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+# ADR-0175 — Calibrated Attempt-and-Eliminate Learning: Two Regimes Under wrong=0
+
+**Status:** Proposed
+**Date:** 2026-05-28
+**Author:** Shay
+**Anchor:** [[thesis-decoding-not-generating]]
+**Discussion / derivation:** [SESSION-2026-05-28 — From "Another Matcher" to a Self-Calibrating Problem Solver](../sessions/SESSION-2026-05-28-risk-reward-learning-architecture.md)
+**Builds on:** [ADR-0174 — Held-Hypothesis Comprehension](./ADR-0174-held-hypothesis-comprehension.md) (the `eliminate_violating` / `reevaluate` / `contemplate` substrate — here **repointed from reading to solving**), the calibration module, capability axes G1–G5, the round-trip filter + multi-branch disagreement rule, teaching-safety (proposal-only / reviewed = the seal)
+**Supersedes:** the matcher-oriented Phase 5b sub-phases (5b.1 single-sentence / 5b.2 cross-sentence / 5b.3 deep) in ADR-0174 — they collapse into *instances* of this architecture.
+
+---
+
+## Context — why per-shape matching cannot compound, and the contradiction underneath
+
+ADR-0163/0164/0174 moved the engine from `0/50/0` to `3/47/0` on GSM8K
+train_sample by building, one at a time, recognizers and injectors for specific
+sentence shapes. Each addition lifts `correct` by 0–2 cases. The 2026-05-28
+measurement (GSM8K's own `<>` calculator annotations over the 47 refused
+cases) explains why this is structural, not incidental:
+
+- **37/47 (79%)** of refused cases need multiplication; **43/47** need mul-or-div;
+ **0/47** are single-step (median 3 steps).
+- The *single-sentence* multiplicative aggregate — the supposed "simplest" target
+ — is exactly **one idiosyncratic case** (0021); the regular in-clause shape
+ already works.
+
+So the needed *operations* are few and general (the solver already supports
+`{add, subtract, transfer, multiply, divide, apply_rate, compare_additive,
+compare_multiplicative}` with pack lemmas), but the *phrasings* are unbounded. A
+matcher per phrasing buys a handful of cases and the curve flattens. **This is
+overfitting by construction** — the library-of-founds trap the thesis forbids
+([[thesis-decoding-not-generating]]).
+
+The deeper problem surfaced once we asked "let the solver *attempt* instead": a
+solver that **refuses whenever uncertain is safe but frozen** — it can only ever
+learn what a human hands it. Autonomous learning *requires* attempting through
+uncertainty, which is exactly where being wrong lives. **You cannot have
+autonomous learning and a global "never be wrong" at the same time.** The current
+design applies `wrong=0` even to gold-labeled *practice* data — i.e. it is built
+to give up on the very corpus meant to teach it.
+
+The full derivation (problem → dead-ends → pivots → solution) is in the
+[session doc](../sessions/SESSION-2026-05-28-risk-reward-learning-architecture.md).
+This ADR records the decision.
+
+## Decision
+
+Relocate the engine's intelligence from front-end pattern-matching into a
+**problem solver that attempts a grounded derivation and learns by elimination**,
+and govern *when it may attempt vs. must refuse* with a **deterministic,
+per-context risk/reward gate grounded in earned calibration** — while preserving
+`wrong=0` on everything served, by construction.
+
+### 1. Two regimes (the "mode")
+
+- **Serving** — anything the engine commits to a consumer who will act on it
+ (chat runtime, held-out/generalization measurement). Cost of error ≈ ∞.
+ **`wrong=0`, absolute, unchanged.** Refuse unless certain. This ADR does not
+ weaken the field/answer-integrity invariant in any way.
+- **Practice** — attempting on material where being wrong is **checkable and not
+ served**. Here **wrong is the elimination signal**, not a failure. This is the
+ only place autonomous learning occurs.
+
+### 2. The proof-carrying seal (why wrong=0 survives a hot practice loop)
+
+Practice never writes to serving. It emits **proposals that carry their own
+proof** (round-trips, forced-unique, introduces zero wrong, replay-stable);
+nothing crosses into the serving path except through the existing **proposal-only,
+reviewed** teaching gate. Practice may be as bold as its calibration allows
+*because* the seal makes its mistakes structurally unable to become a served
+answer. This reuses, and must not bypass, teaching-safety.
