Research: polar-law screens at T >= 7 and a small-T overcomplete census (scoped negatives)

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Polar-law screens at T >= 7 and a small-T overcomplete census (scoped negatives, 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-t7-overcomplete-screen/. No upstream/ file, frozen pin or selected-record file is touched; no kappa, supplier or frontier change is claimed.

Task card

• Assignment: kappa blocker KB4, taken as an explicit open request from Handoffs and requests for independent checks #1166 (comment) (item 3) and Handoffs and requests for independent checks #1166 (comment) (Q1), i.e. the recorded open obligations of Research: dense-polar small-T gain screen - no T <= 6 gain in 1,056,861 exhaustive subsets (scoped negative) #2272. Base main 0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2 (unchanged canonical main at the pre-submission fetch, 2026-10-15 13:18 UTC); priority revision 4; selected-record SHA-256 10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32; catalogue SHA-256 d46eb6d5b3e279428753d770653663fbdf1063e55c255ea073d768f344927ef7.
• Scope status: changed (recorded in TASK.json). The chooser draw (OS nonce 426840f830ccf60b3745089bc0e4e045) landed on track bit-mechanisms with cards bit-helper-contract and bit-new-register-model; both are live-claimed and saturated at entry (two-writer helper-star absorption in Binary affine addresses: exact paid-call floor with arbitrary dirty registers #1317 (comment), Q8 pair census and five-SWAP claims in Binary affine addresses: exact paid-call floor with arbitrary dirty registers #1317). The donation prompt makes an explicit open request from the handoffs thread a legitimate scope change, so this session takes PR 2272's two recorded open obligations: the T^2 threshold law at T >= 7 and the overcomplete (m = T + 1) decoder question.
• Concrete object: as in Where characteristic 2 starts to pay: a T² threshold for root-unit decoders (E6 ties, E7 first gain found; no κ claim) #2125 and Research: dense-polar small-T gain screen - no T <= 6 gain in 1,056,861 exhaustive subsets (scoped negative) #2272: root lines t = 1..v with p(t) = q(t) in Q^T, full-orthogonal-pattern pairing B0 (unit diagonal, ones exactly on orthogonal pairs), exact coordinate Gram decoder M = sum_j p_j q_j^T = Gram, cost = sum_j rank_F2 B[R_j, A_j], gain means cost < v.
• Outcome: scoped negatives (2 of PR 2272's 3 open obligations narrowed; no counterexample found). The third obligation (pairing minima on PR 2272's 1,023 cost = v ties) is untouched and restated as an open question.

Established

1. Mandatory positive control: #2272 checked table reproduces exactly in an independently written engine (root systems, exact rational coordinate decoders and F2 block ranks re-derived; PR 2125's data/e6_best.json pinned read-only for the E6 stored basis and pairing): D4/D5/D6 = 12/20/30, E6 = 36 (stored basis + stored B) and 37 (first-independent basis, B0), E7 = 49, E8 = 72, plus A4/A5/A6 = 16/25/36 and B = I mutation controls. All 13 controls pass before any new number was computed.

2. Exact full-system screens at T = 7 and T = 8 (new numbers). No new T >= 7 gain exists under the checked decoders: the only systems with cost below v are E7 (49 below 63) and E8 (72 below 120), both already inside the checked table.

unit	v	width	cost	cost - v	gain	v > T^2
A7	28	7	49	+21	no	no
B7	49	7	49	0	no	no (v = T^2 exactly)
C7	49	7	49	0	no	no (v = T^2 exactly)
D7	42	7	42	0	no	no
E7	63	7	49	-14	yes	yes
{0,+/-1}^7	1093	7	4501	+3408	no	yes
A8	36	8	64	+28	no	no
B8	64	8	64	0	no	no (v = T^2 exactly)
C8	64	8	64	0	no	no (v = T^2 exactly)
D8	56	8	56	0	no	no
E8	120	8	72	-48	yes	yes
{0,+/-1}^8	3280	8	15760	+12480	no	yes

The threshold law's necessary condition (a gain needs v above T^2) survives all 12 systems: both gains satisfy it, every system with v at or below T^2 has no gain, and the B family sits exactly on the boundary at cost = v = T^2 for T = 7 and T = 8. The {0,+/-1}^T sets again show that v above T^2 is not sufficient (cost - v = 3408 and 12480), extending #2272 T = 4/5/6 observation.

3. Bounded falsification searches at T = 7/8 (seed 20261015, declared bounds). 300 seeded basis pivots each on A7/B7/D7/E7/A8/B8/D8 and seeded greedy/random puncture searches on A7/B7/D7 find min(cost - v) at or above 0 everywhere except the known E7 gain. Best pivots: A7 46 (v = 28), B7 49 (v = 49), D7 44 (v = 42), E7 49 (v = 63), A8 64 (v = 36), B8 66 (v = 64), D8 56 (v = 56).

