From 3f7781efccdf10fa61a1e0a6f11c191579b21280 Mon Sep 17 00:00:00 2001 From: Aaron Wang Date: Fri, 16 Oct 2026 06:35:19 -0400 Subject: [PATCH] KB5g: close the F-star family gap - sub-anchor-box decomposition and fixture-density sub-decoder census Package research/fstar-family-gap-seed1c25f56f. Verdict: the family gap named(M) < cert(M) isolated by PR 2901 is harmless on the entire realized domain and is exactly characterized as the interior of the sub-anchor box. PROVEN: Lemma SB - a0(t) intersect b0(t) is exactly the set of tied totals, so a sub-anchor pair certifies line t iff |a intersect b| is odd; tau = 1 lines force the unique tied bit on both labels, tau = 2 lines exactly one of the two tied bits, and tau = 0 lines have no certifying box pair. Lemma LB - named <= form_r, box <= cert, and named = max(MP, MX) when S_gamma is empty, which makes tied_min's named value exactly 9. CHECKED, stdlib verify.py, deterministic, about 230 s, certificate digest 031927ba33ed45a0: controls 5 groups; recorded-anchor calibration (MP 9/16 exhaustive over 4096/6144 pivot maps, tied_min hill-climb witness 11, cex_min_1 stored witness 18 replayed, chain 22 on the stored pairing, reach 0/12 and 8/19); fixture ladder with per-line label diffs (every attaining label pair lies in the sub-anchor box; box = cert exactly at 11/11 and 18/18 against PR 2111's recorded certificate limits); smallest-first sub-decoder census of both fixtures at fixture density (1410 records at v = 3..6; 21 deep exact records with cert = box = form_r = named exactly, gap_named = 0; 0 falsifiers; 0 below-threshold; min named ratio 2/3). DEMOTED: the matched-pivot floor alone is strictly weaker than cert on the fixtures (9 < 11, 16 < 18); a general additive lifting bound is contradicted by the data shape (gap 0 with up to 3 cancellation lines, gap 2 at 12 of 12); no threshold counterexample was found and tied_min at v = 12 remains the minimal realized gap-2 instance in the searched domain. F-star itself is not proved and not refuted. Predecessor credit: F-star, cert(M), the chain and the fixtures with their recorded certificate limits - DreamingOfClouds, PR 1590 / PR 2111 / PR 2162. Lemmas SF/MP/RK/AN and the exact-completion engine - Aaron Wang, PR 2717. Lemma MX and the exhaustive MP/MX fixture values - Aaron Wang, PR 2747. Zero digraphs and chi(Z) - Aaron Wang, PR 2774 / PR 2890. Lemmas SC/SG - Aaron Wang, PR 2869. The family-gap definition and record - Aaron Wang, PR 2901. Exact cost 12 of tied_min - Aaron Wang, PR 2379; exact cost 21 of cex_min_1 - Nick Petalas, PR 2206. All authors verified at current ground truth on 2026-10-16. Prepared by Aaron Wang with help from opencode - substantial AI assistance, no model named, no independent human review claimed. No kappa claim. --- .../CERTIFICATE.json | 103382 +++++++++++++++ .../fstar-family-gap-seed1c25f56f/README.md | 389 + .../SOURCES.json | 24 + .../TASK-CARD.md | 98 + .../fstar-family-gap-seed1c25f56f/TASK.json | 88 + .../fstar-family-gap-seed1c25f56f/VERIFY.json | 20 + .../data/cex_min_1.B22.json | 1 + .../data/cex_min_1.json | 1 + .../data/cex_min_1.witness.json | 1 + .../data/tied_min.json | 1 + .../fstar-family-gap-seed1c25f56f/verify.py | 1364 + 11 files changed, 105369 insertions(+) create mode 100644 research/fstar-family-gap-seed1c25f56f/CERTIFICATE.json create mode 100644 research/fstar-family-gap-seed1c25f56f/README.md create mode 100644 research/fstar-family-gap-seed1c25f56f/SOURCES.json create mode 100644 research/fstar-family-gap-seed1c25f56f/TASK-CARD.md create mode 100644 research/fstar-family-gap-seed1c25f56f/TASK.json create mode 100644 research/fstar-family-gap-seed1c25f56f/VERIFY.json create mode 100644 research/fstar-family-gap-seed1c25f56f/data/cex_min_1.B22.json create mode 100644 research/fstar-family-gap-seed1c25f56f/data/cex_min_1.json create mode 100644 research/fstar-family-gap-seed1c25f56f/data/cex_min_1.witness.json create mode 100644 research/fstar-family-gap-seed1c25f56f/data/tied_min.json create mode 100644 research/fstar-family-gap-seed1c25f56f/verify.py diff --git a/research/fstar-family-gap-seed1c25f56f/CERTIFICATE.json b/research/fstar-family-gap-seed1c25f56f/CERTIFICATE.json new file mode 100644 index 0000000000..841c7ac86c --- /dev/null +++ b/research/fstar-family-gap-seed1c25f56f/CERTIFICATE.json @@ -0,0 +1,103382 @@ +{ + "calibration": "PASS_FG_CALIBRATION", + "census_domain": [ + { + "enumerated": 220, + "fixture": "tied_min.json", + "size": 3, + "thinned": false, + "total": 220 + }, + { + "enumerated": 240, + "fixture": "tied_min.json", + "size": 4, + "thinned": true, + "total": 495 + }, + { + "enumerated": 240, + "fixture": "tied_min.json", + "size": 5, + "thinned": true, + "total": 792 + }, + { + "enumerated": 60, + 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"PIVOT": 1, + "REPAIR": 10 + }, + "reach": 8, + "threshold": 12, + "v": 19, + "witness": 18 + }, + "tied_min.json": { + "MP": 9, + "box_escape": [], + "cancel": 12, + "diff": [ + { + "D": 3, + "a": 6, + "a0": 7, + "b": 2, + "b0": 3, + "kind": "BOX-INTERIOR", + "t": 11, + "tau": 2 + }, + { + "D": 6, + "a": 2, + "a0": 7, + "b": 6, + "b0": 6, + "kind": "REPAIR", + "t": 6, + "tau": 2 + }, + { + "D": 5, + "a": 7, + "a0": 7, + "b": 1, + "b0": 5, + "kind": "BOX-INTERIOR", + "t": 2, + "tau": 2 + }, + { + "D": 6, + "a": 3, + "a0": 7, + "b": 6, + "b0": 6, + "kind": "REPAIR", + "t": 3, + "tau": 2 + }, + { + "D": 5, + "a": 4, + "a0": 7, + "b": 5, + "b0": 5, + "kind": "REPAIR", + "t": 10, + "tau": 2 + }, + { + "D": 3, + "a": 2, + "a0": 7, + "b": 3, + "b0": 3, + "kind": "REPAIR", + "t": 8, + "tau": 2 + }, + { + "D": 3, + "a": 3, + "a0": 7, + "b": 1, + "b0": 3, + "kind": "BOX-INTERIOR", + "t": 4, + "tau": 2 + }, + { + "D": 3, + "a": 5, + "a0": 7, + "b": 1, + "b0": 3, + "kind": "BOX-INTERIOR", + "t": 1, + "tau": 2 + }, + { + "D": 6, + "a": 7, + "a0": 7, + "b": 4, + "b0": 6, + "kind": "BOX-INTERIOR", + "t": 9, + "tau": 2 + }, + { + "D": 5, + "a": 3, + "a0": 7, + "b": 5, + "b0": 5, + "kind": "REPAIR", + "t": 5, + "tau": 2 + }, + { + "D": 6, + "a": 5, + "a0": 7, + "b": 4, + "b0": 6, + "kind": "BOX-INTERIOR", + "t": 0, + "tau": 2 + } + ], + "in_box": true, + "kinds": { + "BOX-INTERIOR": 6, + "REPAIR": 5 + }, + "reach": 0, + "threshold": 8, + "v": 12, + "witness": 11 + } + }, + "package": "research/fstar-family-gap-seed1c25f56f" +} diff --git a/research/fstar-family-gap-seed1c25f56f/README.md b/research/fstar-family-gap-seed1c25f56f/README.md new file mode 100644 index 0000000000..cb9dcfa38d --- /dev/null +++ b/research/fstar-family-gap-seed1c25f56f/README.md @@ -0,0 +1,389 @@ +# fstar-family-gap-seed1c25f56f + +**Closing the family gap of open problem F-star: the sub-anchor-box +decomposition of the attaining certificates, and a smallest-first +sub-decoder census at fixture density (no kappa, no frontier change).** +Open problem F-star of `research/root-unit-decoder-lemmas` +([PR 1590](https://github.com/CrocSwap/integer-mult-bounds/pull/1590), +section 7, OPEN) asks whether `cert(M) >= floor(2v/3)` holds for every exact +3-total rational decoder `M = sum_{j<=3} p_j q_j^T`. [PR 2901](https://github.com/CrocSwap/integer-mult-bounds/pull/2901) +settled the cancellation-line residue on both canonical fixtures and +isolated the residual obstruction as the **family gap** `named(M) < +cert(M)`: the matched-pivot family certificate reaches 9 and 16 on the +fixtures while `cert` is 11 and 18 (recorded), a gap of exactly 2 on each. +This package (kappa-blocker **KB5g**) attacks that gap: it either lifts the +named families to `cert`, proves the gap harmless at the threshold, or +characterizes exactly what the named families miss and how small the gap +gets before it appears. **F-star itself is not proved and not refuted +here.** Every claim below is labelled PROVEN, CHECKED or DEMOTED. + +## Verdict + +**The family gap is harmless on the entire realized domain, and its exact +content is characterized: it is the interior of the sub-anchor box.** + +1. **CHECKED (exhaustive both sides):** on all 21 exact sub-decoder records + (`v = 3..6`, smallest-first, deep rungs) `cert = box = form_r = named` + **exactly** (gap 0), and `cert = max(MP, |S_gamma|) + 1` exactly. Every + attaining label pair lies in the sub-anchor box. +2. **CHECKED (given PR 2111's recorded certificate limits):** on both full + fixtures the sub-anchor box attains `cert` exactly (`box = 11 = cert` on + `tied_min`, `box = 18 = cert` on `cex_min_1`), and every attaining label + pair lies in the box. The recorded family gap 2 is reproduced against the + named families (`named = 9` and `16`), and its per-line content is + classified: the witnesses use REPAIR-form pairs plus strictly interior + box pairs `a` proper subset of `a0(t)`, `b` proper subset of `b0(t)`, + `K_tt = 1`, that belong to no named form. +3. **CHECKED (scoped negative):** no threshold counterexample and no + sub-threshold record exists in the searched domain: `named_lo = + max(MP, |S_gamma|) >= floor(2v/3)` on all 1410 census records and on both + fixtures (9 >= 8, 16 >= 12), so by the proven chain `cost >= cert` no + falsifier can exist there at all. The falsifier criterion was evaluated + on every record and its decisive branch was never reached. +4. **DEMOTED:** the matched-pivot floor alone (assignment route (a)) is + strictly weaker than `cert` on the fixtures (9 < 11, 16 < 18), so + `named = MP = cert` fails; `MP >= floor(2v/3)` itself is witnessed + everywhere but not proved. A general additive lifting bound (route (b)) + of the form `cert >= named + f(k, v)` is contradicted by the data's + shape: the gap is 0 on records with up to 3 cancellation lines and 2 on + `tied_min` with 12 of 12, so it is not a function of the cancellation + count or of `v`. A realized threshold counterexample (route (c)) was not + found; the minimal realized gap-2 instance in the searched domain remains + `tied_min` at `v = 12` (no gap >= 2 at any `v <= 6` record). + +The exact remaining gap to a proof of F-star: **prove the sub-anchor box +complete** (`cert = box-max`, at least when every line has `tau(t) >= 1`) +and then prove the box floor `box-max >= floor(2v/3)`, or prove the +matched-pivot floor `max_c sum_i MAS(Z[C_i]) >= floor(2v/3)` (which would +close F-star despite being weaker than `cert`). Both remain OPEN here. + +## The object and the ladder + +An exact 3-total rational decoder on `v` lines is `M = sum_{j=1..3} p_j +q_j^T` over Q with `M_tt != 0` on every line. Row support `pi(t) = {j : +p_j(t) != 0}`, column support `sigma(t) = {j : q_j(t) != 0}`, `D(t) = +pi(t) intersect sigma(t)` (nonempty). An L8 labelling is F2 vectors `a(t)` +supported in `pi(t)` and `b(t)` supported in `sigma(t)`, `K_ts = a(t) . b(s) +mod 2`; `cert(M)` is the largest `|S|` admitting an order with `K_tt = 1` on +`S` and `K_ts = 0` at every incidence `M_ts = 0` with `t` before `s`. The +proven chain `cost >= sum_j r_j >= cert(M)` is from +`research/root-unit-decoder-lemmas`; this package does not restate it. + +In the canonical reduced normalization (Lemma SC/SG of +[PR 2869](https://github.com/CrocSwap/integer-mult-bounds/pull/2869), Aaron +Wang; `alpha_j = -m_j`): `a0_j(t) = [v_2(p_j(t)) = m_j]`, `b0_j(t) = +[v_2(q_j(t)) = gamma - m_j]`, with the scope identity `K0_ts = 2^{-gamma} +M_ts mod 2`; `S_gamma = {t : v_2(M_tt) = gamma}` is the anchor reach, the +cancellation lines are `C = {t : v_2(M_tt) > gamma}`, and `tau(t)` is the +tie number of Lemma CT (PR 2901). + +This package measures five rungs per instance, each an exact or bounded +maximum of the best certificate value a label class can build: + +| rung | label class | value function | status | +|---|---|---|---| +| `named_lo` | Lemma SG anchor plus Lemma MP matched pivots | `max(MP, \lvert S_gamma\rvert)` | exact (MP exhaustive) | +| `named` | all four named families (SG, MX repairs, MP, CL3 mixings) | max construction value | exact on deep records; seeded bounded scan on fixtures | +| `form_r` | every named-form pair: `(a0,b0)`, `(a*, b0)` repairs, `(e_c,e_c)` pivots | `ab_value` (exact for fixed labels) | exact on deep records; bounded on fixtures | +| `box` | every sub-anchor pair `a` subset of `a0(t)`, `b` subset of `b0(t)`, `K_tt = 1` | `ab_value` inside `b0` | exact on deep records; witnessed on fixtures | +| `cert` | every L8 labelling | exact-completion DP (PR 2717 engine) | exact on deep records; witnessed/recorded on fixtures | + +Here `ab_value(a, b)` is the exact certificate value of one fixed label +assignment: the largest kept set with `K_tt = 1` whose conflict digraph is +acyclic (subset DP). The ladder facts below hold in general. + +## Lemma SB (the sub-anchor box) - PROVEN + +Write `a0(t)`, `b0(t)` for the anchor labels of the canonical reduced +normalization. + +> **Lemma SB.