Research: exact star-decoder hierarchies, target budgets, and COPY bounds

[the-missionary-company]

This research contribution determines sharp information thresholds for signed proper-star decoders and proves physical COPY obstructions for explicitly defined classes. It adds uniform proofs, exact prescribed-target criteria, and independent finite certificates. There is no new multiplication coefficient or claimed physical supplier.

For q=p-k>=3, the first five generalized weights are now exactly
(2q+2,4q+3,6q+3,8q+2,10q). At the boundary q=2,k>=4, an affine cube gives (6,11,14,15,16): three, four, or five selectable targets all require eleven deletions. These deletion counts measure information recovery; the physical notes separately charge copied frame dimensions and inherited endpoint schedules.

Results and review entry points

Only research/asymmetric-hypercube-decoder/ changes. Start with its README.

• Uniform higher weights: HIGHER-WEIGHTS.md, FOURTH-WEIGHT.md, and FIFTH-WEIGHT.md prove the first five weights for every k>=2,q>=3. Every second-weight word is a sum of two adjacent minimum stars. Every rank-three and rank-four optimum is a star clique; every rank-five optimum is a five-star clique for q>=4. The notes give exact counts and prescribed-target independence criteria. The q=3 rank-five equality classification remains open.
• Exceptional cube geometry: CUBE-OPTIMIZERS.md proves exact ordinary-cube shortening and classifies all rank-three through rank-five optima for q=2,k>=4. Prescribed target sets of those sizes attain eleven deletions exactly when they use the same original pairs and have affinely independent signs.
• Arbitrary graph targets: HIGHER-GRAPH.md gives the complete k=2 generalized-weight hierarchies. GRAPH-TARGETS.md adds exact partition formulations for arbitrary target sets, a capacity-matching formula for every forest in the nonempty-star code, and every prescribed triple budget in both codes. These results go beyond choosing a favorable target set.
• Core shortening: CORE-SHORTENING.md identifies the full unrestricted-sign shortening on k-2 original pairs at q=3, its exact dimension, and the sharp weight-sixteen threshold of the additional polynomial layers. Simply repeating the fixed-sign graph code omits these layers.
• Recoding and sequential copying obstruction: COPY-OBSTRUCTION.md proves that an integral separating pairing with odd diagonal forces total copied frame rank C>=v, including arbitrary linear observation recodings, exact nonlinear postprocessing, individually readable corrections, and chronological copy-to-zero innovations under the stated monotone-frame model. Every even-size paired-port geometry with overlap-one readability satisfies the hypothesis. The inherited odd-stage ledger then has no positive deficit. The selected E8 geometry has even diagonal and is outside this result.
• Native odd-star cost: ODD-NATIVE-COST.md proves a bit-saving ceiling c<107/66606 for every odd k>=5 in the inherited schedule with at least five stages, accounting for source/target climbs, idle hosting, and whole banks even while granting other work free. A separately stated conditional three-stage extension excludes odd k>=7. Both ceilings are below the campaign-start selected coefficient. Arbitrary recoding and COPY-assisted innovations are outside this particular theorem.

Earlier uniform minimum-word rigidity, all optimal two-dimensional subcodes, and exact prescribed-pair budgets remain included. A generalized weight d_r yields d_r-r deletions for some selectable r-target set, not every prescribed set. Code statements cover both proper-star spaces; the original uniform overlap corrections are absorbed only at power-of-two k.

Reproduction and evidence

Python 3.10+; standard library only:

for check in shortening optimizer block copy_obstruction core_shortening graph_targets; do
  python3 -I -O -B "research/asymmetric-hypercube-decoder/${check}_verify.py" --self-test || exit
done
for check in higher cube boundary; do
  python3 -I -O -B "research/asymmetric-hypercube-decoder/${check}_verify.py" --self-test || exit
done
python3 -I -O -B research/asymmetric-hypercube-decoder/global_verify.py --self-test
python3 -I -O -B research/asymmetric-hypercube-decoder/verify.py --negative-controls

The current campaign checkers cover 42 full-generator cube shortenings; all 187 relevant RM(1,4) subspaces and 6,748 target sets; complete small complementary-block models; eight exact binary/rational COPY matrix cases and a sequential-innovation fragment; full-generator original-core shortenings at k=3,...,6 with all 1,048,576 words of the smallest code; and 4,112 complete graph target sets plus 73 triple/quadruple orbits covering 12,650 sets at p=4. The graph oracle directly checks 4,329,472 decoder assignments. Receipts distinguish exhaustive domains, overlapping case instances, and logical meet-in-the-middle candidate counts.

All six campaign self-tests passed isolated optimized execution, relocation, corruption rejection and failure-output preservation. The graph checker also tests atomic replacement failure cleanup. Proofs and implementations received separate AI-assisted reviews; the lead reviewed the resulting proofs and code. Earlier higher-target receipts were reproduced under Python 3.11.13 and 3.14.5; campaign receipts currently record Python 3.14.5. No human or proof-assistant verification is claimed. Finite enumeration supports the separate uniform proofs. Selected supplier cold replay, unrelated regeneration, and hosted CI are not claimed.

Provenance and remaining work

Original base: 66849e1475c930fd7732b1a0067ffee7e1e95f60. Campaign continuation base: 7d526020fc368ce4157cd436995c2f4fcf683318. Latest inspected main snapshot: 007664306fff84fbb1c9a1b4053b69f439764034, whose selected record and actual #890 integration review report conditional κ 1661957774647029/10^18; its #589-on-#555 bit supplier is unchanged. upstream/ and selected certificates are untouched. Seed 48, structural decoder research.

Credit Luke Mansoor (#732) for uniform minimum distance, Sagar Thakrar (#725) for the q=1 exception, Alishan Ladhani (#747) for the finite (6,4) minimum-word classification, ellipsi2 (#772) for uniform ranks, and Joel Pulikkan / GamingPuzzled (#231) with icekylinx/eumemic for the paired-cube decoder lineage. The graph incidence-code/cut bridge is credited to Martínez-Bernal, Valencia-Bucio, and Villarreal, arXiv:1812.04106v2, Corollary 2.13. Physical notes retain the inherited odd-stage/hosting and rank-product credits and immutable source pins.

Overlap was refreshed through #958. Ross Richardson's #958 studies a distinct five-port point-only decoder with quadratic labels and both overlap-one and overlap-three corrections; our native odd-star result uses incidence labels, additional proper-star observations, and overlap-one corrections. #939/#954 concern E8 geometry outside our integral odd-diagonal hypothesis. No literature-wide priority claim is made.

CAMPAIGN-18H.json records the active minimum eighteen-hour campaign, reviewed milestones, exact receipts and remaining obligations. The present checkpoint does not complete that campaign. Higher-rank classification, remaining odd-size observation classes, and fully priced source/helper/frame/restoration/recursive schedules remain under investigation. No deletion count is promoted to a physical supplier invoice.

Prepared by Sam Sung (@the-missionary-company) with substantial OpenAI Codex assistance, including proof development, implementation, and independent AI-assisted review. Apache-2.0.

[Mister-Meeseeks]

The 20-PR pilot triaged 8f4600b41bf1f0f5d07f209e5d3950eb22ef5cad and is leaving this PR open for focused follow-up.

Remaining maintainer review: Dedicated proof review of uniform higher weights, equality classifications, graph targets and COPY model; finite checks must not substitute for those proofs.

This is a bounded-review disposition, not a rejection or a request to repeat unrelated full-suite checks. Pilot record. Your contribution and original attribution remain in the queue ledger.