Research: delivery lower bounds for #968's frame-pair centers (no new κ)

[JosieElliston]

Research note: scoped negative result for one realization class, plus independent verification. There is no κ claim and no supplier word. Main's selected κ (0.001661960547713947) is unchanged. Only research/biframe-decoder-ladder/ is added.

#968 (themilkman216, merged) explains #717's cheap complex decoders (P111/63, P104/56) as frame-pair decoders and proves the exact 8-term floor. It prices P104's favorable budget at +5.51% b on T37, and says the PR589 bit cap leaves about 1.33% of b. It leaves building the asymmetric centers open. This package bounds that construction cost from below, for one explicit P104 decoder:

• The cut is not common. A P104 center gathers in u_b^⊥ while its readers span u_a^⊥. Theorem C's survivor is therefore ⟨u_a⟩: the cheapest copy tree drops u_b^⊥ → ⟨u_a⟩ at cost 7, and every reader must stand at a frame that contains u_a. Targets read 2–6 mixed centers (12/24/32/24/12 targets), and their cut frames have exactly that dimension. The coordinate decoder reads at frame 0.

• Single barrier cut: ≥ 144 delivery copies. A copy that is never dropped again can only climb, so it serves a chain of nested reader frames. Each center's 52 readers need 52 distinct frames (1059 incomparable pairs). Dilworth then gives ≥ 19 chains per center, so ≥ 144 copies beyond the 8 centers.

• Staggered reads in one global order: ≥ 47 copies. This is exhaustive over all 40,320 orders; the minimum is 55 chains.

• Priced against Research: frame-pair decoders and the exact 8-term complex copy-cost floor (no new kappa) #968's budget at Format limits part 2 (no κ claim): erratum to #595, exact invoice prices, frontier-word censuses, decoder floors, spec requirements, centre and fragmentation walls #717 P3's helper prices:

  delivery	w8	w9	w9 merged	reaches the bit cap?
  single cut (≥ 144 copies)	7.3% b	10.1% b	20.2% b	never
  staggered (≥ 47 copies)	2.4% b	3.3% b	6.6% b	only with width-4-priced copies

  At shipped helper widths the single cut exceeds the 5.51% budget, so a barrier-cut realization cannot help. The staggered costs are inside the budget at no-merge prices, which leaves a window of at most about 3% b. Using that window needs a 104-port skeleton whose copies are exact, restored and frame-legal, at no more than ~79 copies at w9 or ~39 merged.

• Independent re-derivation of Research: frame-pair decoders and the exact 8-term complex copy-cost floor (no new kappa) #968's floor, cst ≥ max(v − 48, 56) for v ≥ 96. The verifier was written independently of Research: frame-pair decoders and the exact 8-term complex copy-cost floor (no new kappa) #968 and adds coverage:

  • all 11 Format limits part 2 (no κ claim): erratum to #595, exact invoice prices, frontier-word censuses, decoder floors, spec requirements, centre and fragmentation walls #717 D3 hit files are shown to be exact bi-frames (Research: frame-pair decoders and the exact 8-term complex copy-cost floor (no new kappa) #968 replays 3);
  • the label-span thresholds are enumerated over all 255 W(E8) support classes directly, with no flat-containment citation;
  • W(E8)-transitivity on the 2025 frames is checked;
  • new: the fully mixed frame pairs are exactly the unimodular root bases whose dual bases are roots (|det| = 1 on all 7168).

  Research: frame-pair decoders and the exact 8-term complex copy-cost floor (no new kappa) #968 additionally proves the bound is attained at every p.

python3 -B research/biframe-decoder-ladder/verify.py            # ~35 s, standard library only
python3 -B research/biframe-decoder-ladder/verify.py --orders   # also re-runs the 40,320-order search (~6 min)

The expected last line is PASS_BIFRAME_DECODER_LADDER: P104 centers need >= 144 delivery copies at a single cut, >= 47 staggered; #968 floor cst >= max(v - 48, 56) re-derived; 11 #717 decoders are exact bi-frames; 10 controls.

