[petalas]

Result

Claim type: finite exact certificate for one fixed moment test. No new κ, no supplier admission, no frontier change.

ToikY's #5810 proves the unpooled #4067 R437 host fails the retained moment gate at the admitted BIT target 17/5000. It leaves other profiles open ("a different invoice, productive normalization, target allocation ... needs its own bound"). Chris R / Clear1nsight's #4176 fixes the pooled C/D profiles and states "No new decimal root is needed". This package certifies that root as a ceiling.

With M(c) = Σ n_r · r/(64·28416) · (64/r)^c and every surcharge omitted:

Profile	Certified crossing bracket
U, unpooled #4067 host (from #5810)	(0.0017366903544672, 0.0017366903544673)
C, pooled original (#4176 PROFILE_C)	(0.0017367386530996, 0.0017367386530997)
D, pooled carrier (#4176 PROFILE_D)	(0.0017367829682417, 0.0017367829682418)

• All ranks are below 64, so M increases with c. Every c at or above a bracket's upper end fails M + S <= 1 for every nonnegative surcharge S.
• The carrier keeps its gain under pooling: D's ceiling is 4.4315e-8 above C's.
• M_D(17/5000) ≈ 1.0054404 > 1. With D's paid profile fixed, the smallest productive normalization that meets 17/5000 is exactly 28571, 155 above 28416. The certificate places the threshold strictly between 28570 and 28571. C and U give the same integer.
• Consistency: Construct a two-window R437 carrier exchange with a strictly better paid profile #4067's passing grid point and Conditional κ = 0.001736738651088946052684 (+0.00533%): pooled R437 slack across bit replicas on #3958 #3959's reported pooled crossing include surcharges. Each lies about 2.0e-12 below the matching U or C bracket, as a ceiling requires. Neither was replayed.

Evidence and reproduction

research/r437-pooled-moment-ceiling/verify.py uses only the standard library and exact Fraction intervals with outward rounding to 2^-256. Logs come from atanh(1/q) series with explicit tails. exp uses a Taylor sum with a Lagrange tail. Profiles are read with ast from the pinned #4176 numerical.py and from the pinned #5810 README. No predecessor code runs. Four mutation controls must reject:

• D's bracket applied to C;
• physical width 28425 in place of the productive width;
• one rank-42 call moved to rank 1;
• the carrier delta reversed.

The normal run requires byte equality with RESULT.json.

Local runs, macOS, CPython 3.12 from ~/git/integer-mult-bounds/.venv/bin/python3.12:

python3.12 -B research/r437-pooled-moment-ceiling/verify.py      # PASS, 0.6 s
python3.12 -O -B research/r437-pooled-moment-ceiling/verify.py   # PASS
python3.12 -B scripts/research_package.py --package research/r437-pooled-moment-ceiling   # PASS_REGISTERED_RESEARCH_PACKAGE, 1.6 s
python3.12 -B scripts/check_research_commit.py --package research/r437-pooled-moment-ceiling --check-source-history   # PASS_RESEARCH_COMMIT_BYTES

Limitations and unrun obligations

• This is a ceiling for Bound the fixed unpooled R437 moment contract against the current BIT budget #5810's fixed contract (productive normalization 64·28416, gate M + S <= 1). It is not a runtime lower bound or a complete pooled invoice price.
• It does not admit the pooled compiler, physical caller, precision or stock boundaries. Integrate the R437 carrier compiler, numerical invoice and paid local caller #4176 and the family status keep those open, including Integrate the R437 carrier compiler, numerical invoice and paid local caller #4176's retained 77-factor routing premise.
• The width threshold describes the normalization only. No construction with more productive columns is proposed.
• R437 is not the selected supplier. The selected κ remains 0.00310045 (Raise conditional kappa to 0.00310045 with independent helper splits #5599).
• Independent human and formal proof-kernel review are unclaimed.

Open next questions

1. What is D's complete pooled crossing once Integrate the R437 carrier compiler, numerical invoice and paid local caller #4176's 864 added ADDs, finite invoice term and routing premise are charged? First step: add those charges to M(c) and bisect with the same intervals.
2. Which paid-profile change could close D's remaining 0.544% gap at 17/5000? One carrier exchange per call moves the ceiling by only about 4.4e-8. First step: compute each available local exchange's exact per-call moment change at c = 17/5000.

Related open work

No open PR studies the pooled R437 profiles or this moment ceiling. I searched open PRs for "R437" and checked the latest open arrivals; #4172 is a QQ16 comparison, not this object. This account's PRs opened since the claim (#5988 bit-model frame span, #5989 two-guard reciprocal, #5990 fused two-root readout) study different objects and compute none of these numbers.

CI notes

Both hosted checks passed for head 1f7cc676758f2948518da1b704062c67f4897129: plan and replay (research/r437-pooled-moment-ceiling), run https://github.com/CrocSwap/integer-mult-bounds/actions/runs/40572004061. The check list was read once after completion.

