Scoped negative result: classify the E8 rank-five boundary at 36 punctures

research/e8-rank5-boundary/CLAIM.json

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+{
+  "assignment_seed": 46,
+  "randomness": "Python secrets.randbelow(100), OS randomness, drawn once at 2026-10-13T08:35:58Z",
+  "track": "structural decoder research",
+  "base_commit": "f567884aeb8851b6e794af1184119a3bfbe1e631",
+  "claimed_at": "2026-10-13T08:41:34.289802+00:00",
+  "expected_finish": "2026-10-13T09:35:00Z",
+  "coordination": "GitHub GraphQL pinnedIssues empty; only open issue #650 requests Discussions. No reservation issue created. Claim recorded locally.",
+  "candidates": [
+    {
+      "object": "Hurwitz adaptive nonlinear message supports",
+      "decision": "deferred; fixed-support #785 already complete and physical adaptive metadata pricing is unresolved"
+    },
+    {
+      "object": "E8 first rank-five puncture boundary: exactly 36 deleted root lines",
+      "decision": "selected; #631 explicitly leaves 36 as its first open rank-threshold boundary. Enumerate all equality cases modulo Weyl conjugacy and optimize each star basis exactly."
+    },
+    {
+      "object": "P112 extra-total decoder lift",
+      "decision": "rejected as overlapping #845; multiple lifts too broad for this sprint"
+    }
+  ],
+  "diversification": "46 modulo 3 = 1 selects the second candidate; restrict to the first previously unexcluded rank-five boundary, not a frontier-word retuning.",
+  "hypothesis": "At exactly 36 punctures, no star label decoder beats 63/111 copy cost per live port; cheap-reader compatibility may close the coarse rank-floor gap.",
+  "method": "Generate E8 roots and binary reader code; enumerate standard rank-five parabolic flats and all 511 nonzero reader words with exactly 36 deletions; quotient duplicate punctures, compute exact rational reader ranks and minimum-weight F2 bases. Inherited Weyl/parabolic reduction upgrades this equality census to all 36-puncture sets possessing a rank-five reader. Other sets have rank floor six.",
+  "deliverable": "Scoped negative-result note with written reduction, exact deterministic receipt, source pins, independent verification and mutation controls; no supplier or kappa claim.",
+  "stopping_rule": "Complete all 56 standard rank-five flats times 511 readers and all resulting equality punctures; no general deep-puncture annealing.",
+  "overlap_snapshot": {
+    "open_prs": 660,
+    "max_pr": 857,
+    "inspected": [
+      620,
+      631,
+      669,
+      703,
+      717,
+      738,
+      765,
+      767,
+      785,
+      814,
+      845,
+      848,
+      855
+    ]
+  },
+  "status": "completed",
+  "final_overlap_check": {
+    "checked_at": "2026-10-13T08:56:20.809094+00:00",
+    "open_prs": 667,
+    "max_pr": 874,
+    "main_commit": "f567884aeb8851b6e794af1184119a3bfbe1e631",
+    "pr631_head": "fb893b5ee1317c2da3b297e6fe78eca0857d1a76",
+    "pinned_coordination_issues": 0,
+    "finding": "No overlap with the exact 36-puncture equality classification. New #868 is Gaussian all-clique dual scope; #874 audits X4 ideal gain additivity."
+  },
+  "result_update": {
+    "at": "2026-10-13T08:58:21.815166+00:00",
+    "classification": "scoped negative result; no new kappa",
+    "finding": "Every 36-puncture E8 star label decoder has cost at least 54 on 84 live ports. Exactly 7,560 punctures admitting a rank-five reader form one Weyl orbit and have exact star cost 62. Thus no 36-puncture set improves 63/111, extending PR631 through 36.",
+    "completed_coverage": {
+      "standard_rank_five_pairs": 28616,
+      "all_threshold_pairs": 130305,
+      "standard_equality_representatives": 2,
+      "complete_equality_orbit": 7560,
+      "reader_ranks_per_representative": 511,
+      "single_port_restorations": 72
+    },
+    "result_link": "https://github.com/mightyOil/integer-mult-bounds/tree/research/e8-rank5-boundary-seed46/research/e8-rank5-boundary",
+    "verification": "Both exact implementations pass on CPython 3.14.7 and 3.11.15, including isolated optimized mode; package mutations, predecessor controls and selected-record check pass. See VALIDATION.md."
+  }
+}

