Audit: PR442 hosting windows make the partial width-seven bit rows unreachable for replica-local hosting on the frozen recipe (no new kappa)

[thakrarsagar]

Adversarial audit: no new κ, and the selected record is unchanged. The window-free partial width-seven rows of merged #1023 and merged #1281 cannot be reached under PR442's replica-local hosting windows. This holds on the frozen PR589 stage recipe, with whole residuals and no banks. This PR gives the exact window-respecting optimum for every k = 0..74 instead.

• Track: adversarial audit, seed 72.
• Card: uniform-transfer-audit (one hosting implication), with the scope refined.
• Base: 0b1cccf8.
• Package: research/bit589-width7-windows-seed72/.

Why

#1023 (ours) lists window timing as an unchecked obligation. #1281's open question 1 asks whether its relaxed optimum can be hosted chronologically for an actual width-7 helper. It names PR1179's contract and both per-replica and cross-replica pooling. Both #1023 and #1281 place all 9k width-7 residuals per replica in capacity-7 idle slots, ignoring when each stage's residual can be hosted.

This PR settles the per-replica, frozen-recipe part of that question. The PR1179 contract audit and cross-replica pooling stay open.

The windows

windows.py writes out the nine-stage idle schedule (K = 4, L = 9). At run time it matches that schedule, event by event and field by field, against a reference built from the pinned global_k.py: its own recipe() and idle-insertion block are extracted with ast and executed inside a transcription of the surrounding loop.

On that one schedule, three window predicates agree:

• this code;
• PR442's hosting_core.windows;
• audit_hosted_k.windows, which enforces s in W[i] for every hosted slot.

All 14 frozen selection9-p111 entries lie inside their windows. The selection hosts nothing in children 1, 3 and 5, so the singleton windows rest on the global_k.py match. A test drops one bridge and checks that the mismatch is rejected.

Idle child	0	1	2	3	4	5	6	7	8
Capacity	14	7	28	7	42	7	42	28	14
Admissible stages	0–1	2	0–3	4	0–5	6	3–8	5–8	7–8

Each capacity-7 child admits exactly one stage. So per replica at most 3k of the 9k width-7 residuals can sit in 7-slots.

Results

For every k = 0..74 and both of #1023's local brackets, the package gives:

• the exact window-respecting optimum;
• an explicit residual-level placement attaining it, built and checked at run time (verify.build). Each slot is checked for window, width and offset, cells are disjoint, and every (stage, helper) is hosted exactly once;
• its endpoint, priced with the pinned engines and the inherited surcharge;
• a certificate that no window-feasible replica-local placement reaches the next 1e-24 grid point. The certificate is tight: it equals the engine moment of the explicit histogram to 1e-50.

k = 0 reproduces the selected bit endpoint 0.001733427039589844234749.

k	Bracket	Window-respecting optimum	Window-free optimum (#1281 profile)	Retained
1	optimistic	0.001734347102 (+0.053%)	0.001734846264 (+0.082%)	0.648
2	optimistic	0.001735271920 (+0.106%)	0.001736327170 (+0.167%)	0.636
10	optimistic	0.001743225534 (+0.565%)	0.001748291126 (+0.857%)	0.659
37	optimistic	0.001770809187 (+2.157%)	0.001789671631 (+3.245%)	0.665
37	pessimistic	0.001761884716 (+1.642%)	0.001780556367 (+2.719%)	0.604
74	optimistic	0.001809907119 (+4.412%)	0.001849754238 (+6.711%)	0.657

Selected rows are shown; the README has the full set. Endpoints are truncated and percentages/retained fractions are rounded. The window-free column is this package's pricing of #1281's closed-form profile; #1281 itself prices k = 0, 1, 2, 37 and 74. Across k = 1..74 the windows keep 63.6–67.1% of the window-free gain under the optimistic bracket and 56.3–60.7% under the pessimistic one. So the window-free rows overstate the gain by about 1.5× or 1.7× respectively, roughly flat in k. This assumes #1023's one-unit local pricing.

