diff --git a/research/mode-fiber-radius-bounds/CERTIFICATE.json b/research/mode-fiber-radius-bounds/CERTIFICATE.json new file mode 100644 index 0000000000..f7aa51ad35 --- /dev/null +++ b/research/mode-fiber-radius-bounds/CERTIFICATE.json @@ -0,0 +1,15 @@ +{ + "schema": "mode-fiber-radius-basis-v1", + "basis_rows": [ + 7, + 35, + 57, + 73, + 87, + 90, + 92, + 99 + ], + "tau": 6, + "scope": "Rows of M forming a basis of L = M^T Z^112; every other row has integer coordinates in this basis with l1 norm at most tau. Found by search/basis_search.py; verify.py recomputes everything exactly." +} \ No newline at end of file diff --git a/research/mode-fiber-radius-bounds/PROOF.md b/research/mode-fiber-radius-bounds/PROOF.md new file mode 100644 index 0000000000..ba8a3da373 --- /dev/null +++ b/research/mode-fiber-radius-bounds/PROOF.md @@ -0,0 +1,183 @@ +# Mode-fiber rounding radius: a finite reduction and 1 ≤ R ≤ 2 + +Prepared for apoursh with substantial Anthropic Claude assistance. Apache-2.0. + +## Setting + +M is the 112×8 mode matrix of Dillon Amburgey's +[PR4933](https://github.com/CrocSwap/integer-mult-bounds/pull/4933), read from +`research/coded-native-caller/NORMALIZERS.json` (`mode_columns`), with rows +m_0, …, m_111. Nick Petalas's #6312 (`research/mode-preserving-rounding`, open +at head `46593234bfe658b63fb829f0d22b7eb216e392a2`) studies rounding x ∈ ℝ^112 to +z ∈ ℤ^112 with Mᵀz = Mᵀx, and asks in its first open question for the least +uniform radius + + R = sup { ρ(x) : the fiber F(x) = {z ∈ ℤ^112 : Mᵀz = Mᵀx} is nonempty }, + ρ(x) = min { ‖z − x‖∞ : z ∈ F(x) }. + +The minimum exists because F(x) is discrete. #6312 proves R ≥ 1, using +x = (e_0+e_7+e_20+e_111)/2. This note gives a finite characterisation of R = 1 +and explicit upper bounds: a basis certificate gives R ≤ 6, and the block structure of M gives R ≤ 2. + +Write L = Mᵀℤ^112 ⊂ ℤ^8, V = ker Mᵀ ⊂ ℝ^112, Z = Mᵀ[0,1]^112 (a zonotope) and +S = {Mᵀz : z ∈ {0,1}^112} (subset sums of rows). + +## 1. Structure of M (exact, verify.py) + +* Every row has four nonzero entries, all ±1. The rows are distinct, and MᵀM = 56·I. +* The 14 row supports are exactly the 14 weight-four words of the extended + Hamming code H8 = [8,4,4]. Each support carries 8 rows, so 8 of its 16 sign + patterns occur. The row set is **not** closed under negation. +* L = {x ∈ ℤ^8 : x mod 2 ∈ H8}, the Hamming construction of √2·E8, with + [ℤ^8 : L] = 16. verify.py proves this by comparing Hermite normal forms + of the rows and of H8 ∪ 2ℤ^8. +* For every i, the other 111 rows have rank 8. + +## 2. Theorem A (finite reduction) + +**R ≤ 1 if and only if L ∩ int Z ⊆ S.** If some t ∈ L ∩ int Z is not a 0/1 +subset sum of the rows, then every y ∈ (0,1)^112 with Mᵀy = t has ρ(y) > 1, +so R > 1. With #6312's lower bound: + + R = 1 ⇔ every lattice point of L in the open zonotope int Z is a subset sum of rows. + +*Proof.* + +(a) *Invariance and continuity.* F(x+n) = F(x)+n for n ∈ ℤ^112, so ρ is +ℤ^112-periodic. For v ∈ V, F(x+v) = F(x), so |ρ(x+v) − ρ(x)| ≤ ‖v‖∞. + +(b) *Generic points suffice.* Fix x with F(x) ≠ ∅. The coordinate functional +v ↦ v_i is not identically zero on V. Otherwise e_i ∈ V^⊥ = col span(M), say +e_i = Ma. Then m_k·a = 0 for all k ≠ i, and since those rows have rank 8 +(§1), a = 0, a contradiction. So {v ∈ V : x_i + v_i ∈ ℤ} is a countable union +of proper affine hyperplanes of V for each i. The union over i is a null set, +so there are v ∈ V with arbitrarily small ‖v‖∞ such that x+v has no integral +coordinate. By (a), R is the supremum of ρ over points with no integral coordinate. + +(c) *Reduction to the open cube.* For such x, put y = x − ⌊x⌋ ∈ (0,1)^112. +Then ρ(y) = ρ(x), and t := Mᵀy ∈ L, because Mᵀx ∈ L (the fiber is nonempty) +and Mᵀ⌊x⌋ ∈ L. For y_i ∈ (0,1), an integer z_i satisfies |z_i − y_i| ≤ 1 if +and only if z_i ∈ {0,1}. Hence ρ(y) ≤ 1 ⇔ t ∈ S, and otherwise ρ(y) > 1. + +(d) *Which t occur.* Mᵀ is a surjective linear map ℝ^112 → ℝ^8. The image of the +open cube is the interior of the image of the closed cube, so +{Mᵀy : y ∈ (0,1)^112} = int Z. + +Combining (b)–(d): R ≤ 1 ⇔ L ∩ int Z ⊆ S. If t ∈ (L ∩ int Z) \ S, every y in +the open cube over t has ρ(y) > 1. ∎ + +Consequences. Deciding R = 1 is a **finite** problem, since Z is bounded. A +counterexample has a finite exact certificate: a lattice point t, a rational +y ∈ (0,1)^112 with Mᵀy = t, and infeasibility of the 0/1 system Mᵀz = t. +Conversely, R = 1 needs every one of the finitely many interior lattice points +to be a subset sum. Boundary lattice points of Z impose no condition. + +## 3. Theorem B (explicit upper bound): R ≤ 6 + +Let B be the eight rows `[7, 35, 57, 73, 87, 90, 92, 99]` from CERTIFICATE.json. +verify.py checks |det M_B| = 16 = covol(L). Since the rows lie in L, B is a basis +of L. Each other row m_i (i ∈ N, |N| = 104) has integer coordinates w_i ∈ ℤ^8 +in this basis, and max_i ‖w_i‖₁ = τ = 6. Let W be the 8×104 matrix with columns w_i. + +Take y ∈ (0,1)^112 with Mᵀy = t ∈ L, as in Theorem A(b,c). + +*Step 1 (Beck–Fiala rounding of y_N against W).* Keep x ∈ [0,1]^N, starting +from x = y_N, and let F be its fractional coordinates. Call row b of W +*active* if Σ_{i∈F}|W_{bi}| > τ. Since Σ_b Σ_{i∈F}|W_{bi}| ≤ τ|F|, there are +fewer than |F| active rows. So some nonzero d supported on F satisfies +W_active d = 0. Move x along d until one more coordinate is integral, and +repeat until F = ∅. Each active row value is preserved. Once a row becomes +inactive, only coordinates of F change afterwards, each by less than 1. Its +total later change is therefore less than Σ_{i∈F}|W_{bi}| ≤ τ, or zero. The +result z_N ∈ {0,1}^N has ‖W(y_N − z_N)‖∞ < τ. + +*Step 2 (absorb into the basis).