Format limits part 2 (no κ claim): erratum to #595, exact invoice prices, frontier-word censuses, decoder floors, spec requirements, centre and fragmentation walls

[DreamingOfClouds]

No κ claim. This continues the merged #595. It records what we have proved and measured about the limits of the construction format and of the shipped #656 word, including an erratum to #595. Every result states its scope. Each is marked CHECKED (recomputed by verify.sh; CHECKED (log) = a shipped, sha256-pinned log is compared), ARGUED (proof or file:line argument in the README) or RECORDED (an earlier run that is not repeated; notes are shipped where the row names them).

python3 -m pip install -r research/format-limits-2/requirements.txt          # sections 5-9 only
bash research/format-limits-2/verify.sh /fresh/output/dir                    # 233 PASS lines, ~22 s; step 0 fails hard on any MANIFEST/provenance mismatch
E1_FULL=1 bash research/format-limits-2/verify.sh /fresh/output/dir          # also re-runs #595's a5 unchanged: 228 PASS lines, ~62 s

Results

id	result	status
E1	ERRATUM to #595. The ×1.15–2.15 "schedule/ρ family" multipliers (#595 census item 8, open problem 3) come from dropping idle-frame rules, not from a ρ family: idle changes free ×2.15, one-child batching ×1.55, no split pre-stretch ×1.18 at s = 9 (×1.15 best, s = 5), both sides.	numbers CHECKED; reading ARGUED
E2	ρ is not a verifier input (it is the hard-coded final bank swap). Over all pair-preserving ρ at P ≤ 3 (reachable set exactly Sp(4,2) = 720 and Sp(6,2) = 1,451,520) none beats B_K: multiplier 1.000.	(b) CHECKED (log, re-run and compared); (a) ARGUED
P1	Complex invoice calculus on #626 (reproduced to 16+ digits): centre −1 = +22.6% b; one cst unit +2.81%; one port +2.15%; one w9 helper +0.068% (+0.137% if its width merges).	CHECKED
P2	Bit invoice calculus on #555: c reproduced to 24 digits; cst unit ±2.80% c, port 2.15%. Word-level helper knobs (0.06–0.11% c) not re-run.	root, mass, cst rows CHECKED; rest RECORDED
P3	Helper exchange rate by width on #649: w9 0.070% (no merge) / 0.140% (merged), down to w4 0.026 / 0.072; explains the old 0.027–0.143 spread.	CHECKED
P4	Stage count s = 9 → 7 is worse at every R: hosting overflows 8.7%, b₇ only +0.21% over b₉ at equal R.	CHECKED (model)
P5	The climb mass is a forced cover (mass = mW − deficit): mass padding is illegal, frame annealing on the #626 ck/jp8 words gains 0.0% b (fixed op order; jump-merging frame raises as in #741 were not in the move set), "fewer helpers" is not a gain.	identity CHECKED; SA outcome RECORDED
C1	Provenance census of #656: 492 helpers born at 4 allocation sites: S1 scratch 224, S3 packet copy 156, S4 slot copy 112; κ identity exact.	CHECKED (tags from a recorded rebuild)
C2	The shipped word is a dirty-borrow program, not a clean SLP: 2056 distinct read rows (minimum chain cover 2056), 268 of 422 B-phase helpers are pure dirt, equal to the S3 + S4 sites.	CHECKED
C3	The 112 ports form a cap in F₂⁹ (no three XOR to 0, every hyperplane misses ≥ 48); 8-port blocks sit at dimension ≤ 5.	cap CHECKED; corollary ARGUED
G1	Gauge-free cold-end bound N_cold ≤ Σ_b(|Q_b| − rank C[Q_b]); #645 attains 44.	CHECKED on #645; 504 / K480 values RECORDED
G2	Gauge44 is gauge-optimal at 44 for its frame schedule; read-order ceiling 48 (+0.13–0.16% b), home-first legal ceiling 46.	44 = bound CHECKED; ceilings RECORDED
G3	Theorem. A helper can end at a hyperplane U only if the supports of its row of M⁻¹ and its column of M lie in Q_U; covers every reordering and relay strategy.	ARGUED (Lean on #593's package RECORDED)
