Research: classify 240-port optimal punctures and certify the all-target threshold

research/global-star-punctures/CLAIM.json

@@ -0,0 +1,38 @@
+{
+  "track": "Structural decoder research",
+  "assignment_seed": 42,
+  "random_source": "OS randomness via secrets.randbelow(100), drawn once",
+  "base_commit": "66849e1475c930fd7732b1a0067ffee7e1e95f60",
+  "claimed_at_utc": "2026-10-13T04:43:35.636860+00:00",
+  "expected_finish_utc": "2026-10-13T05:25:00Z",
+  "coordination": "No pinned issue; sole open issue #650 concerns Discussions. Local claim used.",
+  "candidates": [
+    "E8 mixed-port extensions beyond the minimal puncture equality case (adjacent to #715)",
+    "Off-target proper stars after puncturing paired hypercubes (explicitly outside #669 Theorem 2)",
+    "Alternative cross-stage corrector-rank compression beyond the fixed-role format of #446"
+  ],
+  "chosen_object": "One retained target in the (p,k)=(6,4) paired-port family, with arbitrary global proper-star coefficients and uniform overlap-class corrections",
+  "seed_use": "The tens digit selects k=4 and the units digit selects p-k=2; target index 42 in the documented lexicographic port order.",
+  "hypothesis": "Allowing off-target proper stars may lower the target-local deletion threshold 2(p-k)+1=5.",
+  "method": "Exact F2 incidence and dual-certificate calculations; seek a proof or an explicitly bounded finite result with a witness and negative controls.",
+  "stopping_rule": "One question only; finish experiment by 05:14 UTC and reserve remaining time for verification and handoff. No unbounded parameter sweep.",
+  "expected_deliverable": "Self-contained scoped research note, exact verifier, deterministic certificate, immutable source pins and one focused PR; no kappa claim.",
+  "overlap_check": {
+    "open_pr_count": 535,
+    "highest_pr": 716,
+    "relevant_pr": 669,
+    "relevant_head": "308f1e6ec3bb0b6ec0584cf7bfe219addc219b1e"
+  },
+  "scope_refinement": {
+    "reason": "Final metadata refresh found concurrent #725 and #732, absent from the initial snapshot.",
+    "open_pr_count": 551,
+    "highest_pr": 732,
+    "new_emphasis": "Complete (6,4) equality classification (only four target-local optima), plus sharp 85-deletion algebraic all-target threshold and explicit information set. The already-completed rank and minimum-distance work is independent corroboration of #732.",
+    "experiment_changed": false,
+    "predecessors_credited": [
+      669,
+      725,
+      732
+    ]
+  }
+}

research/global-star-punctures/README.md

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+# Exactly four optimal punctures in the 240-port global star code
+
+**Scoped negative result / exact finite classification; no new κ.** For the
+240 paired ports with four coordinates chosen from six pairs, arbitrary
+off-target proper stars do not reduce the five deletions needed to decode one
+retained target. There are exactly four optimal deletion sets per target. Each
+isolates one of its three-coordinate stars; no additional optimal support is
+made possible by off-target stars.
+
+The same certificate gives the exact **algebraic all-target threshold: 85
+deletions**. The proper-star code has dimension 155. An explicit set of 155
+retained ports admits decoders for every target, while no larger surviving
+family does. This is a statement about scalar identities, not their physical
+implementation or cost.
+
+The original question came from Theorem 2 of
+[Sam Sung's PR #669](https://github.com/CrocSwap/integer-mult-bounds/pull/669),
+which leaves punctured global stars outside its scope. Two contributions
+arrived during this experiment: [Luke Mansoor's #732](https://github.com/CrocSwap/integer-mult-bounds/pull/732)
+proves the general minimum threshold and independently checks the same
+240-port rank and distance; [Sagar Thakrar's #725](https://github.com/CrocSwap/integer-mult-bounds/pull/725)
+classifies 67 optimal patterns in the smaller 80-port (5,4) case, only four of
+which are target-local. Those threshold/rank results are credited, not claimed
+as new here.
+
+This contribution adds the **complete equality classification for (6,4)**:
+all four optimal patterns are target-local, and there are no off-target-only
+optima. It also supplies the **sharp algebraic all-target threshold and an
+explicit information set**, outside the one-target scope of both concurrent
+notes. The overlapping minimum-distance calculation is separately implemented
+corroboration. No code or proof from either concurrent package is imported.
+The finite classification does not claim a general equality theorem for other
+p or k.
+
+## Model
+
+There are six pairs `{2i, 2i+1}`, for `0 <= i < 6`. A port S is a four-element
+set using four distinct pairs. Let P be all `2^4 * binom(6,4) = 240` ports.
