ramanujan-challenge/optional/ADVERSARIAL_AUDIT.md

Adversarial Audit — Ramanujan Challenge Problem 2.8

Verdict

The recurrence-specific proof path passes the repaired adversarial audit. Every Ore, differential-gauge, terminating-induction, valuation, convergence, and all-four-column obligation is now displayed as an equation and replayed without a computer-algebra decision procedure. The active all-column proof is an elementary positive-cone contraction; the earlier spectral/stable-graph route remains in the package as a replayed legacy alternative.

The exact trust boundary is important:

• The proof imports the classical Chudnovsky formula as one explicitly named theorem, with a precise citation to a complete modular/CM derivation.
• It also uses foundational results stated with their hypotheses: absolute convergence of power series, the maximum modulus principle, and completeness of bounded monotone real sequences.
• It does not claim to be axiom-free or to reconstruct those foundational theorems from set theory.

Relative to that explicit boundary, no recurrence-specific assumption, vacuous implication, numerical-equality inference, or hidden CAS remainder remains.

Defects found and repaired

Initial defect	Why it failed	Equation-level repair
The deformed transfer was under-defined	Only one substituted coefficient was shown; later notation changed the meaning of the first argument	Displayed all sixteen entries of \mathcal M(u,x), defined M_N(x)=\mathcal M(2N+3,x), and displayed both official seed rows
Three matrix terms lost a plus sign during the first repair	The manuscript matrix then differed from the certified matrix	Restored the three sums in c_1,c_2,c_3; hostile replay caught this before release
Ore divisions used quo_rem	A zero remainder was trusted rather than exhibited	Replaced every division with four direct cleared factorizations D_r=q_rL_+, including the fourth companion closure
“Standard ascension identity” and transformed ODE were named but not derived	The coefficient mechanism was hidden	Added initial coefficient and consecutive-ratio equations; expanded the {}_3F_2 Euler operator explicitly
ODE normalization was claimed to determine the terminating {}_4F_3 uniquely	False: the exponent 2n supplies an additional analytic branch	Replaced uniqueness with base, generic, and top coefficient induction for the actual one-step operator
The scalar one-step operator was not tied to the challenge matrix	Hard-coded d_0,d_1 could have described a surrogate	Added horizontal reconstruction, all sixteen differential-gauge equations, and the exact matrix contraction producing d_0+zd_1
Only the first base component was initially checked	The actual compact seed row was not yet known to be horizontal	Added all four base-row reconstruction equations, the base terminating-operator equation, and all four base adjoint residuals
Two DVR-lemma hypotheses were only implicit	The induction had not displayed the k_{N+1} leading direction or J_N(0)e_1\ne0	Added both expansions and cited them explicitly at the induction step
A transfer norm inequality used an upper bound with exponent -1	The inequality direction was invalid for column four	Split j\le3 and j=4, then used coupled row factors to obtain \|\mathcal B_m\|_\infty\le4981375/512<10000
The maximum-modulus step omitted holomorphy of the quotient	Formal divisibility only supplied a local removable germ	Proved holomorphy on (
Birkhoff–Poincaré was used as a black box for three columns	It hid the exceptional hyperplane, dominant functional, decay, and denominator nonvanishing	First replaced it with an explicit stable graph; the optimized proof now eliminates the spectral layer entirely via T_m=\mathcal P\mathcal B_m\mathcal P^{-1}>0 and a four-weight min/max contraction
The spectral route required a quartic root count, eigenvector, and exceptional-hyperplane analysis	Although repaired, it created unnecessary proof surface	Verified all 285 positive numerator coefficients, the positive limiting transfer, and both positive seeds; all four quotients are now convex averages with uniformly positive weights
Division in columns 2,3,4 preceded an eventual-nonzero proof	The displayed quotients were not yet justified	The positive-cone seed and transfer identities now give Q_{m,j}>0 for every m\ge1, before any quotient is formed
The direct rational differential gauge produced large unreduced intermediates	Correct but slow replay increased resource and serialization risk	Added a separately reconstructed J_0+xJ_1+x^2J_2 decomposition and checked the denominator-cleared polynomial gauge in 176 scalar coefficient obligations
The terminating step polynomial obscured its structure with 21 expanded terms	Large coefficients made transcription review difficult	Rewrote it in u=2n+1,\ q=2n-t, then added a direct coefficient identity against the former expansion
A FAMM SoftScar was initially linked with DerivedFrom	It did not follow the repository’s calibrated Supports parent pattern	Corrected every parent role and added a fail-closed FAMM interchange validator; all scars remain advisory
The checker could succeed while Wolfram/Sage were absent	A stored transcript was being treated as proof evidence	Made standard-library rational-polynomial verifiers mandatory; Wolfram and Sage are now optional independent cross-checks
Metadata called Q/P the requested orientation	The official challenge asks for P/Q	Corrected every release document to state P/Q\to\sqrt{10005}/\pi as the official orientation

Mandatory replay

Run:

./run_checks.sh

The mandatory path executes:

1. p28_rank_ode_bound_verifier.py
2. p28_convergence_constants.py
3. p28_standalone_equations.py
4. p28_optimized_gauge.py
5. p28_positive_cone.py
6. p28_mutation_sensitivity.py
7. p28_famm_scars_validator.py

It then replays p28_dominant_product_algebra.py as a preserved legacy cross-check; that quartic/spectral route is not required by the active proof.

The third verifier checks:

• four cleared tail factorizations;
• lowest and generic tail coefficients;
• horizontal reconstruction;
• the terminating-operator closure;
• all sixteen differential-gauge entries;
• the authoritative matrix-to-scalar contraction;
• the base polynomial and four base-row components;
• the base terminating equation and four base adjoint residuals;
• constant, generic, and top terminating induction;
• ascension and the {}_3F_2 Euler equation.

The positive-cone verifier checks:

• the authoritative M_m, balanced \mathcal B_m, and T_m=\mathcal P\mathcal B_m\mathcal P^{-1};
• all sixteen rational identities T_{m,ij}=N_{ij}/D_{ij};
• all 285 strictly positive coefficients of the N_{ij}(m-1,R-4);
• the exact positive limiting matrix;
• all eight positive coordinates of the two official transformed seeds.

The optimized gauge separately checks 176 scalar coefficients while the original sixteen-entry gauge remains in the standalone checker. These verifiers use fractions.Fraction and explicit coefficient dictionaries. None uses polynomial division, factorization, a simplifier, Gröbner bases, irreducibility, GCD, a root finder, a special-function package, sampling, or a stored transcript.

Forbidden-shortcut search

The mandatory runner rejects these constructs in the proof path:

• quo_rem
• is_irreducible
• polynomial gcd
• Birkhoff/Poincaré delegation
• “standard ascension”
• ODE-normalization uniqueness
• the former dominant-product lemma in the active manuscript

No occurrence of native_decide, axiom, sorry, or admit was found.

Independent hostile replays

Independent reviews and mutation replays targeted:

• logical validity, indexing, vacuity, and denominator domains;
• Ore/special-function and matrix-to-scalar algebra;
• convergence and all-column division;
• one-coefficient corruption of the positive-cone numerator table;
• one-coefficient corruption of the optimized (J)-decomposition.

The defects in the table above were discovered during those loops. The final Ore, gauge, positive-cone, convergence, and logic/vacuity replays return PASS, and both corrupted checkers fail at their intended identities. Release engineering then repeats the mandatory checks in a clean extraction, rebuilds the PDF, and performs page-by-page visual inspection.