Research notes (no κ claim): #109 and lower bounds, the coding gap, a Lorentzian minrank obstruction, and a loss chain

[ShikeChen01]

No κ claim. This PR adds only research/lower-bounds-coding-gap/, which contains four notes, their scripts and
DRUP proofs. No supplier, certificate or existing file changes. Based on main at 17ce0c7.

Summary

#	Result	Status
1	#109 contradicts no unconditional lower bound for multiplication. Each bound fails on a stated hypothesis (on-line, bounded coefficients, well-conditioning, or Li–Li).	audit
1	Lemma B′: #109's machine can be made oblivious at the same κ: fixed-width scalars, compute-both branches, and an oblivious Schönhage–Strassen whose log log factor fits the p^κ margin that 08-assembly.tex already reserves. So if #109's Theorem 1 holds, multiplication has bounded-degree Boolean circuits of size O(n(log n)^{1−κ}).	proved, given a procedure-by-procedure audit (included)
1	If #109 holds, on-line multitape multiplication is slower than off-line by Ω((log n)^κ) (Paterson–Fischer–Meyer). So no relaxed variant of #109's transforms can have (log n)^{o(1)} overhead.	proved, given #109
2	The rank saving is a coding gap. On #109's labelled exchange network, weighted by rank or subspace distance, coding achieves rate 1 while flow ≤ (s + 2W)/(Wm + 2W). This is < 1 exactly when s < Wm. It reproduces the paper's η exactly, and it is the same mechanism as dysfunctor's 33-vertex Li–Li counterexample, whose flow we re-certify as exactly 149/150.	proved (analytic LP dual) + exact
3	I_{11,1} and I_{13,1} (triples + pairs) have fitting minrank n, not n − 1, over every field of characteristic ≠ 2. E8 = I_{9,1} is the case n − 9 = 0.	proved; SAT step exhaustive with drat-trim-checked DRUP proofs; exact ranks over Q
4	Loss chain: a ≈ (1/m)(D/W)/L_ent, with L_ent = ln m − E ln r. It reproduces gen4, #493, #501, #589 and the selected #596 supplier to < 10⁻⁸; the #596 row gives main's κ. There is also a piece-count remark: under H1–H6 no label family reaches κ → 1 (copy-free windows give a ≲ 0.31/(h − 1)).	exact; Theorems 1 and 4 proved; H6 assumed

What is new compared with prior work

• Li–Li is already refuted (2026-10-07), by Zhang–B. Li–Z. Li (arXiv 2610.09367, already cited in
  docs/research/stronger-rank-screens.md), by Braverman–He (arXiv 2610.10108), and by
  dysfunctor/network-coding-counterexample, which
  builds its network from Weighted dual-suffix carriers: conditional κ = 7.9961717358e-5 #109's XOR swap. So "Weighted dual-suffix carriers: conditional κ = 7.9961717358e-5 #109 ⇒ ¬Li–Li" (via AFKL) is moot and gives no error signal in either
  direction. Note 1 says this explicitly. What is new is Lemma B′, the circuit corollary and the on-line/off-line
  separation. We found neither in the preprint, the hub (PRs and comments through Independently replay and audit PR627 factored cold ends #633), openai/math, AFKL,
  HvdH 2021/2025 or the public discussion.
• Note 2 generalises dysfunctor's observation from the XOR swap to the rank-weighted motif with any η > 0, as an
  identity. The 33-vertex flow value is an independent re-check, not a new result.
• Note 1's Ailon row cites Prove uniform prime-power cyclic inverse bounds (no κ claim) #621 and Uniform rounded prime-power cyclic inversion (no κ claim) #628 (chafreaky): their length-independent but exponential-in-r conditioning
  bounds for the prime-power cyclic inverses are consistent with the row.
• Note 3 covers a family that Format obstructions (no new κ): exact bounds on window width, copy cost and label dimension #446, Format limits (no κ claim): certified ceiling κ ≤ 5.1595e-3, spec census, structural lemmas, helper bounds and open problems #595 (C.6), Leech/Turyn research (no κ claim): scoped fitting bounds and cost screens #598 and Niemeier research (no κ claim): operator factors and paid spinor circuits #606 (DominikScholz), Research notes (no κ claim): E8 decoder floors are tight; extension-ring, triple and Lorentzian units lose after helpers; cold-end, dispatch and fusion-search negatives #620 (GamingPuzzled; ADE and number-field label screens, with Correct the archived E6 screen with an exact exceptional-root census #630's E6 → D6 correction by vectorsss), hipotures' lattice screen and
  Sussman's notes do not cover. Unlike Format limits (no κ claim): certified ceiling κ ≤ 5.1595e-3, spec census, structural lemmas, helper bounds and open problems #595 C.6, its SAT step ships proof logs.
• Note 4 is complementary to Format limits (no κ claim): certified ceiling κ ≤ 5.1595e-3, spec census, structural lemmas, helper bounds and open problems #595 (DreamingOfClouds) and Correct format ceiling displays and stage endpoint checks #602 (artuskg). Format limits (no κ claim): certified ceiling κ ≤ 5.1595e-3, spec census, structural lemmas, helper bounds and open problems #595's κ ≤ 5.1595·10⁻³ is much tighter
  inside its scope, and its §A.5/§C.7/§E already price helpers, descents and the log-gap. The note claims only the
  four-factor identity and a hypothesis-light piece-count bound. It also lists the route-specific ceilings it sits
  beside: Add an exact κ-ceiling check for the exponent assembly (no new κ) #183, Conditional κ = 0.001274032589159993: complex kernel with paid residual adapters #458, Research: structural barriers for kappa = 0.01 (no new exponent) #203, A κ ceiling of 2.5592e-3 for every word on #144's paired-cube design (no new κ) #201 and Swapnil Jain's P1/P2.
• Dropped as duplicate: our "a saving needs different scalar and label characteristics" lemma (Swapnil Jain's
  closed ideas; Sussman's mod-2 argument), our frozen-word and E8-architecture ceilings (superseded by Format limits (no κ claim): certified ceiling κ ≤ 5.1595e-3, spec census, structural lemmas, helper bounds and open problems #595), and
  our fixed-word retiming optimality results (Research: projection-cut frame optimizer and eight costed structural transfers (no new kappa) #556, Complete binary-frame ceiling for PR551: no next kappa tick with fixed owners #557).