research/e8-rank5-boundary/NOTICE.md

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+# Attribution and license
+
+This package is contributed under the repository's Apache-2.0 license.
+New work: Philipp (@mightyOil), 2026, with substantial OpenAI Codex assistance
+in mathematical reasoning, code, validation and documentation. Independent
+computational checks were implemented by a separate AI-assisted worker; this
+is not a claim of independent human or proof-assistant review.
+
+The two files under `sources/pr631/` are unchanged sources by **Chi Zhao**, with
+substantial OpenAI Codex assistance, from CrocSwap/integer-mult-bounds PR631,
+commit `fb893b5ee1317c2da3b297e6fe78eca0857d1a76`. Their source notices are
+retained. They are Apache-2.0 per their own notice and the repository license.
+
+Their sharp threshold theorem, parabolic/Weyl reduction, and earlier
+35-puncture ratio theorem remain Chi Zhao's contribution. Their original
+contextual use of "current" is historical and does not refer to the new
+package's base or current selected frontier.
+
+The star label-decoder model and minimum-weight basis formulation come from
+Joel Pulikkan / GamingPuzzled's PR620. DreamingOfClouds' PR595/PR602 and
+eumemic's PR543 supply earlier format and supplier context, retaining their
+own predecessor credit. See README.md and SOURCES.json for exact scope and pins.
+
+No changes are made to `upstream/`, the predecessor files, original supplier
+sources, certificates/selected-result.json or repository-wide review ledgers.

