[petalas]

Result

Claim type: scoped written proof with exact finite checks, conditional on two cited hypotheses. No κ, supplier or frontier change. Not a complete multiplication theorem and not a formal-kernel proof.

Answers open question 3 of #4954 ("Close the six-bit window") for the one ordinary-BIT stage that note analyses (N = 2^25986965159937, atom 1/250). #4954 proved the source borrowed-digit prescription (k < e/2) infeasible for log2 e <= 52182660963729 and fitting from 52182660963735, with c_B and S_old undecided.

• At depth two the prescription fits iff e > 4N²w + 4δ_B + 2k_old, where w = ceil(e^(1/250)) and δ_B ∈ [0, c_B] is the chart excess.
• Every width of depth at least one with log2 e <= t* = 500(H1+1)/249 = 52182660963730.9236947791… fails to fit. This needs nothing beyond B >= 1, q >= 3, and holds for any larger admissible depth.
• Under c_B <= N² (Bound the selected #4470 BIT-stage stock on both sides: every single-cover-level width overdraws, fit from log2 e = 52182660963735 (no new κ) #4954's hypothesis) and k_old <= N² (The old-leaf reserve of the selected BIT stage is one (no new κ) #4970: S_old = 1 gives k_old = 1, fallback at most 103), every width with log2 e >= t* + 2^(-2·10^11) fits.
• So the six-bit window closes to a window of width at most 2·2^(-2·10^11) in log2 e around t*, independent of B and of S_old within those bounds.

At the integer widths e = 2^t that #4954 tabulated:

t	…729	…730	…731	…732	…734	…735
k < e/2 (this PR)	no	no	fits	fits	fits	fits
k_par < e/4 guard	no	no	no	holds	holds	holds
#4954 (proved)	no	open	open	open	open	fits

The fit entries assume c_B <= N² and k_old <= N²; the no entries assume nothing more. The guard of #4954 moves from …735 to …732, which answers the question of #7101 for the e/4 form. With S_old = 1 the guard is not needed: the stated reserve fits from …731. Below the threshold the conversion must still call the retained supplier directly, as #4954 said; only the location is new.

Evidence

New package research/bit-stock-band-threshold (PROOF.md, verify.py, RESULT.json, README.md, TASK.json, SOURCES.json, VERIFY.json). verify.py (stdlib, about 2 s) does the following:

• authenticates the six pinned inputs by SHA-256 and the cited sentences of Bound the selected #4470 BIT-stage stock on both sides: every single-cover-level width overdraws, fit from log2 e = 52182660963735 (no new κ) #4954 and The old-leaf reserve of the selected BIT stage is one (no new κ) #4970;
• checks 170 chart-envelope cases and 336,636 depth-two instances of the source formula against the exact criterion;
• scans the exact-integer translation chain on twelve (N, A) pairs (exhaustive from e = 1 for nine of them, windowed for three);
• computes the exact least fitting width with big integers at six ranks from 2^40 to 2^101, where w is huge as in the real stage (best reading 4N² to 8N² above the infeasible edge, worst 10N² to 18N²);
• does exact exponent arithmetic at the selected rank;
• runs ten rejection controls (tampered rank, three mutated criteria, mutated edges, shifted rank, wrong-side integer claims).

Checks use explicit exceptions on failure, so -O does not strip them. RESULT.json is the receipt that verify.py --write regenerates.

Local runs on commit 27163c173 (CPython 3.12.15):

PASS_BIT_STOCK_BAND_THRESHOLD t*=52182660963730.923694779116465863453815261044 guard_t*=52182660963731.927710843373493975903614457831 depth2_instances=336636 scans=12 large_rank=6 controls=10
PASS_REGISTERED_RESEARCH_PACKAGE research/bit-stock-band-threshold outcome=completed
PASS_RESEARCH_COMMIT_BYTES {"commit": "27163c17372181f430d3a913cbf65130908fdff1", "files": 13, ...}

Fresh direct runs of the two registered VERIFY.json commands passed in 0.579 s and 0.563 s, respectively. Each has a 120 s manifest timeout. The registered package runner passed both commands in 1.581 s; the commit/source-history check passed in 0.254 s.

