Scoped negative result: no nine-stage stage recipe improves PR589's idle hosting (no new kappa)

[ianlibanio]

Scoped negative result (bit supplier); no new κ.

Result. No nine-stage stage recipe in the searched class raises PR #589's bit c. The class is K = 4 exchanged pairs, one rank-1 local per stage, at most one ±1 bridge per gap, and entries |≤ 8|. It contains 2,070,720 valid recipes, which reduce to 1,064 distinct idle-window structures.

• The template recipe attains the maximum, tied by seven other structures with the identical remainder.
• The width-8 hosting wall R ≤ 4v = 444 holds for every recipe in the class.
• The selected Conditional kappa 0.001672660197076918: rare-pattern T16 exchanges #1094 assembly stays complex-bound: complex b is 0.0016755 and bit c is 0.0017334.

What this adds

The merged research/bit589-width-packing (Sagar Thakrar) proved that width-8 re-hosting cannot improve PR #589 with the template's idle windows fixed. This PR varies the stage recipe (recipe(K) of global_k.py), which is what produces those windows. Nothing in the overlap search had searched it.

	result
Recipe → exact c	The template recipe alone reproduces the frozen selection9-p111.json windows (index by index), the remainder {2:435,4:444,6:444,7:666,42:9} and c = 1733427039589844234749/10^24. The vendored hosting_k.solve gives the same objective.
Optimistic ceilings	Any split of the 27 idle level-steps gives c ≤ 0.001927 (+11.2%), but needs only 4–5 advances. With ≥ 9 advances the ceiling is c ≤ 0.0017515 (+1.04%), which needs 8-level advances.
Exhaustive class	9 / 10 / 11 idle segments occur (75 / 513 / 476 structures), with max advance 6. Capacity is 27 − #segments ≤ 18 slots per copy set, at most 3,996 against 3,951 needed. Of the 1,064 structures, 989 fail total capacity, 27 a stage, 2 transport, and 46 are hostable.
Best exact c	Equal to PR #589's (8 structures), with the others 0.044–0.213% lower. None is higher.

Why. For Δ ≤ 8, ⌊7Δ/8⌋ = Δ − 1, so capacity is 27 − #segments. The minimum segment count is 3(K−1): 3, 6 and 9 for K = 2, 3, 4. The longest advance is 2(K−1).

Local-count floor (not new). The 2K+1 floor is first proved uniformly by open #743 (Johnny Franks; archived report M) and is credited there. PROOF.md gives an equivalent invariant formulation, which the search uses for pruning.

Reproduce

python3.12 -m pip install numpy==2.5.4 scipy==1.18.1
python3.12 -B research/bit589-recipe-floor/verify.py --reuse --jobs 2     # registered generic replay
python3.12 -B research/bit589-recipe-floor/verify.py --jobs 12            # + full K=4 search (~25 min)
python3.12 scripts/research_package.py --package research/bit589-recipe-floor

Expected:

PASS_BIT589_RECIPE_FLOOR floor=2K+1 K3_recipes=28464 K4_recipes=2070720 K4_structures=1064 max_capacity=3996 hostable=46 best_c=1733427039589844234749/1000000000000000000000000 equal=8 above_pr589=0 no_new_kappa (…s)
PASS_REGISTERED_RESEARCH_PACKAGE research/bit589-recipe-floor outcome=negative-result

Executed at this head (Linux x86-64; numpy 2.5.4, scipy 1.18.1):

• research_package.py --package (registered --reuse replay) passes in 24 s on Python 3.12.15. The metadata/pin check passes.
• verify.py --record --jobs 12 passes on Python 3.12.15 (26 min 55 s, 165 MB RSS). It wrote data/search-K4.json and EXPECTED.json.
• An independent full replay verify.py --jobs 10 passes on Python 3.14.8 (21 min 56 s). Its freshly rerun K=4 search equals the committed file.
• unittest discover -s tests -p 'test_research_*.py': 20 tests OK (1 skipped).
• The CrocSwap CI has not run on this head.

Controls.

• Positive. The template-recipe reproduction described above.
• Negative (19). Dropping any bridge, dropping any local, flipping a local's type, and removing a stage from every window all fail. The invariants (4,0,0,4) / (0,4,4,0) hold at the identity and the template product.
• Agreement. A separate prototype search gives the same K = 4 counts, and the K = 3 structures agree between prototype and package. The numpy and pure-Python floor checks agree.