+
+### 3. The attempt/refuse gate — a deterministic ratio (NOT reinforcement learning)
+
+Per attempt, the action is licensed iff measured reliability meets the human-set
+ceiling for that action's blast-radius:
+
+```
+license(action, C) := reliability_of_relevant_checker(C) / θ_required(action, C) ≥ 1
+```
+
+- The two-regime structure **collapses the reward side**: in serving only
+ reliability matters (learning value is irrelevant to a served answer); in sealed
+ practice the threshold is "is it checkable?" (`θ_practice = 0`). We therefore
+ never have to assign units to "value" or "learning" — **the only thing
+ quantified is reliability vs. ceiling.**
+- `θ` are **human-set, version-controlled constants** per class and blast-radius
+ (`θ_practice = 0`; `θ_propose`; `θ_serve` strict, e.g. `.99`). Raising autonomy =
+ a human lowering `θ_serve(C)` for a class the ledger has earned. **The engine
+ never sets or raises its own ceiling.**
+
+### 4. The per-class calibration ledger (counts, not learned weights)
+
+Per **class** (= capability axis G1–G5), a **replayable ledger of counts** —
+nothing learned, nothing stochastic, every figure a tally over deterministic
+attempts or an explicit human constant:
+
+- `n(C), correct(C), wrong(C), refused(C)` — already produced by the eval harness.
+- `t2_verified(C), t2_agrees_gold(C)` — on the live gold anchor set.
+- `reliability(C) = conservative_floor(correct(C), n(C))` — a deterministic
+ **lower bound**, pessimistic at small `n`, so luck cannot grant appetite.
+- `t2_precision(C) = conservative_floor(t2_agrees_gold(C), t2_verified(C))` — how
+ trustworthy self-verification is on `C`; the number that licenses widening past
+ gold.
+
+This lives in the **calibration module**; `conservative_floor` is fixed arithmetic
+(see Open Questions for its shape).
+
+### 5. The checkability ladder — privilege ∝ reversibility
+
+Checkability is not a line but a confidence-stratified ladder. **Governing rule:
+require check-strength proportional to the reversibility/blast-radius of the action
+it licenses**, because a false positive in the checker (a wrongly-"verified"
+belief) is a *persistent contaminant* and is far worse than a missed learning
+opportunity.
+
+| Tier | Checker | May change |
+|---|---|---|
+| **1 — External truth** | gold label / known answer | serving-bound knowledge (via ratification) — *anchors* |
+| **2 — Convergent self-verification** | round-trip **∧** ≥2 *structurally-distinct* derivations agree **∧** unit/dimensional consistency **∧** no contradiction with vault/packs (conjunctive) | provisional, **retractable** knowledge (still ratified before serving) |
+| **3 — Consistency-only** | merely no contradiction with the known | **practice-internal pruning only** — never crosses the seal |
+
+**Tier 2 is the operating median**: the widest checkability still strong enough to
+*create* knowledge, needing no human and no label, so the practice arena scales
+toward open-world. Tier 3 keeps the arena wide (attempt anything; learn search
+shape) while being reversible and sealed.
+
+### 6. Provenance + retractability
+
+Every learned belief stores `(tier, n_at_admission)`. Retraction is deterministic:
+a Tier-1 (or stronger Tier-2) contradiction → retract, and decrement
+`t2_agrees_gold` → `t2_precision` falls → the `θ`-gate tightens. Provenance is what
+makes widening past gold *safe* — weak beliefs are quarantined by confidence and
+reversible. Extends CORE's existing provenance / exact-recall discipline to beliefs.
+
+### 7. Gold tether — defense against correlated self-delusion
+
+Tier-2 agreement only helps if derivations are **independent**; a shared wrong
+premise (the engine misunderstands "twice") makes them all agree *and* round-trip
+while all being wrong. Defenses:
+- **Independence is counted, not assumed**: Tier-2 requires **≥2 structurally
+ distinct paths** (different operation multiset or different intermediate
+ quantities).