** (i) `a0(t) intersect b0(t)` is exactly the set of totals +> attaining the tie number `tau(t)`, so `|a0(t) intersect b0(t)| = tau(t)`. +> (ii) A sub-anchor pair (`a` subset of `a0(t)`, `b` subset of `b0(t)`) +> certifies line `t` (`K_tt = 1`) if and only if `|a intersect b|` is odd, +> which requires `tau(t) >= 1`; for `tau(t) = 1` the unique tied bit lies in +> both `a` and `b`; for `tau(t) = 2` exactly one of the two tied bits lies +> in `a intersect b`. (iii) A `tau(t) = 0` line has no certifying +> sub-anchor pair: any witness keeping it must leave the box on that line. + +*Proof.* (i) `j` lies in `a0(t) intersect b0(t)` exactly when `j` is in +`D(t)`, `v_2(p_j(t)) = m_j` and `v_2(q_j(t)) = gamma - m_j`, that is when +`v_2(p_j(t) q_j(t)) = gamma`, which is the definition of being counted by +`tau(t)`. (ii) For sub-anchor labels `a intersect b` is a subset of +`a0(t) intersect b0(t)`, and `K_tt = |a intersect b| mod 2`; odd +intersection needs a tied bit, and the two case statements read off (i). +(iii) is (ii) with `tau(t) = 0`. `QED` + +On `tied_min` every line has `tau = 2` and on `cex_min_1` the tie numbers +are 1, 2 and 3; Lemma SB explains the shape of every witness pair found +below (each certifying box pair picks an odd number of tied bits). + +## Ladder bounds - PROVEN + +> **Lemma LB.** (a) `named(M) <= form_r(M)`: every certificate the named +> families carry uses Form-R label pairs per line, and `ab_value` is the +> exact maximum for fixed labels. (b) `box(M) <= cert(M)` by definition of +> the label classes. (c) On any instance where `S_gamma` is empty the Lemma +> CL3 mixing equals the matched-pivot value (the anchor label pair has +> `K0_tt = 0` on every line and certifies nothing), so `named = max(MP, +> MX)` there. + +*Proof.* (a) Each named family (Lemma SG anchor, Lemma MX repairs, Lemma MP +pivots, Lemma CL3 mixings) assigns each line one of the three Form-R pairs; +its carried certificate is a certificate of that assignment, whose exact +value is `ab_value`. (b) is the identity of label spaces. (c) with +`S_gamma` empty, `A` is empty in every named construction and the remaining +constructions are matched pivots (Lemma MP) and their Lemma MX variants. +`QED` + +Part (c) makes `tied_min`'s named value exact: `named = max(9, 9) = 9` (MP +exhaustive over 4096 pivot maps; MX exhaustive 9 per +[PR 2747](https://github.com/CrocSwap/integer-mult-bounds/pull/2747), Aaron +Wang, reproduced by the seeded scan here). `cex_min_1`'s named value 16 is +PR 2747's exhaustive MP/MX record (reproduced by seeded scans here; the +CL3 rung is a bounded scan, PR 2901's caveat retained). + +## Fixture settlement: the box attains cert exactly - CHECKED + +Both canonical fixtures of [PR 2111](https://github.com/CrocSwap/integer-mult-bounds/pull/2111) +(DreamingOfClouds) settle inside the sub-anchor box. Every number is +recomputed by `verify.py` in one run except the certificate limits 11 and +18, which are PR 2111's recorded CP-SAT values (CHECKED (recorded), not +re-verified here). + +| quantity | `tied_min` (12 lines) | `cex_min_1` (19 lines) | +|---|---|---| +| threshold `floor(2v/3)` | 8 | 12 | +| cancellation lines | 12 of 12 | 11 of 19 | +| anchor reach `S_gamma` | 0 | 8 (lines 0, 4, 5, 6, 8, 11, 14, 15) | +| `named_lo` = max(MP, reach) | 9 (4096 maps, exhaustive) | 16 (6144 maps, exhaustive) | +| `named` (PR 2901 record, reproduced) | 9 | 16 | +| `form_r` (bounded) | 10 (hill-climb) | 15 (hill-climb; not a bound) | +| `box` | 11 (witnessed) | 18 (stored witness) | +| `cert` | 11 (recorded, witnessed) | 18 (recorded, witnessed) | +| family gap `cert - named` | 2 | 2 | +| box gap `cert - box` | 0 | 0 | +| chain engine (stored pairing) | cited 12 (PR 2379) | 22 (reproduced, 10+6+6) | + +So the recorded family gap 2 is exactly the distance from the named +families to the sub-anchor box, and the box closes it on both fixtures +(box >= 11 and 18 by the replayed witnesses; box <= cert = 11 and 18 by +PR 2111's recorded limits). The Form-R rung shows one further split: on +`tied_min` Form-R labels reach 10 of the 11 (bounded hill-climb over a +space of 13,841,287,201 assignments; whether Form-R alone reaches 11 is +OPEN), so at least one of the two missing lines needs strictly interior box +pairs there. + +**What the families miss, per line (PR 2901 open question 2 / PR 2747 open +question 3, first step).** The attaining witnesses classify as follows +(`verify.py` diff tables; ANCHOR = `(a0,b0)`, REPAIR = `(a*, b0)`, PIVOT = +`(e_c,e_c)`, BOX-INTERIOR = strictly interior box pair): + +| witness | ANCHOR | REPAIR | PIVOT | BOX-INTERIOR | outside box | +|---|---|---|---|---|---| +| `tied_min` 11-witness (engine re-found) | 0 | 5 | 1 | 5 | 0 | +| `tied_min` 11-witness (calibration run) | 0 | 5 | 0 | 6 | 0 | +| `cex_min_1` 18-witness (stored, replayed) | 2 | 10 | 1 | 5 | 0 | + +Every attaining label pair lies in the sub-anchor box on both fixtures and +on all 21 exact records (CHECKED). The strictly interior pairs are exactly +the pairs Lemma SB (ii) describes: on a `tau = 2` line they pick one of the +two tied bits in `a intersect b`, dropping the non-tied bits of `a0`/`b0`; +this is the label space no named family enumerates (the named forms fix +`b = b0(t)` on A/U lines or force `a = b` unit vectors on pivots). + +## Smallest-first sub-decoder census - CHECKED (scoped negative) + +Every sub-decoder `M[S,S]` of a fixture is an exact 3-total decoder +(restriction preserves rank <= 3 and the nonzero diagonal; CHECKED control) +and inherits the fixture-density zero digraphs by construction - the exact +recorded generator obstruction of PR 2901 (its k-axis generators cannot +realize them; Lemma RK (c) forces `|pi(t) intersect sigma(s)| >= 2` at +every arc). The census enumerates subsets smallest-first (sizes 3 to 6, +deterministic strides recorded per size in `CERTIFICATE.json`), computes +the cheap rungs on every record, and the exact rungs (`named`, `form_r`, +`box`, `cert`, all exhaustive) on every flagged record until the deep +budget (21 records) is spent. + +| family | domain | records | deep exact | falsifiers | below threshold | +|---|---|---|---|---|---| +| sub-decoders of `tied_min` | `v = 3..6`, sizes 3 complete, 4-6 strided | 760 | 0 flagged | 0 | 0 | +| sub-decoders of `cex_min_1` | `v = 3..6`, strided | 650 | 21 | 0 | 0 | + +Totals: 1410 records, 0 falsifiers, 0 below-threshold records, minimum +`named_lo/v = 2/3` (exactly at the threshold ratio). On all 21 exact deep +records (all of them `cex_min_1` sub-decoders, each with 1 to 3 +cancellation lines): `cert = box = form_r = named` **exactly**, `cert = +named_lo + 1` exactly, every witness in the box. In particular the family +gap is **0** on every exact record and never reaches 2 below `v = 12` in +the searched domain; the gap is not a function of the cancellation count +(gap 0 with `k_cancel` up to 3; gap 2 at `k_cancel = 12`). + +**Falsifier criterion, armed on every record:** 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`). It never fired; in fact its decisive +branch was never reached, because `named_lo >= floor(2v/3)` on every record +and the named witnesses are replayed certificates: by `cost >= cert` no +admissible `B` below threshold can exist on any searched record. This is a +scoped statement about the searched domain, not a non-existence theorem. + +**Exact obstruction of the remaining search space (recorded).** Sizes +`v = 7..11` of both fixtures and all but the recorded strides at sizes 4-6 +were not enumerated (compute cap). The gap-2 phenomenon needs either those +sizes or the full fixtures; no generator is claimed to realize it at +smaller `v` (an unrealizability claim would need a proof this package does +not have). + +## Assignment routes (a), (b), (c) - disposition + +- **(a) the family certificate meets the threshold (`named = MP >= + floor(2v/3)` for every exact 3-total decoder): DEMOTED.** `MP >= + floor(2v/3)` is witnessed on all 1410 census records and both fixtures + (9/12, 16/19) but not proved; and the identity `named = cert` fails on + the fixtures (9 < 11, 16 < 18). A proof of the MP floor would still close + F-star (cert >= MP); the exact obstruction to proving it here: the floor + is a statement about realizable rank-3 patterns (Lemma RK (c) constrains + every arc), the fixture-density zeros resist random construction (PR + 2901's recorded generator obstruction), and no analytic route was found; + the observed feedback-vertex sums `v - MP` are 3 and 3 against ceilings + `ceil(v/3) = 4` and 7, with no monotone argument in sight. +- **(b) the gap is harmless via a certified lifting: CHECKED on the + realized domain, DEMOTED in general.** On all 21 exact records the + lifting is exact (`named = cert`, gap 0); on both fixtures `box = cert` + exactly (given the recorded limits) with the missing labels classified + per line. A proved additive bound of the form `cert >= named + f` is + DEMOTED: the realized gaps (0 at `v <= 6` with cancellation lines, 2 at + `v = 12` with 12 of 12) fit no additive function of `k_cancel` or `v` + the analysis produced, and Lemma SB shows the box can certify nothing on + a `tau = 0` line, so no uniform per-cancellation-line surplus can hold. +- **(c) the first realized threshold counterexample: NOT FOUND.** The + strict form (`cert < floor(2v/3)`) and the threshold form (`named < + floor(2v/3) <= cert`) are both unrealized in the searched domain; the + minimal realized gap-2 instance remains `tied_min` at `v = 12`, and its + gap content is exactly the interior-box labels above. + +## What is proved, what is checked, what is demoted + +- **PROVEN (README, written proofs):** Lemma SB (the sub-anchor box + structure: `a0 intersect b0` = tied totals; certifying box pairs pick an + odd number of tied bits; `tau = 0` lines have no certifying box pair); + Lemma LB (the ladder bounds `named <= form_r`, `box <= cert`, and `named + = max(MP, MX)` when `S_gamma` is empty, making `tied_min`'s named value + exactly 9). +- **CHECKED (machine, `verify.py`, stdlib, deterministic, about 230 s, + certificate digest `031927ba33ed45a0`):** controls 5 groups (brute-force- + from-definition agreement 3/3 at `v = 3`, corrupted-witness rejection + 2/2, `ab_value` soundness against completion 10/10, restriction exactness + with the Lemma CT and scope identities, diagonal-style decoder sanity); + the recorded-anchor calibration (MP 9/16 exhaustive over 4096/6144 pivot + maps; `tied_min` hill-climb witness 11; `cex_min_1` stored witness 18 + replayed; chain 22 on the stored pairing; reach 0/12 and 8/19); the + fixture ladder with the per-line label diffs and `box = cert` at 11/11 + and 18/18; the 1410-record census with 21 exact deep records at + `named = cert` exactly; 0 falsifiers, 0 below-threshold. +- **DEMOTED / OPEN:** F-star for general rational decoders is **not** + proved here and **not** refuted. The matched-pivot floor (PR 2747 open + question 1) is not proved. The sub-anchor-box completeness (`cert = + box-max`) is CHECKED on the realized domain only; `tau = 0` lines are the + only structurally excluded case (Lemma SB (iii)) and no `tau = 0` line + occurs in the fixtures. Form-R completeness on `tied_min` (does Form-R + reach 11?) is OPEN (space of 13,841,287,201 assignments, hill-climb + 10). The fixture certificate limits 11 and 18 rest on PR 2111's recorded + CP-SAT optimality, not re-verified here. The CL3 rung on `cex_min_1` is a + bounded scan (PR 2901's caveat retained). The census thinning is recorded + per size; "no falsifier" is a scoped statement about the declared domain. + +## Open next questions + +1. **Sub-anchor-box completeness.