Research: prove sharp fit for the original selected BIT stock

[petalas]

Summary

Add research/selected-bit-stock-fit/, a source-bound application of #2324's staircase/tail method to the original selected BIT reserve defined in #2084/#2120.

S(e,w) = [28425 |GL_64(F_q)| q^(4096(w-1))]^D(e)
D(e) = 0 below 64, otherwise 1+D(42 floor(e/64))
w = ceil(e^s),  0<s<=1/500,  q prime > 2^80

least uniform integer cutoff for ceil(log_q S(e,w)) < e/2: 311305

Both source-pinned selected atom exponents lie in this domain. This is fit for a prescribed conditional reserve, not supplier admission, a new cutoff technique, a replacement of the selected algorithm cutoffs or a new kappa. The full-width stock is chosen before borrowing; no body-dependent reserve is substituted.

#2324's cutoff 311311 concerns the different #2234 replacement reserve with lcm(28416,28425) in place of 28425. The six-width difference is not a correction to that result. Credit Adam Gall / #2084, Nick Petalas / #2120, Aakash / #2324 for the method, Sagar Thakrar / #2343 for prior use of the sharpness prime, and the retained supplier/stock lineage named in the README. #2547's later-stage completion audit is complementary.

Base and refreshed canonical main are 0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2, priority revision 4. Canonical claim and status. Exact predecessor heads and live source/member pins are recorded in the package.

Evidence

• Before: at width 311304, certified prime 15*2^78+1 makes borrowing exactly 155652 digits, so strict half-width fit fails.
  After: an exact four-interval staircase and proved dyadic tail establish fit at every integer width from 311305 onward, including every later atom-width ceiling jump.
• Ten package test groups pass normally and optimized. An independent all-child depth oracle checks all 1,048,577 widths through 2^20; a separate full exact 64-factor GL product corroborates the sharpness bound without the primary Bernoulli relaxation. Source, receipt, semantic, strict-fit and composite-prime rejection controls pass.
• The 65 repository research-workflow tests, selected binding baseline, warnings-as-errors syntax compilation and whitespace checks pass locally on Linux x86_64 / CPython 3.12.15.

python3.12 -Werror -B scripts/research_package.py \
  --package research/selected-bit-stock-fit

Physical packet/compiler, completed-leaf, H2/time, other size thresholds and precision/tape obligations remain inherited. The witness prime's recipe-specific exclusions are not checked, so sharpness is over the stated all-prime arithmetic domain. No full selected cold replay, formal-kernel proof, independent human review or hosted CI success is claimed.

Merge Danger

Door: two-way

Only one opt-in research package is added. Selected records and frozen predecessors are unchanged.

Blast Radius: research

[ywz65]

Independent source-bound review: PASS, no material issue found at 2876e991263abdfd17450d8a6f14b7c0873eb36c. This corroborates Nick Petalas's sharp cutoff for the prescribed original-width reserve; it does not claim a new cutoff theorem or replace the credited method of PR2324.

I reconstructed the source application from PR2084 at 8001a23b003bec5dff5e61edec907b652c475932 and PR2120 at 378593574261e0b0915e7e8a83147aa640274fc8. They specify the complete 28425-role GL64 packet, fixed atom width, rank 42 floor child, individual child restoration, completed leaves, and stock selected from the original width before borrowing. The selected PR1094 citation, PR589 receipt, and authenticated PR1087 assembler support the numerical parameters and both atom exponents in 0<s<=1/500. The five live source files, catalogue provenance, two predecessor texts and seven submitted files were byte-bound, together with the consumed archive member. No predecessor checker was imported into the independent calculation.

The independent arithmetic route differs from the package's inverse-threshold certificate:

• D(e) is constant on each [64j,64j+63]. Checking the first admitted width of each block covers all 737271 widths from 311305 through 1048575 in 11520 blocks. The complete depth counts are 94007, 212352, 323584 and 107328 for depths 19 through 22. Every block satisfies the strict half-fit bound.
• For the infinite tail, use the exact rational majorant r=501/500, with r^500>2. For 2^t<=e<2^(t+1), the retained two-level contraction gives D(e)<=2(t+1), and the packet bound gives k<A_t+1, where A_t=2*8193*(t+1)*r^(t+1). The exact base is below 2^19; successive growth is at most 1837/1750<2. This proves every later width, including atom-width jumps.
• For the sharpness prime, a separate product inequality gives B=product_(i=1)^64(1-q^-i)>1-1/(q-1), hence B^19>(q-20)/(q-1). The exact sandwich P^19*(q-20)>q^3*(q-1) and P^19<q^4 independently yields k=155652 at width 311304. The modular order certificate proves the witness prime is prime; this is not a probable-prime test.

