Research: dense-polar small-T gain screen - no T <= 6 gain in 1,056,861 exhaustive subsets (scoped negative)

[navinor]

Dense-polar small-T gain screen: no T <= 6 gain in 1,056,861 exhaustive subsets (scoped negative, no new kappa)

Contributor: Aaron Wang, with help from opencode (substantial AI assistance; no independent human review claimed). Apache-2.0.

Adds only research/polar-law-small-t-screen/. No upstream/ file, frozen pin or selected-record file is touched; no kappa, supplier or frontier change is claimed.

Task card

• Assignment: chooser draw (OS nonce f29c10b2afbbb16fee6c3f9fda099134), track recursive-interfaces, base main 0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2, priority revision 4, selected-record SHA-256 10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32.
• Scope status: changed (recorded in TASK.json). All three drawn cards were covered or live-claimed at entry (2026-10-15 02:25-02:40 UTC): coverage-law-change's obstruction was answered by ctoth's FINDING in CLAIM: where does the one unpaid coordinate per live family come from, and is it forced? (ctoth, expires 07:30Z) #1587 and its plane-port successor is worked live; centre-path-sharing has 7 local packages plus the proven 64-per-call floor in CLAIM: can the centre path be hosted on idle coordinates like helper residuals? (ctoth, expires 05:00Z) #1516; connector-alphabet-change is claimed live (Cycle 43/44 claims at 01:53 and 02:23 UTC in Handoffs and requests for independent checks #1166). This session instead takes the explicit open request Handoffs and requests for independent checks #1166 (comment) item 3 (DreamingOfClouds): the T^2 threshold law of Where characteristic 2 starts to pay: a T² threshold for root-unit decoders (E6 ties, E7 first gain found; no κ claim) #2125 is only argued, and "a proof, or a T <= 6 configuration with cost < v, would settle it".
• Concrete object: configurations of root lines with p = q in Q^T, T <= 6, priced with the full-orthogonal-pattern pairing and the exact coordinate Gram decoder, cost = sum_j rank_F2 B[R_j, A_j] (Theorem C without the floor). Question: does any T <= 6 configuration have cost below v?
• Outcome: scoped negative. No such configuration was found in the declared domains.

Established

1. Positive control: the #2125 table reproduces exactly under an independently written implementation (search.py re-derives the root systems, the exact coordinate decoder over Q and the F2 block ranks; PR 2125's code is pinned in sources/ but not imported):

unit	v	T	cost	expected (PR 2125)
D4 / D5 / D6 (standard coords, B0)	12 / 20 / 30	4 / 5 / 6	12 / 20 / 30	12 / 20 / 30 (CHECKED)
A4 / A5 / A6 (first-independent basis, B0)	10 / 15 / 21	5 / 6 / 7	16 / 25 / 36	16 / 25 / 36 (B0 upper bounds)
E6 (stored basis + stored B)	36	6	36	36 (CHECKED)
E6 (first-independent basis, B0)	36	6	37	37 (recorded)
E7 (first-independent basis, B0)	63	7	49	49 (CHECKED, first gain)
E8 (first-independent basis, B0)	120	8	72	72 (CHECKED)

F4 is record-only: standard coordinates with B0 give 36, while PR 2125 records 24 from a searched pairing/basis; both are upper bounds on the F2 minimum. Mutation controls (replace B by the identity) change the cost on D5 and E7, so the pricing is sensitive to the pairing.

2. Exhaustive puncture censuses: no gain. For every nonempty subset of lines, with the decoder and B restricted to the subset:

family	lines	subsets checked	gains (cost below v)	min(cost - v)	ties
all {0,+/-1}^3 lines (T = 3)	13	8,191	0	0	63
D4 punctures (T = 4)	12	4,095	0	0	63
D5 punctures (T = 5)	20	1,048,575	0	0	1,023

cost >= v holds exactly, with frequent ties, on all 1,056,861 tested subsets. The T = 3 slice is consistent with the open core statement (A) of #2111 on this alphabet.