Reproduce

From the repository root, with CPython 3.12.15 at /Users/nick/git/integer-mult-bounds/.venv/bin/python3.12:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/bit-stock-band-threshold/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -W error -B research/bit-stock-band-threshold/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/bit-stock-band-threshold
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/bit-stock-band-threshold --check-source-history

Limitations and unrun obligations

• The result is for the source prescription with the exact depth D = floor(log_N ceil(e/w)). The parameters file allows a larger explicit depth bound; the infeasible side holds for it, the fit side as stated does not.
• c_B <= N² (Bound the selected #4470 BIT-stage stock on both sides: every single-cover-level width overdraws, fit from log2 e = 52182660963735 (no new κ) #4954; derived for this stage in the open Bound the selected BIT role digit by 2^(H1+13) (no new κ) #7101) and k_old <= N² (The old-leaf reserve of the selected BIT stage is one (no new κ) #4970) are cited and not replayed.
• A window of width at most 2·2^(-2·10^11) in log2 e remains around t*; its exact resolution depends on the realised B and the ceiling in w.
• The sources pinned are the Conditional kappa = 0.00302015 (+0.6304% over #4447): joint helper boundaries #4470 stage of Bound the selected #4470 BIT-stage stock on both sides: every single-cover-level width overdraws, fit from log2 e = 52182660963735 (no new κ) #4954. Bound the selected BIT role digit by 2^(H1+13) (no new κ) #7101 states that the same rank is retained by the later frontier6030 stage; that is not checked here.
• The proof's last step (Lemma 4) is real-analysis over 2^(5·10^13)-sized numbers. The verifier checks its integer constants and runs the same chain exactly at small and large-but-evaluable rank; it does not evaluate the selected numbers.
• No Lean, Docker or selected cold replay was run. Independent human and formal proof-kernel review are unclaimed.

Open next questions

1. Compiled depth bound. Which depth bound does the compiled prescription use, and where does the fit start if it exceeds floor(log_N ceil(e/w))? First step: read it in the compiled stock formula; the first edge moves to 250(2H1 + log2(2D'))/249. Blocked by: the bound is not pinned in the sources used here.
2. Last window. Can the window of width 2·2^(-2·10^11) be decided exactly? First step: fix q and B from the compiler realisation. Blocked by: B is not materialised.
3. Current selected record. Does the later frontier6030/frontier7000 BIT stage use the same rank, atom and stock formula? First step: compare its stage record with research/query-thinned-controllers/RESULT.json. Blocked by: that stage's source bindings are not pinned here.

Related open work

• Bound the selected BIT role digit by 2^(H1+13) (no new κ) #7101 (unpaidmoon, open): bounds the role digit c_B <= 2^(H1+13) for the same stage and poses this window as its first open question. This PR uses c_B <= N² as a cited hypothesis and does not replay Bound the selected BIT role digit by 2^(H1+13) (no new κ) #7101; its question for the e/4 guard is answered above. The sufficient guard at ...735 in Bound the selected BIT role digit by 2^(H1+13) (no new κ) #7101 remains sufficient; this PR proves the sharper ...732 threshold under its stated hypotheses.

• Bound retained backing across the selected complex and BIT backend stack with history-safe stock reserves #7151 (open): bounds retained backing across the selected complex/BIT backend stack. This PR decides the borrowed-digit fit threshold for the pinned Bound the selected #4470 BIT-stage stock on both sides: every single-cover-level width overdraws, fit from log2 e = 52182660963735 (no new κ) #4954 ordinary-BIT stage and does not compute the physical backing coefficients of Bound retained backing across the selected complex and BIT backend stack with history-safe stock reserves #7151.

The own-open-PR check found #7133, #7134 and #7135 created after the CLAIM. They concern dyadic carve windows, odd-rectangle residual ranks and the net splitting tax, respectively. None studies this stock-fit threshold, so there is no shared numerical result to compare.

Credit

JoyboyBrian (#4954: stock analysis, band, Lemmas 1 and 2 re-proved here, the open question), James Reed (#4970: S_old = 1), unpaidmoon (#7101: the c_B bound, cited not replayed), Mark (Matarener: the nested-width bound used there). Apache-2.0.

CI notes

For head 27163c17372181f430d3a913cbf65130908fdff1, run 42914704054, plan job 130174025830, failed in Onboarding helper tests. The log reports 760 tests, one failure and 24 skipped. The sole failing test is:

test_research_current_status.CurrentCiNavigationTests.test_current_ci_section_matches_selected_record

Its assertion requires the literal text `make frontier7000-verify` in the current CI verification section and fails because that text is absent.

At the base commit e54523517d0f03e58352e647c403a5af3e525a57, certificates/selected-result.json names make frontier7000-verify, while the current section of docs/ci-verification.md still describes #6030 and make frontier6030-verify. The failing test, selected record and CI document are unchanged by this PR. This is a repository-level navigation mismatch.

I also read the plan-job logs of run 42905094314, run 42903313790, run 42914729845, run 42914997118 and run 42914936342. Each reports the same sole failing test and missing command, with 760 tests and 24 skipped.

CI did not certify this package. Its replay job was skipped after plan failed. The package replay, both direct normal/optimized verifier commands and the commit/source-history check passed locally at the head above. The numerical evidence is local; the cited hypotheses and written real-analysis proof retain the limits listed above. No CI rerun or retrigger commit was made.