F-star family gap: Lemma BPP's without-loss question answered NO - box-completeness fails at tau >= 1, minimal counterexample v = 3 (no kappa)

research/fstar-box-withoutloss-seed7f3a1c/CERTIFICATE.json

@@ -0,0 +1,91 @@
+{
+ "base_commit": "0614abaef427f123d9b844f85a759e64ee3ab29c",
+ "digest": "0b5b55f84375b37c",
+ "files": {
+  "README.md": "582dd8a6651cfe8d4798b3134112c3e1165d0ed3c6d57c2761c58f04265008f4",
+  "TASK-CARD.md": "1e10ab4f4a657e6c72179957957299e9a97267cec22bbd35410a05c6c224b8f2",
+  "TASK.json": "c2aaa7096f8db10ecdd328a6bba2553f774c06df34c6e36ecd141cae290bc509",
+  "box_search_kj.py": "f341925160cc8476dd3484d3ba513a3e1428804fbd9c804e9db62a9c3e3a85f8",
+  "data/box-search-kj-receipt.json": "26dae32a86d1a3fae7f6cecf6c3e7e4b410a2ffe4a70105c1efe88b9fbf7f011",
+  "data/escape-receipt.json": "9ac402092558d149cb18259371712a461a3dc2deefd63b92c070045592f152df",
+  "data/realize-kj-receipt.json": "93ec89e6fc2ce60fdcddd1516fd31fd29686fc5efd4dd7b6831e0ed45e006cce",
+  "escape_classify.py": "e8d2bdadc6da4d30c52866622ef5a161fb83df085969dbfb74500b2499725804",
+  "predecessor/PINS.json": "1a0ff7143be416017f43adf00a28e41b14daf8e67de7a3d318593c16b62c1922",
+  "realize_kj.py": "f08702f9141b900b71bd0d97d466f2d7bebb0ffcda6a93ac7560817765a8033d",
+  "verify.py": "e99b62688b37fa3bca3c79663a2fa9887f39dc65e904bc14ac200e99292e5369"
+ },
+ "headline": {
+  "box_violations": 0,
+  "cex_box": 2,
+  "cex_cert": 3,
+  "cex_r_box": 2,
+  "cex_tau": [
+   2,
+   2,
+   2
+  ],
+  "counterexample_P": [
+   [
+    -1,
+    -3,
+    0
+   ],
+   [
+    1,
+    -1,
+    -2
+   ],
+   [
+    3,
+    3,
+    3
+   ]
+  ],
+  "counterexample_Q": [
+   [
+    1,
+    -3,
+    2
+   ],
+   [
+    3,
+    -1,
+    -2
+   ],
+   [
+    3,
+    -1,
+    2
+   ]
+  ],
+  "escape_classes": {
+   "E1": 466,
+   "E2": 1768,
+   "E3": 6,
+   "E4": 18
+  },
+  "pr2963_pool_escapes": 19,
+  "projection_drops": 1788,
+  "realizable_instances": 360,
+  "realized_counterexamples": 3,
+  "v2_configs": 2842,
+  "v2_proven": "box = cert = 2 at v = 2, tau >= 1",
+  "v3_candidates": 27,
+  "v3_configs": 3023,
+  "verdict": "box-completeness for tau >= 1 REFUTED; minimal counterexample v = 3, cert = 3 > box = 2"
+ },
+ "markers": [
+  "PASS_KJ_PINS",
+  "PASS_FG_CONTROLS",
+  "PASS_FG_CALIBRATION",
+  "PASS_KJ_ANCHORS",
+  "PASS_KJ_TAU0_DEMO",
+  "PASS_KJ_COUNTEREXAMPLE",
+  "PASS_KJ_JOBS",
+  "PASS_KJ_ESCAPE",
+  "PASS_KJ_REALIZE",
+  "PASS_KJ_WITHOUTLOSS"
+ ],
+ "package": "fstar-box-withoutloss-seed7f3a1c",
+ "schema_version": 1
+}

research/fstar-box-withoutloss-seed7f3a1c/SOURCES.json

@@ -0,0 +1,24 @@
+{
+ "base_commit": "0614abaef427f123d9b844f85a759e64ee3ab29c",
+ "files": [
+  {
+   "git_blob": "b1fea0c94436b236cc2b7d41a7be032811b94c81",
+   "path": "certificates/selected-result.json",
+   "sha256": "bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129",
+   "usage": "provenance"
+  },
+  {
+   "git_blob": "889b95d936746df124872133ae1e7cc339adcb92",
+   "path": "docs/research/agent-task-catalog.json",
+   "sha256": "7d185cd8c2af35988d638d969748ad7b670f902606444dbfc748e4402c340df6",
+   "usage": "provenance"
+  },
+  {
+   "git_blob": "de305484c78226a3f8da5f3277ced78974116fbb",
+   "path": "research/root-unit-decoder-lemmas/README.md",
+   "sha256": "e3c856bc361a57cc2424c829001dba94e8f3a1bc8f6616f9c2f740e9665506a9",
+   "usage": "input"
+  }
+ ],
+ "schema_version": 1
+}

research/fstar-box-withoutloss-seed7f3a1c/TASK-CARD.md

@@ -0,0 +1,80 @@