D1	Complex decoder floor on P112: exactly 8 costly terms give cst ≥ 64 (9 label spans over 440,880 supports); all costly terms β ≥ 8 and n ≥ 9 give cst ≥ 72; n = 8 is infinitesimally rigid up to gauge. Open: mixed β ≤ 7 terms with n ≥ 9.	counts, spans, rigidity CHECKED; step (b), (c) ARGUED; flip-walk logs CHECKED (log)
D2	Any n = 8 exact P112 decoder is D = XA, T = XA⁻ᵀ; absorbing one frame line costs exactly +1 cst, a pair +2 (≈ 272k and 951k sampled cases).	ARGUED; census CHECKED (log)
D3	Non-star complex decoders exist: exact cst 63 on all six single-line P111 punctures, cst 56 on P104. Their buildability price is negative (net −0.1 to −10% b).	hits CHECKED (exact); walks CHECKED (log); price RECORDED
D4	Structured classical decoders cost more than coordinate (64): rotations, WHT, halves 72; coset 4-split 192.	CHECKED
S1	In-place elimination is dead under the shipped frame rules: 0 of 12,432 cross-port in-place pairs legal; 33 illegal reads in every row of M (41.2% of the off-diagonal support).	CHECKED
S2	Spec dependencies of the sink wall: the endpoint operator holds on every role, so target sinks cannot be shared; a full sink forces M_in(X) = 0 only for idempotent labels (non-projection labels open).	ARGUED (citations CHECKED)
S3	R2 (one identical exact representative per gate) is specification; relaxing it to a fixed finite set is unsound as a drop-in (h = 3 toy: 4 of 8 addresses moved).	CHECKED (toy)
S4	Centre descents exist only for the 8 barrier centres of the shipped verifier; the "once per word" premise is false (centre −1 = +22.6% b).	ARGUED (file:line CHECKED)
Y1	Centre/cut facts for M = I − G/2 on 112 D₈-root ports: rank G = 8, cut ranks 6/7/7/8, no weight-4 total in the span of the centres; rank 7 occurs at 8 coordinate splits.	CHECKED
Y2	Full centre delivery is frame-illegal (448 of 456 reads); a 7+1 time-split barrier saves 0%.	rank part CHECKED (8 splits); rest ARGUED / RECORDED
F1	Target fragmentation at the shipped op order: legal ceiling 950 of 1096 ops, loose format bound 229 psi/call (HiGHS-proved maximum 36), S3/S4 reader sets admit no modular chain (0/156, 0/112).	CHECKED for the stated parts; rest RECORDED
F2	Tight-frames wall: legal iff G ⊆ U; slack arises only by deleting ops, time shifts, descents or equal-value re-threading; +1 dimension on any helper pushes 2–8 target frames 8 → 9.	ARGUED; instances RECORDED
F3	S1 block scratch cannot be shared or shortened inside the format: 16 distinct sparse forms per block, 0 of 16 equal twins retireable.	RECORDED
F4	Twin substitution for the 73 frame-insulator helpers fails on #656: all 111 twin reads are value-exact, 0 of 111 are legal one at a time (each raises 2–6 final caps), 0 of 73 helpers deletable.	RECORDED (exact oracle, not shipped)
F5	Re-ordering reads of read-only S3 faces fails on #656: for 27 of the 32 structurally clean faces every B slot from the last write to the copy slot (window ≈ 740–860 ops), in given and reversed reader order, gives 0 legal reads.	RECORDED (exact oracle, not shipped)
F6	The opening miner on the rebuilt #664 word finds 2534 disjoint-lifetime helper pairs; all 2534 need frame growth and none is frame-nested, so 0 register-reuse openings: the same rigidity as #656.	CHECKED (log)
O1	Cross-block totals ("X4" faces): 2688 reach-10 quads all span 4 blocks; ledger 12.21% b (hypothetical design) does not close.	faces CHECKED; ledger RECORDED
O2	#617 → #622 K-triangle ledger on the moment engine: ideal gross +0.52% b for 12 helpers, net +0.0146% after 6 lost cold ends and rewired climbs.	RECORDED
App.	Appendix: conditional candidates (5 items, each with one blocking condition; none is a result).	unverified

31 rows by primary status: 21 CHECKED, 6 ARGUED, 4 RECORDED.