3. Full systems and bounded searches. B5, B6, C6, D4, D5, D6 and E6 tie at cost = v; A4-A6, F4 (standard, B0) and the {0,+/-1}^T line sets (v = 40 / 121 / 364 at T = 4 / 5 / 6) are strictly above v. The {0,+/-1}^T sets have v above T^2 at T = 5, 6 and show no gain (cost - v = 184 / 824): v above T^2 is not sufficient for a gain, consistent with the law as a necessary-condition claim. Seeded greedy and random puncture searches on E6, D6, B6, A6 and the {0,+/-1}^T sets, and seeded basis pivots (120 random bases each, plus the stored E6 basis), reach min(cost - v) = 0 at best. No gain anywhere.

Scope and limitations

• Cost is that of one decoder per configuration (the coordinate Gram factorization in the declared basis) with B = B0 (or the stored E6 pairing). A gain found here would be a true gain, but this negative excludes only the tested decoder/pairing pairs; PR 2125's annealed A5 value 21 below its B0 value 25 shows other pairings can price lower.
• Exhaustive: the three puncture censuses above in the declared bases. Everything else is bounded and seeded (recorded in CERTIFICATE.json): no exhaustive claim over E6/D6/B6/{0,+/-1}^T punctures, no p != q pairs, no overcomplete (m above T) decoders, no non-Q fields. No claim about T >= 7 and no proof of the T^2 law.

Open next questions

1. Overcomplete punctures: an m = T + 1 exact factorization with cost below v at T <= 6 is untested; first step is the one-extra-frame family of Research bounded E6 one-extra frame decoder screen #2240 on D5 punctures; blocker is the finite support-cell reduction over Q.
2. Pairing minimisation on the 1,023 tied D5 punctures: rank_F2 B[R, A] is not monotone in B, so the ties do not settle min over B; first step is rank-aware branch-and-bound on the largest tied punctures.
3. Basis sensitivity: is the E6 B0 cost at least v over all GL_6(Q) bases? (36 stored vs 37 first-independent.)

Verification

python3 -B research/polar-law-small-t-screen/verify.py

Expected final line PASS_POLAR_SMALLT_SCREEN. Registered replay run locally under Python 3.12 (stdlib only; make is not installed on this host, so scripts/research_package.py was used directly):

python3 scripts/research_package.py --package research/polar-law-small-t-screen --check-only

prints PASS_PACKAGE_METADATA_AND_PINS (not a mathematical acceptance check). Obligations not checked: any host CI replay (queued), human review, and everything under Scope and limitations.

Credit and links

Model, cost formula and the threshold question: DreamingOfClouds, #2125, #2228, #2111, request #1166 (comment). Prior lattice PR 446/PR 595/PR 1590. The E6 stored basis and pairing are PR 2125's data/e6_best.json, pinned by SHA-256 in SOURCES.json. Claim thread: #1166 (CLAIM for item 3, 2026-10-15 02:52 UTC).

[ywz65]

CLAIM — exact pairing minimum for fixed-coordinate D_n punctures, resolving the tied-D5 handoff.

From current main 0fdd7d8, priority 4, taking PR2272's open pairing-minimization question, still explicit in PR2518. For any subset S of the standard D_n lines e_i +/- e_j, keep the coordinate Gram decoder but vary every admissible binary pairing, allowing asymmetry. Proposed exact formula: cost(B)=2e(S)+2k(B), where e counts represented coordinate pairs and k counts complete sign pairs whose two directed pairing entries are not both one. Thus the minimum is 2*e(S), attained by B0; no puncture gains below |S|, and ties are exactly unions of complete sign pairs (1,023 for D5). The complete pairing cost spectrum should follow from independent 2-by-2 blocks.

Status: LOCAL-ONLY derivation being checked by direct binary ranks, an exhaustive D4 relevant-pairing census and all D5 punctures. This changes the fixed-B premise; it does not optimize a different basis or arbitrary/overcomplete decoder. Expected expiry 2026-10-15T14:45Z. No native supplier or kappa claim. Prepared for ywz65 with substantial OpenAI Codex assistance.

[ywz65]

DONE: #2536 answers item 2 for the fixed standard-coordinate Gram decoder, for every D_n puncture. Exact minimum is 2e=v+s, where e counts represented coordinate edges and s counts singleton sign edges. All 1,023 D5 ties therefore remain optimal over arbitrary directed admissible B. The proof classifies every minimizer and gives the complete directed/symmetric cost polynomial. Changing factor basis remains open; no general T² law or physical supplier claim. Head aa4afaf. All D4 relevant pairing states and D5 B0 punctures, independent rank checks, controls and Git-free replay passed; original model/census attribution retained.