+# Local task card (KB5j) — written before substantial computation
+
+```text
+Task ID / proposed scope key: KB5j / fstar-box-withoutloss (scope key of the drawn
+  card bit-new-register-model: 7084092dd726e7ba1f2f; scope CHANGED, see below)
+Base commit; exact predecessor and dependency heads:
+  base 0614abaef427f123d9b844f85a759e64ee3ab29c (origin/main at worktree creation,
+  2026-10-16); predecessors PR 2963 head ee755e682b3b30d493e764d3be50003b13b69b4a,
+  PR 2936 head f0ef06f13e4de3e50c211c880bdc6891c09620a1, PR 2914 head
+  3f7781efccdf10fa61a1e0a6f11c191579b21280 (vendored byte-exact, pins asserted).
+Concrete object and question:
+  Open problem F-star (research/root-unit-decoder-lemmas section 7, PR 1590):
+  cert(M) >= floor(2v/3) for exact 3-total rational decoders. Object here is the
+  LAST named blocker of the family-gap line (PR 2963's exact statement):
+  Lemma BPP's WITHOUT-LOSS QUESTION at tau >= 1 - may one assume, without loss,
+  that an attaining L8 witness satisfies Lemma BPP conditions (i)+(ii) (odd tied
+  overlap per kept line, even off-anchor shared parity on forward zero arcs),
+  i.e. is box(M) = cert(M) for every exact 3-total decoder with tau(t) >= 1
+  everywhere (box-completeness)?
+Scope status: changed; reason for refinement or change:
+  Authorized "explicit open request" scope change (assignment records this reason):
+  the drawn cards (bit-new-register-model / bit-helper-contract, chooser seed
+  0194e0b2ffbe43b4e67165afcf5ee6cd, track bit-mechanisms) are not this object;
+  discussion 1166 carries the explicit open request "Lemma BPP's without-loss
+  question remains the exact blocker" (PR 2963 body), which this assignment takes.
+Existing restriction being changed (or exact claim being audited):
+  Audit/settle PR 2914 open question 1 + PR 2936 open question 1 as narrowed by
+  PR 2963: box-completeness for tau >= 1 (Lemma SB (iii) PROVEN excludes tau = 0).
+Method; explicit parameter ranges/candidate alphabet:
+  (1) Classify the escape mechanism of PR 2963's 21 escaping witnesses (v = 3..5
+  fixture sub-decoders) exactly; (2) prove or refute the line-by-line re-pairing
+  (repair) lemma - smallest instance first: v = 2 (case analysis on anchor data),
+  then v = 3 exhaustive label-space checks and a smallest-first decoder search
+  (exact cert vs exact box, both sides certified by replayed witnesses or exact
+  enumeration; int-only arithmetic); (3) if neither falls, isolate the precise
+  parity condition that blocks repair with a minimal demo.
+Nearest overlapping PRs and why this adds something:
+  PR 2914 (box, Lemmas SB/LB), PR 2936 (Lemma BPP, tau >= 1 necessary, v = 7..11
+  census), PR 2963 (tied_min v = 11 exact, box weak cell closed, BPP strictness
+  at tau = 0). None settles the without-loss question; this is its named blocker.
+Declared milestone: local module (scoped lemma/obstruction package; no kappa).
+Optimistic full cost budget; required observable improvement:
+  No kappa claim; milestone is a PROVEN lemma, a minimal counterexample, or an
+  exact isolated obstruction. Compute cap: about 25 minutes of heavy search in
+  blocking chunks with receipts; exactness over speed (never scan-and-trust).
+Unresolved consumer/restoration costs; family expansion decision if continuing:
+  Not applicable at this milestone; F-star closure remains conditional on the
+  chain cost >= cert and the box floor (PR 2914), whose hypotheses must be
+  re-verified link by link if box-completeness falls.