Open problems

• Mixed-beta decoders with n >= 9 (D1): a cst below 64 on P112 would have to mix terms with beta <= 7.
• An order-free fragmentation ceiling (F1, F2): the loose order-free bound (2478 psi/call) is uncalibrated.
• Non-projection rational labels (S2) and a frame schedule with |I| > 8 per block (G1, G2).
• Value-exact re-threading beyond the fixed order and DAG (F2).

Appendix

README, "Conditional candidates (unverified: could work IF...)": 5 items, each with one measured fact and one blocking condition (S1 scratch never read, single-packet S4, order-free frame slack, radix-4 first climb, partner-merge of pure-dirt helpers). None is a result. Two earlier candidates were tested and failed (F4, F5).

Update after submission: single-packet S4 (item 2) is answered negatively: 18 of the 28 deferred S4 adds are illegal even when deferred alone, checked on cycles 0–1. The never-read S1 drop (item 1) is mostly answered negatively: given the shipped end-of-A frames, each of the 11 eight-line S1 blocks needs all 16 registers (forced-matching argument). Blocks (0,1,*) and the true-cap case remain open. See the README appendix.

Notes

• Base: cut from main (66849e14); touches only research/format-limits-2/.
• Check: independently run from a fresh git archive of the head commit: 233 PASS in 22 s (Python 3.14.7 and 3.11.17 with the pinned set, 3.14.7 with the latest numpy/scipy); with E1_FULL=1, 228 PASS in 61 s. Tampering with any file stops step 0 with exit 1. Sections 1-4 use the standard library only.
• Inputs: vendored inputs are byte copies pinned by PR blob id and sha256 in SOURCES.json; existing logs and card records are shipped unchanged (six had absolute local paths, a personal name or CRLF endings removed, listed in SOURCES.json). No Lean, SAT, CP-SAT or DRAT artefact is claimed.
• Errata found in our own earlier claims: the "schedule/rho family" of Format limits (no κ claim): certified ceiling κ ≤ 5.1595e-3, spec census, structural lemmas, helper bounds and open problems #595 (census item 8 / open problem 3) (E1); S1's 4164/8960 legal reads (a port-indexing mismatch; the same word gives 5264 with its own ports, 33 illegal reads per row); "rank 8 at every cut" (rank 7 at 8 coordinate splits); "all 56|56 splits" (only the 8 coordinate splits were run); F1's provider words; P3's random-draw rate table (exact means 0.070 / 0.140).
• Overlap: Scoped negative result: asymmetric Hurwitz E8 copy cost is exactly 10 #703, Research: determinant obstruction for rank-homogeneous register paths (no kappa) #705 and Scoped negative result: R420 adjacent A swaps force both frames #711 are related (D1/D3, S2/F2, F2) and duplicate no row; see README Notes. Scoped negative result: selected CK B frames are optimal at fixed boundaries #698 (fixed-order frame optimality) is cited in P5, F1 and F2, which keep only the distinct parts; Scoped negative result: partial K/M face bypass requires four carriers #683 and Scoped negative result: certify one K-triangle relocation neighborhood #687 study different objects and are credited in NOTICE.md.

Credits: see NOTICE.md for every PR whose word, pipeline or result is used, including #531 (icekylinx), #555 (eumemic), #595 and #604 (DreamingOfClouds), #622, #623, #635, #649, #656 (jeremykothe2), #626 (chafreaky), #645 (icekylinx) and #646 (jeremydiviney).

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

🤖 Generated with Claude Code

[Mister-Meeseeks]

100-PR batch review at 4a97676294f9fa413b65986e9984fddeda71e4cc.

This remains open after scoped triage. Next review step: Multi-part format-limit/erratum package requires separate review of checked, argued and recorded claims.

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.