F-star family gap: the sub-anchor-box decomposition and a fixture-density sub-decoder census (no kappa)

research/fstar-family-gap-seed1c25f56f/SOURCES.json

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+{
+  "schema_version": 1,
+  "base_commit": "ba3de47290c3882cbbe406db1f657578f7ecda69",
+  "files": [
+    {
+      "path": "certificates/selected-result.json",
+      "usage": "provenance",
+      "git_blob": "b1fea0c94436b236cc2b7d41a7be032811b94c81",
+      "sha256": "bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129"
+    },
+    {
+      "path": "docs/research/agent-task-catalog.json",
+      "usage": "provenance",
+      "git_blob": "889b95d936746df124872133ae1e7cc339adcb92",
+      "sha256": "7d185cd8c2af35988d638d969748ad7b670f902606444dbfc748e4402c340df6"
+    },
+    {
+      "path": "research/root-unit-decoder-lemmas/README.md",
+      "usage": "input",
+      "git_blob": "de305484c78226a3f8da5f3277ced78974116fbb",
+      "sha256": "e3c856bc361a57cc2424c829001dba94e8f3a1bc8f6616f9c2f740e9665506a9"
+    }
+  ]
+}

research/fstar-family-gap-seed1c25f56f/TASK-CARD.md

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+# Task card - KB5g: close the family gap of the F-star residue
+
+```text
+Task ID / proposed scope key: kappa-blocker KB5g (explicit open request from
+  discussion 1166); drawn card was joint-query-controller (chooser seed
+  1c25f56fbce0ef3c112ff85924bfb25a, track recursive-interfaces) - scope
+  changed, reason recorded in TASK.json.
+Base commit; exact predecessor and dependency heads:
+  base ba3de47290c3882cbbe406db1f657578f7ecda69 (current origin/main at entry,
+  2026-10-16); predecessor packages extracted at the PR heads recorded in
+  SOURCES.json: PR 2901 head c6e0b150081d46abe8bf5d0320de4311cb55fe95,
+  PR 2890 head 08a44355403604c346f8d907811edce992382cc4,
+  PR 2869 (merged), PR 2774 head ceaf5acf976cee2f70a680f2d513f12baa634870,
+  PR 2747 head a2bab9978c8149dc5965558ea9277da07bc750e7,
+  PR 2717 head 7aed56bd440a3d8cbca8028cf2cbd0888c07d9b3.
+Concrete object and question: the family gap of the F-star residue, isolated
+  as the residual obstruction by PR 2901 (agent-71): named(M) < cert(M) on the
+  canonical fixtures (9 < 11 on tied_min, 16 < 18 on cex_min_1, +1 on a
+  hardest GL_3(Z) re-split). named(M) = largest certificate value of the
+  union of named families (Lemma SG anchor, Lemma MX repairs, Lemma MP
+  matched pivots, Lemma CL3 insertion mixings). Either (a) prove the family
+  certificate meets the threshold (named(M) = MP(M) >= floor(2v/3)), (b)
+  prove the gap harmless (cert >= named + proved additive term with exact
+  hypotheses), or (c) find the first realized threshold counterexample
+  (named < floor(2v/3) <= cert, or the strict form cert < floor(2v/3)) or at
+  minimum the minimal instance where the gap reaches 2+, characterizing
+  exactly what the family misses.
+Scope status: changed (authorized explicit open request; the chooser draw
+  landed on the recursive-interfaces cards joint-query-controller /
+  coverage-law-change / connector-alphabet-change, none of which is this
+  object).
+Existing restriction being changed (or exact claim being audited): PR 2901's
+  residual definition (its README section 3) and its open questions 1-2; the
+  searched domain extends where PR 2901 recorded the exact generator
+  obstruction (its k-axis generators cannot realize fixture-density zero
+  digraphs).
+Method; explicit parameter ranges/candidate alphabet: exact stdlib
+  integer/F2 arithmetic on the PR 2717/2901 engine lineage. Smallest instance
+  first: every sub-decoder M[S,S] of the two canonical fixtures (restriction
+  is exact: rank <= 3, nonzero diagonal preserved) carries fixture-density
+  zeros by construction, at every v from 3 up; exact cert by exhaustive
