Format limits (no κ claim): certified ceiling κ ≤ 5.1595e-3, spec census, structural lemmas, helper bounds and open problems

[DreamingOfClouds]

No κ claim. This package records what we have proved and measured about the limits of the current construction format. It is meant to help whoever finds the next structural step. Every result states its scope. Each is marked checked (recomputed by verify.sh), argued (proof or sketch in the README) or recorded (an input from a named run or PR).

bash research/format-limits/verify.sh /fresh/output/dir   # ~2 min; 298 PASS lines; step 0 fails hard on any MANIFEST/provenance mismatch

Results

id	result	status
A	Format ceiling, rev. 2: κ ≤ 5.1595·10⁻³, complex binds. Per side at s = 5: X* = 6.0005·10⁻³ (bit, 111 lines, cst 63) and 5.1863·10⁻³ (complex, 112 ports, cst 64). Rev. 1's scope x = 3/5 is superseded. At s = 9 the frontier sits 2.66× (bit) / 2.50× (complex) below; 87% / 93% of the log-gap is helper cost.	theorem argued; numbers checked
B	Spec vs convention census: 25 format rules classified, with file:line into the upstream paper and the hub's checkers. Nesting, target-perp sinks, subspace frames and one bank pair per invocation are conventions; a common frame per gate, restoration and the κ = min composition are spec.	anchors checked; classification argued
C.1	Frame sandwich: a register's frame lies between the span of what it carries and the orthogonal of what it will serve. The three complex frame walls are its faces.	argued; toy checked
C.2	Lemma N: the rank-1 port factor is a gauge and is always paid; any detour costs exactly 2.	toys checked
C.3	Lemma R: complement sinks w^⊥ (w·u = 1) are legal at zero cost but forced to w = u on all four frontier complex words checked.	checked
C.4	Lemma Q: a degenerate-label unit is dominated by its quotient.	toys checked
C.5	Lemma λ: a phase-frame butterfly needs λ = ±i; b < c is the price of √−1. The outer already uses shift twiddles.	checked
C.6	Pairing uniqueness: at the E8 decoder pattern with window ≤ 9, B = W·Uᵀ is unique, so asymmetric labels are a gauge.	checked (SAT)
C.7	Per-call descent pricing: a rank-r descent costs 2r data mass in every call (≈ −5.6% per rank-1 descent on either side).	checked; dominance argued
C.8	Splitting a residual across hosting streams meets a trace obstruction under every invertible storage change; uncharged ≤ +0.67% b.	argued
D	Design-free helper floors: bit ≥ 212 / 138 vs 439 shipped; complex ≥ 107 / 0 vs 496 (so at most 48% / 22% of today's helpers are certified necessary). Coupling the levels adds 0. The "224 carriers" complex floor is a design choice.	checked
E	Complex helper prices on #569 by class and width (0.039–0.143% b per helper; the hosting offset depends on width only), plus the moment-cost census on #555 and #569.	checked (k = 1); k > 1 recorded

Open problems (where we think the next step has to come from)

• Enablers paid once per word, not per call. A change paid in every local word costs about 2.8% of the supplier value per rank unit, so it pays only if its per-word gain exceeds 2.8·Δ% (README §F). Candidates: a descent or gauge change hoisted out of the local word, or a decoder whose stations serve many calls.
• A non-star decoder. The copy charge is the largest scope multiplier (×6.8 if star scope fails), but the E8 pattern pins the pairing (C.6) and the lossless pattern (C.3).
• The schedule/ρ family (×1.15–2.1). The only spec freedom never exercised (B.3, item 8).
• New skeletons. The design-free floors certify at most 48% / 21% of today's helper count, so nothing here excludes a much better skeleton on either side.

Notes

• Base: this branch is cut from main (7dfa22b) and touches only research/format-limits/.
• Inputs: vendored inputs are pinned by PR blob id in SOURCES.json. Requirements are pinned. Independently re-verified from a fresh git archive of 39112c2: 298/298 on Python 3.11 and 3.14 with the pinned set, on 3.14 with the latest numpy/scipy, and on a dev environment. Tampered or unlisted files stop step 0 with exit 1.
• Errata found: the E7 hyperplane-flat census is 8821 (an earlier note said 2258), and the cover numbers are unchanged. A tail-lemma constant is corrected (Hermite–Hadamard).

Credits: see NOTICE.md for every PR whose word, pipeline or result is used. Among them are #531 and #130 (icekylinx), #543/#555 (eumemic), #569 (DominikScholz), #442, #446, #509 and #516, and the swarm PRs cited in each section.

Prepared by DreamingOfClouds with Claude Opus 5.5 (Anthropic), via Claude Code; Apache-2.0.

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