** Is `cert(M) = box-max(M)` whenever + every line has `tau(t) >= 1`? Why it matters: it reduces F-star to a + finite per-line label space (at most 24 certifying pairs per line) and + makes the family gap exactly the interior of that box. First step: search + for the smallest instance where an attaining certificate must leave the + box (all 21 exact records and both fixtures are box-valued; `tau = 0` + lines are the only excluded case by Lemma SB). Blocker: none; analysis + plus a bounded exact search. +2. **The matched-pivot floor (PR 2747 open question 1).** Is `max_c sum_i + MAS(Z[C_i]) >= floor(2v/3)` for every realizable rank-3 pattern? Why it + matters: Lemma MP alone would prove F-star with no anchor. First step: a + SAT/CP construction of rank-3 patterns minimizing the floor (the + fixture-density zeros resist random construction; Lemma RK (c) forces + `|pi(t) intersect sigma(s)| >= 2` at every arc). Blocker: needs a + constraint solver, not random search. + +## Reproduce + +Python 3.10+, standard library only, from the repository root: + +```sh +python3 -B research/fstar-family-gap-seed1c25f56f/verify.py +python3 scripts/research_package.py --package research/fstar-family-gap-seed1c25f56f --check-only +python3.12 scripts/research_package.py --package research/fstar-family-gap-seed1c25f56f +``` + +Expected markers: `PASS_FG_CONTROLS`, `PASS_FG_CALIBRATION +tied_min=11/12 cex_min_1=18/19 mp=9/16 chain=22 reach=0/12,8/19`, +`PASS_FG_CENSUS records=1410 exact=21 exact-named=21 gap1=49 gap2=0 +below-threshold=0 falsifiers=0 max-gap-named=0`, +`PASS_FSTAR_FAMILY_GAP no-falsifier-in-domain`, and `CERTIFICATE.json` with +digest `031927ba33ed45a0` (observed 217-229 s on the authoring host). +`make` is not installed on the authoring host; the commands above are the +underlying ones. + +## Credit and provenance + +F-star, the `cert(M)` definition, L8 labellings and the proven chain +`cost >= sum_j r_j >= cert(M)` are from the merged +`research/root-unit-decoder-lemmas` package +([PR 1590](https://github.com/CrocSwap/integer-mult-bounds/pull/1590) by +DreamingOfClouds (Claude Opus 5.5 (Anthropic) via Claude Code assistance)). +The rank-3 configuration model, the named instances `tied_min` and +`cex_min_1`, their recorded one-copy certificate limits (11 and 18, CP-SAT, +CHECKED (recorded)) and the tied-minima obstruction are DreamingOfClouds' +work in [PR 2111](https://github.com/CrocSwap/integer-mult-bounds/pull/2111) +and [PR 2162](https://github.com/CrocSwap/integer-mult-bounds/pull/2162); +`data/tied_min.json` and `data/cex_min_1.json` are byte-identical copies of +their data files (sha256 +`670ed90924b601ade22f0d4adaff46d1667e5449326a0f7ba7b88dd8f5906f93` and +`86064701dde94b4f35427bc3e19275eaf1e40e9e90d4fea28e7857f3564d4dd7`); the +stored cost-22 pairing is `data/cex_min_1.B22.json` (sha256 +`15f5039069a175831f92c4ffe023a23a6f6e6bd6665f0ca2c2734bbe672b51ef`) and the +stored 18-witness `data/cex_min_1.witness.json` (sha256 +`153f4d2e8dc262f547a91f2daff9724cab88beb4d45c399497efc68a8ed2c933`) is from +[PR 2717](https://github.com/CrocSwap/integer-mult-bounds/pull/2717)'s +package. The residue statement, Lemmas SF/MP/RK/AN and the exact-completion +engine design this package adapts are Aaron Wang's work in PR 2717; Lemma +MX and the exhaustive MP/MX fixture values (9 and 16) are Aaron Wang's +[PR 2747](https://github.com/CrocSwap/integer-mult-bounds/pull/2747); the +zero-digraph machinery is Aaron Wang's +[PR 2774](https://github.com/CrocSwap/integer-mult-bounds/pull/2774); +Lemmas SC/SG and the canonical reduced normalization are Aaron Wang's +[PR 2869](https://github.com/CrocSwap/integer-mult-bounds/pull/2869); the +chi(Z) machinery is Aaron Wang's +[PR 2890](https://github.com/CrocSwap/integer-mult-bounds/pull/2890); the +family-gap definition and record are Aaron Wang's +[PR 2901](https://github.com/CrocSwap/integer-mult-bounds/pull/2901). The +exact cost 12 of `tied_min` is Aaron Wang's +[PR 2379](https://github.com/CrocSwap/integer-mult-bounds/pull/2379); the +exact cost 21 of `cex_min_1` is Nick Petalas' +[PR 2206](https://github.com/CrocSwap/integer-mult-bounds/pull/2206). All +authors above were verified at current ground truth with `gh pr view +--json author,title` on 2026-10-16. This package does not claim their +results; the new content is Lemma SB, Lemma LB, the sub-anchor-box +machinery and decomposition, the exact named/Form-R/box/cert ladder on +small instances, the per-line classification of the attaining witnesses, +and the sub-decoder census at fixture density. + +Freshness: base `ba3de47290c3882cbbe406db1f657578f7ecda69` is the frozen +assignment pin (kept intentionally). Pair-collision checks +(discussions 1317 + 1166, nested replies, 2026-10-16 08:58-09:00 UTC and +the pre-submission re-run) found no live claim on this object; the last +F-star activity on the board is KB5f's FINDING/DONE/RELEASED at 2026-10-16 +08:53:52 UTC. Entry overlap survey 2026-10-16 09:01 UTC and the refined +pre-submission re-run (complete pagination) found only the known F-star +lineage and unrelated traffic. + +Prepared by **Aaron Wang** with help from opencode (substantial AI +assistance; no model named; no independent human review claimed). New work +in this package is Apache-2.0. diff --git a/research/fstar-family-gap-seed1c25f56f/SOURCES.json b/research/fstar-family-gap-seed1c25f56f/SOURCES.json new file mode 100644 index 0000000000..bd49294825 --- /dev/null +++ b/research/fstar-family-gap-seed1c25f56f/SOURCES.json @@ -0,0 +1,24 @@ +{ + "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" + } + ] +} diff --git a/research/fstar-family-gap-seed1c25f56f/TASK-CARD.md b/research/fstar-family-gap-seed1c25f56f/TASK-CARD.md new file mode 100644 index 0000000000..bcb736f7f2 --- /dev/null +++ b/research/fstar-family-gap-seed1c25f56f/TASK-CARD.md @@ -0,0 +1,98 @@ +# 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. diff --git a/research/fstar-family-gap-seed1c25f56f/TASK.json b/research/fstar-family-gap-seed1c25f56f/TASK.json new file mode 100644 index 0000000000..54a1196a21 --- /dev/null +++ b/research/fstar-family-gap-seed1c25f56f/TASK.json @@ -0,0 +1,88 @@ +{ + "actual_scope": "Kappa-blocker KB5g (authorized explicit open request): CLOSE THE FAMILY GAP of open problem F-star of research/root-unit-decoder-lemmas section 7 (cert(M) >= floor(2v/3) for exact 3-total rational decoders), the precise residual obstruction isolated as named(M) < cert(M) by PR 2901 (README section 3). Either (a) prove the family certificate meets the threshold (named(M) = MP(M) >= floor(2v/3) for every exact 3-total rational decoder), (b) prove the gap harmless with a certified lifting bound cert >= named + at 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 named families miss. The search extends where PR 2901 recorded its exact generator obstruction: its k-axis generators could not realize fixture-density zero digraphs; every sub-decoder M[S,S] of the canonical fixtures is an exact 3-total decoder with fixture-density zeros by construction, so the sub-instance axis attacks the weak spot directly, smallest instance first.", + "ai_assistance": "Substantial AI assistance: this package was produced by Aaron Wang with help from opencode (opencode agent harness). No independent human review claimed. No model is cited per orchestrator attribution rules.", + "author": "Aaron Wang (with help from opencode)", + "base_commit": "ba3de47290c3882cbbe406db1f657578f7ecda69", + "bottleneck_link": "F-star is the named route to closing every open T = 3 window at once (research/root-unit-decoder-lemmas section 7, consequence PROVEN given F-star). After PR 2869 (entry classes collapsed) and PR 2901 (cancellation lines settled on both fixtures) the family gap named(M) < cert(M) is the precise residual obstruction of the residue: a proof that the named families reach floor(2v/3), or a certified lifting from named to cert, closes F-star; a realized threshold counterexample refutes it.", + "evidence": "FAMILY GAP CHARACTERIZED ON THE FIXTURES AND ON A SMALLEST-FIRST SUB-DECODER CENSUS; NO THRESHOLD COUNTEREXAMPLE; F-STAR NOT PROVED AND NOT REFUTED. PROVEN (README.md): Lemma SB (the sub-anchor box structure: a0(t) intersect b0(t) is exactly the set of tied totals, so a box pair certifies t iff |a intersect b| = 1; tau = 1 lines force the unique tied bit on both labels; tau = 2 lines force exactly one of the two tied bits; tau = 0 lines have no certifying box pair) and the ladder bounds named <= form_r (every named-family certificate uses Form-R labels and ab_value is exact for fixed labels) and box <= cert. CHECKED (verify.py, stdlib, deterministic, about 230 s, certificate digest 031927ba33ed45a0): controls 5/5 groups; calibration reproducing every recorded anchor (tied_min MP 9/4096 maps and hill-climb witness 11; cex_min_1 MP 16/6144 maps, stored witness 18 replayed, chain 22 on the stored pairing, reach 0/12 and 8/19); the fixture ladder with per-line label diffs (every attaining label pair lies in the sub-anchor box: tied_min 11-witness mixes 5 REPAIR-form, 5 BOX-INTERIOR and 1 PIVOT pair; cex_min_1 18-witness mixes 10 REPAIR, 5 BOX-INTERIOR, 2 ANCHOR, 1 PIVOT; box reaches cert exactly on both fixtures: 11 = 11 and 18 = 18 against PR 2111's recorded certificate limits); the smallest-first sub-decoder census of both fixtures (1410 records at v = 3..6; 21 deep records with exact Form-R/box/cert/named rungs on which cert = box = form_r = named EXACTLY (gap_named = 0) and cert = max(MP, reach) + 1; all witnesses in the box); falsifier criterion armed on every record (exact cert below threshold where exhaustive, or an explicit admissible B with chain cost below threshold): never fired, 0 falsifiers, 0 below-threshold, min named ratio 2/3 (at threshold). The family gap equals exactly 0 on all 21 exact sub-instance records and exactly 2 on the full fixtures (named 9 < 11 and 16 < 18, reproducing PR 2901's record); its minimal realized instance in the searched domain is tied_min at v = 12.", + "existing_restriction_being_changed": "Nothing is relaxed. Audited and extended: PR 2901's family-gap record and its open questions 1-2. The searched domain extends where PR 2901 recorded its exact generator obstruction (its k-axis generators cannot realize fixture-density zero digraphs): every sub-decoder of the canonical fixtures inherits fixture-density zeros by construction. PR 2901's named(M) is computed here with exact Lemma MX constructions and exact Lemma CL3-form optima on the deep records (their CL3 fixture evaluation was a bounded scan, a caveat retained and stated). Every claim carries its exact status label; no 'exactly when', necessity or exhaustive-without-proof wording is used.", + "falsification_test": "An explicit admissible pairing B with cost(B) = sum_j rank_F2 B[R_j,A_j] < floor(2v/3) refutes F-star on that instance (the chain cost >= cert(M) is proven in research/root-unit-decoder-lemmas), as does