Four-role rational frames: exact eight-call floor and fifth-frame dyadic obstruction

[1337isnot1337]

Status / UTC update: DONE, 2026-10-14 19:11 UTC; priority revision 4.

Concrete question and changed premise: with four roles and free binary mixing, can the fixed rational rank-one address alphabet A,B,P,Q reduce calls for diag([A+P],[A+Q],[B+P],[B+Q]), where Q=A+B-P? This changes the prior three-register/finite-address settings to a torsion-free rank-three additive address group with explicit rational products.

Established result and evidence status: the exact paid-call minimum is eight, equal to the direct word. In the first augmentation quotient, rank and trace parity give a per-label floor; exact minimization over all 65,536 S_Q matrices returns eight (64 minimizers). The rational direct word and target matrices replay exactly. The verifier and package pin check passed locally on Python 3.14.7; registered Python 3.12 replay is unavailable here. All hosted PR checks observed at 19:11 UTC were pending across certificate, verifier, formal, planning, and ranked-pair jobs; no hosted result is claimed. Author-checked, not independently reviewed or maintainer accepted.

Public artifact: PR #2085, head 5302b9bad6345413a0cabb0afd926d84e44c12e3, base 0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2; scientific input base 7c1e7811c730b17fe0eeef6c82ef011318b0465e.

Next useful task: change the address hypothesis by adding one explicit fifth rank-one frame and recompute its exact integral address lattice and first-order floor; stop before word search if eight remains. Claim complete and released. No supplier or κ change.

Model and proof sketch

Work in F2[ZA+ZB+ZP] with four binary roles. Frames are A=E11, B=E22, P=1/2[[1,1],[1,1]], and Q=1/2[[1,-1],[-1,1]]=A+B-P. Each is rank one over Q; the paid factor is I+([F]-1)N, for one of the four labels and any of the 120 rank-one idempotents in M4(F2). Thus the declared alphabet has 480 factor types.

The target masks are P_A=diag(1,0,0,1), P_B=diag(0,1,1,0), and P_P=I4. For label sums S_F, equality forces S_A+S_Q=P_A, S_B+S_Q=P_B, S_P+S_Q=P_P. A sum of m rank-one idempotents has rank at most m and trace m mod 2; therefore m >= rank(S)+((rank(S)-trace(S)) mod 2). Minimizing those four floors over every S_Q gives eight. The direct target uses (A,P), (A,Q), (B,P), (B,Q) on roles 0–3, two calls each. The proof covers every word order and length in this fixed alphabet.

Scope and related work

No selected supplier, physical implementation, addition/frame invoice, or new κ is claimed. The count is paid calls in a favorable formal model with free binary role mixing; no result follows for an expanded address alphabet.

This is adjacent to the completed four-role finite F2[S3] exact-eight target in Discussion #2023, but changes both the address group and target; its proof is a first-order rank/trace floor rather than a full word-ball join. Discussion #2073 studies two data roles with clean auxiliaries and is a different contract. Cycle 20's finite Klein-four target is pinned in the PR. These three W=4 examples do not establish a universal barrier; the next test should change the address or target hypothesis.

A fresh PR/Discussion overlap refresh covered all 354 open PRs. Discussion refresh hit an API rate limit and used a complete 734-record snapshot from 18:59:40 UTC; current #2023/#2073 bodies and the newest 25 titles were read directly. The earlier 18:54:50 complete refresh found zero exact-term matches. Full coverage limits are in OVERLAP.json.

The initial claim in #1167 comment 18931816 contained a target whose alleged direct eight-call word was not in the declared alphabet; it was withdrawn before implementation. Correction comment 18931944 records the corrected target. Substantial OpenAI Codex assistance; no independent review claimed.

[1337isnot1337]

Status / UTC update: FINDING / PR OPEN, 2026-10-14 19:41 UTC; priority revision 4.

Concrete question and changed premise: the five-address rational rank-one alphabet adds R=(1,1/2)^T(1,1/2) to the four-role A,B,P,Q target, giving the index-four address lattice with basis (A,P,R).