* Set z_B = y_B + W(y_N − z_N). Then + + Σ_{b∈B} z_b m_b = Σ_B y_b m_b + Σ_N (y_i − z_i) m_i = t − Σ_N z_i m_i ∈ L, + +and B is a basis of L, so z_B ∈ ℤ^8. Hence z ∈ F(y), with |z_i − y_i| < 1 on N +and |z_b − y_b| < τ on B. + +So ρ(y) < 6 for every generic y, and R ≤ 6 by Theorem A(a,b). ∎ + +Local search over row bases (`search/basis_search.py`, three seeds × 200 restarts) +found no basis with τ < 6. This method therefore stops at 6. Theorem C below +uses the block structure instead. + +## 3′. Theorem C: R ≤ 2 + +*Block structure (exact, verify.py).* For each of the 14 supports s, its 8 rows +are exactly {σ_s e_{p(s)} + Σ_{k∈s∖p(s)} ε_k e_k : ε ∈ {±1}³}. So one *pinned* +coordinate p(s) has a fixed sign σ_s, and the three free coordinates take every +sign pattern. In particular, for every j ∈ s the block contains **four disjoint +row pairs** (a,b) with ±2e_j ∈ {m_a − m_b, m_a + m_b}: + +- if j is free, pair rows that differ only at j, so m_a − m_b = 2(m_a)_j e_j; +- if j = p(s), pair opposite free patterns, so m_a + m_b = 2σ_s e_j. + +Either sign of 2e_j is available with coefficients in {±1} on the pair. + +*Hall lemma (exact, verify.py).* For every set X of at most three supports and +every nonempty J ⊆ {0,…,7}, at least |J| supports outside X meet J. The +minimum slack over all cases is 3. + +*Proof of R ≤ 2.* Let y ∈ [0,1)^112 with t = Mᵀy ∈ L. + +1. *Vertex.* P_t = {y′ ∈ [0,1]^112 : Mᵀy′ = t} is a nonempty polytope. Take a + vertex y*. Its fractional coordinates F index linearly independent rows, so + |F| ≤ 8, and |y*_i − y_i| < 1 for all i. Each block's rows lie in a + four-dimensional coordinate subspace, so F contains at most 4 rows of any block. +2. *Round F.* Let z_F be y*_F rounded to the nearest of {0,1}. Then + r′ = Σ_F (y*_i − z_i) m_i lies in L, because t, Σ_{i∉F} y*_i m_i and Σ_F z_i m_i + do, and |r′_j| ≤ ½·#{i ∈ F : m_ij ≠ 0} ≤ 4. +3. *Parity.* h = r′ mod 2 ∈ H8 (§1). If h is a weight-four word, let U be one + row outside F with support h. If h = 1⁸, let U be two rows outside F with + complementary supports (the complement of a word is a word). If h = 0, let + U = ∅. Step 1 guarantees each block has at least 4 rows outside F. Put + u = Σ_U m_i, so |u_j| ≤ 1 and r′ − u = 2w with w ∈ ℤ^8 and |w_j| ≤ 2. +4. *Distinct blocks.* Call rows in F ∪ U damaged, at most 10 of them. A block with + at most 2 damaged rows keeps at least 2 intact pairs for each of its + coordinates. At most three blocks have three or more damaged rows. By the + Hall lemma, the coordinates with w_j ≠ 0 have distinct undamaged-enough + blocks s_j ∋ j. +5. *Assemble.* Set z_i = y*_i for i ∉ F ∪ U, then add 1 on U and, for each j, + ±1 on the rows of |w_j| intact pairs of s_j representing 2·sign(w_j)·e_j. + All touched rows are distinct. Then Mᵀz = Σ_{i∉F} y*_i m_i + Σ_F z_i m_i + u + 2w = t, + and z ∈ ℤ^112. +6. *Error.* On F, |z_i − y_i| < 1. Elsewhere |z_i − y*_i| ≤ 1 and |y*_i − y_i| < 1, + so |z_i − y_i| < 2. + +Hence ρ(y) < 2 for every y in the open cube. By Theorem A(a,b), R ≤ 2. ∎ + +`rounder.py` executes these six steps literally in exact rational arithmetic and +raises if any step's claim fails. verify.py runs it on a fixed set of exact +inputs. A separate stress run on 12,000 exact random and sparse inputs found +no failure; its worst error was 839/420 ≈ 1.998, so 2 is the limit of this +construction, not evidence about R. + +## 4. Evidence for R = 1 (not a proof) + +`search/evidence.py` (scipy/HiGHS) samples lattice points t ∈ L near the boundary +of Z, where Theorem A's condition is hardest. It does this in two ways: + +- near a vertex (face rank 0): the vertex minus a small inward lattice vector; +- near faces spanned by 1–7 random rows: rows off the face are pinned by the side + of the supporting functional, rows on the face get fractional weights with + denominator 2, 3, 4 or 6, and a small inward lattice vector is subtracted. + +It keeps t with a positive LP interior margin and solves the 0/1 system. All +1,864 interior points found (by face rank 0–7: 401, 350, 402, 372, 175, 60, 72, 32) +are subset sums. verify.py re-checks each witness exactly, t ∈ L and Σ_{i∈z} m_i = t. +Interiority comes only from a floating-point LP and is **not** certified. +This sample supports, but does not prove, the conjecture R = 1. + +## Scope + +R is a property of the stated exact-mode contract for this M, inherited from +#6312. Nothing here shows that an actual caller must preserve these modes, or +supplies a physical rounder, error ledger, tape/precision bill, supplier +admission, new κ or frontier change. The five framework assumptions keep +their status. Theorems A, B and C are written proofs whose finite inputs are +checked exactly; no proof-assistant or independent human review is claimed. +Beck–Fiala's iterative rounding (1981), Hall's marriage theorem and the Hamming +construction of E8 are classical and credited as such. diff --git a/research/mode-fiber-radius-bounds/README.md b/research/mode-fiber-radius-bounds/README.md new file mode 100644 index 0000000000..e06ebfa38f --- /dev/null +++ b/research/mode-fiber-radius-bounds/README.md @@ -0,0 +1,84 @@ +# Mode-fiber rounding radius: 1 ≤ R ≤ 2, and R = 1 is a finite question + +A scoped partial answer to the first open question of Nick Petalas's #6312 +(`research/mode-preserving-rounding`, open at head `46593234`). For the PR4933 +mode matrix M (112×8), let R be the least ℓ∞ radius such that every x with a +nonempty integer fiber {z ∈ ℤ^112 : Mᵀz = Mᵀx} has a point of that fiber within R. +#6312 proves R ≥ 1. This package proves: + +- **Structure.** L = Mᵀℤ^112 = {x ∈ ℤ^8 : x mod 2 ∈ H8}, the Hamming construction + of √2·E8, with index 16. The 14 row supports are the weight-four words of + [8,4,4], with 8 rows each. MᵀM = 56·I. The row set is not closed under negation. +- **Theorem A (finite reduction).** R ≤ 1 if and only if every lattice point of L + in the *open* zonotope Mᵀ(0,1)^112 is a 0/1 subset sum of rows. One interior + lattice point that is not a subset sum forces R > 1. So **R = 1 is a finite + decision problem**, and either answer has a finite exact certificate. +- **Theorem C.