+Success evidence; falsification test; time/compute cap:
+  Success: (a) PROVEN box-completeness at tau >= 1 (written proof + machine
+  checks), or (b) minimal tau >= 1 decoder with cert > box-max, both sides exact
+  with replayed witnesses, or (c) exact obstruction to the repair/projection
+  lemma with a minimal demo and one stated condition. Falsification test of any
+  proof: smallest v = 2/v = 3 instances enumerated from the definition; corrupted
+  witness rejection controls. Time cap: one bounded pivot maximum.
+Predecessor authors; intended artifact and reproduction command:
+  PR 1590 DreamingOfClouds (Claude Opus 5.5 (Anthropic) via Claude Code
+  assistance); PR 2111 DreamingOfClouds; PR 2717/2747/2774/2869/2890/2901/2914/
+  2936/2963 Aaron Wang (navinor) - all verified with gh pr view --json author,title
+  at current ground truth. Artifact: research/fstar-box-withoutloss-seed7f3a1c/
+  with verify.py (python3 -B verify.py).
+```
+
+Entry checks recorded: chooser draw (seed `0194e0b2ffbe43b4e67165afcf5ee6cd`,
+track `bit-mechanisms`, cards `bit-new-register-model` / `bit-helper-contract`) —
+saturated for this object, the recorded reason for taking the assigned explicit
+open request. Pair-collision checks (toolkit `disc-comments --grep`, discussions
+1317 + 1166, nested replies) at entry found no live claim on this object; the
+last family-gap activity is KB5i's FINDING/DONE/RELEASED (PR 2963, 2026-10-16
+14:36:56 UTC), which this assignment extends. Entry overlap survey
+(`find_overlap.py`, object proper names and synonyms, `--refresh`, complete
+pagination: 240 open PRs, 1660 discussion records, 14:44-14:45 UTC) found only
+the known F-star lineage and unrelated traffic. Recorded anchors reproduced from
+pristine predecessor copies before computation (`PASS_FG_CALIBRATION
+tied_min=11/12 cex_min_1=18/19 mp=9/16 chain=22 reach=0/12,8/19`;
+`PASS_KB5I_NAMED records=12 named-exact=12 named=9x12 gap2-proven=12`;
+`PASS_KB5I_BOXREPLAY in-box=12/12`; `PASS_BOX_SEARCH bpp-strict-tau0=yes
+bpp-strict-tau1=no tau-ge1=60 hard=0 box-violations=0`; PR 2936 tau = 0 demo
+cert = 2 > box = 1 asserted value-by-value).

research/fstar-box-withoutloss-seed7f3a1c/TASK.json

@@ -0,0 +1,77 @@
+{
+ "actual_scope": "Kappa-blocker KB5j (authorized explicit open request): SETTLE LEMMA BPP's WITHOUT-LOSS QUESTION of open problem F-star (cert(M) >= floor(2v/3) for exact 3-total rational decoders): may one assume, without loss at tau >= 1, that an attaining L8 witness satisfies Lemma BPP conditions (i)+(ii), i.e. is box(M) = cert(M) for every exact 3-total decoder with tau(t) >= 1? Either prove box-completeness (projection/repair lemma), construct the minimal tau >= 1 counterexample with both sides exact, or isolate the precise parity obstruction to repair with a minimal demo.",
+ "ai_assistance": "Substantial AI assistance: this package was produced by Aaron Wang with help from opencode (opencode agent harness). No model named per orchestrator attribution rules. No independent human review claimed.",
+ "author": "Aaron Wang (with help from opencode)",
+ "base_commit": "0614abaef427f123d9b844f85a759e64ee3ab29c",
+ "bottleneck_link": "F-star (cert(M) >= floor(2v/3)) is the named route to closing every open T = 3 window at once (research/root-unit-decoder-lemmas section 7). PR 2963 named Lemma BPP's without-loss question as the exact blocker of the box-completeness route. This package settles it: box-completeness FAILS at tau >= 1 (minimal counterexample v = 3, cert = 3 > box = 2), so the completeness link of the PR 2914 chain is dead; the box-floor link and cost >= cert survive, and the surviving narrowed question (shareable class c_j = gamma) is stated exactly.",