• Research note: PR589 bit hosting is width-8 optimal; width-7 residuals worth up to +17-23% bit headroom (no new kappa) #1023: all 8 published partial rows (k = 1, 10, 37, 74; both brackets) lie above their certified bounds, so none is reachable.
• Research: exact local and pooled partial width-seven slot optima (no new kappa) #1281, replica-local:
  • Research: exact local and pooled partial width-seven slot optima (no new kappa) #1281's closed-form local optimum is transcribed and checked against its merged EXPECTED.json (digest-pinned, read as data; no Research: exact local and pooled partial width-seven slot optima (no new kappa) #1281 code is run). Its optimality is Research: exact local and pooled partial width-seven slot optima (no new kappa) #1281's result and is not re-derived here.
  • All 75 idle profiles agree.
  • All 9 priced cases are reproduced exactly, in local endpoint and local histogram.
  • Priced here under both brackets, all 148 local rows (k = 1..74) lie above the per-replica bound. 8 of these are also priced by Research: exact local and pooled partial width-seven slot optima (no new kappa) #1281.
• Research: exact local and pooled partial width-seven slot optima (no new kappa) #1281, pooled32: its priced rows with k > 0 also lie above the per-replica bound. Pooling is not replica-local hosting, so this says nothing about cross-replica owner maps.
• Research note: PR589 bit hosting is width-8 optimal; width-7 residuals worth up to +17-23% bit headroom (no new kappa) #1023's all-width-7 optimum {42:340, 21:1} stays window-feasible (104 sextet splits by max-flow). Its +17–23% figure is not affected.

Why it is exact

1. Lemma A (exchange). Copy cost F(r) = 64^a r^(1−a) + λ is subadditive with F(0) = 0. So some optimum puts each stage-2, 4 and 6 width-7 residual alone in a copy of child 1, 3 or 5.

2. Width-8 placement is unconstrained. All interval Hall conditions for width 8 are vacuous; this is checked for every k ≤ 74.

3. Two nested chains. The remaining 6k width-7 residuals are feasible exactly when all of these hold:

   • B0 ≤ 2k, B0 + B2 ≤ 3k, B0 + B2 + B4 ≤ 4k;
   • the mirror chain on children 8, 7 and 6;
   • ΣB = 6k.

   This is derived from the 45 interval Hall constraints, asserted for every k. It is confirmed by max-flow on every distribution of the 6k residuals: 462 at k = 1 and 6188 at k = 2.

4. Lower bound. An integer DP over chain sums and slack mod 5 relaxes the copy limits. It uses outward-rounded lower bounds of F from the pinned engine, so it is a rigorous lower bound.

5. Attainment. The DP minimiser expands into the explicit placement, whose cost equals the DP value exactly.

Scope (what stays open)

• Hosting format. The result covers the PR442/PR589 nine-stage template only, with replica-local hosting, whole residuals, no banks and the frozen stage recipe. Research consolidation: uniform proof of the archived 2P+1 stage bound #743 proves the 2P + 1 stage lower bound that this format attains.
• Other recipes. Open Scoped negative result: no nine-stage stage recipe improves PR589's idle hosting (no new kappa) #1487 (ianlibanio) varies the recipe for width-8 hosting. Its recipes may have other capacity-7 windows, which are not checked here.
• Cross-replica owner maps. T16-style interleaving is not ruled out.
• Windows are necessary, not sufficient. Projector compatibility (Audit: selected PR589 bit-hosting pending projectors across all hosted helpers (no new kappa) #1307) and owner inactivity remain open. So does a physical width-7 helper: Research: two-sided boundary obstruction for R439 width-seven helpers #1128, Finite bit construction: sparse R439 kernel entrance gauge (no new kappa) #1179, Finite bit construction: fixed-prefix helper470 retirement and five cut obstructions (no new kappa) #1256, Scoped negative result: R439 retirement windows across all 713 B cuts #1271, and Scoped obstruction: helper468 must end at rank eight after B88's target MOVE #1393 (discussion R439 helper468: all later B/C cuts for rank-seven retirement (seed17) #1362). So the window optimum is itself only an upper bound for real hosting.
• Finite bit construction: fixed-prefix helper470 retirement and five cut obstructions (no new kappa) #1256's two-slot move. Under this recipe, the move is window-feasible only for residuals hosted at stage 2, 4 or 6.
• Inherited assumptions. All of them remain, and this is not a proof of the multiplication theorem.

