ramanujan-challenge/optional/FAMM_SCARS.md

FAMM scars for Ramanujan Challenge Problem 2.8

Status

This file and certificates/p28_famm_scars.json are advisory discovery artifacts. They do not alter the proof, authorize pruning, or assert membership in a canonical DiscoveryStore.

The formula-optimization rebuild is finalized. SHA-256 pins for the rank/ODE, convergence, standalone-equation, denominator-cleared-gauge, positive-cone, FAMM-interchange, and package-runner checkers are recorded in the JSON, along with the final solution.tex and solution.pdf hashes. The older solution hashes are retained solely as provenance for baseline commit 492c8ab.

The bundle records defects found during the adversarial proof loop so later searches can prioritize equation-level checks without mistaking past failures for universal impossibility results.

The governing rule is:

An observation, failure signature, SoftScar, or blocked promotion idea may change route priority. It may not remove a proof candidate.

The JSON therefore contains no AuthorizedHardScar.

Authoritative FAMM sources

The schema and authority policy were read from allaunthefox/MathPunch-FiniteState at commit 9df0f48576aefce91eb1fc13ff876bec1007162d:

File	Relevant rule
docs/specs/FAMM_REFINED.md	Exact and advisory memory are separate; only exact/formal, replayed, in-scope, instance-matched, version-matched scars may hard-apply.
docs/specs/FAMM_TOPOLOGY_ESCALATION_V1.md	Machine layout and physical observations never change mathematical authority; advisory or unreplayed scars never hard-prune.
src/discovery/node.rs	Defines Observation, FailureSignature, Certificate, SoftScar, ProposedHardScar, AuthorizationCertificate, and AuthorizedHardScar, along with typed parent roles.
src/discovery/authorization.rs	The implemented hard-scar gate requires a typed Boolean linear formula, complete failed assignment, deletion-minimized cube, exact linear-constraint certificate, linear-cube-interval authorization, and replay/reauthorization.
src/discovery/canonical.rs	Canonical bytes sort parents and field payloads and bind kind, payload, parents, scope, and checker version under a domain-separated hash.

The Problem 2.8 failures are polynomial, analytic, asymptotic, and proof-engineering failures. They are not instances of the current Boolean linear TypedFormula/CubeRegion authorization language. Consequently, no entry in this package is promoted to AuthorizedHardScar, even when an exact standalone checker supports the underlying equation.

JSON schema choices

p28_famm_scars.json uses the new interchange identifier mathpunch.p28-famm-scar-bundle.v1.

It mirrors the Rust discovery vocabulary without pretending to be a Rust serialization:

• kind uses exact DiscoveryKind names.
• parents use exact ParentRole names and bundle-local integer node IDs.
• A SoftScar's advisory relationship to its FailureSignature uses ParentRole::Supports, never DerivedFrom; CheckedBy separately links a replay certificate when one exists.
• scope is a bundle-local unsigned integer resolved through scope_registry.
• payload uses the Fields variant as ordered key/value pairs; a future importer must sort them as canonical.rs requires.
• checker_version is an unsigned schema/checker generation.
• replay commands, runtimes, artifact paths, and SHA-256 hashes are declared separately in checker_registry.

The bundle intentionally sets these fields to non-authoritative values:

ingested_into_discovery_store = false
canonical_node_hashes         = null
mmr_commitment                = null
pruning_authority             = false

Local node and scope IDs must be remapped by an importer. Canonical discovery hashes may be assigned only after the nodes are constructed through the repository's canonical Rust path.

Scar catalogue

Every row below corresponds to an Observation -> FailureSignature -> SoftScar chain in the JSON.