+ "evidence": "VERDICT: Lemma BPP's without-loss question is answered NO at tau >= 1. PROVEN: box = cert = 2 at v = 2 for tau >= 1 (written proof + exhaustive machine check of all 2842 scope-consistent patterns), so v = 3 is minimal. PROVEN: minimal tau >= 1 counterexample at v = 3: P = [[-1,-3,0],[1,-1,-2],[3,3,3]], Q = [[1,-3,2],[3,-1,-2],[3,-1,2]], M = diag(8,8,12), tau = (2,2,2): cert = 3 (replayed attaining witness a = b = [1,2,4], order [2,1,0]; independent enumeration) > 2 = box (PROVEN upper bound: Lemma RT triangular rank + shareable totals r_box = 2; explicit box witness). PROVEN: the obstruction's exact form - in the box only totals with c_j = gamma (shareable) can enter K, and when every forward pair is a zero arc K[S,S] is unit triangular, so |S| <= r_box; the third total (c_2 = 1 > gamma = 0) is only usable via non-tied shared bits, i.e. BPP condition (i) fails on every attaining witness. CHECKED: PR 2963's 19 distinct escapes are all cosmetic (E1); 360 realizable tau >= 1 instances show 1788 strict projection drops (E2 1768) with 0 box violations, so projection is not without loss either. CHECKED: 3023 v = 3 combinatorial configs, 27 gap candidates, 3 realized as counterexamples, 3 unrealizable (column-attainment), 21 not realized in cap (alphabets up to {+-1..+-7}/{+-2..+-6,0}).",
+ "existing_restriction_being_changed": "Audits/settles PR 2914 open question 1 + PR 2936 open question 1 as narrowed by PR 2963 (box-completeness for tau >= 1). The PR 2914 route (b) required cert = box-max at tau >= 1 plus a box floor; this package refutes the equality.",
+ "falsification_test": "Any counterexample claim is falsified by replaying the stored witnesses and recomputing both rungs from the definition: verify.py re-enumerates cert and box INDEPENDENTLY of the vendored engine (section-7 enumeration) and asserts 3 and 2; the box upper bound is a written rank proof, not a scan; corrupted-witness rejection and the tie/scope identities are asserted on every instance; the v = 2 completeness claim is falsifiable by any of the 2842 enumerated patterns failing.",
+ "intended_artifact": "research/fstar-box-withoutloss-seed7f3a1c/ with verify.py (python3 -B research/fstar-box-withoutloss-seed7f3a1c/verify.py).",
+ "method": "Smallest instance first. (1) Classify the escape mechanism of PR 2963's 21 escaping witnesses exactly (BPP projection, conditions (i)/(ii), per-escape-line bits). (2) Exhaustive combinatorial reduction: cert and box of a decoder depend only on (pi, sg, zero-digraph Z, a0, b0); cert is over-approximated by the full-label value cert_max(Z) (monotone in label freedom), so cert_max(Z) <= box proves completeness on that datum and cert_max(Z) > box is a candidate. v = 2 all scope-consistent patterns; v = 3 all patterns up to line/total relabeling. (3) Realize candidates by exact 3-total integer decoders via the odd/even valuation recipe (m_j = 0 normalization; odd on anchor bits), per-q-row exhaustive search with seeded P restarts, alphabets {+-1,+-3}/{+-2,0} then {+-1..+-7}/{+-2..+-6,0}. (4) Certify every value by replayed witnesses, exact enumeration, an independent section-7 enumeration, and written rank bounds (Lemma RT); int-only arithmetic.",
+ "nearest_overlapping_prs": "PR 2914 (sub-anchor box, Lemmas SB/LB), PR 2936 (Lemma BPP, tau >= 1 necessary, v = 7..11 census), PR 2963 (tied_min v = 11 exact, box weak cell closed, BPP strictness at tau = 0). None settles the without-loss question; this package does, negatively, with the minimal counterexample.",
+ "open_questions": [
+  {
+   "blocked_by": "No realizability characterization beyond Lemma SG + the column-attainment condition; the search so far is bounded (not a non-existence result).",
+   "first_step": "Characterize the (a0,b0,Z) patterns forced by rank-3 factorizations M = P Q^T with prescribed odd/even valuations (odd-term parity of every M_ts); extend the seeded realization search with a valuation-level enumeration (m_j in {0,1}) instead of random restarts.",
+   "question": "Are the 21 remaining v = 3 combinatorial gap patterns (a0, b0, Z) realizable by exact 3-total decoders?",
+   "why_it_matters": "All 21 lie in the shareable class (r_box = 3), so a realization refutes box-completeness there too, while a proof of unrealizability would prove box-completeness at v = 3 in the shareable class - the class holding all prior evidence."
+  },
+  {
+   "blocked_by": "8^4 label enumeration per configuration times the zero-pattern lattice needs a DP or symmetry reduction before it is tractable.",
+   "first_step": "Extend the job-B combinatorial enumeration to v = 4 with a cert-side subset DP over the same (a0,b0,Z) lattice; then realize any gap datum as in job D.",
+   "question": "Box-completeness in the shareable class (every total has c_j = gamma) at v >= 4: is cert = box there?",
+   "why_it_matters": "It is the exact surviving form of the without-loss question once Lemma RT's obstruction is excluded; every previous box = cert datapoint sits in this class."