exact cert < floor(2v/3) where exhaustive a-enumeration terminates; corrupted witnesses must fail the independent replay checker. Armed on every census and fixture record; never fired.", + "intended_artifact": "research/fstar-family-gap-seed1c25f56f/ with TASK.json, TASK-CARD.md, SOURCES.json pins, VERIFY.json, verify.py, data/ (byte-identical fixture copies, the stored cost-22 pairing and the stored 18-witness, sha256 recorded), CERTIFICATE.json (written by verify.py) and README.md; reproduction: python3 -B research/fstar-family-gap-seed1c25f56f/verify.py then python3 scripts/research_package.py --package research/fstar-family-gap-seed1c25f56f --check-only.", + "method": "Exact stdlib integer/F2 arithmetic on the PR 2717/2747/2869/2901 engine lineage (model, subset-DP acyclic/MAS tables, Lemma SC/SG anchor data, matched pivots, exact-completion DP over b-labels, witness replay, admissible-B chain scan). NEW here: the sub-anchor-box machinery (box_exact: a(t) subset of a0(t), b(t) subset of b0(t); exact by exhaustive a-enumeration with an exact b-completion inside b0), the Form-R label-space maximum (ab_value: the exact certificate value of any fixed label assignment, so Form-R is a rigorous upper bound on the PR 2901 named value), the exact named-family value on small instances (Lemma MX constructions over both hypothesis sets, Lemma CL3-form optima, exhaustive where the construction product allows and a seeded bounded scan otherwise), the per-line label diff of attaining witnesses against the named forms and the box, and the smallest-first sub-decoder census M[S,S] of both canonical fixtures (restriction of an exact decoder is exact).", + "nearest_overlapping_prs": "https://github.com/CrocSwap/integer-mult-bounds/pull/2901 (Aaron Wang: isolated the family gap named(M) < cert(M), recorded 9 < 11 and 16 < 18 and +1 on a re-split, and its open question 2 - what labels the attaining certificates use - is this package's analysis core; its CL3 fixture evaluation is a bounded scan, retained as a caveat). https://github.com/CrocSwap/integer-mult-bounds/pull/2747 (Aaron Wang: Lemma MX, exhaustive MP/MX values 9 and 16 on the fixtures, open questions 1 and 3). https://github.com/CrocSwap/integer-mult-bounds/pull/2717 (Aaron Wang: Lemmas SF/MP/RK/AN and the exact-completion engine design). https://github.com/CrocSwap/integer-mult-bounds/pull/2869 (Aaron Wang: Lemmas SC/SG, the canonical reduced normalization this package's box uses). https://github.com/CrocSwap/integer-mult-bounds/pull/2774 and https://github.com/CrocSwap/integer-mult-bounds/pull/2890 (Aaron Wang: other residue classes, not repeated here). PRs 2111/2162 (DreamingOfClouds: the fixtures and the recorded certificate limits 11/18) and PR 1590 (DreamingOfClouds: F-star, cert(M), the proven chain) are cited, not restated. No other work on the family gap existed at entry (overlap survey 2026-10-16 09:01 UTC and the refined pre-submission re-run, complete pagination).", + "open_questions": [ + { + "blocked_by": "Needs a constraint solver for the construction side and is strictly stronger than needed: the matched-pivot floor alone is weaker than cert on the fixtures (9 < 11, 16 < 18).", + "first_step": "A SAT/CP construction of rank-3 patterns minimizing max_c sum_i MAS(Z[C_i]) (fixture-density zeros resist random construction; Lemma RK (c) forces |pi(t) intersect sigma(s)| >= 2 at every arc).", + "question": "Is max_c sum_i MAS(Z[C_i]) >= floor(2v/3) for every realizable rank-3 pattern (PR 2747 open question 1)?", + "why_it_matters": "Lemma MP alone would prove F-star with no anchor at all; witnessed here on all 1410 census records plus both fixtures (9/12, 16/19) but not proved." + }, + { + "blocked_by": "None for the statement; the proof would need the box-completeness argument.", + "first_step": "Either prove cert(M) = box-max(M) for tau >= 1 decoders (all realized witnesses are box-valued: 21 exact records and both fixtures), or exhibit the smallest instance where an attaining certificate must leave the box (tau = 0 lines are the only structurally excluded case, by Lemma SB).", + "question": "Is the sub-anchor box complete: cert(M) = max over box labels, at least when every line has tau(t) >= 1?", + "why_it_matters": "It reduces F-star to a finite per-line label space (at most 24 pairs per line) whose floor could be attacked combinatorially; the family gap is exactly the interior-box labels the named forms miss." + } + ], + "optimistic_full_cost_budget": "No kappa, supplier, word or moment budget is in play (missing-lemma/audit work). The price is one bounded deterministic stdlib run (registered verify below 600 s; observed 217-229 s) plus written analysis.", + "original_card": { + "acceptance_gate": "Name the changed premise of the fixed F0/B1 observation obstruction. Show exactly which query is recovered, legal native reads and restoration, and a full favorable budget including cover maps, helpers and endpoints. Local inverse identities alone do not close the global controller. 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": [ + "One paid query, one changed first pair, or one precisely specified affine helper state.", + "A counterexample or lower bound for that exact proposed repair.", + "One unresolved obligation of a newer local decoder, with exact source heads and a distinct claim/domain." + ], + "bottleneck_link": "The native local words exist; retained query information and its physical access are the missing link to a composed recursive interface.", + "first_experiment": "Check live claims and pin the newest relevant completed local word (starting lead #2540). Choose one unresolved consumer, retention, whole-cover movement, hosting or precision obligation. Price the complete proposal and test that obligation; do not repeat its claimed local inverse or native reconstruction unless providing distinct independent verification.", + "first_step": { + "command": [ + "python3", + "-B", + "scripts/research_baseline.py" + ], + "expected": "PASS_RESEARCH_BASELINE_BINDINGS_ONLY", + "inspect": "Read the QQ16 row in research-family-status.md. The fixed-first-pair witness remains scoped; pending #2540 adds early records and claims a complete local native decoder, but not a favorable global controller. Read its output-frame and ownership contract before designing another inverse.", + "scope": "Source-bound orientation only. Read the named package and review before relying on its local evidence; this command is not its replay or supplier admission." + }, + "id": "joint-query-controller", + "methods": [ + "Design and price one consumer or retained-state boundary for a pinned local decoder, including final restoration.", + "Audit one claimed physical invoice or prove a target-specific obstruction for its explicit retention/hosting contract." + ], + "overlap_terms": [ + "QQ16", + "joint-query", + "changing-chart controller", + "selected observation", + "1969", + "2540", + "early record", + "output-bank retention", + "complete native decoder" + ], + "question": "Can one unresolved retention, output-consumer, hosting or physical-cost obligation complete a pinned QQ16 controller proposal within budget?", + "scope_change_triggers": [ + "Retaining the fixed first pair and ordinary schedule while allowing only homogeneous selected-only control, already excluded by the witness.", + "Treating cover-map counts as zero-cost physical address operations.", + "Recreating the pending complete local decoder or early-record/output-bank coalescing already under investigation without a distinct audit or changed premise." + ], + "starting_points": [ + "docs/research/research-family-status.md", + "research/joint-query/README.md", + "research/joint-query/evidence/QQ16-SELECTED-WITNESS.md", + "research/joint-query/evidence/JOINT-QUERY-TRANSFER.md", + "docs/research/community-efficient1000-review.md" + ] + }, + "outcome": "completed", + "parameter_ranges": "T = 3 totals exactly; rational decoders with nonzero diagonal; integer factorizations (entry-class-independent by PR 2869). Fixtures: PR 2111's tied_min (v = 12) and cex_min_1 (v = 19), byte-identical data copies (sha256 670ed90924b601ade22f0d4adaff46d1667e5449326a0f7ba7b88dd8f5906f93 and 86064701dde94b4f35427bc3e19275eaf1e40e9e90d4fea28e7857f3564d4dd7), the stored cost-22 pairing (sha256 15f5039069a175831f92c4ffe023a23a6f6e6bd6665f0ca2c2734bbe672b51ef) and the stored 18-witness (sha256 153f4d2e8dc262f547a91f2daff9724cab88beb4d45c399497efc68a8ed2c933). Census: every sub-decoder M[S,S] of the fixtures at v = 3..6, enumerated smallest-first with the declared thinning recorded in CERTIFICATE.json (tied_min: all 220 size-3 subsets plus deterministic strides at sizes 4-6; cex_min_1: deterministic strides at sizes 3-6); deep exact rungs (Form-R, box, cert, named exhaustive) on 21 flagged records at v = 3..6; deterministic seeds in CERTIFICATE.json.", + "predecessor_authors": "F-star, the cert(M) definition, L8 labellings and the proven chain cost >= sum_j r_j >= cert(M): research/root-unit-decoder-lemmas, PR 1590 by DreamingOfClouds (Claude Opus 5.5 (Anthropic) via Claude Code assistance). The rank-3 configuration model, the named instances tied_min and cex_min_1, their recorded one-copy certificate limits (11 and 18, CP-SAT) and the stored cost-22 pairing: DreamingOfClouds, PR 2111 and PR 2162. Lemmas SF/MP/RK/AN, the residue statement and the exact-completion engine design this package adapts: Aaron Wang, PR 2717. Lemma MX and the tied-residue closure: Aaron Wang, PR 2747. The zero-digraph machinery: Aaron Wang, PR 2774. Lemmas SC/SG and the canonical reduced normalization: Aaron Wang, PR 2869. The chi(Z) machinery: Aaron Wang, PR 2890. The family-gap record and definition: Aaron Wang, PR 2901. The falsifier census frontier: Aaron Wang, PR 2335. Exact cost 12 of tied_min: Aaron Wang, PR 2379. Exact cost 21 of cex_min_1: Nick Petalas, PR 2206. All authors above were verified at current ground truth with gh pr view --json author,title on 2026-10-16.", + "priority_revision": 5, + "schema_version": 1, + "scope_change_reason": "Authorized 'explicit open request' scope change to kappa-blocker KB5g, recorded per the assignment (discussion 1166; PR 2901's residual is the named open request). The chooser OS draw (seed 1c25f56fbce0ef3c112ff85924bfb25a, track recursive-interfaces) proposed joint-query-controller, coverage-law-change and connector-alphabet-change; none is this object, and the assignment names the KB5g request as the recorded reason for taking the explicit request instead of the drawn cards. Pair-collision check before computation (toolkit disc-comments --grep on discussions 1317 and 1166 including nested replies, 2026-10-16 08:58-09:00 UTC) found no live claim on the family gap, named(M), the matched-pivot floor or the cert threshold: the last F-star activity is KB5f's FINDING/DONE/RELEASED (PR 2901's cancellation lines, 2026-10-16 08:53:52 UTC). Entry overlap survey (find_overlap.py with explicit object terms, complete pagination: 491 open PRs, 1570 discussions, 2026-10-16 09:01 UTC) found only the known F-star lineage (PRs 2717/2747/2774/2890/2901, research/fstar-entry-class-scope-seed63d5f999, research/fstar-cert-census-seed298df375) and unrelated board traffic.", + "scope_status": "changed", + "stop_rule": "Stop at the first exact falsifier (explicit admissible B with chain cost below floor(2v/3), or exact cert below threshold where exhaustive a-enumeration terminates), or, absent one, at the declared census domain with the exact residual recorded (the family-gap decomposition of the README). One bounded pivot maximum; early kill with the exact obstruction recorded. Compute cap: one session, single-process stdlib Python, the registered verify command below 600 s (observed 217-229 s).", + "success_evidence": "PASS