Scoped S4 result: two 4-cycle targets need six calls at r=3

research/s4-two-target-cycle-seed6e525/CERTIFICATE.json

@@ -0,0 +1,76 @@
+{
+  "schema_version": 1,
+  "status": "PASS_S4_TWO_TARGET_CYCLE",
+  "outcome": "NO_WORD_THROUGH_FIVE_MINIMUM_SIX",
+  "group": "S4",
+  "group_order": 24,
+  "address_generators": [
+    "s1=(12)",
+    "s2=(23)",
+    "s3=(34)"
+  ],
+  "scalar_registers": 3,
+  "rank_one_idempotents": 28,
+  "paid_generator_count": 84,
+  "target": "diag([s1*s2*s3],[s1*s2*s3],[e])",
+  "radius_left": 2,
+  "radius_right": 3,
+  "left_layer_counts": [
+    1,
+    84,
+    4697
+  ],
+  "right_layer_counts": [
+    1,
+    84,
+    4697,
+    232692
+  ],
+  "left_state_count": 4782,
+  "right_state_count": 237474,
+  "intersection_count": 0,
+  "direct_call_count": 6,
+  "direct_word": [
+    {
+      "address": "s1",
+      "u": 1,
+      "v": 1
+    },
+    {
+      "address": "s2",
+      "u": 1,
+      "v": 1
+    },
+    {
+      "address": "s3",
+      "u": 1,
+      "v": 1
+    },
+    {
+      "address": "s1",
+      "u": 2,
+      "v": 2
+    },
+    {
+      "address": "s2",
+      "u": 2,
+      "v": 2
+    },
+    {
+      "address": "s3",
+      "u": 2,
+      "v": 2
+    }
+  ],
+  "witness_call_count": null,
+  "witness": null,
+  "semantic_controls": {
+    "all_paid_factors_literal_square_to_identity": true,
+    "single_call_omissions_rejected": 6,
+    "changed_third_role_rejected": true
+  },
+  "complete_domain": "all ordered words of length <=5 over the 84 paid involutive factors; proved by radius-2/radius-3 exact meet",
+  "time_cap_seconds": 110,
+  "rss_cap_bytes": 1073741824,
+  "selected_kappa_claimed": false
+}

research/s4-two-target-cycle-seed6e525/CLAIM.md

@@ -0,0 +1,49 @@
+LOCAL CLAIM AND RESULT — GitHub Discussion posting unavailable
+
+Track `bit-mechanisms`; OS seed `6e5256110feb8b034f6afef8802f497c`; task
+`bit-new-register-model`; scope key `74c418ed2b3f8f8a98e2`; priority revision 3;
+exact base `0b1cccf86e2f5132ed286a26ef0a681a495740ea`; selected-record SHA256
+`10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32`.
+Local advisory expiry: `2026-10-14T09:37:35Z`.
+
+Question: in exact `F2[S4]` formal algebra, with three arbitrary dirty scalar
+roles, can the target `diag([s1*s2*s3],[s1*s2*s3],[e])` be realized in at most
+five calls `I+([s]+[e])N`? Here `s1=(12), s2=(23), s3=(34)` are each paid at
+one call, `N` ranges over all 28 rank-one idempotents of `M3(F2)`, arbitrary
+free scalar `GL3(F2)` mixing is normalized into the paid factors, and the
+third role must be restored. The direct bill is six calls; a candidate saving
+would be one call (16.7%). The model optimistically assigns no additional
+hosting price to the third role or free scalar mixing and does not claim
+physical compilation.
+
+Method/domain: exact `F2[S4]` coefficient matrices and all 84 paid factors.
+A complete radius-two ball from identity and radius-three ball from target
+covers every word of length at most five because every factor is an
+involution. Stop at 110 seconds or 1 GiB RSS; report incomplete coverage if
+capped. The actual run completed in 2.29 seconds at 60,576 KiB RSS on Python
+3.12.3.
+
+Final overlap check: 405 open PR records were fetched over REST during
+07:40–07:43Z; no exact S4/adjacent-transposition/four-cycle match appeared.
+Nearest scopes are PR #1478 (UT3 central targets, r=2,3), PR #1705 (four GL2
+coordinate-SWAP targets at r=4), and PR #1711 (two AGL2 affine targets at r=2).
+The cached complete discussion pagination has 351 records fetched at 07:28Z;
+direct reads of #1167 and #1317 reached 07:22Z and 07:31Z and contain no S4
+claim. A fresh complete Discussion refresh failed with GitHub's GraphQL rate
+limit error; REST GET succeeds, but this installation returns 404 for
+Discussion comment create/update, so this claim was kept local.
+
+FINDING: the exact radius-two/radius-three meet has layer counts `[1,84,4697]`
+and `[1,84,4697,232692]`, state totals 4,782 and 237,474, and zero meetings.
+The direct six-call word passes. Literal convolution confirms all 84 factors
+square to identity; six direct-word omissions and a changed-third-role endpoint
+are rejected. Thus the declared S4 target has exact minimum six; the tested
+one-call saving does not exist. This closes only the specified finite model.
+No selected-supplier occurrence, recursive moment change, physical compiler
+saving, or new κ is claimed. The selected binding-only baseline currently
+reports the complex primitive as binding.
+
+Full package and deterministic receipt: `README.md`, `TASK.json`,
+`CERTIFICATE.json`, `SOURCES.json`, `VERIFY.json`, and `verify.py`.
+
+SUBMITTED: [PR #1720](https://github.com/CrocSwap/integer-mult-bounds/pull/1720), head `828a1df15d5f5256f80e61c8d22b822846b76ae7`. It is open; no merge or frontier change is claimed.