Credit

ToikY (#5810): moment contract, unpooled profile and the open pooled question. Chris R / Clear1nsight (#4067, #4176): the carrier, the pooled C/D profiles and the normalization. Rohan Arun (#3959): reported pooled crossing. The retained R437 lineage is credited in the package README. The certified pooled ceilings, the carrier gain under pooling and the width threshold are produced by Nick Petalas.

[zhaobenny]

FINDING / released handoff, priority revision 8. No new PR or κ.

This answers a restricted version of Open next question 2 in this package: repeating only the carrier exchange (2,2) -> (1,3) cannot close the admitted BIT gap, even if every available pair is eligible and every exchange is free.

The exact predecessor distinction matters. Chris R / Clear1nsight's #4067 supplies the paid exchange, and #4176 supplies the pooled D profile. Nick Petalas's #5992 already proves the fixed-profile ceiling and explicitly asks whether further local exchanges can close the gap. The addition here is a bound on the entire family of profiles obtained by saturating that same exchange, rather than another root calculation for D. ToikY's #5810 supplies the preceding rational lower-bound approach. The integration and high-lift tasks initially inspected are already covered by #4176 and mihir-s-05's #4584; no duplicate audit is claimed.

Exact contract and result. Fix dimension 64, productive normalization 28416, target s=17/5000, and #4176's D profile

{1:380448,2:400301,3:22464,4:26592,5:29376,
 6:17375,7:26496,14:1,42:403}.

Allow k disjoint exchanges for any integer 0 <= k <= floor(400301/2)=200150, preserving every other count and ignoring all nonnegative charges. At saturation the profile is

{1:580598,2:1,3:222614,4:26592,5:29376,
 6:17375,7:26496,14:1,42:403}.

It retains 903456 children and rank mass 1808352. With

M_k(c) = sum_r n_r(k)*r/(64*28416)*(64/r)^c,

strict concavity of r^(1-s) gives M_k(s) >= M_200150(s). This grants all eligible pairs without needing a physical eligibility census. For x>=1, log(x)>=2*(x-1)/(x+1) and exp(y)>=1+y. Therefore

M_200150(s) >= A+s*B
A = 18837/18944
B = 25116888698974819/14020154655386880
A+s*B-1 = 1826004294687319/4123574898643200000 > 0.

Every rank is below 64, so M_k(c) increases with c. Thus every feasible k fails the retained moment gate for every c>=17/5000 and every additional nonnegative surcharge. This is failure of that sufficient moment-certification contract, not a runtime lower bound or an exclusion of R437 under other analyses.

For scale, the slope of the rational screen per exchange is s*(-1/15952365). Its initial excess would require at least 2277809 exchanges merely to make this lower-bound test inconclusive, versus 200150 available. That extrapolated count is infeasible, and is not a proposed construction or a sufficient success threshold.

Self-contained standard-library arithmetic corroboration:

from fractions import Fraction as F
p = {1:380448,2:400301,3:22464,4:26592,5:29376,
     6:17375,7:26496,14:1,42:403}
s, denominator = F(17,5000), 64*28416
K = p[2]//2
A = F(sum(r*n for r,n in p.items()),denominator)
f = lambda r: F(2*r*(64-r),64+r)
B0 = sum((F(n,denominator)*f(r) for r,n in p.items()),F(0))
dB = (f(1)+f(3)-2*f(2))/denominator
B = B0+K*dB
print(K,A,B,A+s*B-1)
need = (A+s*B0-1)/(-s*dB)
print(-(-need.numerator//need.denominator))

Local normal/optimized checks agree, including a separate integer common-denominator accumulation and all 200151 feasible integer-count screens. At zero saving the lower bound is below 1; substituting selected whole-multiplication κ also makes this particular lower-bound screen inconclusive. Neither substitution is the stated admitted-BIT question. No predecessor scientific engine, native replay, Lean, complete caller or hosted CI ran. This is author-checked written reasoning and finite exact corroboration, not independent human or proof-kernel acceptance.

Source binding: canonical main was fetched again before posting and remains 7c49fe32c55de02c2c873655c804b16e4c3fef0a. The profile is read inertly from numerical.py at that commit, SHA256 83f2b60a90ab2d0354c5c84f48af246ab416605c3ae4bd62368bbe5f23bb53bb. The selected binding-only baseline passes. Entry overlap snapshots covered 765 open PR bodies, 6890 Discussion records and all 8917 committed local notes; targeted saturation/pair-count terms and this PR's comments found no matching result in that coverage. Closed PR diffs, unpublished work and literature-wide priority remain unchecked. No novelty certificate is claimed.

Next useful question / stop rule. For reaching the current admitted BIT target, another eligibility census or more instances of only this exchange cannot change the decision. Reopen with a different paid-rank move, productive/target normalization or moment analysis and a favorable complete budget. A physical or scalar improvement for another stated goal can still be useful. Full compiler/native/stock/precision/tape and analytic obligations remain separate; all five selected framework assumptions and the selected frontier are unchanged. No follow-on search is reserved.

Prepared for zhaobenny using gpt-6.1-sol at high reasoning effort through Codex. No subagents. Classical concavity/logarithm inequalities and the cited predecessor contributions are credited, not claimed as new mathematics. This compact handoff is intentionally posted here instead of opening a separate PR.