** **R ≤ 2.** Each support block is "one pinned coordinate, three + free," so every block holds four disjoint row pairs equal to ±2eⱼ for each of + its coordinates. Rounding at a vertex of the fiber polytope leaves a small + lattice residual. A Hall-theorem block assignment then pays that residual with + intact pairs, moving every coordinate by less than 2. `rounder.py` runs the + construction exactly. Theorem B, a simpler basis certificate with Beck–Fiala + rounding, gives R ≤ 6. +- **Evidence, not proof.** All 1,864 sampled interior lattice points near vertices + and faces (face rank 0–7) are subset sums. This supports the conjecture R = 1. + +[PROOF.md](PROOF.md) has the arguments and exact scope. The constructive rounder is +[rounder.py](rounder.py), the basis is in [CERTIFICATE.json](CERTIFICATE.json), the witnesses in [SEARCH.json](SEARCH.json), +and the pins in [SOURCES.json](SOURCES.json). Complete κ and the selected frontier +are unchanged. This is a property of the exact-mode contract. It does not claim +that any caller needs the contract, and supplies no physical rounder or error/tape bill. + +## Reproduce + +From the repository root; the standard library suffices: + +```sh +python3 -I -W error -B research/mode-fiber-radius-bounds/verify.py +python3 -I -O -W error -B research/mode-fiber-radius-bounds/verify.py +python3.12 -B scripts/research_package.py --package research/mode-fiber-radius-bounds +``` + +Each prints `PASS_MODE_FIBER_RADIUS_BOUNDS` in about five seconds. Checked exactly: + +- the structure facts, the HNF lattice identity and the rank condition Theorem A needs; +- the #6312 witness relation; +- the τ = 6 basis (|det| = 16, integral coordinates, maximum ℓ₁ norm exactly 6); +- the 14 pinned blocks and their 2eⱼ pairs; +- the Hall lemma, exhaustively over every set of at most three damaged blocks (minimum slack 3); +- 120 exact runs of the Theorem C rounder (worst error 279/140); +- every subset-sum witness. + +Nine rejection controls fail closed: a flipped matrix entry (in both the +structure and block checks), a repeated basis row, a claimed τ = 5, a non-basis, +a wrong witness mask, an off-lattice target, a broken witness relation and +non-integral modes passed to the rounder. Interiority of the sampled targets came from a floating-point +LP and is not certified. + +The non-certifying search scripts in `search/` need numpy and scipy: +`evidence.py SEED TRIALS` regenerates one evidence run (seeds 11–14, 320 trials +each), and `basis_search.py SEED RESTARTS` repeats the basis search. + +## Open next questions + +1. **Decide R = 1.** Is every lattice point of L in the open zonotope a subset + sum? This would settle #6312's question exactly. First step: reduce the + near-boundary cases with the signed-permutation symmetry of the rows, and try + a face recursion, where deep points are handled by discrepancy rounding and + points near a face reduce to the rows in that flat. Blocker: about 10¹² + lattice points rule out brute force, and the recursion is unproved. +2. **Close the gap between 1 and 2.** Theorem C loses a full unit through the + triangle inequality |z − y| ≤ |z − y*| + |y* − y|. An exact stress run of + 12,000 inputs reaches 1.998 with this construction, so improving the bound needs + a different first step. First step: replace the vertex y* by a point of the + fiber chosen near y, for example by Beck–Fiala rounding inside each block + against the 8 mode constraints. Then reuse the pair-and-Hall repair. + Blocker: the repair must stay within the remaining error budget. + +## Credit + +Dillon Amburgey (PR4933) supplies M. Nick Petalas (#6312) supplies the contract, +the radius-one witness and the question. Its predecessor credits (JoyboyBrian, +Dugongue, Ali Romman) carry over. Beck–Fiala rounding and the Hamming +construction of E8 are classical. Prepared by apoursh with substantial Anthropic +Claude (Claude Code) assistance. New work is Apache-2.0. diff --git a/research/mode-fiber-radius-bounds/SEARCH.json b/research/mode-fiber-radius-bounds/SEARCH.json new file mode 100644 index 0000000000..0171a06e79 --- /dev/null +++ b/research/mode-fiber-radius-bounds/SEARCH.json @@ -0,0 +1 @@ +{"schema":"mode-fiber-subset-sum-evidence-v1","scope":"Search output from search/evidence.py (scipy HiGHS, non-certifying). Each record is [t, face_rank, denominator, z_mask]; bit i of z_mask selects row i. verify.py checks t in L and t = sum of selected rows exactly. Interiority of t was established only by floating LP margin and is not 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\ No newline at end of file diff --git a/research/mode-fiber-radius-bounds/SOURCES.json b/research/mode-fiber-radius-bounds/SOURCES.json new file mode 100644 index 0000000000..c87d43b6f4 --- /dev/null +++ b/research/mode-fiber-radius-bounds/SOURCES.json @@ -0,0 +1,24 @@ +{ + "schema_version": 1, + "base_commit": "2485805a6d47f36ea83ddde6327d453ac82d44dc", + "files": [ + { + "path": "certificates/selected-result.json", + "usage": "provenance", + "git_blob": "d59b5035e7b4cbaa9d688a1d32dd0d6eb79c466a", + "sha256": "9b8d0eea95b3bf0dfcb7938a732a6439b87a2685dec2a5ab9e525b3b34debbfa" + }, + { + "path": "docs/research/agent-task-catalog.json", + "usage": "provenance", + "git_blob": "3404fbdd5e7529a0674eef26114b5071e4586cec", + "sha256": "ead6cdc9d0af6d58e8a49744d8ec8c7c15e13d2ba93c276c434bcf9b51b93a81" + }, + { + "path": "research/coded-native-caller/NORMALIZERS.json", + "usage": "input", + "git_blob": "bca92c222c395d3d75c047fbb678f3cd0947c60f", + "sha256": "ae8aad06aa5d1811d725895b3ab5a4f4daa9dcdd88294aa0d5f723e78bb1e0da" + } + ] +} diff --git a/research/mode-fiber-radius-bounds/TASK.json b/research/mode-fiber-radius-bounds/TASK.json new file mode 100644 index 0000000000..d7c7861bae --- /dev/null +++ b/research/mode-fiber-radius-bounds/TASK.json @@ -0,0 +1,89 @@ +{ + "schema_version": 