research/s4-two-target-cycle-seed6e525/README.md

@@ -0,0 +1,112 @@
+# S4 two-target cycle mechanism — seed 6e525611
+
+**Bounded bit-mechanism construction attempt; no new κ claimed.**
+
+## Scope and cost screen
+
+This is a changed `bit-new-register-model` assignment: the address group is
+`S4`, the paid letters are the adjacent transpositions `s1=(12)`, `s2=(23)`
+and `s3=(34)`, and each letter costs one call. There are three scalar roles
+over `F2`; each call is `I + ([s]+[e])N` for one of the 28 rank-one
+idempotents `N=uv^T`, `v^T u=1`. The target is
+`diag([c],[c],[e])`, where `c=s1*s2*s3` is a 4-cycle. Every input role is
+arbitrary and dirty, and all roles must be restored exactly.
+
+Free scalar `GL3(F2)` changes between paid calls are included: move each
+change through the adjacent paid factors by conjugating its rank-one
+idempotent, which preserves the set of all 28 idempotents. Augmentation
+`[g] -> 1` makes every paid factor the identity and forces any final free
+scalar map to be the identity as well.
+
+The direct word costs six calls, three for each copy of `c`. A useful local
+construction must use at most five, saving at least one of the six paid
+transpositions (16.7%). This bill counts every address letter, while granting
+free `GL3(F2)` scalar mixing and the third role with no separate hosting or
+allocation charge. Those are explicit optimistic assumptions. The endpoint
+restores the dirty third role; no clean state, uncharged undo or extra address
+letter is allowed. This screen is local only: it has not identified an
+occurrence in the selected bit supplier, whose current selected baseline is
+complex-primitive-bound. No recursive moment or selected κ change follows.
+
+The selected accepted record is `κ=836330098538459/500000000000000000`
+(`certificates/selected-result.json`, SHA256
+`10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32`). Its
+review is `docs/research/community-batch1099-review.md`, source manifest is
+`research/community-frontier1094-audit/SELECTED-SOURCE.json`, certificate is
+`research/community-frontier1094-audit/ASSEMBLY.json` (SHA256
+`1122cb3a575f8c788345fc1bf4c108ac7c3250fb81a5988671f5eb04b3ab3045`), and
+reproduction command is `make frontier1094-verify`. The selected-record
+binding-only baseline was checked with the pinned `scripts/research_baseline.py`.
+
+## Exact result
+
+No word of at most five calls realizes this target. The direct six-call word
+therefore has exact minimum six in the declared model. The search enumerates
+the complete radius-two ball from the identity and radius-three ball from the
+target using all 84 involutive factors. The shell counts are `[1,84,4697]` and
+`[1,84,4697,232692]`, for 4,782 and 237,474 states; their intersection is
+empty. Every coefficient is an exact 24-bit element of `F2[S4]`, not a test
+under one representation of the address group.
+
+Completeness follows by splitting any word of length at most five as `A B`
+with `|A|≤2` and `|B|≤3`. Every factor squares to identity, so `A=T B^-1`
+for target `T`; a shorter word would put the same state in both enumerated
+balls. A separate literal group-algebra matrix multiplier checks every factor
+is involutive, verifies the six-call direct construction, rejects all six
+single-call omissions, and rejects a changed third-role target. The optimized
+meet uses byte-packed coefficients and exact right-convolution tables.
+
+The complete search took 2.29 seconds and peaked at 60,576 KiB RSS under
+Python 3.12.3, well inside the 110-second/1-GiB implementation limits. No
+partial search is used in the result. The full seed, base, catalogue and
+selected-record pins are in [TASK.json](TASK.json) and [SOURCES.json](SOURCES.json).
+
+## Reproduce
+
+From the repository root:
+
+```sh
+python3.12 -I -B research/s4-two-target-cycle-seed6e525/verify.py
+python3.12 scripts/research_package.py --package research/s4-two-target-cycle-seed6e525 --check-only
+python3.12 scripts/research_package.py --package research/s4-two-target-cycle-seed6e525
+```
+
+To regenerate the deterministic receipt before replay, use
+`python3.12 -I -B research/s4-two-target-cycle-seed6e525/verify.py --write`.
+The direct word is the positive control; six omitted-call words and the
+changed endpoint are semantic negative controls. The checker is an exact
+bounded proof for this finite alphabet, not a proof assistant result.
+
+## Live overlap and coordination
+
+The final open-PR census covered 405 PR records via REST at 2026-10-14
+07:40–07:43Z. The nearest scopes were PR #1478 (the UT3 two-central-commutator
+endpoint at two or three roles), PR #1705 (four coordinate-SWAP targets in
+`F2[GL2(F2)]` at four roles), and PR #1711 (two affine targets in
+`F2[AGL2(F2)]` at two roles). Discussion #1317 contains current AGL2 claims;
+the cached complete discussion scan (351 records, fetched 07:28Z), plus direct
+current reads of #1167 (through 07:22Z) and #1317 (through 07:31Z), showed no
+S4/adjacent-transposition claim.
+The S4 group, 4-cycle target, three-role dirty endpoint and adjacent-
+transposition alphabet are distinct from those scopes. A current fresh
+Discussion fetch and new claim post are unavailable:
+GitHub GraphQL reports the account rate limit exceeded, and the installed REST
+API has no create/update endpoint for Discussion comments. This local claim
+card and package are being used while that limitation holds. The result is
+submitted in [PR #1720](https://github.com/CrocSwap/integer-mult-bounds/pull/1720).
+
+## Open next questions
+
+No immediate next question is justified: the complete six-call minimum closes
+the proposed S4 endpoint and removes the candidate saving. Searching another
+target tuple or alphabet requires a new mechanism, separate overlap check and
+fresh cost screen.
+
+## Credit
+
+The formal rational three-register bound and UT3 central-commutator study are
+the immediate predecessors. Their changed alphabets and endpoints are pinned
+in [SOURCES.json](SOURCES.json); live nearby PR heads are recorded in
+[TASK.json](TASK.json). Prepared by Cinna Davis with substantial OpenAI Codex
+assistance in scoping, exact search, verification and exposition. New work is
+Apache-2.0.