+  }
+ ],
+ "optimistic_full_cost_budget": "No kappa claim; milestone is the settled question itself. Heavy compute capped: combinatorial enumeration v = 2..3 (seconds), 360-instance sweep (seconds), realization caps 4000-6000 P-restarts per candidate (about 26 s total), all blocking with receipts.",
+ "original_card": {
+  "acceptance_gate": "Name the changed hypothesis, all counted roles and a uniform identity; finite column tests alone do not certify it. A scoped local result may be retained without a complete supplier. Expanding it into further construction rounds requires the family-level budget and obligation gate in research-family-status.md; an optimistic passing relaxation alone is not acceptance.",
+  "allowed_variations": [
+   "Choose a fixed larger register count or a specified nonabelian operation family."
+  ],
+  "bottleneck_link": "A small-width completed saving changes the bit primitive; a finite-address identity alone is insufficient.",
+  "first_experiment": "First compare the alphabet, role count and length with the fully replayed #1974 census and #2027 mixed-root tariff. Identify a changed premise and a legal native implementation; then price one smallest completed instance before searching larger words.",
+  "first_step": {
+   "command": [
+    "python3",
+    "-B",
+    "scripts/research_baseline.py"
+   ],
+   "expected": "PASS_RESEARCH_BASELINE_BINDINGS_ONLY",
+   "inspect": "Read Model section 1 of the three-register bound and identify the exact excluded free operations.",
+   "scope": "Positive control / source-bound orientation only; this is not a new construction or full cold replay."
+  },
+  "id": "bit-new-register-model",
+  "methods": [
+   "Construct the smallest exact identity in a newly specified legal operation alphabet.",
+   "Derive a target-specific relaxation bound for that same explicit alphabet."
+  ],
+  "overlap_terms": [
+   "three-register",
+   "nonabelian",
+   "group algebra"
+  ],
+  "question": "Can an explicitly larger or nonabelian register model escape the scoped three-register obstruction?",
+  "scope_change_triggers": [
+   "Remaining inside the already-excluded three-register abelian model.",
+   "Repeating the two-register GL3 census or a covered three-register length without changing a hypothesis; a formal group-algebra word without native conversion is a partial result."
+  ],
+  "starting_points": [
+   "docs/research/research-family-status.md",
+   "research/bit-three-register-lower-bound/README.md",
+   "certificates/selected-result.json",
+   "research/bit-gl3-mixed-targets-length10/README.md",
+   "docs/research/community-efficient1000-review.md"
+  ]
+ },
+ "outcome": "completed",
+ "parameter_ranges": "v = 2..4; three totals; label space F2^3 bit masks; integer P, Q entries in {-7..7}; 2-adic valuations 0..3; zero digraphs all patterns at v = 2 and all Lemma-SG-consistent patterns at v = 3 (up to line and total relabeling); 360 realizable instances (320 random valuation-tuned v = 3..4 + 40 GL_3(Z) fixture transforms at v = 3); realization caps 4000-6000 P-restarts per candidate with exhaustive q-rows.",
+ "predecessor_authors": "PR 1590 DreamingOfClouds (Claude Opus 5.5 (Anthropic) via Claude Code assistance); PR 2111 DreamingOfClouds; PRs 2717/2747/2774/2869/2890/2901/2914/2936/2963 Aaron Wang (navinor); all verified at current ground truth with gh pr view --json author,title on 2026-10-16.",
+ "schema_version": 1,
+ "scope_change_reason": "Authorized 'explicit open request' scope change to kappa-blocker KB5j, recorded per the assignment (discussion 1166; PR 2963's exact statement - 'Lemma BPP's without-loss question remains the exact blocker' for box-completeness - is the named open request). The chooser OS draw (seed 0194e0b2ffbe43b4e67165afcf5ee6cd, track bit-mechanisms) proposed bit-new-register-model and bit-helper-contract; neither is this object, and the assignment takes the KB5j request instead.",
+ "scope_status": "changed",
+ "stop_rule": "Stop when the without-loss question is settled (proved, counterexample, or exact obstruction with minimal demo). Achieved: counterexample, PROVEN minimal at v = 3. No further search without a new hypothesis.",
+ "task_id": "bit-new-register-model",
+ "time_compute_cap": "One bounded pivot maximum; registered replay under 600 s (observed under 2 min); realization search capped per candidate and recorded as not-found-in-cap, never as unrealizable."
+}