1, + "task_id": "uniform-transfer-audit", + "priority_revision": 8, + "base_commit": "2485805a6d47f36ea83ddde6327d453ac82d44dc", + "original_card": { + "id": "uniform-transfer-audit", + "question": "Can one of the selected record's five inherited assumptions be proved at its actual caller?", + "starting_points": [ + "docs/research/current-priorities.md", + "docs/research/research-family-status.md", + "certificates/selected-result.json", + "docs/research/pr-intake-evidence/pr5599-completion/PROOF-REVIEW.md" + ], + "methods": [ + "Construct and independently check one source-bound instance of the stated first experiment.", + "Seek an exact counterexample or target-specific complete-budget obstruction to that same proposed change." + ], + "acceptance_gate": "Identify the proposition and what would change downstream; do not equate a finite replay with a universal proof.", + "first_step": { + "command": [ + "python3", + "-B", + "scripts/research_selected_baseline.py" + ], + "expected": "PASS_RESEARCH_BASELINE_BINDINGS_ONLY", + "inspect": "Read revision 8, the selected review and live exact-source handoffs. Then follow this card's bounded first_experiment.", + "scope": "Source-bound orientation only, not a supplier replay or theorem proof." + }, + "first_experiment": "Choose one assumption ID (native_all_columns, owned_erasure, compiler, precision_tape, analytic_recovery), one quantified implication and exact dependency closure. Check current handoffs, including #5625 row reentry and #5707 root-prefix lowering, without treating their partial scope as the whole theorem. Prove or falsify that implication with a caller-bound argument; stop at the chosen obligation. Finite tests or root digits do not discharge an all-size statement.", + "bottleneck_link": "Judge the actual current supplier contract and complete signed target/paid bill; frozen-profile ceilings require an applicability proof.", + "allowed_variations": [ + "One exact source head, helper/cut region or quantified proposition and changed hypothesis; check live handoffs first. Different seeds are not different scopes." + ], + "scope_change_triggers": [ + "Changing source head, helper/cut domain, operation alphabet or caller boundary without recording the new scope.", + "Treating an archived proposal or local numerical pass as complete supplier admission." + ], + "overlap_terms": [ + "native_all_columns", + "owned_erasure", + "compiler", + "precision_tape", + "analytic_recovery" + ], + "existing_checks": [ + { + "scope": "PR #4100: proposed unconditional construction and finite checks, without independent acceptance.", + "evidence": "research/unconditional-batched-crt/README.md", + "remaining": "Independent proof audit of one caller-bound all-size implication; all other premises remain explicit.", + "status": "author-checked", + "source_commit": "337d04939571a637aeab54a0f58f54fde811663a", + "selected_record_sha256": "bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129" + }, + { + "scope": "Selected PR5599: complete helper-subspace construction and independent caller replay; five framework assumptions remain.", + "evidence": "docs/research/pr-intake-evidence/pr5599-completion/PROOF-REVIEW.md", + "remaining": "The next experiment must identify uncovered source-specific work. Successor archive intake does not transfer this admission.", + "status": "reviewed", + "source_commit": "09dd0965711ca7eeb169057a420abab1c9052137", + "selected_record_sha256": "2a0220e866417774606e0f991e41b9c1d26b1f43b97857238cf478dbc7d330b5" + } + ] + }, + "scope_status": "changed", + "actual_scope": "Open question 1 of #6312 (research/mode-preserving-rounding, head 46593234bfe658b63fb829f0d22b7eb216e392a2): the least uniform l_inf radius R for exact mode-preserving integer rounding on all nonempty fibers of the PR4933 mode map M.", + "scope_change_reason": "The session drew recursive-interfaces cards needing a full construction context. #6312 had just changed this card to a precision side result and left its first open question with only a lower bound. That self-contained question is answered here in part.", + "bottleneck_link": "Side result on the precision contract of the coded inverse. It bounds what an exact mode-preserving rounder can achieve on this M; it does not show that any caller needs one.", + "budget_minutes": 90, + "stop_rule": "Stop at a rigorous characterisation plus explicit upper bounds, or at one exact counterexample t in L cap int Z outside S. Do not claim R = 1 from search evidence.", + "outcome": "partial", + "evidence": "PROOF.md Theorems A, B and C, with verify.py exact checks: L is the Hamming E8 lattice; the genericity rank condition; the tau = 6 basis; the pinned blocks and 2e_j pairs; the exhaustive Hall lemma; 120 exact runs of the Theorem C rounder; 1,864 exact subset-sum witnesses.", + "author": "apoursh", + "ai_assistance": "Anthropic Claude (Claude Code) designed the argument, wrote the code and ran the searches under apoursh's direction.", + "open_questions": [ + { + "question": "Is every lattice point of L in the open zonotope int Z a 0/1 subset sum of rows (equivalently, R = 1)?", + "why_it_matters": "It settles #6312's open question exactly; the search sample supports yes.", + "first_step": "Use the signed-permutation automorphism group of the 112 rows to reduce the finitely many near-boundary cases. Try a face recursion: points deep in Z are handled by discrepancy rounding, and points near a face reduce to the rows lying in that flat.", + "blocked_by": "Brute force is infeasible (roughly 1e12 lattice points), and the face recursion is not proved." + }, + { + "question": "Can R <= 2 be lowered towards 1?", + "why_it_matters": "Theorem C loses one unit through the vertex step; an exact stress test of 12,000 inputs reaches 1.998, so the construction itself is tight at 2.", + "first_step": "Replace the vertex y* with a fiber point near y, for example by blockwise Beck-Fiala rounding