+Let D be the deleted ports, F=P\D the surviving ports, and T a retained target.
+All input symbols x_S, for S in F, are independent over F₂.
+
+For every nonempty compatible coordinate set L of size less than four, define
+
+```text
+star_L(F) = sum_{S in F, L subset S} x_S.
+C_j(T,F)  = sum_{S in F, |S intersect T| = j} x_S,  j=1,2,3.
+```
+
+The allowed decoder is
+
+```text
+x_T = sum_L a_(T,L) star_L(F) + sum_{j=1}^3 b_(T,j) C_j(T,F).
+```
+
+The 232 star coefficients may vary arbitrarily with both T and L, including
+L outside T. The three correction coefficients may vary with T, but remain
+uniform within each overlap class. No degree-four star or correction C_4 is
+allowed: either would directly supply x_T. The global sum is optional here;
+the certificate proves that it already belongs to the proper-star span for
+this instance. Both stars and corrections are recomputed on F after deletion.
+
+## Why the problem reduces to one binary code
+
+Put `A_l(T,F) = sum_{L subset T, |L|=l} star_L(F)`. Coefficient comparison at
+every surviving source gives
+
+```text
+C_1 = A_1 + A_3,    C_2 = A_2 + A_3,    C_3 = A_3.
+```
+
+For a source intersecting T in r coordinates, A_l has coefficient
+`binom(r,l) mod 2`. Substitution for r=0,...,4 proves these identities,
+including r=4 where each right-hand side is zero. Deleting sources just
+restricts these pointwise identities. Thus the correction columns add no
+freedom, even after puncturing.
+
+Let C be the span in F₂^240 of all proper-star incidence vectors on P. A
+decoder on F exists for T exactly when some c in C has c_T=1 and
+`support(c) subset {T} union D`. This equivalence does not assume a symmetric
+decoder, a bound on the number of totals, or an efficient implementation.
+
+## Certificate and exhaustive finite argument
+
+[CERTIFICATE.json](CERTIFICATE.json) contains 155 independent proper stars,
+85 independent parity checks H, an information set of 155 ports, and the four
+optimal puncture witnesses. The builder and verifier are separate programs;
+the verifier does not import or execute the builder or a predecessor checker.
+
+The verifier independently constructs the coordinate sets and incidence
+supports. It checks that every parity row annihilates **every** proper star
+and overlap correction, and checks the two ranks using set elimination.
+Since `155 + 85 = 240`, this proves `C = kernel(H)` and `dim(C)=155`.
+It also checks H against the global sum. No solver status is trusted.
+
+Let h_i be column i of H. The assignment seed selects target index 42 in the
+specified ordering (pair combinations, then sign tuples). Define A to contain
+zero, every non-target h_i, and every sum h_i+h_j of distinct non-target
+columns. Its **28,681 elements are all distinct**:
+
+```text
+1 + 239 + binom(239,2) = 28,681.
+```
+
+Every set D of at most four non-target ports splits into two parts of size at
+most two. Therefore `h_T = sum_{i in D} h_i` would imply
+`A intersect (h_T+A)` is nonempty. The checker exhausts this intersection and
+finds it empty. This excludes every allowed decoder with at most four
+deletions, across all `2^155` codewords, without enumerating their coefficients.
+
+At the boundary, the checker exhausts all **2,246,839 triples** of non-target
+ports and looks up the complementary pair syndrome. Unique pair syndromes
+make this lookup lossless. Exactly 40 hits yield four distinct five-element
+sets, each appearing `binom(5,2)=10` times. Overlapping pairs and triples are
+handled by symmetric difference; they would produce a smaller support and
+are rejected. Every five-element support has such a partition, so the census
+is complete. Its fixed stopping rule is exhaustion of these lists, not a time
+limit or a heuristic search.
+
+For each resulting set D, a three-coordinate J contained in T satisfies
+`star_J(F)=x_T`. The verifier checks every surviving source coefficient. These
+four J are exactly the four triples in T. Equivalently, D consists of the five
+other ports containing J. The fifth deletion is necessary: restoring any one
+of them breaks the identity, and the direct equation solver also rejects it.
+
+Permutations of the six coordinate pairs and independent swaps within pairs
+act transitively on ports and preserve the full star family and overlap
+classes. Hence the result for index 42 applies to **every target** in this
+fixed geometry. Every nonzero codeword contains a target, so C has minimum
+distance six; its minimum words are precisely the 160 triple-star supports
+(`240 * 4 / 6 = 160`, also `2^3 * binom(6,3)`). This is a finite,
+computer-assisted classification, not a formalized proof of multiplication.
