[NicholasStone]

PR #5193 leaves explicit q-series for the four projection modes and exact parity-sensitive leading error constants open. This adds a self-contained solution in research/projection-parity-constants/, with a proof, standard-library exact verifier, frozen receipt, source pins and independent AI review record.

For the unstrengthened recurrence with seed a_4=(0,27456,48576,68320,86592) and T_m=a_m(m)/a_m(1), the result is

T_m = 439277/149175
    + 2^(-m) [815083/596700 + (-1)^m (323/28) K]
    + O(4^(-m)),

K = product_{i>=1} ((1-2^(-i))/(1+2^(-i)))^2 > 0.

The even and odd limits of the scaled error are respectively 1.535225897075989125... and 1.196743266657880490...; the full scaled sequence has no limit. The proof derives an explicit convergent q-series for every fixed mode, applies classical q-Gauss summation, and subtracts the poles through +/-4 to justify the remainder. The rational limit is inherited from #5151; the four-mode reduction and sharp geometric order retain #5193's credit.

Validation

CPython 3.12.14, standard library only:

python3 -B scripts/research_package.py --package research/projection-parity-constants
python3 -B scripts/check_research_commit.py --package research/projection-parity-constants --check-source-history

Registered replay passed in normal and optimized, isolated, warning-strict Python, with identical outputs: 2,691 raw-recurrence/profile equalities, 276 finite-product identities, 520 eigenmode transformations, 32 generating-series enclosure comparisons, 16 residue checks and six rejected semantic perturbations. The committed-byte/source-history preflight passed for all eight package files; the GitHub publication tree 3222b7447ab0e7e1d5fe8ecc6a5b1e81fb440429 exactly matches that checked local tree. Finite checks corroborate the analytic argument; they are not a formal proof. Hosted CI remains separate.

REVIEW.md binds a separate Codex agent's mathematical/code review to exact hashes and records an independently written raw recurrence check through block 100. This is AI review, not human or proof-kernel verification.

Scope, sources and credit

Publication base: canonical main 1ec1119972c9aefe42525172d07b332fcff81cb4, priority revision 7. The proof/review source snapshot is ff3cc58e370d0f678e529246f8c73e8094e18fcf; the relevant #5193 sources are unchanged. TASK.json explicitly declares the changed scope from the catalogue's independent-moment card to this user-selected arithmetic side result. Targeted projection PR searches and nearby source/handoff checks were performed; this is bounded overlap coverage, not a global novelty guarantee.

Prepared for Nicholas Stone / NicholasStone with substantial OpenAI Codex assistance, including a separate AI review agent. Credit Jacob Alan Sussman (#4628), Maximilian Lutz (#4643), Nick Petalas (#5151), JoyboyBrian (#5184), Krzysztof Wojt (#5193), Eumemic (#288), Swapnil Jain and the retained lineage. New text/code: Apache-2.0; frozen predecessor files are unchanged.

No new kappa, selected-frontier change, supplier admission, physical caller, hook-rank certificate or full multiplication theorem is claimed. Open next questions are monotone-seed classification, further finite-coefficient simplification and the independent inherited physical/hook-rank obligations; concrete first steps and blockers are in TASK.json.

Merge scope: one isolated opt-in eight-file research package; no shared code or selected-result changes.

[jwr404]

Independent review (jwr404). Base: upstream/main 1ec1119, head f0cdec0.

Replay. python3.12 -B scripts/research_package.py --package research/projection-parity-constants gives PASS_REGISTERED_RESEARCH_PACKAGE … outcome=completed. Hosted CI was still at plan: in progress when I checked.

Headline (1), checked from the raw recurrence with none of the package's code. My script iterates the stated N_k from the a₄ seed in exact integers up to m = 120. It forms s_m = 2^m (T_m − 439277/149175) as an exact rational and compares it with the claimed limits. K comes from 400 factors at 60 digits: K = 0.014671073764252386633631502280977…

m	s_m	2^m (s_m − limit)
20	1.535240402…	15.2096
21	1.196747364…	8.5932
60	1.53522589707598913854…	15.20949201510080
61	1.19674326665788049460…	8.59318437169323
120	1.53522589707598912534924210082669782…	15.20949201510080295499796

At m = 120 the even subsequence agrees with 815083/596700 + (323/28)K to about 36 digits. At m = 101, the last odd value I printed, the odd subsequence agrees with 815083/596700 − (323/28)K to about 30 digits. The scaled next-order error settles to parity-dependent constants, ≈ 15.2095 (even) and ≈ 8.5932 (odd). That directly confirms an O(4^{−m}) remainder with its own parity term, consistent with the poles at ±4 in §4.

Per-mode asymptotic (12), checked from recurrence (5). I took the closed forms A_j (9), D_j (10) and E_j⁻ (11) and iterated v_n by (5) at 160 digits for j = 1..4. For n = 30, 31, 60, 61, 90, 91, the remainder 4^n (v_n − A_j − D_j 2^{−n} − E_j⁻ (−2)^{−n}) is bounded and 2-periodic:

• j = 1: 0.03177 / 0.02031
• j = 4: 0.1326 / 0.04078

The values repeat to all shown digits, as the ±4 poles predict. A₁ = 5/6 as stated.

Proof, checked by hand.

• §3. The prefactor ratio in (4) and the functional equation (6) are consistent. The poles of (7) are simple and lie only at z ∈ {2^i} ∪ {−2^{i+1}}.
• §4. Subtracting the poles at 1, ±2 and ±4 leaves a function analytic on |z| < 8, and the Cauchy bound O(5^{−n}) is valid.
• §6 enclosure. Each 1 − d_i ≥ 0 with d_i ≤ 4·2^{−i}, and Σ_{i>N} 4·2^{−i} = 2^{2−N}, which gives K_N(1 − 2^{2−N}) ≤ K ≤ K_N.

I did not re-derive the q-Gauss evaluations symbolically. The numerical agreement of A_j and E_j⁻ with the actual v_n, for j = 1..4, is strong evidence that (9) and (11) are right where they are used.

Verdict: correct as a scoped arithmetic theorem about this recurrence. The headline constants are confirmed to about 36 digits. No κ is claimed and none is affected.