against the 8 mode constraints, then reuse the pair-and-Hall repair.", + "blocked_by": "The repair must fit in the remaining error budget per coordinate." + } + ] +} diff --git a/research/mode-fiber-radius-bounds/VERIFY.json b/research/mode-fiber-radius-bounds/VERIFY.json new file mode 100644 index 0000000000..f0114f74f0 --- /dev/null +++ b/research/mode-fiber-radius-bounds/VERIFY.json @@ -0,0 +1,36 @@ +{ + "schema_version": 1, + "status": "ready", + "python": "3.12", + "requirements": [], + "inputs": [ + "research/coded-native-caller/NORMALIZERS.json" + ], + "commands": [ + { + "argv": [ + "{python}", + "-I", + "-W", + "error", + "-B", + "{package}/verify.py" + ], + "timeout_seconds": 60, + "checks": "Mode-matrix structure, L = {x : x mod 2 in H8} by HNF, genericity ranks, PR6312 witness relation, tau = 6 basis, pinned blocks and 2e_j pairs, exhaustive Hall lemma, 120 exact radius-below-two rounder runs, 1,864 subset-sum witnesses and nine rejection controls." + }, + { + "argv": [ + "{python}", + "-I", + "-O", + "-W", + "error", + "-B", + "{package}/verify.py" + ], + "timeout_seconds": 60, + "checks": "The same explicit checks under optimized Python." + } + ] +} diff --git a/research/mode-fiber-radius-bounds/rounder.py b/research/mode-fiber-radius-bounds/rounder.py new file mode 100644 index 0000000000..5f41704000 --- /dev/null +++ b/research/mode-fiber-radius-bounds/rounder.py @@ -0,0 +1,194 @@ +"""Constructive radius-below-two rounder for the PR4933 mode map (PROOF.md, Theorem C). + +Exact rational arithmetic, standard library only. Given y in [0,1)^112 whose +mode vector t = M^T y lies in L, returns an integer z with M^T z = t and +max_i |z_i - y_i| < 2. Every step of Theorem C is executed literally, so a +failure of any lemma raises instead of being silently repaired. +Prepared for apoursh with substantial Anthropic Claude assistance. Apache-2.0. +""" +from fractions import Fraction as Q +import itertools + + +def need(condition, message): + if not condition: + raise ValueError(message) + + +def blocks_of(rows): + """Support blocks: support -> (row indices, pinned coordinate, pin sign, free coordinates).""" + out = {} + for i, r in enumerate(rows): + out.setdefault(tuple(j for j in range(8) if r[j]), []).append(i) + info = {} + for s, ix in out.items(): + fixed = [j for j in s if len({rows[i][j] for i in ix}) == 1] + need(len(ix) == 8 and len(fixed) == 1, 'block has one pinned coordinate') + p, free = fixed[0], [j for j in s if j != fixed[0]] + need({tuple(rows[i][j] for j in free) for i in ix} == set(itertools.product((-1, 1), repeat=3)), + 'block rows are every sign pattern on the free coordinates') + info[s] = (ix, p, rows[ix[0]][p], free) + return info + + +def pairs_for(rows, block, j): + """Four disjoint row pairs (a, b) of the block with m_a - m_b = 2 e_j or m_a + m_b = 2 e_j (signed).""" + ix, p, sign, free = block + out, used = [], set() + for a in ix: + if a in used: + continue + if j == p: # antipodal on the free coordinates: m_a + m_b = 2 sign e_p + b = next(b for b in ix if all(rows[b][k] == -rows[a][k] for k in free)) + out.append(('sum', a, b)) + else: # flip coordinate j only: m_a - m_b = 2 rows[a][j] e_j + b = next(b for b in ix if b != a and all(rows[b][k] == rows[a][k] for k in range(8) if k != j)) + out.append(('diff', a, b)) + used.update((a, b)) + need(len(out) == 4, 'four disjoint pairs') + return out + + +def null_vector(vectors): + """A nonzero rational dependency among the given 8-dimensional vectors (len(vectors) == 9).""" + n = len(vectors) + a = [[Q(vectors[k][j]) for k in range(n)] for j in range(8)] + pivots, r = [], 0 + for c in range(n): + piv = next((i for i in range(r, 8) if a[i][c]), None) + if piv is None: + continue + a[r], a[piv] = a[piv], a[r] + a[r] = [x / a[r][c] for x in a[r]] + for i in range(8): + if i != r and a[i][c]: + f = a[i][c] + a[i] = [x - f * y for x, y in zip(a[i], a[r])] + pivots.append(c) + r += 1 + freec = next(c for c in range(n) if c not in pivots) + d = [Q(0)] * n + d[freec] = Q(1) + for row, c in enumerate(pivots): + d[c] = -a[row][freec] + return d + + +def vertex(rows, y): + """Move y within {y' in [0,1]^112 : M^T y' = M^T y} until the fractional rows are independent.""" + y = list(y) + while True: + frac = [i for i in range(112) if y[i].denominator != 1] + if len(frac) <= 8: + if rank_of([rows[i] for i in frac]) == len(frac): + return y, frac + sample, d = frac, small_dependency(rows, frac) + else: + sample = frac[:9] + d = null_vector([rows[i] for i in sample]) + steps = [] + for i, di in zip(sample, d): + if di > 0: + steps.append((1 - y[i]) / di) + elif di < 0: + steps.append(-y[i] / di) + step = min(steps) + for i, di in zip(sample, d): + y[i] += step * di + + +def rank_of(vectors): + rows, r = [[Q(v) for v in vec] for vec in vectors], 0 + for c in range(8): + piv = next((i for i in range(r, len(rows)) if rows[i][c]), None) + if piv is None: + continue + rows[r], rows[piv] = rows[piv], rows[r] + for i in range(len(rows)): + if i != r and rows[i][c]: + f = rows[i][c] / rows[r][c] + rows[i] = [a - f * b for a, b in zip(rows[i], rows[r])] + r += 1 + return r + + +def small_dependency(rows, frac): + for k in range(2, len(frac) + 1): + for sub in itertools.combinations(frac, k): + if rank_of([rows[i] for i in sub]) < k: + vecs = [rows[i] for i in sub] + [(0,) * 8] * (9 - k) + d = null_vector(vecs)[:k] + if any(d): + full = {i: Q(0) for i in frac} + full.update(dict(zip(sub, d))) + return [full[i] for i in frac] + raise ValueError('no dependency') + + +def matching(demand, good): + """Distinct good blocks for each coordinate with nonzero demand (augmenting paths).""" + owner = {} + + def augment(j, seen): + for s in good: + if j in s and s not in seen: + seen.add(s) + if s not in owner or augment(owner[s], seen): + owner[s] = j + return True + return False + + for j in [j for j in range(8) if demand[j]]: + need(augment(j, set()), 'Hall