Research: frame-pair decoders and the exact 8-term complex copy-cost floor (no new kappa)

[themilkman216]

Research note / scoped structural result; no new κ or supplier. The selected κ = 0.001661960547713947 (PR942, T37), the existing generators and upstream/ are unchanged. Only research/complex-frame-pair-decoders/ is added.

The binding complex supplier's copied-information cost is its decoder constant: 64 on 112 ports, giving the deficit 37·(8·112 − 9·64). PR717 (DreamingOfClouds) used random walks to find non-star complex decoders of cost 63 on P111 and 56 on one P104. This package explains, bounds and prices them, in PR717's own B14 model, for exact 8-term decoders (D = RA, T = R·2A^{-T}):

1. Mechanism. A term whose normals are roots α and β with ⟨α,β⟩ odd and both lines punctured costs exactly 7. If either line is live, or α = β, it costs 8. The reason: the labels of N(α) span u_α^⊥ ⊂ F2^9, and the pairing H_α × H_β has rank 8 − [⟨α,β⟩ odd]. All three PR717 hits have exactly this structure, and their per-term costs match PR717's recorded values.

2. Family. Frame-pair decoders on P120 − F0 − F1 have cost 56. Of the 1260 frames F1 inside P112, 896 admit them (exactly 8 each; all 7168 are checked). The other 364 admit none. PR717's random clique is one of the 896.

3. Theorem. This is a finite exhaustive census plus a written proof. For every puncture of p ≤ 24 lines, every exact 8-term decoder has cst ≥ 72 − p (p ≤ 15) and cst ≥ 56. Both bounds are attained at every p, so cst/v ≥ 7/13, with equality only at p = 16 (the shipped unit is 4/7). The census covers:

   • all 8 W(E8) hyperplane-flat types × 511 hyperplanes and 130,305 codimension-2 pairs. A root support drops to 8 dimensions with one puncture (its own line); every other type needs ≥ 14; any 7-dimensional span needs ≥ 25.
   • the 1080-direction D7 orbit, enumerated in full.

4. Favorable budget. PR717's P1 knob model is transported to the selected T37 profile and priced with the same hash-pinned engines. The gate reproduces the selected b and κ exactly.

   • P111: b +0.67%, κ 0.001673104139787043.
   • P104: b +5.51%, which exceeds the bit supplier. κ is capped by PR589 at 0.001730427469823602 (+4.12%), and that cap applies to any complex-only change.

   So any realization of the asymmetric centres must cost less than the 5.51% b gain to help at all, and less than about 1.3% of b to reach the cap.

Deeper-puncture searches for exact 8-term complex decoders (p ≤ 24) are therefore closed. Still open:

• decoders with n ≥ 9 terms, including PR717's mixed-β problem;
• decoders exact only on the E-graph;
• punctures with p > 24;
• building the asymmetric centres. PR717 recorded, without shipping, a negative price for this.

Reproduce

Python 3.10+, standard library only, about 30 s, from the repository root:

python3 -I -B research/complex-frame-pair-decoders/verify.py

Expected: 48 PASS lines, no FAIL, exit 0, and finally
PASS_COMPLEX_FRAME_PAIR_DECODERS: types 8; D7 orbit 1080; frames 1260 (admissible 896); chain p=0..24; P104 7/13; bit cap kappa 0.001730427469823602; controls 6.

Negative controls:

• a mutated decoder loses exactness;
• restoring one self-line gives 57 = 72 − 15;
• the even coordinate decoder on the same P104 stays at 64;
• the 364 inadmissible frames admit no decoder;
• swapped partners break duality;
• a 25-line set does reach a 7-dimensional span, so the threshold is sharp;
• a tampered vendored file fails both its pin and the cost cross-check (checked separately).

Validation and scope

• Run and passed:
  • verify.py: 48 PASS, on Windows 11 with Python 3.11.9, under -I -B.
  • scripts/verify_selected_result.py, which is make selected-record-check.
  • git diff --check.
• Not run:
  • repository-wide make verify;
  • the selected cold replay, which is not affected because no supplier or assembly changes;
  • Lean;
  • hosted CI.
• Cited, not recomputed: the standard fact that every hyperplane flat of the E8 arrangement is W-conjugate to one of 8 standard flats (Bourbaki, Ch. V). The only non-root type below the thresholds is D7, and its orbit is enumerated directly.
• Pricing scope: the knob pricing is a favorable budget, not a construction. The model removes a port as one modal source and target climb per call plus nine idle rank-48 children per replica, exactly as in PR717 P1.
• Not a formal proof: the finite checks are not a formal proof of the multiplication theorem. The written proof has not had independent human review.

Provenance and credit

• Base and assignment. Base bbc600db2fc15007b7c914f4f2b2ed55fa909c87. The work started at 47481c89 with T21 selected and was rebased when T37 was published; the theorem is unchanged. Sprint track 40–49 (structural decoder research), seed 47 from OS randomness. No pinned coordination issue exists, so the claim and its result are recorded in CLAIM.md.
• Predecessors:
  • DreamingOfClouds: PR717 (B14 model, D1–D3, P1 knobs). Three decoder files are vendored with blob and SHA-256 pins, under Apache-2.0. Also PR595 (format limits, the (Dfc) identity).
  • Joel Pulikkan / GamingPuzzled: PR620 (complex frame decoder of cost 64).
  • PR845: the one-added-total exclusion.
  • icekylinx: the P112 puncture.
  • The selected chain: Jonathan Capalbo (PR942), Chafik Boukhalfa (PR890), Jeremy Kothe (PR869/870), Hao Yan (PR858), DreamingOfClouds (PR589) and eumemic (PR555).
• Overlap check. Checked through Research: exact prescribed three-target deletion thresholds (no new kappa) #966. Prove E8 star-decoder puncture thresholds through 35 deleted lines #631, Scoped negative result: algebraic P111 bit pairing-cost floor of 63 #761, Scoped negative result: classify the E8 rank-five boundary at 36 punctures #875, Scoped negative result: E8 star-copy ratio cannot improve through 55 punctures (no new kappa) #939 and Scoped negative result: classify E8's first decoder rank drop (no new kappa) #954 concern the bit-side star model. Scoped negative result: Gaussian E8 copy floor 40 excludes three stages #738, Research: Gaussian E8 weighted maximal-clique ceiling 40 and independent replay #770, Research note: Eisenstein E8 copy cost ≥22 and sharp clique-method limit #798, Research: Gaussian E8 all-clique certificate ceiling is 40 (no new kappa) #868 and Research: Gaussian nonlinear message copy floor 40b under explicit width pricing #888 concern Gaussian/Eisenstein units. Scoped negative result: five-port quadratic decoder cost ceiling (no new kappa) #958 concerns a five-port XOR decoder. None addresses exact 8-term complex decoders under punctures.
• AI assistance. Prepared with substantial assistance from Claude Opus 5.5 (Anthropic) via Claude Code: research, proof, implementation and documentation. Apache-2.0.

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