+
+## All-target corollary
+
+For a surviving family F, the available decoder vectors are the projection
+`C|F`, including all the correction classes by the identities above. Decoding
+every target requires all unit vectors and thus `C|F = F₂^F`. Its dimension is
+at most 155, so `|F| <= 155`: at least 85 ports must be deleted.
+
+The certificate's `information_set` attains this. Restricting its 155
+independent star rows to those 155 coordinates has rank 155. The independent
+tests additionally solve for all 155 unit vectors and check their literal
+incidences. These are algebraic decoders with unrestricted star use. No
+favorable copy cost, frame schedule, or restoration circuit is inferred.
+
+## Reproduce
+
+Python 3.9 or newer, standard library only; no downloads or external solvers:
+
+```sh
+python3 -I -B research/global-star-punctures/verify.py
+```
+
+Expected final line:
+
+```text
+PASS_GLOBAL_STAR_PUNCTURES ports=240 rank=155 dual=85 minimum_deletions=5 optimal_sets=4 all_target_deletions=85 triples=2246839 controls=8 restored_ports=20 kappa=unchanged
+```
+
+Eight default negative-control categories cover a changed parity bit,
+dependent dual rows, duplicate primal rows, omitted minimum supports,
+incomplete coverage receipts, deletion of the target, all 20 one-port
+restorations, and a wrong coordinate sign. Checks remain active under `-O`.
+
+Additional direct equation checks and certificate regeneration:
+
+```sh
+python3 -I -O -B research/global-star-punctures/test_verify.py
+python3 -I -B research/global-star-punctures/build_certificate.py --output /tmp/global-star-certificate.json
+cmp research/global-star-punctures/CERTIFICATE.json /tmp/global-star-certificate.json
+```
+
+The five test groups use an independent ternary-word port enumeration and
+solve the actual decoder equations, without parity certificates, for all four
+optima, their 20 one-port restorations, and 20 wrong five-port punctures. They
+also check all 155 information-set targets, positive controls outside the
+obstruction (degree-four stars, k=3, and p=k=4 with one puncture), the vacuity
+of dropping the target equation, and optimized subprocess rejection of four
+corrupted certificates. See [VALIDATION.md](VALIDATION.md) for actual runs.
+
+## Scope, budget, provenance and credit
+
+The proposed escape was to allow off-target stars to beat the known
+target-local puncture threshold. It fails here already at scalar exactness.
+Giving helpers, source access, frame changes, copies, restoration and recursive
+children zero cost cannot make the excluded identities hold. The attaining
+five-deletion witnesses are the predecessor's isolated-star mechanism; their
+paid implementation is not supplied. The 85-deletion information set is not
+a proposed improved supplier, and no complete-cost or κ improvement is claimed.
+
+Other p or k, higher-degree totals, source-dependent correction coefficients,
+other fields/geometries, nonlinear decoders and alternative stage structures
+remain outside this result. No global bound on integer multiplication follows.
+
+Exact base: `66849e1475c930fd7732b1a0067ffee7e1e95f60`.
+[SOURCES.json](SOURCES.json) pins the predecessor heads and inspected source
+bytes. [CLAIM.json](CLAIM.json) records the OS-random seed **42**, three candidate
+tasks, selected object, stopping rule, and overlap checks. No pinned sprint
+issue existed. Only this package is added; `upstream/` and the frontier are
+unchanged. The accepted conditional κ remains
+`51110230636731/31250000000000000 = 0.001635527380375392`, from #626/#622/#623
+with #543. Its construction was read for context, not re-certified here.
+All inherited compiler, common-label/mutable-basis, hosting, restoration,
+precision/tape/recovery and analytic obligations remain.
+
+Credit: Joel Pulikkan / GamingPuzzled's
+[#231](https://github.com/CrocSwap/integer-mult-bounds/pull/231) supplied the
+symmetric parity obstruction, following icekylinx's #144 and eumemic's paired
+cube identity. Sam Sung / @the-missionary-company's
+[#669](https://github.com/CrocSwap/integer-mult-bounds/pull/669), pinned at
+`308f1e6ec3bb0b6ec0584cf7bfe219addc219b1e`, supplied the global unpunctured
+obstruction, target-local puncture theorem, and explicit remaining question.
+No predecessor code is copied or executed. E8 puncture and extension work in
+#620/#631/#715 has a different geometry and was checked for overlap.
+
+Concurrent credit: **Luke Mansoor (@lukemansoor), #732**, at
+`80431a74c31e807581f5c2258c87abe2084caacd`, supplies the general star-code
+distance theorem and the same (6,4) rank/distance. **Sagar Thakrar
+(@thakrarsagar), #725**, at `6eadc4eb4ebc355b13fa36a3138d97a583ea7f37`,
+supplies the (5,4) equality classification. Their source README hashes are
+pinned in SOURCES.json. The change in emphasis after their arrival is recorded
+in CLAIM.json; the scientific certificate and its enumeration domain did not
+change.