markers from verify.py (PASS_FG_CONTROLS, PASS_FG_CALIBRATION tied_min=11/12 cex_min_1=18/19 mp=9/16 chain=22 reach=0/12,8/19, PASS_FG_CENSUS, PASS_FSTAR_FAMILY_GAP) plus README.md analysis: Lemma SB with a written proof; the ladder bounds named <= form_r <= cert and box <= cert with proofs; the fixture settlement (box = cert exactly at 11/11 and 18/18; every attaining label pair in the sub-anchor box with the exact per-line classification); the sub-decoder census (1410 records, 21 exact deep records with gap_named = 0, 0 falsifiers, 0 below-threshold); every accepted value carrying a replayed L8 witness.", + "task_id": "joint-query-controller", + "time_compute_cap": "One session; single-process stdlib Python; no SAT solver, no third-party packages; the registered verify command below 600 s (observed 217-229 s on the authoring host); package registered with timeout_seconds 600." +} diff --git a/research/fstar-family-gap-seed1c25f56f/VERIFY.json b/research/fstar-family-gap-seed1c25f56f/VERIFY.json new file mode 100644 index 0000000000..fb1e4f5005 --- /dev/null +++ b/research/fstar-family-gap-seed1c25f56f/VERIFY.json @@ -0,0 +1,20 @@ +{ + "schema_version": 1, + "status": "ready", + "python": "3.12", + "requirements": [], + "inputs": [ + "research/root-unit-decoder-lemmas/README.md" + ], + "commands": [ + { + "argv": [ + "{python}", + "-B", + "{package}/verify.py" + ], + "timeout_seconds": 600, + "checks": "KB5g family-gap package: controls 5 groups (brute-force-from-definition agreement at v = 3 on 3 sparse decoders, corrupted-witness rejection 2/2, ab_value soundness against completion on 10 random label pairs, restriction exactness with the Lemma CT/scope identities on a sub-decoder, diagonal-style decoder sanity); fixture calibration reproducing every recorded anchor (tied_min MP 9 over 4096 pivot maps, hill-climb witness 11; cex_min_1 MP 16 over 6144 maps, stored witness 18 replayed, chain 22 on the stored pairing, anchor reach 0/12 and 8/19); the fixture ladder with the sub-anchor-box decomposition and per-line label diffs; the smallest-first sub-decoder census of both fixtures (1410 records at v = 3..6, exact Form-R/box/cert/named rungs on 21 flagged deep records, falsifier criterion armed everywhere); prints PASS_FG_CONTROLS, PASS_FG_CALIBRATION, PASS_FG_CENSUS, PASS_FSTAR_FAMILY_GAP and writes CERTIFICATE.json (digest 031927ba33ed45a0)" + } + ] +} diff --git a/research/fstar-family-gap-seed1c25f56f/data/cex_min_1.B22.json b/research/fstar-family-gap-seed1c25f56f/data/cex_min_1.B22.json new file mode 100644 index 0000000000..0161536922 --- /dev/null +++ b/research/fstar-family-gap-seed1c25f56f/data/cex_min_1.B22.json @@ -0,0 +1 @@ +[[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1], [0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0], [0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0], [0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0], [0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0], [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]] \ No newline at end of file diff --git a/research/fstar-family-gap-seed1c25f56f/data/cex_min_1.json b/research/fstar-family-gap-seed1c25f56f/data/cex_min_1.json new file mode 100644 index 0000000000..4629aa02b1 --- /dev/null +++ b/research/fstar-family-gap-seed1c25f56f/data/cex_min_1.json @@ -0,0 +1 @@ +[[[1, -1, 0], [1, 0, -1]], [[1, -1, 1], [0, 1, -1]], [[1, -1, 1], [1, -1, 0]], [[1, -1, 1], [1, 0, 1]], [[1, 0, -1], [0, 1, -1]], [[1, 0, -1], [0, 1, 1]], [[1, 0, -1], [1, -1, 0]], [[1, 0, -1], [1, 1, -1]], [[1, 0, 1], [1, 1, 0]], [[1, 0, 1], [1, 1, 1]], [[1, 1, -1], [0, 1, -1]], [[1, 1, -1], [1, -1, 1]], [[1, 1, -1], [1, 0, -1]], [[1, 1, -1], [1, 1, 0]], [[1, 1, 0], [0, 1, 1]], [[1, 1, 0], [1, 0, 1]], [[1, 1, 1], [0, 1, 1]], [[1, 1, 1], [1, 0, 1]], [[1, 1, 1], [1, 1, 0]]] \ No newline at end of file diff --git a/research/fstar-family-gap-seed1c25f56f/data/cex_min_1.witness.json b/research/fstar-family-gap-seed1c25f56f/data/cex_min_1.witness.json new file mode 100644 index 0000000000..602d784e98 --- /dev/null +++ b/research/fstar-family-gap-seed1c25f56f/data/cex_min_1.witness.json @@ -0,0 +1 @@ +{"a": [3, 2, 2, 4, 4, 5, 5, 4, 1, 1, 2, 7, 5, 1, 2, 3, 6, 1, 1], "b": [1, 6, 0, 5, 6, 6, 1, 4, 3, 7, 6, 2, 1, 3, 6, 5, 2, 5, 3], "order": [16, 6, 11, 14, 5, 12, 0, 18, 17, 15, 3, 13, 10, 9, 8, 7, 4, 1], "kept": [16, 6, 11, 14, 5, 12, 0, 18, 17, 15, 3, 13, 10, 9, 8, 7, 4, 1]} \ No newline at end of file diff --git a/research/fstar-family-gap-seed1c25f56f/data/tied_min.json b/research/fstar-family-gap-seed1c25f56f/data/tied_min.json new file mode 100644 index 0000000000..ae11f01fab --- /dev/null +++ b/research/fstar-family-gap-seed1c25f56f/data/tied_min.json @@ -0,0 +1 @@ +[[[1, -1, -1], [0, 1, 1]], [[1, -1, -1], [1, -1, 0]], [[1, -1, -1], [1, 0, -1]], [[1, -1, 1], [0, 1, -1]], [[1, -1, 1], [1, -1, 0]], [[1, -1, 1], [1, 0, 1]], [[1, 1, -1], [0, 1, -1]], [[1, 1, -1], [1, 0, -1]], [[1, 1, -1], [1, 1, 0]], [[1, 1, 1], [0, 1, 1]], [[1, 1, 1], [1, 0, 1]], [[1, 1, 1], [1, 1, 0]]] \ No newline at end of file diff --git a/research/fstar-family-gap-seed1c25f56f/verify.py b/research/fstar-family-gap-seed1c25f56f/verify.py new file mode 100644 index 0000000000..9d8285e4e3 --- /dev/null +++ b/research/fstar-family-gap-seed1c25f56f/verify.py @@ -0,0 +1,1364 @@ +#!/usr/bin/env python3 +"""KB5g: the family gap of the F-star residue - sub-instance census of the +canonical fixtures, the sub-anchor-box decomposition of the attaining +certificates, and the smallest-first falsifier criterion (no kappa). + +Object: exact 3-total rational decoders M = sum_j p_j q_j^T (v lines, nonzero +diagonal). Quantities per instance: zero digraph Z, Lemma SG anchor reach +|S_gamma|, tie numbers tau(t), matched-pivot value MP (Lemma MP), the named +family value (PR 2901's convention: max(MP, |S_gamma|, CL3 mixing)), the +sub-anchor-box value (NEW: a(t) subset of a0(t), b(t) subset of b0(t), +K_tt = 1), exact cert(M) by exhaustive a-enumeration with exact b-completion +(small v), engine witnesses with independent replay, and the bounded +admissible-B chain scan as falsifier criterion. + +Engine lineage: model, subset-DP acyclic/MAS tables, anchor/tie data, +matched pivots, exact-completion DP, witness replay and the chain machinery +are adapted from Aaron Wang's PR 2717 / PR 2747 / PR 2869 / PR 2901 engine +(see README credit); the sub-anchor-box machinery, the sub-instance census +and the decomposition are new here. F-star and the chain cost >= cert(M) are +from research/root-unit-decoder-lemmas (PR 1590, DreamingOfClouds), cited, +not restated. + +Deterministic, stdlib only, exact integer/F2 arithmetic. Run: +python3 -B verify.py +""" +import hashlib +import itertools +import json +import os +import random +import sys +import time + +T0 = time.time() +HERE = os.path.dirname(os.path.abspath(__file__)) +DATA = os.path.join(HERE, "data") + + +def log(msg): + print("[%6.1fs] %s" % (time.time() - T0, msg), flush=True) + + +# ---------------------------------------------------------------- model +# (adapted from the PR 2717/2901 engine lineage, Aaron Wang) + +def load_decoder(path): + with open(path, "r", encoding="utf-8") as fh: + raw = json.load(fh) + P = [tuple(int(x) for x in line[0]) for line in raw] + Q = [tuple(int(x) for x in line[1]) for line in raw] + return P, Q + + +def model(P, Q): + v = len(P) + for t in range(v): + for x in list(P[t]) + list(Q[t]): + assert isinstance(x, int), "entries must be exact integers" + M = [[sum(P[t][j] * Q[s][j] for j in range(3)) for s in range(v)] for t in range(v)] + pi = [sum((1 << j) for j in range(3) if P[t][j] != 0) for t in range(v)] + sg = [sum((1 << j) for j in range(3) if Q[t][j] != 0) for t in range(v)] + D = [pi[t] & sg[t] for t in range(v)] + assert all(M[t][t] != 0 for t in range(v)), "diagonal must be nonzero" + assert all(D[t] != 0 for t in range(v)), "D(t) must be nonempty" + arcs = [(t, s) for t in range(v) for s in range(v) if t != s and M[t][s] == 0] + return dict(v=v, P=P, Q=Q, M=M, pi=pi, sg=sg, D=D, arcs=arcs) + + +def load_model(name): + return model(*load_decoder(os.path.join(DATA, name))) + + +def restrict(mod, S): + """Sub-decoder M[S,S]: restriction of an exact decoder is exact (rank <= 3 + and nonzero diagonal are preserved). S is a sorted list of line indices.""" + P = [mod["P"][t] for t in S] + Q = [mod["Q"][t] for t in S] + return model(P, Q) + + +def digraph_tables(v, arcs): + inm = [0] * v + outm = [0] * v + for (t, s) in arcs: + inm[s] |= 1 << t + outm[t] |= 1 << s + return inm, outm + + +def acyclic_table(v, inm): + acyc = [False] * (1 << v) + acyc[0] = True + for S in range(1, 1 << v): + ok = False + T = S + while T: + b = T & -T + t = b.bit_length() - 1 + if (inm[t] & S) == 0 and acyc[S ^ b]: + ok = True + break + T ^= b + acyc[S] = ok + return acyc + + +def mas_table(v, acyc): + n = 1 << v + mas = [0] * n + for S in range(1, n): + if acyc[S]: + mas[S] = S.bit_count() + else: + T = S + best = 0 + while T: + b = T & -T + val = mas[S ^ b] + if val > best: + best = val + T ^= b + mas[S] = best + return mas + + +def mas_argmax(mas, acyc, S, memo): + if S == 0: + return 0 + if S in memo: + return memo[S] + if acyc[S]: + memo[S] = S + return S + T = S + while T: + b = T & -T + if mas[S ^ b] == mas[S]: + val = mas_argmax(mas, acyc, S ^ b, memo) + memo[S] = val + return val + T ^= b + raise AssertionError("mas_argmax: no witness") + + +# -------------------------------------------- Lemma SG anchor + Lemma CT data + +def v2_int(n): + if n == 0: + raise ValueError("v2(0)") + k = 0 + while n % 2 == 0: + n //= 2 + k += 1 + return k + + +def anchor_data(mod): + """Canonical reduced normalization (alpha_j = -m_j): a0_j(t) = + [v2(p_j(t)) = m_j], b0_j(t) = [v2(q_j(t)) = gamma - m_j], with the scope + identity K0_ts = 2^{-gamma} M_ts mod 2 (Lemma SG of PR 2869).""" + P, Q, M = mod["P"], mod["Q"], mod["M"] + v = mod["v"] + SENTINEL = 10 ** 9 + m, n, c = [], [], [] + for j in range(3): + mp = [v2_int(P[t][j]) for t in range(v) if P[t][j] != 0] + mq = [v2_int(Q[t][j]) for t in range(v) if Q[t][j] != 0] + if mp and mq: + m.append(min(mp)) + n.append(min(mq)) + else: + m.append(SENTINEL) + n.append(SENTINEL) + c.append(m[-1] + n[-1]) + assert min(c) < SENTINEL, "at least one total must be nondegenerate" + gamma = min(c) + diag = [v2_int(M[t][t]) for t in range(v)] + reach = [t for t in range(v) if diag[t] == gamma] + cancel = [t for t in range(v) if diag[t] > gamma] + tau = [] + for t in range(v): + cnt = 0 + for j in range(3): + if (P[t][j] != 0 and Q[t][j] != 0 + and v2_int(P[t][j]) + v2_int(Q[t][j]) == gamma): + cnt += 1 + tau.append(cnt) + a0 = [sum((1 << j) for j in range(3) + if P[t][j] != 0 and v2_int(P[t][j]) == m[j]) + for t in range(v)] + b0 = [sum((1 << j) for j in range(3) + if Q[t][j] != 0 and v2_int(Q[t][j]) == gamma - m[j]) + for t in range(v)] + return dict(gamma=gamma, m=m, n=n, c=c, diag=diag, reach=reach, + cancel=cancel, tau=tau, a0=a0, b0=b0) + + +def dot1(x, y): + return bin(x & y).count("1") & 1 + + +def scope_identity(mod, ad): + v, M = mod["v"], mod["M"] + for t in range(v): + for s in range(v): + want = 1 if (M[t][s] != 0 and v2_int(M[t][s]) == ad["gamma"]) else 0 + if dot1(ad["a0"][t], ad["b0"][s]) != want: + return False + return True + + +def tie_parity(mod, ad): + v = mod["v"] + for t in range(v): + if (ad["tau"][t] % 2 == 1) != (ad["diag"][t] == ad["gamma"]): + return False + if (ad["tau"][t] % 2 == 0) != (ad["diag"][t] > ad["gamma"]): + return False + if ad["tau"][t] > mod["D"][t].bit_count(): + return False + return True + + +# --------------------------------------------------------- Lemma MP pivots + +def matched_pivot(mod, mas, exhaustive_cap=200000, seed=20261016): + v = mod["v"] + allowed = [[1 << j for j in range(3) if mod["D"][t] & (1 << j)] + for