research/e8-rank5-boundary/README.md

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+# The first E8 rank-five puncture boundary
+
+**Scoped negative result; no new supplier or κ.** At exactly 36 deleted
+E8 root lines, no bit-side star label decoder improves the copy-rank/live-port
+ratio `63/111`. Together with Chi Zhao's [PR631](https://github.com/CrocSwap/integer-mult-bounds/pull/631),
+this extends its exclusion from at most 35 to **at most 36 punctures**.
+
+The new evidence explains why the first opening in that predecessor's coarse
+rank bound does not help. Exactly **7,560** 36-puncture sets admit a reader of
+rational rank five. They form **one Weyl orbit**. Every such set has the same
+reader-rank histogram and exact minimum star cost:
+
+| Reader rank | 5 | 6 | 7 | 8 |
+|---|---:|---:|---:|---:|
+| Number of nonzero readers | 1 | 0 | 42 | 468 |
+| Binary dimension spanned by readers up to this rank | 1 | 1 | 8 | 9 |
+
+An optimal nine-total basis therefore has weights `5,7,7,7,7,7,7,7,8`, costing
+**62** on 84 live ports. Having one rank-five reader does not provide nine
+independent cheap readers. All other 36-puncture sets have cost at least 54.
+Both ratios exceed `63/111`:
+
+```
+62/84 = 31/42 > 21/37 = 63/111,
+54/84 =  9/14 > 21/37.
+```
+
+This is an exhaustive equality-class calculation with a written coverage
+argument, not a failed random search. The minimum cost across *all*
+36-puncture sets is not determined; 54 is only the lower bound used for the
+nonexceptional cases.
+
+## Reproduce
+
+From the repository root, Python 3.10+ and its standard library suffice:
+
+```sh
+python3 -B research/e8-rank5-boundary/verify.py
+```
+
+Expected final line:
+
+```
+PASS_E8_RANK5_BOUNDARY: 28616 rank-five pairs; 2 representatives; 7560 conjugate punctures; cost 62; 72 restorations; 5 negative controls; no new kappa
+```
+
+The default command checks the entire committed `CERTIFICATE.json`, including
+the predecessor source hashes. It writes nothing. `--write /tmp/e8-rank5.json`
+regenerates the deterministic certificate after all checks pass.
+
+A separate implementation uses normalized Fraction RREF, low-pivot binary
+elimination, direct span-membership tests instead of simple-coordinate masks,
+and the cumulative-rank cost formula instead of the greedy basis algorithm:
+
+```sh
+python3 -I -O -B research/e8-rank5-boundary/independent_check.py \
+  --check research/e8-rank5-boundary/CERTIFICATE.json
+```
+
+It imports neither the primary verifier nor predecessor code. It checks all
+28,616 rank-five pairs and both complete 511-reader rank tables. Its scope is
+the `independent_summary` portion of the certificate; the orbit traversal and
+other-dimensional threshold census are checked by the primary verifier.
+Both new implementations retain all checks under optimized Python.
+
+The predecessor is retained byte-for-byte and can also be replayed:
+
+```sh
+python3 -B research/e8-rank5-boundary/sources/pr631/verify.py --negative-controls
+```
+
+That older script intentionally refuses `-O`. See [VALIDATION.md](VALIDATION.md)
+for the actual local validation record and its limits.
+
+## Model and coverage proof
+
+Use the standard 120 real E8 root **lines**, with representatives `L_t` in
+doubled coordinates, squared norm eight. The binary decoder matrix is
+`B_st = (1 + L_s · L_t / 4) mod 2`. Its row space `C` has dimension nine.
+Its nonzero words have weights 56 (255 words), 64 (255 words), and 120 (one).
+For a puncture set `P`, the star weight of a reader `c` is
+
+```
+s_P(c) = rank_Q {L_t : c_t = 1 and t not in P}.
+```
+
+We use exactly the star model in Joel Pulikkan / GamingPuzzled's PR620
+`s2-decoder/bitp.py`, inherited by PR631: minimize the sum of these weights
+over a basis of the nine-dimensional reader space. A collection of additional
+star totals must span that space and contains a basis; nonnegative weights
+cannot make that collection cheaper than the minimum basis. This does not
+cover general source/reader pairing-rank factorizations.
+
+**1. The code still has nine dimensions after 36 deletions.** A nonzero word
+can vanish on all live ports only if its entire support was deleted. Its
+minimum weight is 56, greater than 36. Restriction to the live ports is
+injective, so independence before and after puncturing agrees. Invertible
+changes of label coordinates also preserve it.
+
+**2. Any rank-five case is an equality case.** PR631's sharp deletion
+thresholds for reader ranks at most `0,...,7` are
+`[56,55,53,50,44,36,26,1]`. The primary verifier regenerates all of them over
+255 standard flats and 511 readers. Since 44 deletions are needed for rank
+four, a reader of rank at most five at our boundary has rank exactly five.
+Let `F` be the span of its surviving roots. All reader roots outside `F`
+were deleted. At least 36 deletions are needed to achieve this, and `|P|=36`,
+so necessarily
+
+```
+P = support(c) \ F.
+```
+
+There are no leftover arbitrary deletions in this case. This equality is
+the new search restriction; we are not reopening PR620's annealing search.
+
+**3. Standard flats cover every equality case.** We inherit PR631's
+parabolic-flat reduction, repeated here for clarity. Choose a generic normal
+in `F` perpendicular space, perpendicular to exactly the roots in `F`.
+Move it to the closed fundamental Weyl chamber. Every positive root has
+nonnegative simple-root coordinates, so the perpendicular roots are precisely
+the roots supported on those simple roots perpendicular to the normal. Their
+span is a standard rank-five parabolic flat. Thus it suffices to inspect
+all `binom(8,5)=56` standard flats and all 511 reader words. The main verifier
+checks integral, common-sign simple coordinates and preservation of the full
+reader code by each of the eight simple reflections. Physical ranks are
+preserved because these reflections are orthogonal. This supplies the
+invariance needed to transport the cost calculation back to any original
+puncture set.
+
+**4. The complete equality census has two standard representatives.** The
+two `(simple subset, reader)` pairs are recorded in the certificate, with all
+reader and deleted root indices. The simple subsets are `[0,1,2,3,4]` and
+`[1,2,3,4,5]` in the explicitly generated simple system. These are *two
+standard-flat representatives*, not two distinct Weyl orbits. The following
+word in the eight simple reflections carries the first puncture set to the
+second (apply from left to right):
+
+```
+5,4,3,1,2,0,3,2,4,3,1,5,4,3,2,0
+```
+
+Breadth-first closure of the first set under all eight reflections has exactly
+7,560 distinct masks. Every newly encountered image is enqueued and all eight
+outgoing images are inspected before completion. Combined with the covering
+argument and the explicit connection, this proves the asserted single orbit
+and its exact size. Root numbering belongs to this standard-coordinate
+package, not the Hamming-coordinate numbering of a shipped supplier.
+
+**5. Exact minimum cost.** For both representatives, compute all 511 rational
+reader ranks. The histogram is shown above. Let `r_k` be the binary rank of
+the span of all readers of weight at most `k`. For integer weights in `[0,8]`,
+the minimum basis cost is `sum_{k=0}^7 (9-r_k)`: a greedy basis contains a
+maximum independent set from each nested weight threshold. Here
+`(r_0,...,r_7)=(0,0,0,0,0,1,1,8)`, yielding 62. The certificate also gives
+an explicit attaining basis, whose weights are recomputed from roots.
+
+**6. The other cases.** If a 36-puncture set has no rank-five reader, Step 2
+also excludes lower ranks. Every basis reader has rank at least six, hence
+the star cost is at least `9*6=54`. This completes the stated ratio exclusion.
+Combining it with PR631's earlier theorem gives the at-most-36 corollary.
+
+## Controls and scope
+
+Every default primary replay includes:
+
+- Corrupted root geometry, binary decoder and flat-membership rejection.
+- A tempting set of **nine distinct** cheap readers with invoice 61, rejected
+  because its binary span has dimension eight.
+- A wrong-characteristic control: rational physical rank is not binary rank.
+- All 72 single-port restorations across the two representatives. Each raises
+  the unique rank-five reader to rank six, confirming the deletion boundary.
+
+The final item is a positive boundary control, not 72 malformed-file rejections.
+The independent implementation supplies separate rational, binary and geometry
+controls. Receipt and source mutations are tested during packaging.
+
+The result concerns root-only E8 ports, unrestricted independent input bits,
+the fixed label decoder, and additive **star** copy-rank accounting. It does
+not rule out general pairing-rank decoders, extra/non-root ports, helper
+savings, correlated inputs, changed stage geometry, or other mechanisms.
+Puncture counts 37 and above are outside the new theorem. There is no new
+global puncture optimum, formal proof-assistant theorem or full multiplication
+theorem here.
+
+The hypothesis tested was a reduction of this copy-rank ratio at its first
+unexcluded boundary. That necessary arithmetic opportunity fails even if
+additional helper, source-access, frame, restoration and recursive-child
+expenses are optimistically assigned zero. This says nothing about improving
+another part of a full supplier invoice at an unchanged or worse copy ratio.
+No new supplier or outer assembly is proposed or priced. The selected
+construction's all-size/compiler, hosting, precision/tape and analytic
+assumptions remain separate and unresolved by this note.
+
+## Provenance, prior work and assignment
+
+Base commit: `f567884aeb8851b6e794af1184119a3bfbe1e631`. At that base the
+selected conditional saving is `1661634653669567/10^18` (PR799 with PR589 on
+PR555), as recorded in `certificates/selected-result.json`. Its actual
+construction, maintainer review and the PR ledger were inspected. This package
+does not modify the selected record or any existing repository file.
+
+Assignment seed **46** was drawn once from operating-system randomness.
+[CLAIM.json](CLAIM.json) records the three candidates, scope, stopping rule
+and overlap checks. No pinned sprint coordination issue existed; the claim
+and its result update are local. Completed coverage is 28,616 boundary pairs,
+130,305 total threshold pairs, both equality representatives, both 511-reader
+rank tables and the complete 7,560-mask equality orbit. There is no random
+search or unreported timeout in these domains.
+
+The predecessor is pinned to PR631 head
+`fb893b5ee1317c2da3b297e6fe78eca0857d1a76`; its original README and verifier
+are retained under `sources/pr631/` without changes. [SOURCES.json](SOURCES.json)
+records their hashes and exact paths, plus base-context hashes. Context hashes
+describe the inspected base and are not a claim that future selected records
+must stay unchanged. New scripts import no predecessor implementation.
+
+Credit **Chi Zhao / PR631** for the all-flat reduction, sharp single-reader
+thresholds and earlier at-most-35 theorem; **Joel Pulikkan / GamingPuzzled /
+PR620** for the model, optimum-basis method and puncture investigations;
+**DreamingOfClouds / PR595 and PR602** for format-limit context; and **eumemic /
+PR543** and the prior E8 supplier lineage for the original construction
+context. These inherited results are not claimed as new. The new contribution
+is the equality classification, its exact orbit and invoice, and the
+36-puncture exclusion with independent arithmetic evidence.
+
+Prepared by **Philipp (@mightyOil)** with **substantial OpenAI Codex assistance**
+in proof development, implementation, independent computational checking and
+writing. The separate checks were also AI-assisted; no independent human
+review is claimed. Apache-2.0; see [NOTICE.md](NOTICE.md).