+- **A live Tier-1 anchor set always runs**, measuring `t2_precision(C)` per class.
+ When it drifts below a floor, appetite contracts. Gold doesn't just teach — it
+ *audits whether self-verification is trustworthy*, which is the calibration loop
+ closing.
+
+### 8. Diagnostic refusal — the router between skill and knowledge
+
+Every refusal must **name the missing piece**, so effort routes to the right axis:
+- *"quantities extracted, units consistent, no grounded derivation reaches target"*
+ → **skill** gap → solver search / elimination practice.
+- *"unknown relation / unit relationship"* → **knowledge** gap → acquire a
+ world-fact.
+- *"two grounded derivations disagree"* → **genuine ambiguity** → stay refused.
+
+Quantified: if `reliability(C)` still climbs with practice → skill gap (keep
+practicing); if it has stalled → knowledge gap (needs a new world-fact). Extends
+typed refusals + the OOV gradient + the math-reader-refusal audit corridor.
+
+### 9. Three compounding stores
+
+Each diagnosis routes to where learning accumulates: **experience → vault** (exact,
+deterministic recall), **world-knowledge → ratified packs**, **skill → the
+solver's elimination-learned pruning**. The flywheel: stronger solver → more vault
+experience + sharper pruning → fewer knowledge gaps to ask about → less
+contemplation per problem → fewer hand-authored packs → compounds.
+
+### 10. Self-proving acquisition and the narrowing of human input
+
+Autonomy does **not** mean "no human input" — the engine cannot conjure world-facts
+from nothing; facts enter from an ingested data/experience stream. The human role
+*shifts* from hand-authoring meaning to **curating what it ingests + ratifying
+what it has already self-proven**. The bridge is **self-proving acquisition**: new
+knowledge is proposed *with a mechanical proof attached* — the schema-proof-
+obligation discipline (CLAUDE.md) pointed at learning. The ratification gate never
+opens to ungrounded learning; it simply has less to do as the engine's proofs get
+stronger.
+
+### 11. "Creative," defined for a deterministic engine
+
+Not stochastic invention. **A willingness to leap a gap in known structure** — a
+recombination no stored pattern directly licenses — always a step from given
+ground, never from the void. The checkability tier is the leap dial: a Tier-3 leap
+stays a hypothesis; a Tier-2 leap becomes provisional knowledge; a Tier-1-confirmed
+leap becomes an anchor.
+
+## Non-negotiable invariants (must be *proven*, not asserted)
+
+Per CLAUDE.md §Schema-Defined Proof Obligations, each of these requires a test that
+**fails** under the violation it names:
+
+1. **wrong=0 on serving is untouched.** A test must fail if any practice-regime
+ artifact reaches a served answer without crossing the ratification gate.
+2. **The search cannot bank a spurious answer.** A test must fail if a derivation
+ that is *not* grounded+unique+round-tripping is admitted as knowledge (the
+ `20/5` coincidence class).
+3. **Determinism / replay.** All ledger counts, the `conservative_floor`, the gate,
+ and the search are deterministic and replayable; a test must fail on any
+ run-to-run divergence. **No learned weights, no stochastic sampling, no
+ approximate recall** — the vault stays exact.
+4. **No self-authorization.** A test must fail if the engine mutates any `θ`
+ ceiling. Ceilings are human-set config only.
+5. **Retractability.** A test must fail if a Tier-1 contradiction does not retract
+ the contradicted Tier-2 belief and tighten the gate.
+
+## Consequences
+
+- **What it collapses:** the per-shape matcher backlog. Multiplicative/comparative/
+ fraction cases become the **first practice arena** where attempt-and-eliminate is
+ proven, not a set of shapes to hand-match. ADR-0174's 5b sub-phases are
+ superseded.
+- **The train_sample double-duty is resolved.** It currently serves as *both*
+ practice arena and serving-regression canary. Decouple: practice may attempt all
+ 47 (scored correct/wrong/refused, wrongs feed elimination) while `wrong=0` stays
+ absolute on the serving contract + held-out generalization, and the hazard
+ canaries (0050) keep guarding serving.