Scope

• Notes 1–2 concern consequences of Weighted dual-suffix carriers: conditional κ = 7.9961717358e-5 #109's Theorem 1 if it holds. They neither support nor refute it.
• Note 3 is a minrank statement about two graphs. It bounds no supplier, and the general odd-n claim is a conjecture.
• Note 4's remark assumes H1–H6. H6 is checked on the certified words but not proved in general, and the remark is
  not a universal ceiling.
• "Proved" means a written argument checked only by its authors. Nothing here is formalised in Lean, and certificate
  success is not a formal verification of the multiplication theorem.

Reproduce

python3 -m pip install -r research/lower-bounds-coding-gap/requirements.txt
DYSFUNCTOR=/path/to/network-coding-counterexample DRAT_TRIM=/path/to/drat-trim \
  bash research/lower-bounds-coding-gap/verify.sh /fresh/out

The run took about 75 s from a fresh venv (Python 3.14.4, WSL Ubuntu) on the rebased branch, and every step printed PASS:

• manifest;
• η exact;
• flow 149/150 exact;
• 10,395 cliques;
• final span 7/10/12;
• σ-model 0/130 violations;
• N = 6, 8, 10, 12 UNSAT with drat-trim s VERIFIED (fresh and shipped proofs);
• loss chain roots;
• piece-count certificates.

Without DYSFUNCTOR the flow check prints SKIP. Without DRAT_TRIM the proofs are written and the CNFs compared
byte-for-byte, but the proofs are not checked.

Attribution and AI disclosure

Prepared by @ShikeChen01 with substantial assistance from Claude Opus 5.5 (Anthropic), via Claude Code. AI agents
drafted the arguments, the audit and the scripts. Apache-2.0. Upstream: OpenAI #109. Prior work used or cited:
dysfunctor; Zhang, B. Li and Z. Li; Braverman and He; Afshani, Freksen, Kamma and Larsen; Harvey and van der Hoeven;
Jacob Sussman (wht-power-saving-lean); Swapnil Jain; hipotures; DreamingOfClouds (#446, #589, #595); artuskg (#602);
DominikScholz (#598, #606); GamingPuzzled (#620); vectorsss (#630); chafreaky (#621, #628); jeremykothe2 (#596); eumemic (#493, #501). Details are in NOTICE.md.

🤖 Generated with Claude Code

[Mister-Meeseeks]

Thank you @ShikeChen01. This contribution is now retained on main in the scoped 100-PR review and contributor credits.

Contribution: Lower-bound, coding-gap, Lorentzian and loss-chain research notes.

Review and limits: Four shipped SAT proofs pass the pinned drat-trim checker; the optional external 33-vertex instance is not replayed. Retain written conditional arguments separately from computational certificates. Oblivious compilation and broad literature implications are not established by a SAT run or profile calculation alone.

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: 189b5771bc1792d5038105463e476e30cb6eb58c. Publication: b3c9e5c269fcfb25dfb4eda4ebc419c052d4c93d.

[philippeVV]

Follow-up and credit: #2838 builds on this PR's Lemma B′ and HvdH Lemma 2.1–2.5 audit. It completes the HvdH §§2–5 obliviousness audit (including a clamp-and-select fix for Lemma 2.2's scaling branch), so no Schönhage–Strassen replacement is needed. It also gives your open "κ ≤ γ" item a quantitative form, via AFKL with flow ≥ coding/γ. Thanks for the groundwork.