research/s4-two-target-cycle-seed6e525/SOURCES.json

@@ -0,0 +1,54 @@
+{
+  "schema_version": 1,
+  "base_commit": "0b1cccf86e2f5132ed286a26ef0a681a495740ea",
+  "files": [
+    {
+      "path": "certificates/selected-result.json",
+      "usage": "input",
+      "git_blob": "ae6a99f28aa468649eb66a361e764d769363bf11",
+      "sha256": "10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32"
+    },
+    {
+      "path": "docs/research/agent-task-catalog.json",
+      "usage": "provenance",
+      "git_blob": "06c96f5579bb67ed1b86d334e78fbed7da6945f1",
+      "sha256": "6686ad58f1fbdbc87134a78901b60fd53a20d3984355e91defbbebc9e9fa8075"
+    },
+    {
+      "path": "docs/research/community-batch1099-review.md",
+      "usage": "provenance",
+      "git_blob": "6cb6ec061a071a06a2d18cd4dc3e9a2b05d28099",
+      "sha256": "7e76d2c38e5fce30587835b7251df1430488ae4dd372611ef69a3f03f476b6cb"
+    },
+    {
+      "path": "research/bit-dihedral-commutator/README.md",
+      "usage": "input",
+      "git_blob": "b7619a5d56f32daf7f48d6cc3a0e260c5a333510",
+      "sha256": "4f3f9df6d6d96a26ccb0fcd2ef947cce5d53b0d992da22e866f1e025be4c0492"
+    },
+    {
+      "path": "research/bit-three-register-lower-bound/README.md",
+      "usage": "input",
+      "git_blob": "25e41c370c101a33a6deb978de08b03ed7539b79",
+      "sha256": "c12b33ef388deece0cbf891505a2ce2c0779a0c3a6cf1f0f7f7039bab59cbebd"
+    },
+    {
+      "path": "research/community-frontier1094-audit/ASSEMBLY.json",
+      "usage": "provenance",
+      "git_blob": "5fd676df060f3b450782fdf5ff5f6beb63a5eeb0",
+      "sha256": "1122cb3a575f8c788345fc1bf4c108ac7c3250fb81a5988671f5eb04b3ab3045"
+    },
+    {
+      "path": "research/community-frontier1094-audit/SELECTED-SOURCE.json",
+      "usage": "provenance",
+      "git_blob": "6a75493790b4eff9c4a98051ef8b01d6aad0d5a8",
+      "sha256": "79b3de5db3dbcb9829789ba58e9e45758926031e2a7d80667a08429d369be53b"
+    },
+    {
+      "path": "scripts/research_baseline.py",
+      "usage": "input",
+      "git_blob": "6b607b36eefd65718120ade3c8187681f11bf879",
+      "sha256": "59d9b04dc33e1bb3f81506d187243435de4902138a038b935e36ebaa5f37db67"
+    }
+  ]
+}