+- **Risk concentrates in the search + checker.** This is where the project's
+ correctness mandate is most stressed; invariants #1–#2 are the load-bearing work.
+- **Mostly composition, not new machinery:** classes = capability axes; counts =
+ eval harness; reliability + lower bound = calibration module; replay guarantees
+ reproducibility; `θ` = a small config table; seal = teaching-safety; elimination
+ = ADR-0174's `eliminate_violating`/`reevaluate`/`contemplate` repointed to
+ solving.
+
+## Phasing (wrong=0-first; each phase ships its proof obligations)
+
+1. **Ledger + gate substrate.** Per-class calibration ledger, `conservative_floor`,
+ the ratio gate, `θ` config table. Invariants #3–#4 proven. Zero behavior change
+ to serving.
+2. **Sealed practice regime on GSM8K train.** Run attempt-and-eliminate over the 47
+ (Tier-1 gold checkable); score correct/wrong/refused as *practice* metrics;
+ wrongs produce elimination records. Invariant #1 proven (nothing leaks to
+ serving). Diagnostic refusal (§8) emitted.
+3. **Grounded derivation search.** Bounded, deterministic operation-chain search
+ over extracted quantities, gated by grounding + unit + unique + round-trip.
+ Invariant #2 proven (the spurious-answer test). Measure the flip-curve on the
+ multiplicative chunk; require it to hold under ADR-0114a perturbation.
+4. **Tier-2 self-verification + provenance + tether.** Convergent self-verification,
+ per-belief provenance, the live gold tether + `t2_precision`. Invariant #5
+ proven. Widen the arena past gold-labeled material.
+5. **Self-proving proposals into the ratification corridor.** Practice emits
+ proof-carrying proposals; the (narrowing) HITL gate admits to serving. Measure
+ the serving-`correct` lift this produces with `wrong=0` held.
+
+## Acceptance criteria (Proposed → Accepted)
+
+1. Phase 1 substrate lands; invariants #3–#4 proven; serving byte-identical.
+2. A prototype grounded search demonstrably **refuses** the `20/5`-class spurious
+ derivation (invariant #2) on a curated case.
+3. The practice regime is provably sealed from serving (invariant #1).
+4. Capability-axis lanes G1–G5, S1 remain 100% `wrong=0`; pinned lane SHAs pass.
+5. Cross-references to ADR-0174 (substrate), teaching-safety (seal), and the
+ thesis reviewed and confirmed consistent.
+
+## Open questions
+
+1. **Shape of `conservative_floor` and `N_min`** — how pessimistic at small `n`.
+ This single choice sets how cautiously the system earns autonomy. Candidate:
+ a deterministic Wilson/Wald-style lower bound, or a simpler `require n ≥ N_min
+ AND wrong ≤ W_max` rule. Resolve before Phase 1 PR.
+2. **First practice-arena home.** GSM8K train (gold-labeled, checkable, already
+ wired) is the obvious Phase 2/3 home; confirm no serving-path coupling remains
+ after the train_sample double-duty decoupling.
+3. **Search bound + determinism budget.** The operation-chain search must be
+ bounded and replay-stable; fix the enumeration order and depth cap before
+ Phase 3.
+
+## Cross-references
+
+- **Derivation:** [SESSION-2026-05-28](../sessions/SESSION-2026-05-28-risk-reward-learning-architecture.md).
+- **Substrate repointed:** [ADR-0174](./ADR-0174-held-hypothesis-comprehension.md)
+ (`eliminate_violating` / `reevaluate` / `contemplate`), calibration module,
+ capability axes G1–G5, round-trip filter + disagreement rule, teaching-safety.
+- **Anti-overfitting obligations:** ADR-0114a (perturbation / OOD / depth / adversarial axes apply to every flipped case).