research/e8-rank5-boundary/SOURCES.json

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+{
+  "base_commit": "f567884aeb8851b6e794af1184119a3bfbe1e631",
+  "predecessor_pr": 631,
+  "predecessor_head": "fb893b5ee1317c2da3b297e6fe78eca0857d1a76",
+  "files": {
+    "sources/pr631/README.md": {
+      "sha256": "281f706bf29d1f91dd64ba096f9d66d622f49e9d82d0ff306fb9374f77ba7333",
+      "original_path": "research/e8-continuation-20261013/punctures/README.md"
+    },
+    "sources/pr631/verify.py": {
+      "sha256": "9add2f1d71466f00963652b818e168fe2e2a4fa6e62cafa9811389fdba7000c2",
+      "original_path": "research/e8-continuation-20261013/punctures/verify.py"
+    }
+  },
+  "context_files": {
+    "certificates/selected-result.json": "9873519224e2db72d6a8c926f51e3e75129671e467948c6f6b03744f8bcfc699",
+    "docs/research/community-frontier799-review.md": "bd91c10c67f2a873fd456b95cbb43f715a43dac0e0a66938737d591c6b754052",
+    "research/e8-decoder-floors-and-cold14-notes/s2-decoder/bitp.py": "a692723e1844cb4d3a6c4a346e31b1d707133a1e086176fda6a3fec32d108637",
+    "research/e8-decoder-floors-and-cold14-notes/s2-decoder/REPORT.md": "16c53bf8afd9f91eca02b2af9c9f5f09385823fc705dd1a48153508cfdff28b6"
+  }
+}