Scope and obligations

• Searched class only. Not searched:
  • gaps with ≥ 2 bridges (a K = 3 probe did not finish in 20 min);
  • banks, and split or cross-copy residuals;
  • other stage counts, words or replica schemes;
  • width-7 helpers.
• Hosting relaxes geometry. "Hostable" is PR Conditional κ ≈ 1.2375517e-3: idle-axis hosting of helper residuals (+9.2% over #434) #442's count/window programme. Open Audit: selected PR589 bit-hosting pending projectors across all hosted helpers (no new kappa) #1307 shows that this is weaker than disjoint physical axes, so stricter geometry could only remove candidates; the negative conclusion stands.
• Proof status. The K = 4 result is exhaustive finite enumeration, not a formal proof.
• Inherited assumptions. All inherited bit assumptions remain: mutable-source realization, whole-call hosting, all-size compiler, precision and tape, and H1–H5. This is not a Lean multiplication theorem.
• Files. Only research/bit589-recipe-floor/ is added. The selected record, PR Bit c = 0.001733427039589844: two A-section transports on #555's punctured R439 word (+0.0216%, κ unchanged) #589's package and upstream/ are unchanged.
• Open next questions are in README.md and TASK.json:
  • the multi-bridge class;
  • a 3(K−1) segment theorem;
  • the recipe choice once width-7 helpers exist.

Provenance and credit

• Base and assignment. Base 410026dc5513b87db92ae98571d1d5984c69a983 (the work started on 8bddbee3; every pinned input is byte-identical on both). OS-random seed 18, bit-supplier track, card selected-packing (scope refined).
• Claim. Discussion #1427.
• Overlap. find_overlap.py --refresh over all open PRs and discussions, refreshed just before submission (742 PRs, 171 discussions; complete pagination). Concurrent open Independent verification: PR589 paid rank ledger by exact rational projectors (no new kappa) #1421 (commdata2338) independently reconstructs PR Bit c = 0.001733427039589844: two A-section transports on #555's punctured R439 word (+0.0216%, κ unchanged) #589's paid rank ledger by projectors. It corroborates, by a different route, the profile and remainder that the positive control here derives from the recipe.
• Credit.
  • Template and recipe: Zhihao Chen and eumemic (Compressed nine-stage E8 windows and whole-block banking for a conditional κ = 9.98243154231047e-4 > 2^-10 bound #386/Conditional κ = 1.128983115440183e-3: an E8 bit word with 740 helpers under #433's complex supplier #436), chafreaky (Conditional κ = 0.001133928043872860: R740 shared-donor kernel with explicit residual bank routes #439/Conditional κ = 0.001138823042306561: retain rank138 with complete 32-replica banks #443), DreamingOfClouds (Conditional κ = 0.001283479303751122: E8 bit word R715 (carrier lower bound + F2 overlap cancellation) with #458's complex supplier (+0.711% over #460) #463/Bit word R480 at nine stages: c = 0.001587447106092548137108 (+1.41% over #493) #510/Bit c = 0.001733427039589844: two A-section transports on #555's punctured R439 word (+0.0216%, κ unchanged) #589).
  • Hosting: DaysSky (Conditional κ ≈ 1.2375517e-3: idle-axis hosting of helper residuals (+9.2% over #434) #442).
  • Word: eumemic (Conditional κ = 0.001587467473762886: an E8 bit word with 420 helpers, c = 0.001653826988 (+4.18% over #510) #543/Bit c = 0.001733052823: #545's punctured unit on the R420 skeleton (111 live ports, +4.79% over #543) #555) and Jacob Sussman (E8 unit, borrowing chain).
  • 2K+1 floor: report M and Johnny Franks (Research consolidation: uniform proof of the archived 2P+1 stage bound #743).
  • Fixed-window bound: Sagar Thakrar (bit589-width-packing).
  • Geometric audit: commdata2338 (Audit: selected PR589 bit-hosting pending projectors across all hosted helpers (no new kappa) #1307).
  • Paid-profile predecessors: SethBurkart123 (Scoped negative result: PR589's bit word is exactly frame-optimal (no new kappa) #886) and Scoped negative result: no relocation or Steinberg rewrite with exact reframing raises PR589's bit c (no new kappa) #933.
• AI assistance. Prepared by Ian Libânio with substantial AI assistance (Claude Opus 5.5, Anthropic, via Claude Code). Apache-2.0.

Workflow feedback (optional). find_overlap.py surfaced bit589-width-packing immediately. That package landed after this session's original base, so the refresh-before-submit step was essential. The main friction was that the generic 600 s / 1200 s replay budget cannot hold the full K=4 search; a hash-pinned committed search plus an optional long replay was the workaround.

🤖 Generated with Claude Code

[Mister-Meeseeks]

Batch review at cfd97ca1e9d332e497bb29ba9cdbdeb594e23800.

This remains open after scoped triage. Next review step: The primary reuse replay passed. The required full K=4 census hit the 2,400-second supplemental limit after the last recorded progress of 960/1,922 branches. Complete that census in a dedicated replay before incorporation; partial progress is not an exhaustive certificate.

This is a specific continuation point, not a rejection or a request to redo already recorded checks. Completed local evidence is retained separately from outstanding whole-head obligations.

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