+- **Thesis:** [[thesis-decoding-not-generating]] — find, comprehend, rationalize; not a library of founds.
diff --git a/docs/sessions/SESSION-2026-05-28-risk-reward-learning-architecture.md b/docs/sessions/SESSION-2026-05-28-risk-reward-learning-architecture.md
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+# Session 2026-05-28 — From "Another Matcher" to a Self-Calibrating Problem Solver
+
+**Status:** Discussion captured ✓ — ADR-0175 to follow
+**Headline:** A scoping session for GSM8K Phase 5b turned into the discovery that the
+whole "build another recognizer per problem shape" strategy is structurally
+self-limiting (overfitting by construction), and converged on a different
+architecture: **move the intelligence into the solver (attempt + eliminate),
+separate a sealed practice regime from the wrong=0 serving regime, and gate
+attempts with a deterministic risk/reward ratio grounded in earned per-class
+calibration.** This document is the *journey* — the problem, the dead-ends, the
+pivots, and how we arrived at the solution. The formal decision is **ADR-0175**.
+
+---
+
+## TL;DR
+
+We started trying to scope "Phase 5b.1 — single-sentence multiplicative
+aggregate." Measurement showed the target was **one idiosyncratic case (0021)**,
+and that the broader per-shape-matcher approach can never compound: GSM8K
+phrasings are unbounded, so each matcher buys a handful of cases and the curve
+flattens. That is overfitting wearing an engineering hat.
+
+The turn (driven by Shay): stop asking the *front-end* to recognize every shape;
+give the *solver* room to **attempt** a derivation over extracted quantities and
+**learn by elimination** from what's wrong. The solver already supports the
+operations; the front-end was the bottleneck and the source of brittleness.
+
+That exposed the real contradiction: a system that **refuses whenever uncertain
+is safe but frozen** — it can only ever learn what a human hands it. Autonomous
+learning *requires* attempting through uncertainty, which is exactly where being
+wrong lives. You cannot have autonomous learning and a global "never be wrong" at
+once.
+
+Resolution: **two regimes** — *serving* (wrong=0, refuse-unless-certain,
+unchanged) and *sealed practice* (attempt-and-eliminate, where wrong is the
+learning signal, not a failure) — separated by a **proof-carrying seal**. Which
+regime applies is decided per-attempt by a **deterministic risk/reward ratio**:
+`measured_reliability(class) / required_reliability(action) ≥ 1`. Reliability is
+an **earned, per-class, conservatively-bounded count** from a live gold tether —
+not a learned weight, not a probability from nowhere. Checkability is a **tiered
+ladder** (gold > convergent self-verification > consistency-only) where the
+*privilege a passed check earns* scales with the check's strength, and everything
+weak is reversible. Humans raise the per-class ceiling on evidence; the engine
+never self-authorizes.
+
+---
+
+## Where we started: scoping Phase 5b.1
+
+Phase 5a had just shipped (PR #430): the inert GSM8K reader retired, single
+canonical parse path, `3/47/0` byte-identical, net −1,038 LOC. We turned to
+Phase 5b — the *semantic* lift — and began with what the scope called the
+cleanest first slice: "single-sentence multiplicative aggregate" (e.g. case 0021,
+"He bench presses 15 pounds for 10 reps and does 3 sets" → 15×10×3 = 450).
+
+## Dead-end #1 — the per-shape matcher reflex
+
+Every instinct pulled toward *building another recognizer*: detect this shape,
+emit that primitive. Two measurements stopped that cold.
+
+**The brute-force measurement that lied.** A bounded expression search over each
+refused case's numbers found "matches" — but several were *coincidences*
+(`20/5 = 4` for a case whose answer was 4 for unrelated reasons; `5−3 = 2`). The
+measurement itself overfit. Lesson banked: **number-matching without grounding
+manufactures spurious answers** — the same hazard a naive solver search would
+have.
+
+**The ground-truth measurement.** Parsing GSM8K's own `<>` calculator
+annotations (real operations, not guesses) over the 47 refused cases:
+
+| Profile | Count | Share |
+|---|---|---|
+| Uses multiplication (`*`) | **37/47** | 79% |
+| Uses mul **or** div | 43/47 | 91% |
+| Pure add/subtract | 4/47 | 9% |
+| **Single-step solvable** | **0/47** | **0%** |
+
+Step-count histogram: `2:13 · 3:12 · 4:8 · 5:8 · 6:3 · 7:3`.