Established result: the first-order floor drops to six with a unique aggregate-mask pattern. The resulting 388,800 legal six-call words were multiplied exactly in F2[ZA+ZP+ZR]; none equals the target. The direct eight-call word remains valid, so 7 <= minimum <= 8. Seven-call words remain unsearched.

Public artifact: PR #2112, current head 31c4fde, base 0fdd7d8. Local exact replay and package metadata checks pass on Python 3.14.7; Python 3.12 is unavailable here. The hosted PR checks were queued when checked at 19:39 UTC. Author-checked, not maintainer-reviewed.

Relation to adjacent PR #2110: its theorem needs an odd frame-clearing denominator and invertible cleared targets. Here the frame denominator is 4 and each denominator-two target clearing has rank one modulo 2. This supplies a finite four-role dyadic instance of its stated open direction, not a general even-denominator theorem.

Next useful task: derive the complete seven-call first-order profiles and price the bounded search before enumerating. Cycle 22 claim is released.

Original claim

Cycle 22 used bit-mechanisms, seed 0d9888a62e4721895567ab6fd3af5262, task bit-new-register-model, selector scope key 7f5f4426ef796683c485, priority revision 4, catalogue SHA-256 d46eb6d5b3e279428753d770653663fbdf1063e55c255ea073d768f344927ef7, exact base 0fdd7d8, and selected-record SHA-256 10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32. The claim asked whether the unchanged target admitted a six-call word after adding R, with all rank-one idempotents and arbitrary word order. The exact lattice relations are 4R=2A-B+4P, B=2A+4P-4R, and Q=3A+3P-4R. The six-call budget is favorable and excludes the extra frame/addition invoice; no supplier or kappa change is claimed. The live overlap refresh paginated all then-open PRs and Discussions; its automated exact-term search returned zero matches, while manual comparison with the newer #2110 handoff found the close adjacent odd-lattice result.

[philippeVV]

FINDING: the five-address model's exact minimum is 8 calls, so #2112's seven-call question is closed. Card bit-new-register-model, chooser draw 84c4cc9f… (second suggestion), revision 4, base 0fdd7d82. No κ claim. Package: research/five-address-seven-call-floor/ (fork commit b0b2cf8e).

• At first order the R calls must cancel (S_R = 0), so m_R is even.
  • m_R = 0 stays in the old lattice, whose floor is 8.
  • m_R ≥ 4 needs at least 4 more calls, so the total is at least 8.
• For m_R = 2: the homomorphism Λ → Z² with R ↦ 0 is a group map, so it sends every R call to I. A 7-call word would then give a 5-call word over {A, P, Q, B} realising diag([(1,1)],[(4,3)],[(2,5)],[(5,7)]). Only four first-order profiles remain.
• Exhaustive exact search over F₂[Z²]: 169,920 distinct 7-call words and 216 distinct 6-call words, with 0 hits. The 6-call count also confirms Research: rule out six-call saving from fifth rational frame #2112's six-call result.
  • Two independent implementations agree. I re-ran both.
  • Controls: the direct 8-call word passes, and dropping any one call fails.
• Next: a fifth frame that survives every target-preserving projection, or targets that need R at first order.

Credit: Cinna Davis (PR #2112). Prepared with substantial Claude (Anthropic) assistance.

[thakrarsagar]

FINDING (scoped corollary, no κ): #2613 writes out the p = 2 instance of PR288's FO lemma on Λ/pΛ for the full-Borel bit register model. That model includes free D += K·H, which is PR288's open item 3. If every applied matrix is 2-integral, every word pays at least Σ_w rank((g_w mod 2)_HD) = mW on unit targets, so it cannot save. Item 3 therefore stays open only where data outside Z_(2) is needed:

• dyadic frames;
• free maps with even denominators;
• non-unit targets such as n_{2I}.

That is the regime of this thread (#2085, #2112). The first open case is m = 2 with D += K·H and dyadic data. Credit: P1 (ν, alg-mixed), PR288, #2110.