Here is the independent arithmetic core, using only the standard library and explicit gates that remain active under -O:

from fractions import Fraction as F
from math import gcd

def need(ok):
    if not ok:
        raise ValueError("independent stock review failed")

def depth(e):
    d = 0
    while e >= 64:
        e = 42*(e//64)
        d += 1
    return d

P, cutoff, end = 28425, 311305, 2**20
need(P < 2**15 and 2*42**2 < 64**2)
blocks = covered = 0
for lo in range((cutoff//64)*64, end, 64):
    first, last = max(cutoff, lo), min(end-1, lo+63)
    d = depth(lo)
    need(depth(first) == depth(last) == d)
    need(2*(8192*d + (3*d+15)//16) < first)
    blocks += 1
    covered += last-first+1
need((blocks, covered) == (11520, 737271))
r = F(501, 500)
need(501**500 > 2*500**500)
need(2*8193*21*r**21 + 1 < 2**19)
need(F(22,21)*r == F(1837,1750) < 2)
q = 15*2**78 + 1
need(q > 2**80 and 2**156 > q)
need(pow(11,q-1,q) == 1)
need(gcd(pow(11,(q-1)//2,q)-1,q) == 1)
need(P**19*(q-20) > q**3*(q-1))
need(P**19 < q**4 and depth(cutoff-1) == 19)
k = 19*8192+4
need(2*k == cutoff-1)
# Failure controls: weak fit, a later w jump, and a composite witness.
need(not (2*k < cutoff-1) and 2*k <= cutoff-1)
need(2**500 < 2**500+1 < 3**500)
need(15*2**3+1 == 11**2 and pow(11,120,121) != 1)
print("PASS independent blocks/tail/strict sharpness")

The full independent checker passed under native CPython 3.12.7 with -I -O -B. The unchanged package also passed its optimized primary and all ten test groups in a Git-free source export (4.383 seconds for the tests), using writable TEMP/TMP. Those runs are corroboration of the written all-width proof, not a finite extrapolation.

The all-prime domain and its sharpness must remain attached to the result: the witness's recipe-specific prime exclusions are untested, and stock fit does not establish an actual algorithm cutoff. Full-cover/compiler realization, completed physical leaves and spectators, H2 time, later-stage composition, precision/tape and other size guards remain inherited. PR2324's 311311 concerns a different replacement reserve, so the six-width difference is not a correction. No supplier, selected exponent, formal-kernel or human-review claim follows. This review and its independent checks were prepared for ywz65 with substantial OpenAI Codex assistance.

Freshness: canonical main and all three reviewed heads were unchanged at 2026-10-15 15:06:38 UTC. Overlap uses the last complete cached Discussion snapshot plus a fresh 50-PR REST supplement and targeted comments; it is not a complete fresh Discussion scan or a novelty certificate.

Separate root review reproduced the full independent receipt and public inline code under native CPython3.14.5 with -I -O -B, rehashed all15 source files and the consumed archive member, and checked the floor-block, tail-induction, strict product sandwich and prime-order arguments. No material issue was found within the stated scope.

Canonical main and source head were rechecked unchanged at 2026-10-15T15:10:55.825724+00:00.

Original advance claim preserved

CLAIM (ywz65, LOCAL-ONLY, expires 2026-10-15 16:00 UTC): independent source-contract and strict-fit review of this exact head 2876e99. I will reconstruct the original-width reserve premises from pinned PR2084/2120, check selected parameter bindings, and independently certify the finite depth/strict-ceiling boundary and an all-size rational tail bound without importing this checker. The goal is an inspectable review result, not a replacement stock, new cutoff claim, eligible-prime minimality, compiler admission or new kappa. Canonical main refreshed at 0fdd7d8, priority revision 4; consolidation picker scope narrowed to this proof/source handoff. Cached complete Discussion coverage plus fresh bounded REST PR/comment screening; no complete fresh Discussion claim. Final review will state exact source pins, controls and limits. Substantial OpenAI Codex assistance.