Research exact p=25 root-pair decoder boundary

[1337isnot1337]

Change and mathematical scope

Adds research/p25-root-pair-bound-seed632639/, a scoped exact obstruction for eight-term, all-pairs decoders whose row and column normals are E8 roots, at exactly 25 punctures. Each root support has 63 distinct extremal 25-line masks that first lower its live label span to dimension seven. Across the 120 root directions, all 7,560 masks are distinct, so one puncture set can lower at most one root-normal support.

The rank inequality rank(X,Y) >= dim(X)+dim(Y)-9 then gives a total separate COPY bill of at least 54: at most one term can use the exceptional support, costing at least five even when the same root appears on both sides of that term; the other seven terms cost at least seven each. This is a new p=25 boundary result beyond the p<=24 floor. It is not a decoder supplier or a new exponent.

At 95 live ports, the best compatible center profile (5,7,7,7,7,7,7,7) has exact pinned ceiling interval [0.001538546310035, 0.001538546310036], 8.965% below the selected local-word ceiling. Added setup, frames, transitions, hosting, precision and restoration are all set to zero. The result closes this root-pair route against the 5% task gate under the unchanged coverage law; arbitrary dual bases, masked identities and nine or more terms remain outside scope.

Task and overlap

• Exact base: 0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2.
• Track/card: complex-mechanisms / complex-rank-shape-decoder; seed 632639f1d387026a8b953fa82c286d82; selector scope key 02b31ebd160542c86299; priority revision 4; catalogue SHA-256 d46eb6d5b3e279428753d770653663fbdf1063e55c255ea073d768f344927ef7.
• Selected-record SHA-256 10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32; certificate SHA-256 1122cb3a575f8c788345fc1bf4c108ac7c3250fb81a5988671f5eb04b3ab3045.
• Scope status: changed from block-frame climbs to the exact p=25 support boundary in the root-pair dual-basis family. The new parameter boundary is the first one beyond the p<=24 theorem and directly tests whether lower copy rank can change the paid local profile.
• Predecessors: PR #717, PR #968, PR #1619, and PR #1624. PR Research: fixed paired26 completion minimum 62 with exact certificates #1624 treats one fixed p=26 object; this package covers all p=25 masks for the root-pair family.
• Active neighboring work checked: Discussion #1622 studies ten-term P112 gauges; Discussion #1952 studies cyclic masked rank-three profiles; Discussion #2119 studies rank-six phase schedules. Open PR #2232, head 3391f27dd0f226f0e1ead8b2b829eb6d3ec567d3, screens four-term E6 projected-Gram decoders with an adapted pairing; its 36 E6 lines and projection model differ from this E8 p=25 family. Open PR #2228, head ad1757e319225f3a5b86e3d3fa72897bd9f6db40, provides a broader dense root-code floor and lists E8, P112 and P104 cases. This package supplies the exact p=25 extremal-mask census and an eight-term dual-root bound; it does not claim to exceed that broader theorem.
• Claim and result: Discussion #1167 comment 18937692.

Validation

Reproduce from repository root:

python3 -I -B research/p25-root-pair-bound-seed632639/verify.py
python3 -I -O -B research/p25-root-pair-bound-seed632639/verify.py
python3 scripts/research_package.py --package research/p25-root-pair-bound-seed632639 --check-only
python3.12 scripts/research_package.py --package research/p25-root-pair-bound-seed632639

Both exact verifier modes pass under CPython 3.14.7; the metadata and source-pin check passes. The registered replay could not run locally because Python 3.12 is not installed; GitHub CI will run the declared Python 3.12 commands. The source manifest pins the base, task catalogue, selected result, review, source manifest, certificate, predecessor model, and ceiling engine. The selected record's reproduction command remains make frontier1094-verify; it was not rerun because this PR does not modify that source chain.

The finite checker reconstructs the 120 root lines and rank-nine label pairing, tests all 130,305 functional pairs for every root support, verifies 7,560 distinct extremal masks and their exact residual ranks, checks the 24-puncture and second-root controls, and recomputes the rational ceiling. The written argument in the README derives the decoder COPY floor from the exact support census and the binary rank inequality. This is not a formal proof of the full multiplication theorem.

Credit

Retains the PR717/PR968 E8 model and p<=24 theorem, PR1619's favorable port/copy price, the selected #1094 source chain, and PR1624's distinct fixed p=26 work. All are pinned in SOURCES.json and the task card. New work is Apache-2.0. Prepared by Cinna Davis with substantial OpenAI Codex assistance in scope selection, exact enumeration, proof development, pricing, verifier implementation and writing; no independent human or proof-assistant review is claimed.

Open next questions

Can a non-root dual basis or an overcomplete/masked decoder at p=25 avoid the one-exceptional-root restriction and pass the fully charged local moment gate? The root-pair result does not cover those alphabets. The smallest next step is to choose one explicit non-root support alphabet, derive its exact support-rank floor and screen the complete local profile before synthesis. A source-aligned frame and full setup/hosting/precision/restoration invoice are currently missing.

[DreamingOfClouds]

A related data point on the open question (non-root dual bases at p = 25), in case it saves someone a screen.

We ran an exhaustive check of the same eight-term, all-pairs E8 model with arbitrary biorthogonal normals (x_i·y_j = 0 for i ≠ j), not only roots, over the whole window 25 ≤ p ≤ 44. The comparison point was P102's ratio 54/102 rather than the local-word ceiling. Result: for every puncture set S in that window, cost ≥ (54/102)(120 − |S|), so no strict win in this class.

How it goes, briefly:

• The per-term bound is the same rank inequality you use: r_j ≥ 9 − d(x_j) − d(y_j).
• Each side's deficiency is bounded by its parabolic flat. All 440,880 hyperplane flats of E8 (8 Weyl types) are enumerated.
• Only root flats reach deficiency 2 with at most 44 punctures (25 lines for d = 2, 37 for d = 3). D7 flats need exactly 42, with at most one extra root kernel nearby. The other six types need more than 44. This matches your 25-line extremal masks.
• An exact search over families of root kernels (union ≤ 44, W(E8)-reduced) closes every p except 36 and 44. There, a biorthogonality refinement on the 1,323 remaining unions caps the total deficiency at 22 < 28 and 24 < 32.

Scope and limits:

• Covered: dual-basis / biorthogonal eight-term decoders in this model.
• Not covered: the bit model, nine or more terms, and non-diagonal N. Without biorthogonality the relaxation leaves very thin margins: cost 44 against 44.47 at p = 36, and 40 against 40.24 at p = 44. So coupled or overcomplete decoders at p = 36 or 44 are the place your open question still has room.
• Beating only P104 (x between .5294 and .5385) is not covered either.
• This is an unaudited computation, not a registered package.

Scripts and notes: https://github.com/DreamingOfClouds/integer-mult-bounds/tree/f8136a7c0e7c148eacf6f2d04e1dd02769f126b8/research/notes/punct2-window (NOTES.md has the argument and the holes).

Prepared by DreamingOfClouds with Claude Opus 5.5 (Anthropic), via Claude Code; Apache-2.0.