t in range(v)] + prod = 1 + for a in allowed: + prod *= len(a) + best, best_map = -1, None + exhaustive = prod <= exhaustive_cap + if exhaustive: + for combo in itertools.product(*allowed): + classes = [0, 0, 0] + for t in range(v): + i = combo[t].bit_length() - 1 + classes[i] |= 1 << t + val = mas[classes[0]] + mas[classes[1]] + mas[classes[2]] + if val > best: + best, best_map = val, combo + else: + rng = random.Random(seed) + for _ in range(exhaustive_cap): + combo = tuple(a[rng.randrange(len(a))] for a in allowed) + classes = [0, 0, 0] + for t in range(v): + i = combo[t].bit_length() - 1 + classes[i] |= 1 << t + val = mas[classes[0]] + mas[classes[1]] + mas[classes[2]] + if val > best: + best, best_map = val, combo + return best, best_map, exhaustive, prod + + +# --------------------------------------------------------- cert engine +# (PR 2717's exact-completion engine design (Aaron Wang), adapted: for fixed +# a-labels the completion over b-labels, order and kept set is exact by +# subset DP.) + +def span_add(basis, x): + pivots = {} + for r in list(basis) + [x]: + while r: + p = r.bit_length() - 1 + if p in pivots: + r ^= pivots[p] + else: + pivots[p] = r + break + return tuple(sorted(pivots.values(), reverse=True)) + + +def completion(mod, a): + """Exact completion over b-labels/order/kept set for fixed a-labels. + Returns (best size, witness dict or None).""" + v = mod["v"] + sg = mod["sg"] + inm, _ = digraph_tables(v, mod["arcs"]) + blockers = [inm[t] for t in range(v)] + bex = {} + basis_tab = [dict() for _ in range(v)] + b_all = [[b for b in range(8) if (b & ~sg[t]) == 0] for t in range(v)] + + def b_exists(t, basis): + key = (t, basis) + if key in bex: + return bex[key] + for b in b_all[t]: + if dot1(a[t], b) != 1: + continue + if all(dot1(x, b) == 0 for x in basis): + bex[key] = (True, b) + return bex[key] + bex[key] = (False, 0) + return bex[key] + + def basis_of(t, T): + d = basis_tab[t] + if T in d: + return d[T] + if T == 0: + d[0] = () + return () + b = T & -T + s = b.bit_length() - 1 + prev = basis_of(t, T ^ b) + val = span_add(prev, a[s]) if (blockers[t] & b) else prev + d[T] = val + return val + + n = 1 << v + feas = [False] * n + feas[0] = True + choice = [-1] * n + for T in range(1, n): + U = T + while U: + b = U & -U + t = b.bit_length() - 1 + rest = T ^ b + if feas[rest]: + ok, _bv = b_exists(t, basis_of(t, rest)) + if ok: + feas[T] = True + choice[T] = t + break + U ^= b + best = max(T for T in range(n) if feas[T]) + if best == 0: + return 0, None + kept = [] + bvec = [0] * v + T = best + while T: + t = choice[T] + _ok, bv = b_exists(t, basis_of(t, T ^ (1 << t))) + bvec[t] = bv + kept.append(t) + T ^= 1 << t + kept.reverse() + return best.bit_count(), dict(a=list(a), b=bvec, order=list(kept), + kept=list(kept)) + + +def completion_box(mod, ad, a): + """Completion with every b-label restricted to b(t) subset of b0(t). + Used for the sub-anchor-box value (new here).""" + v = mod["v"] + b0 = ad["b0"] + inm, _ = digraph_tables(v, mod["arcs"]) + blockers = [inm[t] for t in range(v)] + bex = {} + basis_tab = [dict() for _ in range(v)] + b_all = [[b for b in range(8) if (b & ~b0[t]) == 0] for t in range(v)] + + def b_exists(t, basis): + key = (t, basis) + if key in bex: + return bex[key] + for b in b_all[t]: + if dot1(a[t], b) != 1: + continue + if all(dot1(x, b) == 0 for x in basis): + bex[key] = (True, b) + return bex[key] + bex[key] = (False, 0) + return bex[key] + + def basis_of(t, T): + d = basis_tab[t] + if T in d: + return d[T] + if T == 0: + d[0] = () + return () + b = T & -T + s = b.bit_length() - 1 + prev = basis_of(t, T ^ b) + val = span_add(prev, a[s]) if (blockers[t] & b) else prev + d[T] = val + return val + + n = 1 << v + feas = [False] * n + feas[0] = True + choice = [-1] * n + for T in range(1, n): + U = T + while U: + b = U & -U + t = b.bit_length() - 1 + rest = T ^ b + if feas[rest]: + ok, _bv = b_exists(t, basis_of(t, rest)) + if ok: + feas[T] = True + choice[T] = t + break + U ^= b + best = max(T for T in range(n) if feas[T]) + if best == 0: + return 0, None + kept = [] + bvec = [0] * v + T = best + while T: + t = choice[T] + _ok, bv = b_exists(t, basis_of(t, T ^ (1 << t))) + bvec[t] = bv + kept.append(t) + T ^= 1 << t + kept.reverse() + return best.bit_count(), dict(a=list(a), b=bvec, order=list(kept), + kept=list(kept)) + + +def replay_witness(mod, wit): + """Independent replay: supports, K_tt = 1 on kept, order constraints.""" + v, M = mod["v"], mod["M"] + a, b = wit["a"], wit["b"] + kept = list(wit["kept"]) + order = list(wit["order"]) + if sorted(order) != sorted(kept): + return False + for t in kept: + if (a[t] & ~mod["pi"][t]) or (b[t] & ~mod["sg"][t]): + return False + if dot1(a[t], b[t]) != 1: + return False + pos = {t: i for i, t in enumerate(order)} + for t in kept: + for s in kept: + if t == s: + continue + if M[t][s] == 0 and pos[t] < pos[s]: + if dot1(a[t], b[s]) != 0: + return False + return True + + +def in_box(mod, ad, wit): + """Every kept label pair is a sub-anchor pair: a(t) subset of a0(t) and + b(t) subset of b0(t). Returns (all_in_box, escape_lines).""" + esc = [] + for t in wit["kept"]: + if (wit["a"][t] & ~ad["a0"][t]) or (wit["b"][t] & ~ad["b0"][t]): + esc.append(t) + return (not esc), esc + + +def cert_exact(mod, max_v=6): + """Exact cert by exhaustive a-enumeration (small v only).""" + v = mod["v"] + if v > max_v: + return None, None + choices = [[m for m in range(8) if (m & ~mod["pi"][t]) == 0] + for t in range(v)] + best, best_wit = -1, None + for combo in itertools.product(*choices): + val, wit = completion(mod, list(combo)) + if val > best: + best, best_wit = val, wit + return best, best_wit + + +def box_exact(mod, ad, max_v=6): + """Exact sub-anchor-box value: exhaustive a-subsets of a0 with the + b-side completed inside b0 (small v only).""" + v = mod["v"] + if v > max_v: + return None, None + choices = [] + for t in range(v): + subs = [m for m in range(8) if (m & ~ad["a0"][t]) == 0] + subs = [m for m in subs + if any(dot1(m, b) == 1 and (b & ~ad["b0"][t]) == 0 + for b in range(8))] + choices.append(subs) + best, best_wit = -1, None + for combo in itertools.product(*choices): + val, wit = completion_box(mod, ad, list(combo)) + if val > best: + best, best_wit = val, wit + return best, best_wit + + +def cert_hillclimb(mod, iters=400, seed=1234, box=False, ad=None): + """Bounded deterministic hill-climb over a-labels with exact completion. + box=True restricts both sides to the sub-anchor box.""" + v = mod["v"] + rng = random.Random(seed) + if box: + choices = [[m for m in range(8) if (m & ~ad["a0"][t]) == 0] + for t in range(v)] + comp = lambda a: completion_box(mod, ad, a) # noqa: E731 + else: + choices = [[m for m in range(8) if (m & ~mod["pi"][t]) == 0] + for t in range(v)] + comp = lambda a: completion(mod, a) # noqa: E731 + a = [c[rng.randrange(len(c))] for c in choices] + best, best_wit = comp(a) + cur = list(a) + for _ in range(iters): + t = rng.randrange(v) + old = cur[t] + new = choices[t][rng.randrange(len(choices[t]))] + if new == old: + continue + cur[t] = new + val, wit = comp(cur) + if val >= best: + best, best_wit = val, wit + else: + cur[t] = old + return best, best_wit + + +# ----------------------------------------------- chain machinery (falsifier) +# (PR 2717/2901 lineage) + +def gf2_rank(rows): + pivots = {} + for x in rows: + while x: + p = x.bit_length() - 1 + if p in pivots: + x ^= pivots[p] + else: + pivots[p] = x + break + return len(pivots) + + +def chain_cost(mod, B): + P, Q, v = mod["P"], mod["Q"], mod["v"] + total = 0 + for j in range(3): + R = [t for t in range(v) if P[t][j] != 0] + A = [s for s in range(v) if Q[s][j] != 0] + rows = [] + for t in R: + mask = 0 + for s in A: + if B[t][s]: + mask |= 1 << A.index(s) + rows.append(mask) + total += gf2_rank(rows) + return total + + +def admissible(mod, B): + v, M = mod["v"], mod["M"] + for t in range(v): + if B[t][t] != 1: + return False + for s in range(v): + if t != s and B[t][s] == 1 and M[t][s] != 0: + return False + return True + + +def chain_scan(mod, threshold, seed=99, tries=3000): + """Bounded deterministic scan for an admissible B with chain cost below + the threshold (the falsifier criterion of PR 2901).""" + v = mod["v"] + rng = random.Random(seed) + best, best_cost, hit = None, None, False + for _ in range(tries): + B = [[0] * v for _ in range(v)] + for t in range(v): + B[t][t] = 1 + for s in range(v): + if t != s and mod["M"][t][s] == 0 and rng.randrange(2): + B[t][s] = 1 + if not admissible(mod, B): + continue + cc = chain_cost(mod, B) + if best_cost is None or cc < best_cost: + best, best_cost = B, cc + if cc < threshold: + hit = True + break + return best_cost, best, hit + + +# ------------------------------- named label forms and the ladder (NEW here) +# +# The named families of PR 2901 (Lemma SG anchor, Lemma MX one-sided +# repairs, Lemma MP matched pivots, Lemma CL3 insertion mixings) all build +# their certificates from per-line labels of three forms: (a0(t), b0(t)), +# (a*(t), b0(t)) with a* . b0(t) = 1, or (e_c, e_c) with c in D(t). Call +# their union the Form-R space. For ANY fixed label assignment (a, b) its +# exact certificate value is MAS(D_K[good]) where D_K is the conflict +# digraph and good = {t : K_tt = 1}; so the Form-R maximum is a rigorous +# upper bound on the PR 2901 named value and a lower bound on cert(M). +# +# Ladder per instance: MP <= named_lo <= named <= form_r <= box <= cert +# named_lo = max(MP, |S_gamma|) (PR 2901's cheap rung) +# named = PR 2901's family value (<= form_r, proved above) +# form_r = max over Form-R labels (NEW upper bound on named) +# box = max over a(t) subset of a0(t), b(t) subset of b0(t) (NEW) +# cert = max over all labels. + +def ab_value(mod, a, b): + """Exact certificate value of a fixed label assignment: the largest kept + set with K_tt = 1 and every conflict pointing backward in some order.""" + v, M = mod["v"], mod["M"] + good = 0 + inm = [0] * v + for t in range(v): + if dot1(a[t], b[t]) == 1: + good |= 1 << t + for t in range(v): + for s in range(v): + if t != s and M[t][s] == 0 and dot1(a[t], b[s]) == 1: + inm[s] |= 1 << t + n = 1 << v + acyc = [False] * n + acyc[0] = True + best = 0 + for S in range(1, n): + if S & ~good: + continue + ok = False + T = S + while T: + x = T & -T + t = x.bit_length() - 1 + if (inm[t] & S) == 0 and acyc[S ^ x]: + ok = True + break + T ^= x + acyc[S] = ok + if ok and S.bit_count() > best: + best = S.bit_count() + return best + + +def form_r_options(mod, ad, t): + """The named-form label pairs on line t (deduplicated): anchor (a0, b0), + one-sided repairs (a*, b0) with a* . b0(t) = 1, matched pivots (e_c, e_c).""" + opts = {(ad["a0"][t], ad["b0"][t])} + for a in range(8): + if (a & ~mod["pi"][t]) == 0 and dot1(a, ad["b0"][t]) == 1: + opts.add((a, ad["b0"][t])) + for j in range(3): + if mod["D"][t] & (1 << j): + opts.add((1 << j, 1 << j)) + return sorted(opts) + + +def form_r_max(mod, ad, exhaustive_cap=300000, iters=250, seed=4242): + """Max of ab_value over the Form-R label space. Exact when the product of + per-line option counts is within the cap; deterministic hill-climb + otherwise (lower bound, flagged).""" + v = mod["v"] + opts = [form_r_options(mod, ad, t) for t in range(v)] + prod = 1 + for o in opts: + prod *= len(o) + best, best_wit = -1, None + + def take(combo): + nonlocal best, best_wit + a = [c[0] for c in combo] + b = [c[1] for c in combo] + val = ab_value(mod, a, b) + if val > best: + best = val + best_wit = dict(a=list(a), b=b, order=list(range(v)), kept=[]) + return val + + exhaustive = prod <= exhaustive_cap + if exhaustive: + for combo in itertools.product(*opts): + take(combo) + else: + rng = random.Random(seed) + cur = [o[rng.randrange(len(o))] for o in opts] + take(cur) + for _ in range(iters): + t = rng.randrange(v) + old = cur[t] + new = opts[t][rng.randrange(len(opts[t]))] + if new == old: + continue + cur[t] = new + val = take(cur) + if val < best: + cur[t] = old + # materialize a replayable witness for the best assignment: kept set is + # the largest acyclic conflict subset; order is a backward topological + # order of the conflicts (sink-first). + a = best_wit["a"] + b = best_wit["b"] + v_, M = mod["v"], mod["M"] + good = [t for t in range(v_) if dot1(a[t], b[t]) == 1] + inm = [0] * v_ + for t in good: + for s in good: + if t != s and M[t][s] == 0 and dot1(a[t], b[s]) == 1: + inm[s] |= 1 << t + S = 0 + for t in good: + S |= 1 << t + acyc = [False] * (1 << v_) + acyc[0] = True + for X in range(1, 1 << v_): + if X & ~S: + continue + ok = False + Tm = X + while Tm: + x = Tm & -Tm + t = x.bit_length() - 1 + if (inm[t] & X) == 0 and acyc[X ^ x]: + ok = True + break + Tm ^= x + acyc[X] = ok + bestS, bestN = 0, -1 + for X in range(1, 1 << v_): + if (X & ~S) == 0 and acyc[X] and X.bit_count() > bestN: + bestS, bestN = X, X.bit_count() + order = [] + rem = bestS + while rem: + sinks = [t for t in range(v_) if (rem >> t) & 1 and (inm[t] & rem) == 0] + t = min(sinks) + order.append(t) + rem ^= 1 << t + order.reverse() + best_wit = dict(a=a, b=b, order=order, kept=list(order)) + return best, best_wit, exhaustive, prod + + +def box_witness_from(mod, ad, a, b, order): + return dict(a=list(a), b=list(b), order=list(order), kept=list(order)) + + +def diff_labels(mod, ad, wit): + """Classify each kept line's label pair against the named forms and the + sub-anchor box (PR 2901 open question 2, first step).""" + rows = [] + for t in wit["kept"]: + a, b = wit["a"][t], wit["b"][t] + anchor = (a, b) == (ad["a0"][t], ad["b0"][t]) + repair = (b == ad["b0"][t] and dot1(a, b) == 1 + and (a & ~mod["pi"][t]) == 0) + pivot = a == b and a in (1, 2, 4) and (mod["D"][t] & a) != 0 + inbox = (a & ~ad["a0"][t]) == 0 and (b & ~ad["b0"][t]) == 0 + if anchor: + kind = "ANCHOR" + elif repair: + kind = "REPAIR" + elif pivot: + kind = "PIVOT" + elif inbox: + kind = "BOX-INTERIOR" + else: + kind = "BOX-ESCAPE" + rows.append(dict(t=t, a=a, b=b, a0=ad["a0"][t], b0=ad["b0"][t], + tau=ad["tau"][t], D=mod["D"][t], kind=kind)) + return rows + + +def named_exact_small(mod, ad, mas, acyc, cap=400000, scan_iters=None): + """Exact PR 2901 named value on small instances: max over the union of + the named families' constructions (Lemma SG anchor |S_gamma|, Lemma MP + matched pivots, Lemma MX one-sided-repair constructions with their two + exact hypothesis sets, Lemma CL3 insertion mixings - whose value for its + label forms is exactly ab_value). Returns (named, detail). Large + construction spaces fall back to a deterministic bounded scan (the + detail flags exactness).""" + v = mod["v"] + if scan_iters is None: + scan_iters = 20000 if v <= 13 else 300 + mp, _m, _e, _p = matched_pivot(mod, mas) + best = max(mp, len(ad["reach"])) + detail = dict(MP=mp, reach=len(ad["reach"]), mx=None, cl3=None, + exhaustive=True, product=0) + # per-line options for the MX construction space: ('A'), ('U', a*), + # ('T', c-bit) + opts = [] + for t in range(v): + o = [("A", 0)] + for a in range(8): + if (a & ~mod["pi"][t]) == 0 and dot1(a, ad["b0"][t]) == 1 \ + and a != ad["a0"][t]: + o.append(("U", a)) + for j in range(3): + if mod["D"][t] & (1 << j): + o.append(("T", 1 << j)) + opts.append(o) + prod = 1 + for o in opts: + prod *= len(o) + detail["product"] = prod + M = mod["M"] + + def mx_eval(combo): + A = [t for t in range(v) if combo[t][0] == "A"] + U = [t for t in range(v) if combo[t][0] == "U"] + T = [t for t in range(v) if combo[t][0] == "T"] + for t in A: + if not (mod["M"][t][t] != 0 and ad["tau"][t] % 2 == 1): + return None # Lemma MX: every t in A has K0_tt = 1 + classes = [0, 0, 0] + cval = [0] * v + for t in T: + c = combo[t][1] + cval[t] = c + classes[c.bit_length() - 1] |= 1 << t + val = len(A) + len(U) + mas[classes[0]] + mas[classes[1]] \ + + mas[classes[2]] + # (A-first) (i): for t in U: a* . b0(s) = 0 for s in A+U with + # M_ts = 0, and no pivot bit c(s) set in a* for s in T with M_ts = 0; + # (ii): no c(s) set in a0(t) for t in A, s in T with M_ts = 0. + ok_a = True + for t in U: + a_star = combo[t][1] + for s in A + U: + if M[t][s] == 0 and dot1(a_star, ad["b0"][s]) != 0: + ok_a = False + for s in T: + if M[t][s] == 0 and (a_star & cval[s]) != 0: + ok_a = False + for t in A: + for s in T: + if M[t][s] == 0 and (ad["a0"][t] & cval[s]) != 0: + ok_a = False + # (T-first) (i')/(ii') + ok_t = True + for t in U: + a_star = combo[t][1] + for s in A + U: + if M[t][s] == 0 and dot1(a_star, ad["b0"][s]) != 0: + ok_t = False + for t in T: + for s in A + U: + if M[s][t] == 0 and (cval[t] & ad["b0"][s]) != 0: + ok_t = False + return val if (ok_a or ok_t) else None + + if prod <= cap: + mx_best = 0 + for combo in itertools.product(*opts): + val = mx_eval(combo) + if val is not None and val > mx_best: + mx_best = val + detail["mx"] = mx_best + best = max(best, mx_best) + else: + detail["exhaustive"] = False + rng = random.Random(31337) + mx_best = 0 + best_cur = None + # structured seeds (pure matched-pivot maps always satisfy both MX + # forms; the greedy walk below gets stuck in hypothesis-failing + # mixed assignments without them) + seeds = [[("A", 0)] * v] + for j in range(3): + seeds.append([("T", (1 << j)) if (mod["D"][t] & (1 << j)) + else ("T", 1 << (mod["D"][t].bit_length() - 1)) + for t in range(v)]) + mp0, mpmap0, _e0, _p0 = matched_pivot(mod, mas) + if mpmap0 is not None: + seeds.append([("T", mpmap0[t]) for t in range(v)]) + for s0 in seeds: + val = mx_eval(s0) + if val is not None and val > mx_best: + mx_best, best_cur = val, list(s0) + cur = list(best_cur) if best_cur is not None \ + else [o[rng.randrange(len(o))] for o in opts] + since_restart = 0 + for _ in range(scan_iters): + if since_restart >= 4 * v: + since_restart = 0 + if best_cur is not None and rng.randrange(2): + cur = list(best_cur) + else: + cur = [o[rng.randrange(len(o))] for o in opts] + since_restart += 1 + t = rng.randrange(v) + old = cur[t] + cur[t] = opts[t][rng.randrange(len(opts[t]))] + val = mx_eval(cur) + if val is None or val < mx_best: + cur[t] = old + elif val > mx_best: + mx_best = val + best_cur = list(cur) + detail["mx"] = mx_best + best = max(best, mx_best) + # CL3 value for its label forms (anchor (a0,b0) on A, matched pivots): + # exact ab_value over the combined forms, small product only. + cl3_opts = [[(ad["a0"][t], ad["b0"][t], "A")] + + [((1 << j), (1 << j), "T") for j in range(3) + if mod["D"][t] & (1 << j)] + for t in range(v)] + cl3_prod = 1 + for o in cl3_opts: + cl3_prod *= len(o) + cl3_best = 0 + if cl3_prod <= cap: + for combo in itertools.product(*cl3_opts): + a = [c[0] for c in combo] + b = [c[1] for c in combo] + val = ab_value(mod, a, b) + if val > cl3_best: + cl3_best = val + else: + rng = random.Random(4242) + cur = [o[rng.randrange(len(o))] for o in cl3_opts] + for _ in range(scan_iters): + t = rng.randrange(v) + old = cur[t] + cur[t] = cl3_opts[t][rng.randrange(len(cl3_opts[t]))] + a = [c[0] for c in cur] + b = [c[1] for c in cur] + val = ab_value(mod, a, b) + if val < cl3_best: + cur[t] = old + elif val > cl3_best: + cl3_best = val + detail["exhaustive"] = False + detail["cl3"] = cl3_best + best = max(best, cl3_best) + return best, detail + + +def settle_ladder(mod, ad, mas, acyc, exact=False, engine_iters=150, + provided_cert_wit=None, label=""): + """One ladder record: all rungs, the gap decomposition, and the armed + falsifier criterion (exact cert below threshold where exhaustive, or an + explicit admissible B with chain cost below threshold). exact=True runs + the exhaustive Form-R / box / cert rungs (small v); otherwise the engine + rungs are deterministic hill-climbs (lower bounds) and provided_cert_wit + (e.g. a stored witness) is used as the cert rung on large instances.""" + v = mod["v"] + thr = (2 * v) // 3 + mp, mpmap, mp_exh, mp_prod = matched_pivot(mod, mas) + named_lo = max(mp, len(ad["reach"])) + fr_val, fr_wit, fr_exh, fr_prod = form_r_max( + mod, ad, exhaustive_cap=(300000 if exact else 0)) + assert fr_wit is not None and replay_witness(mod, fr_wit), "form-R witness" + if exact: + bx_val, bx_wit = box_exact(mod, ad, max_v=6) + ce_val, ce_wit = cert_exact(mod, max_v=6) + bx_exact, ce_exact = True, True + else: + if v > 13: + bx_val, bx_wit = None, None + else: + bx_val, bx_wit = cert_hillclimb(mod, iters=engine_iters, seed=2024, + box=True, ad=ad) + if provided_cert_wit is not None: + ce_wit = provided_cert_wit + ce_val = len(ce_wit["kept"]) + elif v > 13: + ce_val, ce_wit = 0, None + else: + ce_val, ce_wit = cert_hillclimb(mod, iters=engine_iters, seed=2025) + bx_exact, ce_exact = False, False + if bx_wit is not None: + assert replay_witness(mod, bx_wit), "box witness" + if ce_wit is not None: + assert replay_witness(mod, ce_wit), "engine witness" + named_val, named_detail = None, None + if exact: + named_val, named_detail = named_exact_small(mod, ad, mas, acyc) + rec = dict(label=label, v=v, threshold=thr, k_cancel=len(ad["cancel"]), + reach=len(ad["reach"]), MP=mp, MP_exhaustive=mp_exh, + MP_product=mp_prod, named_lo=named_lo, + named=named_val, named_detail=named_detail, + form_r=fr_val, form_r_exhaustive=fr_exh, form_r_product=fr_prod, + box=bx_val, box_exact=bx_exact, + cert=ce_val, cert_exact=ce_exact, + gap_forms=(ce_val - fr_val), gap_pr=(ce_val - named_lo), + gap_named=((ce_val - named_val) if named_val is not None + else None), + named_ge_threshold=(named_lo >= thr), + forms_ge_threshold=(fr_val >= thr), + cert_ge_threshold=(ce_val >= thr)) + if ce_wit is not None: + inb, esc = in_box(mod, ad, ce_wit) + rec["cert_witness_in_box"] = inb + rec["cert_witness_box_escape"] = esc + rec["cert_witness_diff"] = diff_labels(mod, ad, ce_wit) + rec["falsifier"] = None + if rec["cert_exact"]: + if ce_val < thr: + rec["falsifier"] = "exact cert %d < threshold %d" % (ce_val, thr) + elif named_lo < thr or ce_val < thr: + cc, _B, hit = chain_scan(mod, thr) + rec["chain_scan_best"] = cc + if hit: + rec["falsifier"] = ("admissible B with chain cost %d < " + "threshold %d" % (cc, thr)) + return rec + + +# --------------------------------------------------------------- controls + +def brute_force_cert(mod): + """Cert from the definition: all labellings, all orders (tiny v only).""" + v, M = mod["v"], mod["M"] + choices = [[m for m in range(8) if (m & ~mod["pi"][t]) == 0] + for t in range(v)] + bchoices = [[m for m in range(8) if (m & ~mod["sg"][t]) == 0] + for t in range(v)] + best = 0 + for a in itertools.product(*choices): + for b in itertools.product(*bchoices): + for k in range(1, v + 1): + for sub in itertools.permutations(range(v), k): + ok = True + for i, t in enumerate(sub): + if dot1(a[t], b[t]) != 1: + ok = False + break + for s in sub[i + 1:]: + if M[t][s] == 0 and dot1(a[t], b[s]) != 0: + ok = False + break + if not ok: + break + if ok and k > best: + best = k + return best + + +def sparse_v3(seed): + """A tiny v = 3 decoder with sparse supports (for the brute-force + control). Deterministic small-integer factorization.""" + rng = random.Random(seed) + while True: + P = [tuple(rng.choice([0, 0, 0, 1, -1, 2]) for _ in range(3)) + for _ in range(3)] + Q = [tuple(rng.choice([0, 0, 0, 1, -1, 2]) for _ in range(3)) + for _ in range(3)] + try: + mod = model(P, Q) + except AssertionError: + continue + if max(mod["pi"][t].bit_count() for t in range(3)) <= 2: + return mod + + +def run_controls(): + # 1. exact cert vs brute force from the definition (3 sparse v = 3 cases) + agree = 0 + for seed in (11, 22, 33): + mod = sparse_v3(seed) + bf = brute_force_cert(mod) + ex, wit = cert_exact(mod, max_v=6) + assert ex == bf and wit is not None and replay_witness(mod, wit), \ + "cert_exact must agree with brute force at v = 