Checks

• Bindings. The frontier1094 certificate, named by the selected record (or by a checkpoint record once the frontier moves), binds the PR589 record and the three engines that Research note: PR589 bit hosting is width-8 optimal; width-7 residuals worth up to +17-23% bit headroom (no new kappa) #1023's packing.py pins.
• Controls (14 tests).
  • The schedule equals the global_k.py reference. A dropped bridge and six single-field mutations are rejected, and the unanchored three-way path is shown to accept the dropped bridge.
  • Research note: PR589 bit hosting is width-8 optimal; width-7 residuals worth up to +17-23% bit headroom (no new kappa) #1023's k = 1 placement puts 6 residuals outside their windows.
  • The k = 0 certificate is recomputed and checked for tightness.
  • Window-free rows lie above the bound, and window-respecting rows below it.
  • Research: exact local and pooled partial width-seven slot optima (no new kappa) #1281's local and pooled32 rows lie above the bound.
  • The all-width-7 optimum is window-feasible.
  • Dropping the chain constraint B0 ≤ 2k breaks the brute-force agreement.
  • Widening child 0's window breaks the chain structure.
  • Width-8 Hall conditions are vacuous.
• Replay. research_package.py --check-only passes, and RESULT.json is reproduced byte for byte. The registered python3.12 replay passes (PASS_REGISTERED_RESEARCH_PACKAGE, 80 s of wall time on an unloaded host). An earlier attempt timed out only because the host was heavily loaded.
• Environment. Run on Python 3.12.15 and 3.14.8 (macOS 26.5, arm64). verify.py needs about 140 s of CPU.
• CI. The red verify (…, tests) job comes from main, not from this PR. It fails identically on unmodified 0b1cccf8. The repair is CI repair: refresh the single-intersection spectrum audit pin after #778 (tests job red on main) #1752.

Reproduce

python3 -I -B research/bit589-width7-windows-seed72/verify.py        # about 140 s of CPU; PASS_BIT589_WIDTH7_WINDOWS
python3 -I -B -m unittest discover -s research/bit589-width7-windows-seed72 -p test_package.py
python3.12 scripts/research_package.py --package research/bit589-width7-windows-seed72

Overlap

• Research note: PR589 bit hosting is width-8 optimal; width-7 residuals worth up to +17-23% bit headroom (no new kappa) #1023 (ours, merged): its frozen files are left untouched.
• Research: exact local and pooled partial width-seven slot optima (no new kappa) #1281 (mightyOil, merged): this settles the per-replica, frozen-recipe part of its open question 1.
• Scoped negative result: no nine-stage stage recipe improves PR589's idle hosting (no new kappa) #1487 (ianlibanio, open): complementary; it covers width-8 hosting under other recipes.
• Research consolidation: uniform proof of the archived 2P+1 stage bound #743 (0xjohnnydev).
• Audit: selected PR589 bit-hosting pending projectors across all hosted helpers (no new kappa) #1307 (commdata2338).
• Research: two-sided boundary obstruction for R439 width-seven helpers #1128 (aliromman) and research/bit-width-seven-boundary.
• Finite bit construction: sparse R439 kernel entrance gauge (no new kappa) #1179 (adamgall), Finite bit construction: fixed-prefix helper470 retirement and five cut obstructions (no new kappa) #1256 (RossRichardson), Scoped negative result: R439 retirement windows across all 713 B cuts #1271 (mightyOil) and Scoped obstruction: helper468 must end at rank eight after B88's target MOVE #1393 (1337isnot1337).

Credit also goes to:

• PR442 (DaysSky);
• PR444, PR443 (the global_kernel.py recipe) and PR439 (explicit residual bank routes) (chafreaky);
• PR386 and PR436 (Zhihao Chen, eumemic);
• PR463, PR510 and PR589 (DreamingOfClouds).

find_overlap.py found no duplicate of the window audit.

Disclosure

Prepared by Sagar Thakrar (thakrarsagar) with substantial AI assistance from Claude (Anthropic) via Claude Code. Parallel read-only scout and judge subagents proposed and ranked the audit candidates and found the window obstruction. A post-PR adversarial review (three reviewers and a verifier) found overclaims and a missing runtime anchor for the schedule; the second commit fixes them, and a recheck's smaller gaps are closed in the third. No claim was posted to Discussions. New material is Apache-2.0.

[Mister-Meeseeks]

Batch review at b4fd72163234c164d9a05f2f7dac9c57dcd7fb66.

Merged to main, preserving the original contributor commits and package attribution.

Reviewed scope: Fixed nine-stage replica-local width7 windows admit only3k of9k such children in rank7 slots. Interval-Hall chain reduction and exact relaxed DP checked against attained placements for all k=0..74; full replay and14 tests pass on extended budget. Frozen windows and recipe only, not a new legal width7 supplier or global barrier.

The pinned scoped command passed; the batch ledger contains its command, input hashes, environment and log. This incorporates the research/tooling contribution and does not change the selected κ or discharge the inherited multiplication-theorem assumptions.

Batch review, attribution and evidence. Thank you for the contribution.