Negative results from a search for a new coding mechanism: m = 2 closed on both sides, flag theorems, closed routes (no new κ)

[eumemic]

Scoping note, no new κ: one README, six standard-library checkers and one data file under
research/negative-results-paradigm-search/. No existing file changes.

Why. Every frontier witness (7.43e-4, #285, at the time of writing) gets its saving from the same label
(index-coding) mechanism, with a per-level coding gain of about 1.003 at m ≈ 110–120. We searched for a different
mechanism that could at least double the saving exponent on both the bit and the complex side. We found none. This PR
records what we tried and why each route fails, so others need not repeat the work. It complements Swapnil Jain's
closed-ideas list and its papers P1 and P2, and notes the overlaps.

Main new results. Each is stated in its model; nothing here bounds integer multiplication itself.

Side	Result	Status
Complex	m = 2 never saves for any frames in GL₄(C). All monomial maps are free (arbitrary phases, non-Clifford, approximate-arithmetic and "y-phase" frames included), gate coefficients are arbitrary complex numbers, and gates may join registers at different frames. Exact FO certificate on Λ²C⁴.	[cert]
Complex	m = 3: no torus-invariant FO certificate can exist in characteristic ≠ 2 (Fano matroid).	[proof]
Complex	m = 3, W = 2 is closed in the dilation normal form for every s ≤ 5: window lemmas plus Gröbner-decided coefficient systems. In the full register model, inserting one free element anywhere still gives no saving.	[proof]/[cert]
Complex	Exact non-Clifford diagonal frames in C[y]/(y^r+1) reduce to Clifford frames plus 2^k-th roots of unity. Approximate arithmetic is legitimate but does not relax the exact group-ring identity.	[proof]/[arg]
Bit	Flag theorem: with any per-role Levi targets (invertible lower triangular, mixed or not), any rational frames and any payload field, W·m − s ≤ Σ_{up frames} (m/2 − rank A). So m = 2 never saves.	[proof]
Bit	Theorem G₀: the flag bound survives mid-network Levi maps, the non-abelian model where every shear is a commutator and all character bounds fail. The proof uses Sylvester rank functions on group rings of solvable quotients; a separate adversarial review found it correct.	[proof]
Bit	Abelian m = 3: s ≥ 2W + 1 for every W (tight-frame lemma).	[proof]
Machine operations outside the frame model	Masked gates reduce to clean ancillas; clean ancillas are blocked; Fourier-duality retirement of the bit primitive is circular; multiplicative index maps obey a valuation lemma; carry-free maps are not linear-time; nonlinear controlled-shift frames keep target cost m and make a nonlinear shear cost 2.	[proof]/[arg]/[cert]
Coding-gap seeds	Per-session sign gauges do not rescue an isometric frame embedding of Braverman–He's PG(2,9) code: the signed incidence system is unbalanced. Cayley seeds have gap ≤ D/|S|. Common-frame networks with s ≤ 8 never save, for any frames.	[cert]/[proof]/[arg]

Section 2 of the README also records earlier routes we tried and discarded on 8–9 October:

• in-class ideas on the paired-cube design: 5-pair ports, register recycling, source-assisted words, nested-chain
  modules, centre restructuring, the bridged schedule and others;
• new-primitive ideas: different-frame gates, characteristic-0 Tanner codes, small instances, centre decoders, more
  than one saved dimension per role, cover structure, cross-level schemes.

Section 3 lists what is still open: complex m = 3 at W = 2 (two or more insertions) and W = 3; bit m = 3 at small W;
the full non-abelian bit model at m = 2; non-smooth nonlinear frames in the register model; and large m, where no
construction exists but the bounds allow up to about m/2.

cd research/negative-results-paradigm-search/checks
python3 -B complex_m2_lambda2_certificate.py   # m = 2 complex certificate, < 1 s
python3 -B multiplicative_index_maps.py
python3 -B fourier_duality_f3.py
python3 -B nonlinear_frames_m1.py              # about 2 s
python3 -B flag_model_cliques.py 2             # (3: about 10 s)
python3 -B bh_sign_gauges.py

Scope.

• The packaged checkers cover the m = 2 complex certificate, the Fourier-duality identity, the multiplicative BFS,
  the nonlinear m = 1 census, the flag-model cliques and the PG(2,9) sign test.
• The flag theorem, Theorem G₀ and the m = 3 lemmas are written proofs. Their large searches (SAT, MITM,
  Gröbner) are reported, not packaged.
• Section 2's rows summarise earlier exploration notes.

Credits.

• Paired cubes: icekylinx (Paired cubes and shared completed cores for conditional kappa = 4.609169e-4 #144). Source-assisted compression: icekylinx (Source-assisted common-frame compression for conditional kappa = 6.566436e-4 #184). Pair donors: ikeboy (Pair donors to avoid parity erasure on the source-assisted v4 word: conditional κ = 6.647872e-4 #193).
• Design ceiling: DaysSky (A κ ceiling of 2.5592e-3 for every word on #144's paired-cube design (no new κ) #201). Structural barriers: huxint (Research: structural barriers for kappa = 0.01 (no new exponent) #203). Lucas-parity obstruction: GamingPuzzled (The paired-cube decoder does not extend to 4-dimensional ports: a Lucas-parity obstruction (no new κ) #231).
• Bridged schedule: J. A. Sussman (Parallel-effort entry: Lean proof of #130's frame lemma and check of #193's complex word, Fourier side (no κ claim) #209). Five-stage suppliers: hcg890 (Conditional κ ≈ 7.037017435 × 10⁻⁴ > 2⁻¹¹: five-stage bit and complex suppliers #234).
• Closed-ideas list and P1/P2: Swapnil Jain.
• The PG(2,9) instance: Braverman and He.

Prepared by eumemic with substantial Anthropic Claude assistance. Apache-2.0.

🤖 Generated with Claude Code

[Mister-Meeseeks]

Thank you @eumemic. This contribution is now retained on main in the scoped 100-PR review and contributor credits.

Contribution: Scoped negative-results and new-mechanism survey.

Review and limits: All six packaged exact checkers freshly passed, including q3 flags. Preserve separately the written arguments, reported unbundled searches and conjectural interface claims; those are not upgraded to independently verified universal barriers.

The complete original source, notices and relevant execution evidence are retained through the pinned source archive. The review distinguishes fresh local runs, source-bound historical Linux CI and prior exact-head review; it does not claim 100 fresh full mathematical audits. Closing this historical PR as reconciled. This preserves parallel and cumulative credit without replacing the selected frontier or promoting priced research targets to completed constructions.

Reviewed head: 475a088b5862f75f0b10596cfd6842d19531a5ec. Publication: 50d768818f579d4e165b4c257c6ee25bca300802.