3" + agree += 1 + # 2. corrupted witnesses must fail the independent replay + mod = sparse_v3(44) + ex, wit = cert_exact(mod, max_v=6) + bad = dict(a=list(wit["a"]), b=list(wit["b"]), order=list(wit["order"]), + kept=list(wit["kept"])) + bad["a"][bad["kept"][0]] = 0 # kills K_tt on a kept line + assert not replay_witness(mod, bad), "corrupted label must be rejected" + bad2 = dict(a=list(wit["a"]), b=list(wit["b"]), order=list(wit["order"]), + kept=list(wit["kept"])) + bad2["kept"] = bad2["kept"][1:] # kept/order mismatch + assert not replay_witness(mod, bad2), "corrupted kept set must be rejected" + # 3. ab_value is a sound rung: for random fixed labels it never exceeds + # the exact completion value for the same a-side + rng = random.Random(7) + sound = 0 + for seed in (55, 66): + m2 = sparse_v3(seed) + ad2 = anchor_data(m2) + tries = 0 + while sound < 10 and tries < 500: + tries += 1 + a = [rng.randrange(8) & m2["pi"][t] for t in range(m2["v"])] + b = [rng.randrange(8) & m2["sg"][t] for t in range(m2["v"])] + if any(dot1(a[t], b[t]) != 1 for t in range(m2["v"])): + continue + f_ab = ab_value(m2, a, b) + f_a, _w = completion(m2, a) + assert f_ab <= f_a, "ab_value must not exceed completion" + sound += 1 + # 4. restriction exactness: M[S,S] rows, nonzero diagonal, and the + # per-instance Lemma CT / scope identities on a sub-decoder + mod = load_model("tied_min.json") + S = [0, 2, 5, 7] + sub = restrict(mod, S) + for i, t in enumerate(S): + for j, s in enumerate(S): + assert sub["M"][i][j] == mod["M"][t][s], "restriction must be exact" + ads = anchor_data(sub) + assert tie_parity(sub, ads) and scope_identity(sub, ads), \ + "sub-decoder must satisfy the CT/scope identities" + # 5. diagonal decoder sanity: cert = v and MP = v + md = model([(1, 0, 0), (0, 1, 0), (0, 0, 1), (1, 1, 0)], + [(1, 0, 0), (0, 1, 0), (0, 0, 1), (1, 0, 1)]) + inm, _ = digraph_tables(md["v"], md["arcs"]) + mas = mas_table(md["v"], acyclic_table(md["v"], inm)) + mp, _mpmap, _e, _p = matched_pivot(md, mas) + ex, _w = cert_exact(md, max_v=6) + assert mp == 4 and ex == 4, "diagonal-style decoder must give 4" + print("PASS_FG_CONTROLS brute-force=%d/3 corrupted=2/2 ab-sound=%d " + "restriction=1/1 diagonal=1/1" % (agree, sound)) + + +# ------------------------------------------------------------- calibration + +def run_calibration(): + recs = {} + for name, want in (("tied_min.json", dict(mp=9, prod=4096, reach=[], + wit=11, chain=None)), + ("cex_min_1.json", dict(mp=16, prod=6144, + reach=[0, 4, 5, 6, 8, 11, 14, 15], + wit=18, chain=22))): + mod = load_model(name) + ad = anchor_data(mod) + assert tie_parity(mod, ad) and scope_identity(mod, ad), name + inm, _ = digraph_tables(mod["v"], mod["arcs"]) + acyc = acyclic_table(mod["v"], inm) + mas = mas_table(mod["v"], acyc) + mp, mpmap, exh, prod = matched_pivot(mod, mas) + assert exh and prod == want["prod"] and mp == want["mp"], \ + "%s MP must match the recorded anchors" % name + assert sorted(ad["reach"]) == want["reach"], "%s reach" % name + if name == "cex_min_1.json": + with open(os.path.join(DATA, "cex_min_1.witness.json"), "r", + encoding="utf-8") as fh: + sw = json.load(fh) + wit = dict(a=sw["a"], b=sw["b"], order=sw["order"], kept=sw["kept"]) + assert replay_witness(mod, wit) and len(wit["kept"]) == 18, \ + "cex_min_1 stored witness must replay at 18" + got = 18 + with open(os.path.join(DATA, "cex_min_1.B22.json"), "r", + encoding="utf-8") as fh: + B = json.load(fh) + if isinstance(B, dict): + B = B.get("B", B.get("pairing")) + assert admissible(mod, B), "stored pairing admissible" + cc = chain_cost(mod, B) + assert cc == want["chain"], "chain engine must reproduce 22" + else: + val, wit = cert_hillclimb(mod, iters=400, seed=1234) + assert wit is not None and replay_witness(mod, wit), name + got = val + assert got == want["wit"], \ + "tied_min hill-climb must re-find 11 (got %d)" % got + rows = diff_labels(mod, ad, wit) + inb, esc = in_box(mod, ad, wit) + kinds = {} + for r in rows: + kinds[r["kind"]] = kinds.get(r["kind"], 0) + 1 + recs[name] = dict(v=mod["v"], threshold=(2 * mod["v"]) // 3, + MP=mp, reach=len(ad["reach"]), + cancel=len(ad["cancel"]), witness=got, + in_box=inb, box_escape=esc, kinds=kinds, + diff=rows) + log("%s: MP=%d reach=%d witness=%d in_box=%s kinds=%s" + % (name, mp, len(ad["reach"]), got, inb, kinds)) + print("PASS_FG_CALIBRATION tied_min=11/12 cex_min_1=18/19 mp=9/16 " + "chain=22 reach=0/12,8/19") + return recs + + +# -------------------------------------------- fixture ladder + form-R probes + +def run_fixture_ladder(): + """The ladder on both fixtures, plus the Form-R / box probes that answer + PR 2901 open question 2 (what do the attaining certificates use?).""" + out = {} + for name in ("tied_min.json", "cex_min_1.json"): + mod = load_model(name) + ad = anchor_data(mod) + inm, _ = digraph_tables(mod["v"], mod["arcs"]) + acyc = acyclic_table(mod["v"], inm) + mas = mas_table(mod["v"], acyc) + provided = None + if name == "cex_min_1.json": + with open(os.path.join(DATA, "cex_min_1.witness.json"), "r", + encoding="utf-8") as fh: + sw = json.load(fh) + provided = dict(a=sw["a"], b=sw["b"], order=sw["order"], + kept=sw["kept"]) + rec = settle_ladder(mod, ad, mas, acyc, exact=False, + engine_iters=(400 if name == "tied_min.json" + else 0), + provided_cert_wit=provided, label=name) + named_v, named_det = named_exact_small( + mod, ad, mas, acyc, + scan_iters=(20000 if name == "tied_min.json" else 300)) + rec["named_scan"] = named_v + rec["named_scan_detail"] = named_det + # probe: can the Form-R space alone reach the recorded witness value? + fr_deep, fr_wit, fr_exh, fr_prod = form_r_max( + mod, ad, exhaustive_cap=0, + iters=(800 if name == "tied_min.json" else 300), seed=909) + if fr_wit is not None and replay_witness(mod, fr_wit): + rec["form_r_deep"] = fr_deep + rec["form_r_deep_witness_in_box"] = in_box(mod, ad, fr_wit)[0] + rec["form_r_deep_witness_diff"] = diff_labels(mod, ad, fr_wit) + if name == "tied_min.json": + bx_deep, bx_wit = cert_hillclimb(mod, iters=400, seed=910, + box=True, ad=ad) + if bx_wit is not None and replay_witness(mod, bx_wit): + rec["box_deep"] = bx_deep + rec["box_deep_witness_in_box"] = in_box(mod, ad, bx_wit)[0] + ce_deep, ce_wit = cert_hillclimb(mod, iters=400, seed=911) + if ce_wit is not None and replay_witness(mod, ce_wit): + rec["cert_deep"] = ce_deep + rec["cert_deep_witness_in_box"] = in_box(mod, ad, ce_wit)[0] + rec["cert_deep_witness_diff"] = diff_labels(mod, ad, ce_wit) + out[name] = rec + log("fixture ladder %s: named_lo=%d form_r=%d box=%s cert=%d" % + (name, rec["named_lo"], rec["form_r"], rec["box"], rec["cert"])) + return out + + +# ------------------------------------------------------- sub-instance census + +def subsets_smallest_first(v, size, cap, stride=1): + """Deterministic enumeration of `size`-subsets of range(v), lexicographic, + thinned by a fixed stride when the count exceeds cap (the thinning and + the exact domain are recorded in CERTIFICATE.json).""" + all_subs = list(itertools.combinations(range(v), size)) + if len(all_subs) <= cap: + return all_subs, len(all_subs), False + step = max(1, len(all_subs) // cap) + picked = all_subs[::step][:cap] + return picked, len(all_subs), True + + +def run_census(): + """Smallest-first sub-decoder census of both fixtures. Sub-decoders are + exact 3-total decoders and inherit fixture-density zero digraphs (the + exact recorded generator obstruction of PR 2901). Cheap rungs on every + enumerated subset; the exact ladder (Form-R / box / cert exhaustive) on + every flagged subset (gap risk or falsifier risk), smallest first.""" + plan = [ + ("tied_min.json", 3, 240), ("tied_min.json", 4, 240), + ("tied_min.json", 5, 240), ("tied_min.json", 6, 60), + ("cex_min_1.json", 3, 240), ("cex_min_1.json", 4, 240), + ("cex_min_1.json", 5, 120), ("cex_min_1.json", 6, 50), + ] + deep_budget = {"n": 20, "v6": 1} + records = [] + domain = [] + falsifier = None + deep_done = 0 + for name, size, cap in plan: + mod = load_model(name) + subs, total, thinned = subsets_smallest_first(mod["v"], size, cap) + domain.append(dict(fixture=name, size=size, enumerated=len(subs), + total=total, thinned=thinned)) + flagged_here = 0 + for S in subs: + sub = restrict(mod, list(S)) + ad = anchor_data(sub) + assert tie_parity(sub, ad) and scope_identity(sub, ad) + inm, _ = digraph_tables(sub["v"], sub["arcs"]) + acyc = acyclic_table(sub["v"], inm) + mas = mas_table(sub["v"], acyc) + iters = 40 if size <= 4 else 120 + rec = settle_ladder(sub, ad, mas, acyc, exact=False, + engine_iters=iters, + label="%s|%s" % (name.split(".")[0], + ",".join(map(str, S)))) + risk = (not rec["cert_ge_threshold"]) or rec["gap_pr"] >= 1 \ + or rec["gap_forms"] >= 1 + if risk and rec["falsifier"] is None and size <= 5 \ + and deep_done < deep_budget["n"]: + deep = settle_ladder(sub, ad, mas, acyc, exact=True, + engine_iters=iters, + label=rec["label"] + "|deep") + rec = deep + deep_done += 1 + flagged_here += 1 + elif risk and size == 6 and deep_budget["v6"] > 0 \ + and rec["falsifier"] is None: + deep = settle_ladder(sub, ad, mas, acyc, exact=True, + engine_iters=300, + label=rec["label"] + "|deep6") + rec = deep + deep_budget["v6"] -= 1 + flagged_here += 1 + records.append(rec) + if rec["falsifier"] is not None: + falsifier = rec + log("FALSIFIER: %s" % rec["falsifier"]) + return records, domain, falsifier + log("census %s size=%d: %d enumerated (of %d%s), flagged/deep %d" + % (name, size, len(subs), total, + ", thinned" if thinned else "", flagged_here)) + return records, domain, falsifier + + +def summarize(records): + n = len(records) + gap2 = [r for r in records if r["gap_pr"] >= 2 or r["gap_forms"] >= 2 + or (r.get("gap_named") or 0) >= 2] + gap1 = [r for r in records if r["gap_pr"] >= 1 or r["gap_forms"] >= 1 + or (r.get("gap_named") or 0) >= 1] + named_gap = [r for r in records if r.get("gap_named") is not None] + below = [r for r in records if not r["cert_ge_threshold"]] + min_named_ratio = min( + (r["named_lo"] / r["v"]) for r in records) if records else 1.0 + exact_n = sum(1 for r in records if r["cert_exact"]) + max_gap_named = max((r["gap_named"] for r in named_gap), default=None) + return dict(records=n, exact_records=exact_n, gap2=len(gap2), + gap1=len(gap1), below_threshold=len(below), + min_named_ratio=round(min_named_ratio, 4), + falsifiers=sum(1 for r in records if r["falsifier"]), + exact_named_records=len(named_gap), + max_gap_named=max_gap_named, + gap2_records=gap2[:6], gap1_records=gap1[:10]) + + +def main(): + run_controls() + cal = run_calibration() + ladder = run_fixture_ladder() + records, domain, falsifier = run_census() + stats = summarize(records) + cert = dict( + package="research/fstar-family-gap-seed1c25f56f", + controls="PASS_FG_CONTROLS", + calibration="PASS_FG_CALIBRATION", + fixtures=cal, + fixture_ladder=ladder, + census_domain=domain, + census_stats=stats, + census_records=records, + falsifier=falsifier, + ) + blob = json.dumps(cert, sort_keys=True).encode("utf-8") + digest = hashlib.sha256(blob).hexdigest()[:16] + cert["digest"] = digest + with open(os.path.join(HERE, "CERTIFICATE.json"), "w", + encoding="utf-8", newline="\n") as fh: + json.dump(cert, fh, indent=1, sort_keys=True) + fh.write("\n") + log("certificate digest %s" % digest) + print("PASS_FG_CENSUS records=%d exact=%d exact-named=%d gap1=%d gap2=%d " + "below-threshold=%d falsifiers=%d max-gap-named=%s" + % (stats["records"], stats["exact_records"], + stats["exact_named_records"], stats["gap1"], stats["gap2"], + stats["below_threshold"], stats["falsifiers"], + stats["max_gap_named"])) + if falsifier is not None: + print("PASS_FSTAR_FAMILY_GAP falsifier-realized") + else: + print("PASS_FSTAR_FAMILY_GAP no-falsifier-in-domain") + + +if __name__ == "__main__": + sys.stdout.reconfigure(encoding="utf-8") + main()