SoftScar	Exact scope	Failure signature	Assumption avoided	Replay support
12	Pinned transfer and package	An under-defined or transcription-divergent matrix is used by later identities	Omitted coefficients are harmless	Dependency-free equation replay
22	Tail contiguity for the displayed M_N(x) and shifted {}_4F_3 jet	CAS Ore division is cited without four cleared residual identities	A zero-remainder routine is itself an inspectable certificate	Dependency-free equation replay
32	Terminating denominator, n\ge1, 0\le k\le n	Fourth-order uniqueness is inferred from normalization at z=0	One datum determines a fourth-order analytic solution	Base/generic/top coefficient replay
42	Official matrix-to-scalar bridge	A scalar recurrence is accepted without an exact intertwiner	Sample agreement identifies the official module	Sixteen gauge entries and contraction replay
52	Official R,x_0, (	x	=1/4), m\ge1	An inequality is inverted without reversing its direction
62	Official seed rows, four columns, and positive cone	A named transport theorem hides the hypotheses or denominator conclusion	Spectral machinery is necessary for all-column transport	Exact Pascal-conjugated positive transfer and elementary min/max contraction
72	Historical characteristic quartic and displayed eigenvector	Native factor/GCD/root decisions are used as portable exact proof	CAS decisions carry proof authority by default	Exact coefficient homotopy and polynomial eigenvector replay for the retained legacy route
82	Official four columns in the proved positive cone	A quotient is formed before denominator positivity	Formal ratio notation guarantees a nonzero denominator	Exact cone entry and strictly positive transfer entries
92	Mandatory/optional checker split	A stored PASS transcript substitutes for live replay	A receipt proves the current bytes were executed	Mandatory standard-library runner
102	Release metadata	The reciprocal limit is labelled as the official orientation	Equivalent formulas have interchangeable submission labels	Boxed manuscript theorem and official-scope review
112	Wolfram source serialization	A line break terminates an assignment before leading-plus continuation terms	Printed multiline equality equals parsed equality	Parser round-trip is required; current Wolfram run is optional
122	Pinned rational gauge after denominator clearing	Raw rational expansion produces avoidable expression swell or resource failure	Raw rational normal form is required, or capacity failure falsifies the identity	176 cleared polynomial obligations
132	Official Pascal-conjugated positive cone	Spectral machinery is introduced before testing an elementary positive transport	Eigenvalues and a stable graph are necessary for the official columns	100 exact positive-cone obligations
142	Advisory FAMM interchange bundle	A SoftScar is linked as an exact derivation rather than advisory support	Advisory diagnosis has exact derivational authority	Structural validator requiring Supports and zero hard authority

These scars are deliberately narrow:

• They apply only to the pinned Problem 2.8 objects and proof routes.
• They do not assert that Ore methods, scalar recurrences, asymptotic theorems, CAS tools, or reciprocal formulations are invalid in general.
• They do not rule out a repaired candidate satisfying the missing equation or hypothesis.

Exact replay links

The advisory scars point to these replayable local artifacts:

python3 certificates/p28_rank_ode_bound_verifier.py
python3 certificates/p28_convergence_constants.py
python3 certificates/p28_standalone_equations.py
python3 certificates/p28_dominant_product_algebra.py
python3 certificates/p28_optimized_gauge.py
python3 certificates/p28_positive_cone.py
python3 certificates/p28_famm_scars_validator.py

The complete mandatory path is:

bash run_checks.sh

Sage and Wolfram files remain optional independent cross-checks. Their absence does not convert a stored transcript into proof evidence.

Finalized artifact hashes and checker identifiers are in the JSON. Changing a finalized checker, manuscript source, or PDF requires a new replay and a new bundle version.

Formula-optimization loop

Two optimization results change route priority without changing mathematical authority:

1. The rational gauge is replayed after the diagonal scaling D=\operatorname{diag}(x,1,1,1) and common clearing by (1-z)^2. The resulting companion matrices have bounded polynomial degree, and the checker expands the claim into 176 scalar polynomial obligations. A timeout, capacity rejection, or expression explosion in the unreduced route is a proof-engineering failure, not evidence that the rational identity is false.
2. The current all-column proof conjugates the balanced transfer by the exact Pascal matrix, places both official seed rows in a strictly positive cone, and uses the elementary min/max contraction of positive weighted averages. The earlier spectral and stable-graph argument remains an audited historical route, but it is no longer an active prerequisite for the four-column transport or denominator nonvanishing.

The interchange validator records the corresponding route scars and checks that each SoftScar is advisory: it must have a Supports edge from a FailureSignature, may have a separate CheckedBy certificate, has no hard authority, and cannot prune.

Why no hard scars were emitted

Three exact-certificate-linked promotion ideas are recorded under blocked_promotion_ideas:

1. nonzero cleared Ore residuals;
2. reversed negative-exponent inequalities;
3. nonzero matrix-to-scalar intertwiner residuals.

They are not ProposedHardScar or AuthorizedHardScar nodes. The present authorizer cannot express their formula domain, region semantics, or minimization rule. Promoting any of them requires all of:

1. a versioned typed proof-domain formula;
2. canonical coefficient or inequality encoding;
3. exact applicability-scope semantics;
4. a replayable witness;
5. a sound minimization rule;
6. an authorization certificate;
7. reauthorization after persistence;
8. hostile tests for forged witness, broadened scope, stale version, altered parent, and valid-candidate pruning attacks.

Until that machinery exists, the exact certificates support diagnosis and priority only.

Import requirements

A future importer into DiscoveryStore must:

1. register canonical problem, instance, and scope objects;
2. run certificates/p28_famm_scars_validator.py and reject a malformed role, scope, count, hash pin, or hard-authority claim;
3. verify every declared artifact hash;
4. execute the mandatory checker commands against those exact bytes;
5. translate local IDs to store NodeId values;
6. construct nodes through the Rust API;
7. recompute canonical discovery hashes;
8. replay the resulting store and MMR;
9. retain every SoftScar as non-pruning;
10. leave blocked_promotion_ideas outside HardIndex.

Failure at any step is a typed import or replay failure, not evidence that a mathematical proof candidate is impossible.