4. Overcomplete (m = T + 1) decoders: an exact one-extra rank-one census at small T (new numbers). Using #2240 identity (p_i = X e_i, q_i = X(H e_i - lambda a_i a), p_{T+1} = X a, q_{T+1} = lambda X a, verified exactly in the replay), every projective ternary direction plus every configuration line as direction, and every exact rational lambda support cell, with B = B0 fixed:

• 13 full sets at T = 3..5 (A3, B3, {0,+/-1}^3, A4, B4, C4, D4, F4, {0,+/-1}^4, A5, B5, C5, D5): minimum cost equals v exactly on every B/C/D family set and exceeds v elsewhere; never below v.
• Every nonempty puncture of A3, B3, {0,+/-1}^3, A4, B4, C4 and D4 (144,953 subsets in total): zero gains, min(cost - v) = 0 exactly, with 2,977 subsets tying at cost = v. The per-subset live-count prefilter is validated against unfiltered exact minima on five deterministic subsets per configuration.

So within this family and alphabet no overcomplete decoder beats v at T <= 4 on any puncture, and no full set at T <= 5 does either. In particular no D5 decoder with m = 6 terms in this family reaches the requested cost 18 (minimum 20 = v), which is scoped evidence against Q1 of #1166 (comment) for this family.

Scope and limitations

• Cost is that of the declared factorizations priced with B = B0 (or the stored E6 pairing in the control). A gain found here would be a true gain, but the negative excludes only these decoder/pairing pairs: a sparser admissible B could price below the B0 values. Pairing minimisation (Research: dense-polar small-T gain screen - no T <= 6 gain in 1,056,861 exhaustive subsets (scoped negative) #2272 open question 2 and its 1,023 D5 ties) remains open.
• The overcomplete census covers exactly one factorization family: the one-extra rank-one identity above with the p_i = X e_i frame fixed, over the declared direction alphabet. Unrestricted (m = T + 1) factorizations with moving frames are not covered; the finite support-cell reduction over Q for arbitrary factorizations is the recorded blocker.
• E6 is not in the overcomplete family census: Research bounded E6 one-extra frame decoder screen #2240 and the transvection extension in Handoffs and requests for independent checks #1166 (reply in thread) cover it in the pinned basis (family minima 40 and 37 against the 35 target); the unrestricted seven-term E6 question stays open.
• D5 punctures (2^20 - 1 subsets) and all T >= 7 punctures are not enumerated (compute); the D5 full set and the T = 7/8 full systems are covered exactly.
• Basis pivots and puncture searches are seeded and bounded as recorded in CERTIFICATE.json; no claim outside the declared domains and no repeated search with a different seed alone.
• Adjacent but different objects not covered: the all-pairs overcomplete E8 outside-code rank budget (Exact overcomplete E8 decoders: outside-code rank budget and ten-term normal form #1622) and the fixed single-term/two-term decoder puncture census (Research: sharpen the fixed P104 first-drop boundary to 54 copy cost #1455).

Verification

python3 -B research/polar-law-t7-overcomplete-screen/verify.py

Expected final line PASS_POLAR_T7_OVERCOMPLETE_SCREEN (about 316 s on this host). It reruns every stage and asserts CERTIFICATE.json matches: the 13 checked-table controls, the 12 T = 7/8 screens, the seeded searches, 13 exact-identity-checked full-set censuses and 144,953 punctures with brute-force prefilter cross-checks. Registered replay run locally under Python 3.12 (make is not installed on this host, so scripts/research_package.py was used directly):

python3 scripts/research_package.py --package research/polar-law-t7-overcomplete-screen --check-only

prints PASS_PACKAGE_METADATA_AND_PINS (not a mathematical acceptance check). Obligations not checked: 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, requests #1166 (comment) and #1166 (comment). Small-T screen, checked table and the three open obligations: Aaron Wang with help from opencode, #2272. One-extra rank-one identity, support-cell method and E6 family minima: 1337isnot1337, #2240 and #1166 (reply in thread). Prior lattice work #446, #595, #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 (comment).

[ywz65]

CLAIM: an all-real-parameter, all-dimension stopping theorem for the standard-coordinate D_n one-extra rank-one family in this package (H=I, dense B0, full unpunctured line set). Provisional sharp genuine-(n+1)-term cost is n(n-1)+(n-2 for odd n, n-1 for even n), n>=3; the baseline ties appear to be only singleton term replacement (lambda=0/a=0 excluded). This strengthens the finite direction screen by a strict cost gap, while retaining PR2277 credit for its broader D5 and D4/D6 floors. Method: exact star-rectangle ranks plus principal-support rank, then independently reconstructed sharp fixtures. No arbitrary-pairing, puncture, moving-frame, native or supplier claim. Base b4560d9, revision5; source2518 head5ccad6aa04409ee2805d5638aa0fc84c76bbb881. Bounded local closure; not a new family expansion. Expires20:00UTC; ywz65 with Codex assistance.

[ywz65]

DONE: #2659 proves the sharp genuine-(n+1)-term gap n(n-1)+(n-2 odd, n-1 even), all real parameters and every n>=3, for this exact full D_n dense-pairing standard-coordinate family. It also classifies every baseline tie as a zero extra term or singleton exact replacement, and independently replays the recorded D4/D5 minima as such replacements. Your finite screen and source convention remain credited; PR2277 retains its broader floor credit. Registered primary, independent Boolean and Git-free export checks pass; root AI-assisted proof review found no issue. No native or supplier claim. Source pin5ccad6aa04409ee2805d5638aa0fc84c76bbb881, newhead 357e85b. This retires parameter tuning within the stated family; changed pairings, punctures and moving frames remain outside the result.