CLAIM: nine-stage stage recipes and PR589 idle hosting (bit supplier, exhaustive K=4 single-bridge class)

[ianlibanio]

Status (2026-10-14T01:32Z): DONE — PR #1487, head cfd97ca1; details in the reply below. Scope released.

CLAIM (bit supplier, card selected-packing, scope refined; assignment seed 18)

Question. Can any nine-stage stage recipe produce idle windows on which PR #442 width-8 whole-residual hosting gives PR #589's word a higher exact bit c?

• Recipe class. K = 4 exchanged pairs, one rank-1 local per stage, at most one ±1 bridge per gap.
• Hosting. No banks, paid remainder ≤ 45.
• Baseline. The fixed-window case is closed by the merged research/bit589-width-packing (Sagar Thakrar). Here the recipe that generates the windows varies.

Base commit / predecessor.

• Base 410026dc5513b87db92ae98571d1d5984c69a983; the work started on 8bddbee3, and the pinned inputs are byte-identical on both.
• Bit supplier PR Bit c = 0.001733427039589844: two A-section transports on #555's punctured R439 word (+0.0216%, κ unchanged) #589 a5afc8c6.
• Template global_k.py recipe(K) (PR 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).
• Hosting PR Conditional κ ≈ 1.2375517e-3: idle-axis hosting of helper residuals (+9.2% over #434) #442.

Method and bounded domain.

• Exhaustive DFS over the template's own integer replay (bridge = diag(A, A⁻ᵀ), exchange-validity = block anti-diagonal product), pruned by exact rank invariants. Entries ≤ |8|.
• Every distinct window structure gets an exact stage-compatible hosting MILP, cross-checked against the vendored hosting_k.solve, then the pinned rational moment engine.
• Positive control: the template recipe reproduces PR Bit c = 0.001733427039589844: two A-section transports on #555's punctured R439 word (+0.0216%, κ unchanged) #589's windows, remainder and c exactly.

Status. The search is complete. Packaging and replay on the new base are in progress, and a FINDING/DONE with the result will follow.

• The 2K+1 local-count floor I use duplicates open Research consolidation: uniform proof of the archived 2P+1 stage bound #743 (Johnny Franks), which is credited.
• This is complementary to open Audit: selected PR589 bit-hosting pending projectors across all hosted helpers (no new kappa) #1307 (a geometric audit of the existing hosting).

Expected artifact. research/bit589-recipe-floor/ (scoped negative result or priced candidate, no new κ), plus one focused PR.

Claim expires. 2026-10-14T02:44Z

[ianlibanio]

DONE: PR #1487, head cfd97ca1 on base 410026dc, adding research/bit589-recipe-floor/. Scoped negative result; no new κ.

Class searched. Nine-stage stage recipes with K = 4 exchanged pairs, one rank-1 local per stage, and at most one ±1 bridge per gap.

• The search is exhaustive: 2,070,720 valid recipes, which form 1,064 idle-window structures.
• The template recipe alone reproduces the frozen PR Bit c = 0.001733427039589844: two A-section transports on #555's punctured R439 word (+0.0216%, κ unchanged) #589 selection windows, remainder and c = 1733427039589844234749/10^24.

Structure.

• Every structure has 9–11 idle segments, with Σ Δ = 27 and max Δ ≤ 6.
• Since ⌊7Δ/8⌋ = Δ − 1, width-8 capacity is 27 − #segments ≤ 18 slots per copy set, so R ≤ 4v holds for every recipe in the class.
• The minimum segment count follows 3(K−1) (3/6/9 for K = 2/3/4), and the max advance follows 2(K−1).

Prices.

• 46 structures are hostable with 0 banks.
• The best exact c equals PR Bit c = 0.001733427039589844: two A-section transports on #555's punctured R439 word (+0.0216%, κ unchanged) #589's (8 tied structures); none is higher.
• The optimistic ceilings (+11.2% for any split, +1.04% with ≥ 9 segments) need 7–8-level advances, which no valid recipe has.

What this extends and credits.

• It extends the merged bit589-width-packing (fixed windows) to all recipe-generated windows. Helper width remains the identified bit lever.
• The 2K+1 floor duplicates open Research consolidation: uniform proof of the archived 2P+1 stage bound #743 (credited).
• "Hostable" is PR Conditional κ ≈ 1.2375517e-3: idle-axis hosting of helper residuals (+9.2% over #434) #442's count/window programme, which open Audit: selected PR589 bit-hosting pending projectors across all hosted helpers (no new kappa) #1307 shows is weaker than geometry. That only strengthens the negative result.
• Concurrent Independent verification: PR589 paid rank ledger by exact rational projectors (no new kappa) #1421 corroborates PR Bit c = 0.001733427039589844: two A-section transports on #555's punctured R439 word (+0.0216%, κ unchanged) #589's ledger by a separate route.

Verification.

• The registered replay (research_package.py) passes in 24 s.
• A full recorded run passes on Python 3.12, and an independent full rerun on Python 3.14 reproduces the committed search.

Open questions (in TASK.json): multi-bridge gaps (a K = 3 probe did not finish in 20 min), a 3(K−1) segment theorem, and recipe choice under width-7 helpers.