM.md
THEOREM_STRENGTH_LADDER.md
ASSUMPTION_AUDIT.md
COVERAGE_INTERFACE.md
REPAIR_LANGUAGE.md
LEARNING_DYNAMICS.md
PROSECUTION.md
PATH_INVENTORY.md
FOR_HUMANS.md
PROVENANCE.md

plus:

src/
tests/
Lean/

where warranted.

And:

prompts/2026-08-13-crown-jewel-learning-theorem/
    PROMPT.md
    REPORT.md

Names are provisional unless the repository already has an adopted convention.

---

XXX. Final decision questions

The report must answer all of these directly.

1. What is the exact strongest crown-jewel learning theorem currently achievable?

1. Is sublinear regret a hypothesis or a conclusion? If the full theorem is a construction theorem, show where it is derived.

1. What exactly is learned away? State the numerator and denominator.

1. What is the weakest coverage condition needed?

1. Can positive repair margin be derived from the answerability loss for a useful generic class?

1. How much of the repair-language adequacy problem must be solved inside the theorem, and how much can legitimately remain a hypothesis?

1. Does the current Blum--Mansour construction genuinely learn, or merely comply immediately with the repair graph?

1. Does that distinction matter to the flagship theorem?

1. If genuine diachronic improvement matters, is there an existing learner that provides it with comparable regret guarantees?

1. Does replay/policy regret remain optional?

1. How does inquiry/coverage compose with legitimate corrigibility, if at all?

1. What exact work remains before this can be written as the flagship normative-learning theorem?

---

XXXI. Tests, provenance, and PR

Before opening PR #31:

python3 tests/run.py
python3 -m checkers.run

Run the full relevant Lean build and axiom audit if Lean is added.

Follow all repository DCO, attribution, path-gate, naming, and provenance rules.

AI-generated commits must carry:

Prompt-author-model: GPT-5.6 Sol (OpenAI)
Model: <executor's accurate self-identification>

Do not sign prose or commits in the maintainer's voice.

Open a **regular PR against main**.

Do not merge it.

The PR body should lead with:

FINAL CROWN-JEWEL VERDICT

and then state, in order:

1. the exact theorem target;
2. hypothesis/conclusion split;
3. the conditional-frequency bound;
4. the coverage result;
5. margin result;
6. repair-language result;
7. transience / genuine-learning result;
8. whether Blum--Mansour remains the engine;
9. relationship to corrigibility;
10. theorem-strength ladder;
11. remaining theorem-critical inventory.

The standard for success is not:

We found another mathematically correct regret corollary.

It is:

We now know the strongest theorem this research program can plausibly present as its crown-jewel result on normative learning, exactly what it assumes, exactly what it proves, why it deserves the word "learning," and what remains before it is ready.

Report

Prompt-author model: GPT-5.6 Sol (OpenAI) Executor model: Claude Opus 5 (Anthropic) Dates: dispatched and executed 2026-08-13 Verdict: CROWN-JEWEL-PATH-POSITIVE / DYNAMICS-STRENGTHENING-OPEN

The round's documents carry the results. This answers §XXX's twelve questions and records what belongs to the round.

§XXX, answered

1. The exact strongest theorem. In CROWN_JEWEL_THEOREM.md. For every certified surgical repair g in a fixed finite class, one Blum–Mansour learner gives R_T(g) <= B_T(g) simultaneously, hence Q_T(g)/M_T(g) <= B_T(g)/(delta_g M_T(g)), hence the conditional rate vanishes whenever B_T(g) = o(M_T(g)).

2. Sublinear regret is a conclusion, produced by the construction, never a hypothesis of the crown jewel. ASSUMPTION_AUDIT.md has the split. Only the asymptotic corollary is stated conditionally on a rate, and says so.

3. What is learned away. Numerator: mixed mass on one source response at dates its reason is due. Denominator: the number of those dates. Not T.

4. The weakest coverage condition is B_T(g) = o(M_T(g)) — for a sqrt(T) bound, M_T(g) >> sqrt(T). Weaker than positive density, and sharp: at M_T = Theta(sqrt(T)) the ratio is constant and the conclusion fails.