condition failed') + return {j: s for s, j in owner.items()} + + +def round_within_two(rows, y, blocks=None): + blocks = blocks or blocks_of(rows) + t = [sum(rows[i][j] * y[i] for i in range(112)) for j in range(8)] + need(all(v.denominator == 1 for v in t), 'integral mode vector') + ystar, F = vertex(rows, y) + need(rank_of([rows[i] for i in F]) == len(F) <= 8, 'independent fractional rows') + z = list(ystar) + for i in F: # nearest rounding on F + z[i] = Q(1) if ystar[i] >= Q(1, 2) else Q(0) + rprime = [sum((rows[i][j] * (ystar[i] - z[i]) for i in F), Q(0)) for j in range(8)] + need(all(v.denominator == 1 and abs(v) <= 4 for v in rprime), 'residual in L with |r_j| <= 4') + h = tuple(int(v) % 2 for v in rprime) + damaged, used = set(F), set(F) + supports = {s: tuple(int(j in s) for j in range(8)) for s in blocks} + u = [0] * 8 + if any(h): + parts = [s for s, code in supports.items() if code == h] + if not parts: # all-ones parity: two complementary supports + parts = next([s, c] for s in blocks for c in blocks + if supports[c] == tuple(1 - v for v in supports[s])) + for s in parts: + a = next(i for i in blocks[s][0] if i not in damaged) + z[a] += 1 + damaged.add(a) + u = [x + y for x, y in zip(u, rows[a])] + w = [(rprime[j] - u[j]) / 2 for j in range(8)] + need(all(v.denominator == 1 and abs(v) <= 2 for v in w), 'even remainder with |w_j| <= 2') + count = {s: sum(i in damaged for i in blocks[s][0]) for s in blocks} + good = [s for s in blocks if count[s] <= 2] + need(len(blocks) - len(good) <= 3, 'at most three heavily damaged blocks') + assign = matching([int(v) for v in w], good) + for j, s in assign.items(): + k = int(w[j]) + intact = [pr for pr in pairs_for(rows, blocks[s], j) if pr[1] not in damaged and pr[2] not in damaged] + need(len(intact) >= abs(k), 'enough intact pairs') + for kind, a, b in intact[:abs(k)]: + sign = 1 if k > 0 else -1 + if kind == 'sum': # m_a + m_b = 2 pin_sign e_j + ca = cb = sign * blocks[s][2] + else: # m_a - m_b = 2 rows[a][j] e_j + ca, cb = sign * rows[a][j], -sign * rows[a][j] + z[a] += ca + z[b] += cb + damaged.update((a, b)) + need(all(v.denominator == 1 for v in z), 'integral output') + need([sum(rows[i][j] * z[i] for i in range(112)) for j in range(8)] == t, 'modes preserved') + err = max(abs(z[i] - y[i]) for i in range(112)) + need(err < 2, 'radius below two') + return [int(v) for v in z], err diff --git a/research/mode-fiber-radius-bounds/search/basis_search.py b/research/mode-fiber-radius-bounds/search/basis_search.py new file mode 100644 index 0000000000..1bb3ad147e --- /dev/null +++ b/research/mode-fiber-radius-bounds/search/basis_search.py @@ -0,0 +1,27 @@ +"""Non-certifying local search (numpy) for a row basis of L minimizing the max column l1 norm tau. +Usage: python basis_search.py SEED RESTARTS""" +import json, sys +from pathlib import Path +import numpy as np +M = np.array([[c[i] for c in json.load(open(Path(__file__).resolve().parents[3] / 'research/coded-native-caller/NORMALIZERS.json'))['mode_columns']] for i in range(112)]) +def score(B): + MB = M[B].T.astype(float) + if round(abs(np.linalg.det(MB))) != 16: return None + N = np.setdiff1d(np.arange(112), B) + W = np.round(np.linalg.solve(MB, M[N].T.astype(float))).astype(int) + c = np.abs(W).sum(0); return (int(c.max()), int((c == c.max()).sum())) +rng = np.random.default_rng(int(sys.argv[1])); best = None +for restart in range(int(sys.argv[2])): + while True: + B = list(rng.choice(112, 8, replace=False)); s = score(B) + if s: break + improved = True + while improved: + improved = False + for pos in rng.permutation(8): + for cand in rng.permutation(112): + if cand in B: continue + C = B.copy(); C[pos] = cand; sc = score(C) + if sc and sc < s: B, s, improved = C, sc, True + if best is None or s < best[0]: + best = (s, sorted(int(b) for b in B)); print(restart, best, flush=True) diff --git a/research/mode-fiber-radius-bounds/search/evidence.py b/research/mode-fiber-radius-bounds/search/evidence.py new file mode 100644 index 0000000000..ddb39f8d48 --- /dev/null +++ b/research/mode-fiber-radius-bounds/search/evidence.py @@ -0,0 +1,48 @@ +"""Non-certifying search (scipy/HiGHS): interior lattice points near zonotope faces and vertices, +each recorded with an exact 0/1 subset-sum witness z. Output: evidence_.json.""" +import itertools, json, sys +from pathlib import Path +import numpy as np +from scipy.optimize import milp, linprog, LinearConstraint, Bounds +M = np.array([[c[i] for c in json.load(open(Path(__file__).resolve().parents[3] / 'research/coded-native-caller/NORMALIZERS.json'))['mode_columns']] for i in range(112)], dtype=float); n = 112 +H8 = {tuple(int(v) for v in (np.abs(r) % 2)) for r in M} | {(0,) * 8, (1,) * 8} +in_L = lambda t: tuple(int(v) % 2 for v in t) in H8 +SMALL = [np.array(u, float) for u in itertools.product(range(-2, 3), repeat=8) + if any(u) and sum(map(abs, u)) <= 8 and in_L(u)] +def margin(t): + c = np.zeros(n + 1); c[-1] = -1 + A_ub = np.vstack([np.hstack([-np.eye(n), np.ones((n, 1))]), np.hstack([np.eye(n), np.ones((n, 1))])]) + r = linprog(c, A_ub=A_ub, b_ub=np.r_[np.zeros(n), np.ones(n)], A_eq=np.hstack([M.T, np.zeros((8, 1))]), + b_eq=t, bounds=[(None, None)] * n + [(None, 0.5)]) + return -r.fun if r.status == 0 else -1 +def witness(t): + r = milp(np.zeros(n), constraints=[LinearConstraint(M.T, t, t)], integrality=np.ones(n), + bounds=Bounds(0, 1), options={'time_limit': 30}) + return None if r.status != 0 else [i for i in range(n) if r.x[i] > 0.5] +seed, trials = int(sys.argv[1]), int(sys.argv[2]); rng = np.random.default_rng(seed) +out = []; failures = [] +for trial in range(trials): + k = int(rng.integers(0, 8)) + if k: + T = rng.choice(n, k, replace=False); Q, _ = np.linalg.qr(M[T].T, mode='complete') + f = Q[:, k:] @ rng.normal(size=8 - k) + else: + f = rng.normal(size=8) + s = M @ f; W = np.where(np.abs(s) < 1e-9)[0]; y = (s > 0).astype(float) + q = int(rng.choice([2, 3, 