+
+Prepared by **Alishan Ladhani (@aladh)** with substantial **OpenAI Codex
+assistance** in research, proofs, implementation, verification and writing.
+No independent human peer review is claimed. New files are under the
+repository's [Apache-2.0 license](../../LICENSE).

research/global-star-punctures/SOURCES.json

@@ -0,0 +1,65 @@
+{
+  "repository": "https://github.com/CrocSwap/integer-mult-bounds",
+  "base_commit": "66849e1475c930fd7732b1a0067ffee7e1e95f60",
+  "context_sha256_at_base": {
+    "CONTRIBUTING.md": "a10dd640ec6f4b1d8b6acc31220af512335dc093f634cd25f601acb07ac53d53",
+    "certificates/selected-result.json": "b3bf402f31a69bde0629936a9b569bed48281568d7b16d5fba120b5311ea1f53",
+    "docs/research/current-status.md": "197319a6edf04340ec3623d09d03714fcd6a981359f22bdcefdd06e7e2f4174d",
+    "docs/research/pr-ledger.md": "2e811684201e08007377ee97b1c35c0f0b3c026dd8207e7b0de3ff7cd1e13a91",
+    "docs/research/community-k492-review.md": "3b731d30e80010b15408a289b0b32192700885474c8b36d48ee3320a7a8c5c47",
+    "research/hypercube-decoder-parity/README.md": "2dc89289d1d979aec6709b8c55a2af7cd39d97297bc20fb94adc7d4cdf867cd1",
+    "research/hypercube-decoder-parity/check.py": "82b7ad8d54662db3e159301aaf3cf87edfbe934b6ed5bdc01243812966b2ea40",
+    "research/community-k492-audit/pr626/replay/ASSEMBLY.json": "6127732e2fcdd470dab349c1c51d03cd9f0787ad0309b1b4a7c811a908028f4e"
+  },
+  "predecessors": [
+    {
+      "pr": 231,
+      "head": "3c11e696a2fb2c454585d72710de27cd6bd3be68",
+      "author": "Joel Pulikkan / GamingPuzzled",
+      "contribution": "Symmetric paired-hypercube parity obstruction; retained in main."
+    },
+    {
+      "pr": 669,
+      "head": "308f1e6ec3bb0b6ec0584cf7bfe219addc219b1e",
+      "author": "Sam Sung / the-missionary-company",
+      "contribution": "Global full-family obstruction and target-local puncture theorem; explicitly leaves global punctures outside its scope.",
+      "path": "research/asymmetric-hypercube-decoder/README.md",
+      "sha256": "69b624112d6294f519682fdf762d7f670063ea3f2b05ec0f25ce26e93c481c17"
+    }
+  ],
+  "selected_context": {
+    "kappa": "51110230636731/31250000000000000",
+    "decimal": "0.001635527380375392",
+    "source_prs": [
+      626,
+      623,
+      622,
+      543
+    ],
+    "pr626_head": "660de893d27bb7abd1d5b01b1640207528a5c851",
+    "scope": "Context only; this package does not replay the selected supplier."
+  },
+  "dependencies": "Python >=3.9 standard library. No predecessor code, external solver, or downloaded data is imported at runtime.",
+  "pin_scope": "Historical source/context pins, not a requirement that later main retain these selected-status bytes. The complete scientific instance is specified in README.md and CERTIFICATE.json.",
+  "contributor": "Alishan Ladhani (@aladh)",
+  "ai_assistance": "Substantial OpenAI Codex assistance in research, proof development, implementation, verification and documentation; no independent human peer review claimed.",
+  "license": "Apache-2.0",
+  "concurrent_contributions": [
+    {
+      "pr": 725,
+      "head": "6eadc4eb4ebc355b13fa36a3138d97a583ea7f37",
+      "author": "Sagar Thakrar / thakrarsagar",
+      "contribution": "67 equality cases at (5,4); distinct smaller geometry.",
+      "path": "research/global-star-puncture/README.md",
+      "sha256": "b50aa319fff5f22a8b29745d17bac0b9f650f1129310722deea6dbae1d82c7fa"
+    },
+    {
+      "pr": 732,
+      "head": "80431a74c31e807581f5c2258c87abe2084caacd",
+      "author": "Luke Mansoor / lukemansoor",
+      "contribution": "General minimum-distance theorem and corroborated (6,4) rank/distance; no (6,4) equality classification or all-target information set.",
+      "path": "research/global-star-puncture/README.md",
+      "sha256": "87825a4dd666d8c32df025fe838aca4cdc23a5d3965f79c9aca4913567dba6d4"
+    }
+  ]
+}