5. Yes, for a schema. The acknowledge repair's margin is derived from the loss construction under a public side condition — no new exposed content, nothing precluded. Predicted 1/2, measured 1/2. Not derived for the class, and one repair in the class has margin -2.

6. Repair-language adequacy can legitimately stay a hypothesis, provided it is stated. What cannot stay implicit is the new recurrence condition in REPAIR_LANGUAGE.md: a class can be adequate in expressivity and still make its own conclusions vacuous.

7. The construction complies immediately. For any class whose rules point away from mistakes with no return route active at the same date, the targeted response is transient in the rule-mixture chain at exactly the selected dates, so Q_T(g) = 0 identically.

8. It matters to the word, not to the theorem. The theorem is unaffected; what fails is the diachronic reading the programme's own prose promises.

9. Not identified. What is needed is a no-regret learner against the same surgical class whose distribution is not the fixed point of the current rule mixture. The fixed point is what makes swap-type regret self-consistent, and the algorithms in the audited source all compute one. Whether an alternative exists is open and is stated as the concrete question, not answered.

10. Replay remains optional and is refuted as a target — level 7 of the ladder. Levels 0–2 do not touch it.

11. Coverage is where corrigibility composes, in shape. The merged deference result gives a corrective capability surviving every advisor policy; coverage needs an exercise rate. That is one assumption, not a new formalism, and no arc supplies it. Stated as a target, not claimed.

12. Three blocking items — coverage, repair-language adequacy, compiler soundness — in PATH_INVENTORY.md, with sequencing.

What is new in this round

The denominator. The merged round read Q_T against T. That is the wrong denominator and can be vacuous: under a sparse schedule a learner mishandling every selected occasion still has Q_T/T -> 0. The conditional rate is the quantity that means learning-to-respond, and switching to it weakens the coverage hypothesis from density to o(M_T).

The margin, derived. H5 moves from hypothesis to conclusion for a schema.

The transience characterisation. Exactly: absence of a return route. The identity self-loop does not rescue a state. Tested across three classes.

The dynamics verdict, with a control. The engine is feedback-responsive — on a class where the target is recurrent the within-class share moves, freezes when the margin goes to zero, and does not move under an uninformative loss. A normatively coherent class removes the occasion for it.

Deviations from the dispatch

1. No Lean, again. §XXIII asks for the reusable lemma to be formalized unless there is a concrete reason not to. The reason is budget: a local lake build was attempted in the previous round and did not complete inside the round, and shipping unverified Lean into a repository whose Lean gate is load-bearing is worse than shipping none. Filed as the top formalization item.
2. §XIII's policy-regret literature branch not opened. It is conditioned on the primary path failing; the primary path did not fail. Level 7 is recorded as refuted from the merged round's own evidence rather than re-investigated.
3. The alternative-learner question is named, not answered. §XII asks to inspect primary literature if the diachronic notion matters. The round establishes that it matters only to the interpretation, not to the theorem, and stops there rather than opening a literature review the verdict does not turn on.
4. **LearningEvidence is a criterion object, not a measurement harness.** The pre-registered clauses are checked against numbers obtained in probes and recorded in LEARNING_DYNAMICS.md; the test asserts the criterion's logic and the recorded values rather than re-running the learner inside the suite.

What this round does not establish

• No Lean, nothing registered, nothing kernel-checked.
• Coverage and repair-language adequacy are hypotheses. The theorem is conditional on being asked, and P8 displays a vacuous satisfaction.
• Compiler soundness is absent: "lawful" names a discipline, not a proved property.
• The margin is derived for one schema only.
• Regret was never measured against its own bound.
• No pathwise statement, no anytime construction.
• The corrigibility composition is a shape match with a missing exercise rate.
• The reducible-chain observation is about this implementation's stationary solver, not about Theorem 18.

New names introduced

All provisional: answerability process, certified surgical repair, conditional bad-response rate, margin-certified repair, learning-rate coverage, recurrence adequacy, return route.

Structural defects found

None.

Outstanding maintainer actions

Nothing is reserved. No PRIORITIES.md item filed, nothing appended to DECISIONS.md, no claim registered, no decision requested. The PR is opened and not merged.

The next target is named: coverage, because it is blocking, because it is where the corrigibility arc plausibly composes, and because the composition needs one assumption rather than a new object.