+
+And a scan for *single-sentence* multiplicative aggregates (all operands in one
+clause, gold = their product) returned exactly **one** case — 0021 — whose
+surface ("for 10 reps and does 3 sets", pronoun subject, three different units)
+is the *least* general multiplicative shape in the set, not the simplest.
+
+**The conclusion that reframed everything:** multiplication is needed by 79% of
+the corpus (maximally general — a foundational capability, not a niche), but
+**no case is solvable in one step**, and the regular in-clause shape *already
+works* via existing Wave-A scaffolding. So a "single-sentence multiplicative
+matcher" would flip ~1 case via near-bespoke pattern matching — the textbook
+definition of overfitting the project has repeatedly ruled out ("lifting a few at
+a time → simply overfitting"). The per-shape path is unbounded because phrasings
+are unbounded. It cannot compound.
+
+## Pivot #1 (Shay) — breadth-of-impact is the overfitting test
+
+> "If solutions result in generally large chunks of failed results turning green,
+> then it's more likely that we are getting GENERALLY smarter... if tweaking the
+> injector fixes a big chunk or percentage then it COULD be a solution."
+
+This corrected a sloppy framing ("tweaking the injector = overfitting"). The
+overfitting test isn't *where* the fix lives — it's **breadth**: a change that
+flips a large *general* chunk (and holds under perturbation) is a real capability;
+a per-phrasing patch that flips one case is overfitting. The measurement said the
+biggest, most general chunk is *multiplication itself* — so the question became
+"how do we get general multiplicative capability," not "how do we match 0021."
+
+## Pivot #2 (Shay) — give the solver room to attempt; move the intelligence
+
+> "Maybe the solution is to give it more room to attempt to solve the problem...
+> the better we make the problem solver, the better the model will be at
+> anything/everything... the less we will have to put into packs and the less it
+> will have to contemplate over time, and the faster it will learn what works and
+> it compounds. That's what intelligence does."
+
+This is the architectural turn. Today the front-end (recognizer/injector) must
+hand the solver a *fully-typed, correct* graph or the solver does nothing — so
+all intelligence is forced into surface pattern-matching, and surface is infinite.
+The capable part sits idle:
+
+- The **solver already supports** `{add, subtract, transfer, multiply, divide,
+ apply_rate, compare_additive, compare_multiplicative}` with pack lemmas for
+ each. It just waits passively.
+- The gap is entirely the **reader → injector → Operation** front-end (the
+ recognizer matches a shape but extracts *zero* anchors on real sentences).
+
+The redirection: shrink the front-end to **extract quantities + the relations the
+text licenses**, and grow the solver to **attempt** — search the bounded space of
+operation-chains for a derivation that reaches a *verifiable* answer. This
+generalizes across all phrasings because it never looks at phrasing — it looks at
+quantities and the goal. The compounding the project wants comes from the solver,
+not from a growing library of founds (the [[thesis-decoding-not-generating]]).
+
+**The honest crux we named:** search makes wrong=0 *harder*, not easier — an
+active searcher has more chances to land on a spurious-but-valid answer (cf. the
+brute-force `20/5`). The whole approach rests on the derivation being **grounded**
+(each step licensed by text + unit-consistent), **unique-or-refuse** (the existing
+disagreement rule), and **round-trip-checked** (realize back to language, reject
+if it doesn't match the problem). Those gates already exist in CORE.
+
+## Insight #3 — two failure modes, and the diagnosis that routes them
+
+> "Either it has enough information... to work the problem with masterfully built
+> problem-solving skill, or it needs a wider understanding of what it's trying to
+> decode... it needs more pieces to the puzzle."
+
+When the engine can't decode a problem, exactly one of three things is true:
+
+1. **Skill gap** — has the pieces, can't compose the derivation → improve the
+ reasoner.
+2. **Knowledge gap** — missing a world-fact (what "twice" means, that "each basket
+ holds 50" is a per-container rate) → acquire knowledge.
+3. **Genuine ambiguity** — under-determined → refuse, correctly, forever.