Validation

• Fresh git archive replays of this head pass on Python 3.10.21 and 3.14.7, and the receipt is byte-identical on both.
• --orders passed on Python 3.12.14. That run was on the pre-reframe commit, whose order search is unchanged.
• scripts/verify_selected_result.py passes.
• Tampering is rejected: changing a vendored byte or the receipt makes the check fail.
• Controls:
  • negative: a corrupted coefficient, swapped columns, an un-punctured root, a flipped label, tampered bytes, and a frame of the wrong class;
  • sharpness: the 25- and 14-line thresholds are attained;
  • equality: the coordinate decoder costs 64 on P112 and 72 on P120.
• Not run: repository-wide make verify (no existing file changes), hosted CI, and Lean.

Scope

• The delivery bounds hold in the copy-chain model, for one explicit P104 decoder, under a single cut or staggered global phases.
• Not covered: the other mixed P104 decoders, per-target schedules, copies that are dropped again (which add copy rank), E-graph-only exactness, n ≥ 9 terms, |Π| ≥ 25, and the bit side.
• The helper prices are Format limits part 2 (no κ claim): erratum to #595, exact invoice prices, frontier-word censuses, decoder floors, spec requirements, centre and fragmentation walls #717's measurements at Conditional κ = 0.001652528731617094: PR589 bit supplier and dissolved owner map on PR645's gauge44 construction #649, not re-derived on T37. The budget figures are Research: frame-pair decoders and the exact 8-term complex copy-cost floor (no new kappa) #968's; its README is pinned by sha256.
• All inherited conditions remain. This is not a formal proof of the multiplication theorem.

Pins and overlap

• Developed on 66849e1475c930fd7732b1a0067ffee7e1e95f60 and rebased without conflicts onto main 2921874911956718fcaaaf9396fe8267b3cfa9f0. The package shares no files with main.
• Format limits part 2 (no κ claim): erratum to #595, exact invoice prices, frontier-word censuses, decoder floors, spec requirements, centre and fragmentation walls #717's 11 hit files are pinned at 67e9e846 (identical at 07f4671a) by git blob id and sha256. Main's Format limits (no κ claim): certified ceiling κ ≤ 5.1595e-3, spec census, structural lemmas, helper bounds and open problems #595 cert.py/skel.py and Research: frame-pair decoders and the exact 8-term complex copy-cost floor (no new kappa) #968's README are pinned by sha256.
• Assignment seed 47 (structural decoder research). There was no pinned coordination issue, so the claim is recorded locally in CLAIM.md.
• Research: frame-pair decoders and the exact 8-term complex copy-cost floor (no new kappa) #968 drew the same seed and reached the same floor. It was opened at 11:17 EDT and merged at 11:57 EDT on 2026-10-13. This package's floor, census and delivery bounds were committed locally at 01:53 EDT, but submitted only now, so the floor's credit goes to Research: frame-pair decoders and the exact 8-term complex copy-cost floor (no new kappa) #968.
• Overlap was checked through Independent verification: code-disjoint pricing of all 16 pending kappa claims; every one has deficit W - 64N (no new kappa) #1069. Scoped negative result: complex single lifts cannot lower P112 copy cost #998 (complex single lifts on P112) and Finite complex construction: defer historical swaps for 560 further primitives after clear elision (no new kappa) #969 cite Research: frame-pair decoders and the exact 8-term complex copy-cost floor (no new kappa) #968 for other objects, and Helper minimality (no κ claim): the shipped helper layers of #656 / #799 are minimal for their op multiset #899 and Research theorem: common-flag rational-frame obstruction (no new kappa) #825 cite Format limits part 2 (no κ claim): erratum to #595, exact invoice prices, frontier-word censuses, decoder floors, spec requirements, centre and fragmentation walls #717 for other objects. No PR prices the construction of Research: frame-pair decoders and the exact 8-term complex copy-cost floor (no new kappa) #968's centers.

Credit and disclosure. The floor and the family are #968's. The decoders and the P104/56 observation are #717's (DreamingOfClouds), along with the B14 model, the S4 center semantics and the P3 prices. This package also builds on #595 (ceiling code), #446 (Theorem C), #620 (GamingPuzzled) and the E8 complex-unit lineage. It is submitted from @JosieElliston's account; the research, proofs, code and write-up were produced by Claude Opus 5.5 (Anthropic) via Claude Code. Apache-2.0.

🤖 Generated with Claude Code

[Mister-Meeseeks]

100-PR batch review at ce9531beba1ce115fdcdc44925ded995a29b8bac.

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

Reviewed scope: Frame-pair center delivery costs in specified realization classes and independent predecessor-floor reconstruction; no supplier.

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.