4, 6])) + for _ in range(30): + y[W] = rng.integers(0, q + 1, len(W)) / q; t0 = M.T @ y + if np.allclose(t0, np.round(t0)) and in_L(np.round(t0)): break + else: + continue + t0 = np.round(t0); inward = [u for u in SMALL if f @ u > 1e-9] + for j in rng.permutation(len(inward))[:6]: + t = t0 - inward[j]; m = margin(t) + if m <= 1e-9: continue + z = witness(t) + rec = {'t': [int(v) for v in t], 'face_rank': k, 'q': q, 'lp_margin': round(m, 9)} + if z is None: failures.append(rec); print('NO WITNESS', rec, flush=True) + else: rec['z'] = z; out.append(rec) +json.dump({'seed': seed, 'trials': trials, 'records': out, 'no_witness': failures}, + open('evidence_%d.json' % seed, 'w'), separators=(',', ':')) +print('seed', seed, 'interior points', len(out) + len(failures), 'without witness', len(failures)) diff --git a/research/mode-fiber-radius-bounds/verify.py b/research/mode-fiber-radius-bounds/verify.py new file mode 100644 index 0000000000..566e15f6c8 --- /dev/null +++ b/research/mode-fiber-radius-bounds/verify.py @@ -0,0 +1,280 @@ +#!/usr/bin/env python3 +"""Exact checks for the mode-fiber rounding radius bounds. Standard library only. + +Checks, in exact integer/rational arithmetic, every finite fact the proof uses: +the mode matrix structure, L = M^T Z^112 = {x : x mod 2 in H8}, the genericity +hypothesis, the PR6312 lower-bound witness, the tau = 6 basis certificate, and +the subset-sum witnesses recorded in SEARCH.json. Rejection controls follow. +Prepared for apoursh with substantial Anthropic Claude assistance. Apache-2.0. +""" +from fractions import Fraction as Q +import hashlib +import itertools +import json +from pathlib import Path +import random +import sys + +sys.path.insert(0, str(Path(__file__).resolve().parent)) +import rounder # noqa: E402 + +HERE = Path(__file__).resolve().parent +ROOT = HERE.parents[1] +SOURCE = 'research/coded-native-caller/NORMALIZERS.json' +SOURCE_SHA256 = 'ae8aad06aa5d1811d725895b3ab5a4f4daa9dcdd88294aa0d5f723e78bb1e0da' +WITNESS_SUPPORT, WITNESS_TARGET = (0, 7, 20, 111), 48 + + +def need(condition, message): + if not condition: + raise ValueError(message) + + +def read(path, limit=1 << 20): + need(path.is_file() and not path.is_symlink() and path.stat().st_size <= limit, 'bounded regular file: %s' % path) + return path.read_bytes() + + +def load_rows(raw): + need(hashlib.sha256(raw).hexdigest() == SOURCE_SHA256, 'source hash') + cols = json.loads(raw)['mode_columns'] + need(type(cols) is list and len(cols) == 8 and all(len(c) == 112 for c in cols), 'mode shape') + rows = [tuple(c[i] for c in cols) for i in range(112)] + need(all(type(v) is int and v in (-1, 0, 1) for r in rows for v in r), 'signed entries') + return rows + + +def hnf(vectors): + """Row-style Hermite normal form of the integer lattice spanned by vectors (dimension 8).""" + rows, out = [list(v) for v in vectors], [] + for c in range(8): + while True: + piv = [r for r in rows if r[c]] + if len(piv) <= 1: + break + piv.sort(key=lambda r: abs(r[c])) + for r in piv[1:]: + q = r[c] // piv[0][c] + r[:] = [a - q * b for a, b in zip(r, piv[0])] + if piv: + p = piv[0] + rows.remove(p) + out.append([-v for v in p] if p[c] < 0 else p) + for i in range(len(out)): + for k in range(i): + q = out[k][i] // out[i][i] + out[k] = [a - q * b for a, b in zip(out[k], out[i])] + return out + + +def rank(vectors): + rows, r = [[Q(v) for v in vec] for vec in vectors], 0 + for c in range(len(rows[0]) if rows else 0): + piv = next((i for i in range(r, len(rows)) if rows[i][c]), None) + if piv is None: + continue + rows[r], rows[piv] = rows[piv], rows[r] + for i in range(len(rows)): + if i != r and rows[i][c]: + f = rows[i][c] / rows[r][c] + rows[i] = [a - f * b for a, b in zip(rows[i], rows[r])] + r += 1 + return r + + +def solve(columns, target): + """Solve sum_k x_k columns[k] = target exactly for square nonsingular columns.""" + n = len(columns) + a = [[Q(columns[k][j]) for k in range(n)] + [Q(target[j])] for j in range(n)] + for c in range(n): + piv = next(i for i in range(c, n) if a[i][c]) + a[c], a[piv] = a[piv], a[c] + for i in range(n): + if i != c and a[i][c]: + f = a[i][c] / a[c][c] + a[i] = [x - f * y for x, y in zip(a[i], a[c])] + return [a[i][n] / a[i][i] for i in range(n)] + + +def det(columns): + n, d = len(columns), Q(1) + a = [[Q(columns[k][j]) for k in range(n)] for j in range(n)] + for c in range(n): + piv = next((i for i in range(c, n) if a[i][c]), None) + if piv is None: + return Q(0) + if piv != c: + a[c], a[piv], d = a[piv], a[c], -d + d *= a[c][c] + for i in range(c + 1, n): + f = a[i][c] / a[c][c] + a[i] = [x - f * y for x, y in zip(a[i], a[c])] + return d + + +def hamming_code(rows): + supports = {tuple(int(v != 0) for v in r) for r in rows} + code = supports | {(0,) * 8, (1,) * 8} + need(all(tuple(x ^ y for x, y in zip(a, b)) in code for a in code for b in code), 'linear code') + return supports, code + + +def in_lattice(t, code): + return all(type(v) is int for v in t) and tuple(v % 2 for v in t) in code + + +def check_structure(rows): + need(len(set(rows)) == 112 and all(sum(map(abs, r)) == 4 for r in rows), 'distinct weight-four rows') + gram = [[sum(r[a] * r[b] for r in rows) for b in range(8)] for a in range(8)] + need(gram == [[56 * (a == b) for b in range(8)] for a in range(8)], 'M^T M = 56 I') + supports, code = hamming_code(rows) + need(len(supports) == 14 and len(code) == 16, 'supports are the 14 weight-four words of [8,4,4]') + need(all(sum(tuple(int(v != 0) for v in r) == s for r in rows) == 8 for s in supports), 'eight rows per support') + negation_closed = all(tuple(-v for v in r) in set(rows) for r in rows) + generators = [list(c) for c in code] + [[2 * (j == k) for j in range(8)] for k in range(8)] + lattice = hnf(rows) + need(lattice == hnf(generators), 'L equals the Hamming construction {x : x mod 2 in H8}') + index = 1 + for i, r in enumerate(lattice): + index *= r[i] + need(index == 16, 'index