+
+The mechanism that makes "finding the pieces" efficient is **diagnostic refusal**:
+a refusal that *names the missing piece* routes itself to the right axis
+("quantities extracted but no grounded derivation" = skill; "unknown relation:
+twice" = knowledge; "two grounded derivations disagree" = ambiguity). CORE already
+has typed refusals, the OOV honesty gradient, and the math-reader-refusal audit
+corridor. Knowing what you don't know is most of knowing how to find it.
+
+**Three compounding stores**, each fed by the matching diagnosis:
+- **experience → vault** (exact, deterministic recall),
+- **world-knowledge → ratified packs**,
+- **skill → solver's elimination-learned pruning**.
+
+**The floor (Shay):** "it cannot conjure world-facts from nothing. not even a
+human can. only God himself." Autonomy doesn't mean "no input" — facts about the
+world must enter from a data/experience stream. The human role *shifts* from
+hand-authoring meaning to **curating what it ingests + ratifying what it has
+already self-proven**. The bridge is **self-proving acquisition**: a new fact is
+admitted with a *mechanical proof attached* (round-trips, forced-unique, zero
+wrong, replay-stable) — the schema-proof-obligation discipline, pointed at
+learning. Knowledge acquired with a proof needs less judgment to admit.
+
+## Pivot #4 (Shay) — the contradiction, and the "mode"
+
+> "If we intentionally build around it always 'giving up' whenever it doesn't
+> know, it will never learn without humans. Maybe there should be a 'mode' for
+> when it's allowed to work on a problem... a type of risk-to-reward RATIO system
+> (not... reinforcement learning; not that kind of risk/reward)."
+
+This is the central contradiction stated exactly: **refuse-when-uncertain is safe
+and frozen.** The resolution can't be a global setting — it must be contextual.
+
+**Two regimes (the concrete "mode"):**
+- **Serving** — committed to someone who will act on it; cost of error ≈ ∞;
+ refuse unless certain; **wrong=0, absolute, unchanged.**
+- **Practice** — attempting where being wrong is *checkable and not served*; here
+ **wrong is the elimination signal**, the most informative event possible. The
+ only place autonomous learning can happen.
+
+What keeps wrong=0 intact while practice runs hot: the **proof-carrying seal** —
+practice emits *proposals carrying their own proof*; nothing reaches serving
+except via the (narrowing) ratification gate. CORE already has this seal
+(proposal-only, reviewed teaching).
+
+**"Creative," defined for a deterministic engine:** not stochastic invention — a
+**willingness to leap a gap in known structure** (a recombination no stored
+pattern directly licenses). The risk is the leap is unwarranted; the reward is
+that a *verified* leap becomes new structure. It is always a step from solid
+ground, never from the void — the engineering form of "only God conjures from
+nothing." The risk/reward gate decides *how far across an unlit gap calibration
+permits a step.*
+
+## The checkability question — and the tiered answer
+
+If practice can attempt without a human or a gold label, *what counts as checkable
+enough to learn from?* The asymmetry that governs the answer: **a false positive
+in the checker is far worse than a missed learning opportunity** — a wrongly
+"verified" belief is a persistent contaminant; a refusal is just a missed chance.
+
+So the boundary isn't a line — it's a **confidence-stratified ladder**, with the
+rule: *require check-strength proportional to the reversibility/blast-radius of
+the action it licenses.*
+
+| Tier | Checker | May change |
+|---|---|---|
+| **1 — External truth** | gold label / known answer | serving-bound knowledge (via ratification) — anchors |
+| **2 — Convergent self-verification** | round-trip **∧** ≥2 *structurally-distinct* derivations agree **∧** unit-consistent **∧** no contradiction with vault/packs | provisional, **retractable** knowledge (still ratified) |
+| **3 — Consistency-only** | merely no contradiction | **practice-internal pruning only** — never crosses the seal |
+
+**Tier 2 is the median** — the widest checkability still strong enough to *create*
+knowledge, needing no label, so the arena scales toward open-world. Two
+safeguards make widening safe:
+- **Provenance per belief** `(tier, n_at_admission)` → Tier-2 beliefs are
+ *retractable* when a stronger check later contradicts them.
+- **A live gold tether** measuring `t2_precision(class)` — catches *correlated
+ self-delusion* (a shared wrong premise makes all derivations agree); when it
+ drifts, appetite contracts. Independence is *counted* (≥2 structurally-distinct
+ paths), not assumed.