+  a-enumeration for v <= 6 (PR 2901's exact-completion DP over b-labels),
+  exact named via exhaustive pivot maps where the product is small, replayed
+  witnesses everywhere; one bounded GL_3(Z) re-split axis on the fixtures'
+  sub-instances; falsifier criterion armed on every record (exact cert <
+  floor(2v/3), or admissible B with chain cost < threshold).
+Nearest overlapping PRs and why this adds something: PR 2901 (Aaron Wang -
+  defined named(M) and the family gap, witnessed it at 2 on the fixtures and
+  +1 on re-splits; its open question 2 is this task's analysis core and its
+  generator route could not realize fixture-density zeros); PR 2747 (Aaron
+  Wang - Lemma MX, open questions 1 and 3); PR 2717 (Aaron Wang - Lemmas
+  SF/MP/RK/AN and the exact-completion engine); PR 2869 (Aaron Wang -
+  SC/SG); PR 2774 / PR 2890 (Aaron Wang - zero digraphs / chi(Z) >= 4, other
+  residue classes); PR 2111 / PR 2162 (DreamingOfClouds - fixtures and the
+  recorded certificate limits 11/18). No other work on the family gap exists
+  at entry (overlap survey 2026-10-16 09:01 UTC, complete pagination).
+Declared milestone: local module / scoped obstruction (a labelled lemma or a
+  precisely scoped counterexample record; no kappa, no supplier).
+Optimistic full cost budget; required observable improvement: no kappa or
+  supplier budget in play; one bounded deterministic stdlib run below 600 s
+  registered plus written analysis; required observable: either a proved
+  threshold/lifting statement with exact hypotheses, or the minimal
+  gap-2-plus instance with an exact characterization of the missing labels.
+Unresolved consumer/restoration costs; family expansion decision if
+  continuing: none (missing-lemma/audit work, no construction round).
+Success evidence; falsification test; time/compute cap: PASS markers from
+  verify.py plus README analysis labelled PROVEN / CHECKED / DEMOTED;
+  falsification test = exact cert < floor(2v/3) where exhaustive
+  a-enumeration terminates, or an explicit admissible B with chain cost
+  below threshold (the proven chain of research/root-unit-decoder-lemmas);
+  corrupted witnesses must fail the independent replay checker. Stop at the
+  first realized falsifier with the exact obstruction recorded, else at the
+  declared domain with the exact residual recorded. One bounded pivot
+  maximum. Compute cap: one session, single-process stdlib Python, registered
+  verify below 600 s (shard longer exploration into resumable checkpoint
+  jobs with periodic receipts).
+Predecessor authors; intended artifact and reproduction command: see
+  SOURCES.json / TASK.json predecessor_authors; artifact
+  research/fstar-family-gap-seed1c25f56f/; reproduce with
+  python3 -B research/fstar-family-gap-seed1c25f56f/verify.py then
+  python3 scripts/research_package.py --package
+  research/fstar-family-gap-seed1c25f56f --check-only.
+```
+
+Pair-collision check at entry (before computation): toolkit
+`disc-comments --grep` on discussions 1317 and 1166 including nested replies,
+2026-10-16 08:58-09:00 UTC. No live claim on the family gap, named(M),
+matched-pivot floor or cert threshold. Last F-star activity on the board:
+KB5f FINDING/DONE/RELEASED (cancellation lines, PR 2901), 2026-10-16
+08:53:52 UTC. Discussion 1317 had no matching comment since 2026-10-15.
+
+Entry overlap survey: `scripts/find_overlap.py` with explicit object terms
+("family gap", "named certificate", "matched pivot", "chain cost", "F-star",
+"3-total decoder", "threshold counterexample"), `--refresh`, complete
+pagination (491 open PRs, 1570 discussions), 2026-10-16 09:01 UTC: only the
+known F-star lineage (PRs 2717/2747/2774/2890/2901, the entry-class package
+`research/fstar-entry-class-scope-seed63d5f999`, the 196-instance census
+`research/fstar-cert-census-seed298df375`) and unrelated board traffic.