of L in Z^8') + need(all(rank(rows[:i] + rows[i + 1:]) == 8 for i in range(112)), 'no coordinate is constant on ker M^T') + return code, negation_closed + + +def check_witness(rows): + # PR6312 lower bound: x = (e_0 + e_7 + e_20 + e_111)/2 has the modes of e_48 (radius-one fiber). + half = [sum(Q(rows[i][j], 2) for i in WITNESS_SUPPORT) for j in range(8)] + need(half == [Q(v) for v in rows[WITNESS_TARGET]], 'PR6312 witness relation') + + +def check_basis(rows, certificate): + basis, tau = certificate['basis_rows'], certificate['tau'] + need(len(set(basis)) == 8 and all(0 <= b < 112 for b in basis), 'eight distinct basis rows') + cols = [rows[b] for b in basis] + need(abs(det(cols)) == 16, 'basis covolume equals covolume of L') + worst = 0 + for i in range(112): + if i in basis: + continue + w = solve(cols, rows[i]) + need(all(x.denominator == 1 for x in w), 'integral coordinates in the basis') + worst = max(worst, sum(abs(x) for x in w)) + need(worst == tau, 'maximum column l1 norm equals the certified tau') + return tau + + +def check_search(rows, code, search): + need(search['schema'] == 'mode-fiber-subset-sum-evidence-v1', 'search schema') + faces = {} + for t, face_rank, q, mask in search['records']: + need(in_lattice(t, code), 'search target lies in L') + z = int(mask, 16) + need(0 <= z < 1 << 112, 'mask range') + total = [sum(rows[i][j] for i in range(112) if z >> i & 1) for j in range(8)] + need(total == t, 'subset-sum witness') + faces[face_rank] = faces.get(face_rank, 0) + 1 + return len(search['records']), dict(sorted(faces.items())) + + +def check_blocks(rows): + """Theorem C finite inputs: pinned blocks, four signed 2e_j pairs per (block, j), and the Hall lemma.""" + blocks = rounder.blocks_of(rows) + for s, block in blocks.items(): + for j in s: + pairs = rounder.pairs_for(rows, block, j) + need(len({i for _, a, b in pairs for i in (a, b)}) == 8, 'pairs are disjoint') + for kind, a, b in pairs: + combo = [x + y if kind == 'sum' else x - y for x, y in zip(rows[a], rows[b])] + need(combo[j] in (2, -2) and all(combo[k] == 0 for k in range(8) if k != j), 'pair gives 2e_j') + supports = list(blocks) + slack = min(sum(1 for s in supports if s not in bad and set(s) & set(J)) - len(J) + for k in range(4) for bad in itertools.combinations(supports, k) + for size in range(1, 9) for J in itertools.combinations(range(8), size)) + need(slack >= 0, 'Hall lemma for at most three damaged blocks') + return blocks, slack + + +def fixtures(rows, count, seed=20261031): + """Exact y in [0,1)^112 with M^T y in L: the PR6312 witness, then projected random rationals.""" + rng = random.Random(seed) + witness = [Q(1, 2) if i in WITNESS_SUPPORT else Q(0) for i in range(112)] + yield witness + for n in range(count - 1): + den = rng.choice([2, 3, 4, 5, 6, 7, 12, 30]) + v = [Q(rng.randint(-3 * den, 3 * den), den) for _ in range(112)] + if n % 3: + keep = set(rng.sample(range(112), rng.randint(4, 20))) + v = [x if i in keep else Q(0) for i, x in enumerate(v)] + mt = [sum(rows[i][j] * v[i] for i in range(112)) for j in range(8)] + w = [v[i] - sum(rows[i][j] * mt[j] for j in range(8)) / 56 for i in range(112)] + yield [x - (x.numerator // x.denominator) for x in w] + + +def check_rounder(rows, blocks, count=120): + worst = Q(0) + for y in fixtures(rows, count): + _, err = rounder.round_within_two(rows, y, blocks) + worst = max(worst, err) + need(worst < 2, 'every fixture rounded within radius two') + return count, worst + + +def rejects(function, *args): + try: + function(*args) + except (ValueError, StopIteration, ZeroDivisionError): + return True + return False + + +def controls(rows, code, certificate, search): + flipped = [list(r) for r in rows] + flipped[3][0] = -flipped[3][0] + repeated = {**certificate, 'basis_rows': certificate['basis_rows'][:7] + [certificate['basis_rows'][0]]} + stronger = {**certificate, 'tau': certificate['tau'] - 1} + non_basis = {**certificate, 'basis_rows': list(range(8))} + t, f, q, mask = search['records'][0] + wrong_sum = {**search, 'records': [[t, f, q, format(int(mask, 16) ^ 1, '028x')]]} + off_lattice = {**search, 'records': [[[t[0] + 1] + t[1:], f, q, mask]]} + results = [ + rejects(check_structure, [tuple(r) for r in flipped]), + rejects(check_basis, rows, repeated), + rejects(check_basis, rows, stronger), + rejects(check_basis, rows, non_basis), + rejects(check_search, rows, code, wrong_sum), + rejects(check_search, rows, code, off_lattice), + rejects(check_witness, rows[:48] + [rows[49]] + rows[49:]), + rejects(rounder.blocks_of, [tuple(r) for r in flipped]), + rejects(rounder.round_within_two, rows, [Q(1, 2)] + [Q(0)] * 111), + ] + need(all(results), 'every rejection control must fail closed: %s' % results) + return len(results) + + +def main(): + rows = load_rows(read(ROOT / SOURCE)) + code, negation_closed = check_structure(rows) + check_witness(rows) + certificate = json.loads(read(HERE / 'CERTIFICATE.json')) + tau = check_basis(rows, certificate) + blocks, slack = check_blocks(rows) + rounded, worst = check_rounder(rows, blocks) + search = json.loads(read(HERE / 'SEARCH.json')) + count, faces = check_search(rows, code, search) + n_controls = controls(rows, code, certificate, search) + print(json.dumps({ + 'lattice': 'L = M^T Z^112 = {x in Z^8 : x mod 2 in H8}, index 16', + 'rows_negation_closed': negation_closed, + 'lower_bound': 1, 'basis_upper_bound_theorem_B': tau, 'upper_bound_theorem_C': 2, + 'pinned_blocks': len(blocks), 'hall_min_slack_three_damaged': slack, + 'rounder_fixtures': rounded, 'rounder_worst_error': str(worst), + 'subset_sum_witnesses': count, 'witnesses_by_face_rank': faces, + 'rejection_controls': n_controls, + 'scope': 'Exact finite checks only; the reduction, Beck-Fiala and radius-two arguments are in PROOF.md. ' + 'Interiority of search targets is not certified.'}, indent=1)) + print('PASS_MODE_FIBER_RADIUS_BOUNDS') + + +if __name__ == '__main__': + main()