+
+## Quantifying it — the part that makes it engineering
+
+The naive ratio needs units for "value" and "learning" — a quagmire. The regime
+structure **collapses the reward side**: in serving only reliability matters
+(learning is irrelevant to a served answer); in sealed practice the threshold is
+just "is it checkable?". So **the entire numeric system reduces to: measure
+reliability, compare to a ceiling.** Everything else is counting.
+
+Per **class** (= capability axis G1–G5), a replayable **ledger of counts** —
+nothing learned, nothing stochastic:
+
+- `n(C), correct(C), wrong(C), refused(C)` — the eval already counts these.
+- `t2_verified(C), t2_agrees_gold(C)` — on the live anchor set.
+- `reliability(C) = conservative_floor(correct, n)` — a deterministic **lower
+ bound** (pessimistic at small n, so luck can't grant appetite).
+- `t2_precision(C) = conservative_floor(t2_agrees_gold, t2_verified)` — how
+ trustworthy self-verification is on C; the number that licenses widening past
+ gold.
+
+**The gate is a literal ratio:**
+```
+license(action, C) := reliability_of_relevant_checker(C) / θ_required(action, C) ≥ 1
+```
+`θ` are **human-set, version-controlled constants** per class and blast-radius
+(`θ_practice = 0`; `θ_propose`; `θ_serve` strict, e.g. .99). "Humans ready to let
+it" = a human lowering `θ_serve(C)` for a class the ledger has earned.
+
+**Compounding becomes a measurable curve:** `refused(C)` ↓ and `reliability(C)` ↑
+with practice volume, `wrong` on serving pinned at 0 throughout. And the
+**skill-vs-knowledge diagnosis is quantified**: if reliability still climbs with
+practice → skill gap (keep practicing); if it has stalled → knowledge gap (needs a
+new world-fact). Learning-rate = reliability gain per unit practice.
+
+Almost none of this is new machinery — it's composition: classes = capability
+axes; counts = the eval harness; reliability + lower bound = the calibration
+module; replay guarantees reproducibility; `θ` = a small config table; seal +
+ratification = existing teaching-safety.
+
+## The converged model (one line)
+
+> attempt-or-refuse is a **per-context risk/reward decision**, not a global rule →
+> **two regimes** (serving wrong=0 / sealed practice attempt-and-eliminate)
+> separated by a **proof-carrying seal** → the ratio is
+> `measured_reliability / required_reliability ≥ 1` → reliability is an **earned,
+> conservatively-bounded, per-class count** from a live gold tether → checkability
+> is a **tiered ladder** with privilege ∝ reversibility and provenance making weak
+> beliefs retractable → humans **raise the ceiling on evidence**; the engine never
+> self-authorizes → "creative" = a calibrated leap across a gap, always from given
+> ground.
+
+## Open question carried into the ADR
+
+The one genuinely new number to pin: **the shape of the `conservative_floor`
+function and `N_min`** — how pessimistic to be at small n. That single choice sets
+how cautiously the whole system earns its autonomy.
+
+## What this supersedes
+
+The matcher-oriented Phase 5b sub-phases (5b.1 single-sentence / 5b.2
+cross-sentence / 5b.3 deep) in ADR-0174 collapse into *instances* of this
+architecture rather than standalone work. The multiplicative cases become the
+first **practice arena** where attempt-and-eliminate is proven, not a set of
+shapes to hand-match.
+
+## Cross-references
+
+- **Decision:** ADR-0175 (to be written).
+- **Builds on:** ADR-0174 (held-hypothesis / `eliminate_violating` / `reevaluate`
+ / `contemplate` — the elimination substrate, here repointed from *reading* to
+ *solving*); the calibration module; capability axes G1–G5; the round-trip
+ filter and multi-branch disagreement rule (the wrong=0 gates that make search
+ safe); teaching-safety (proposal-only, reviewed) = the seal.
+- **Thesis anchor:** [[thesis-decoding-not-generating]] — find/comprehend/
+ rationalize, not a library of founds.
+- **Phase 5a (shipped):** PR #430.