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10, 4, 1], "target_basis": [522]}, + {"frames": [{"characteristic": 973, "extra": 34, "source_bad": [], "target_bad": []}], "minimum_dimension": 9, "node": 1939, "source_basis": [546, 290, 162, 98, 18, 8, 4, 1], "target_basis": [50]}, + {"frames": [{"characteristic": 941, "extra": 32, "source_bad": [], "target_bad": []}], "minimum_dimension": 9, "node": 1944, "source_basis": [544, 288, 160, 98, 18, 8, 4, 1], "target_basis": [82]}, + {"frames": [{"characteristic": 877, "extra": 32, "source_bad": [], "target_bad": []}], "minimum_dimension": 9, "node": 1949, "source_basis": [544, 288, 162, 96, 18, 8, 4, 1], "target_basis": [146]}, + {"frames": [{"characteristic": 749, "extra": 32, "source_bad": [], "target_bad": []}], "minimum_dimension": 9, "node": 1954, "source_basis": [544, 290, 160, 96, 18, 8, 4, 1], "target_basis": [274]}, + {"frames": [{"characteristic": 493, "extra": 32, "source_bad": [], "target_bad": []}], "minimum_dimension": 9, "node": 1959, "source_basis": [546, 288, 160, 96, 18, 8, 4, 1], "target_basis": [530]}, + {"frames": [{"characteristic": 979, "extra": 36, "source_bad": [], "target_bad": []}], "minimum_dimension": 9, "node": 2017, "source_basis": [548, 292, 164, 100, 16, 12, 2, 1], "target_basis": [44]}, + {"frames": [{"characteristic": 947, "extra": 32, "source_bad": [], "target_bad": []}], "minimum_dimension": 9, "node": 2022, "source_basis": [544, 288, 160, 100, 16, 12, 2, 1], "target_basis": [76]}, + {"frames": [{"characteristic": 883, "extra": 32, "source_bad": [], "target_bad": []}], "minimum_dimension": 9, "node": 2027, "source_basis": [544, 288, 164, 96, 16, 12, 2, 1], "target_basis": [140]}, + {"frames": [{"characteristic": 755, "extra": 32, "source_bad": [], "target_bad": []}], "minimum_dimension": 9, "node": 2032, "source_basis": [544, 292, 160, 96, 16, 12, 2, 1], "target_basis": [268]}, + {"frames": [{"characteristic": 499, "extra": 32, "source_bad": [], "target_bad": []}], "minimum_dimension": 9, "node": 2037, "source_basis": [548, 288, 160, 96, 16, 12, 2, 1], "target_basis": [524]}, + {"frames": [{"characteristic": 971, "extra": 36, "source_bad": [], "target_bad": []}], "minimum_dimension": 9, "node": 2042, "source_basis": [548, 292, 164, 100, 20, 8, 2, 1], "target_basis": [52]}, + {"frames": [{"characteristic": 939, "extra": 32, "source_bad": [], "target_bad": []}], "minimum_dimension": 9, "node": 2047, "source_basis": [544, 288, 160, 100, 20, 8, 2, 1], "target_basis": [84]}, + {"frames": [{"characteristic": 875, "extra": 32, "source_bad": [], "target_bad": []}], "minimum_dimension": 9, "node": 2052, "source_basis": [544, 288, 164, 96, 20, 8, 2, 1], "target_basis": [148]}, + {"frames": [{"characteristic": 747, "extra": 32, "source_bad": [], "target_bad": []}], "minimum_dimension": 9, "node": 2057, "source_basis": [544, 292, 160, 96, 20, 8, 2, 1], "target_basis": [276]}, + {"frames": [{"characteristic": 491, "extra": 32, "source_bad": [], "target_bad": []}], "minimum_dimension": 9, "node": 2062, "source_basis": [548, 288, 160, 96, 20, 8, 2, 1], "target_basis": [532]}, + {"frames": [{"characteristic": 967, "extra": 40, "source_bad": [], "target_bad": []}], "minimum_dimension": 9, "node": 2117, "source_basis": [552, 296, 168, 104, 24, 4, 2, 1], "target_basis": [56]}, + {"frames": [{"characteristic": 935, "extra": 32, "source_bad": [], "target_bad": []}], "minimum_dimension": 9, "node": 2122, "source_basis": [544, 288, 160, 104, 24, 4, 2, 1], "target_basis": [88]}, + {"frames": [{"characteristic": 871, "extra": 32, "source_bad": [], "target_bad": []}], "minimum_dimension": 9, "node": 2127, "source_basis": [544, 288, 168, 96, 24, 4, 2, 1], "target_basis": [152]}, + {"frames": [{"characteristic": 743, "extra": 32, "source_bad": [], "target_bad": []}], "minimum_dimension": 9, "node": 2132, "source_basis": [544, 296, 160, 96, 24, 4, 2, 1], "target_basis": [280]}, + {"frames": [{"characteristic": 487, "extra": 32, "source_bad": [], "target_bad": []}], "minimum_dimension": 9, "node": 2137, "source_basis": [552, 288, 160, 96, 24, 4, 2, 1], "target_basis": [536]} + ], + "seed": 47, + "source_sha256": {"scripts/binary_phase_frames.py": "4148cfa313cf050ed25296e8db36d35871520b8318fd086c7d1facf0c6af67d7", "scripts/certify.py": "2348d95008276c5d5764f7e028c3918e484582cd589bd4d9b82919161745e53e", "scripts/exclusion_circuit.py": "b253f9d7318becda1c1f3b24e2287ed44a9c27fcea00cf9d531afb950d0457a0", "scripts/experiments/disjoint_tensor.py": "102dd59bf96033f96ba0dab4c2aba77b3a7a73b67b0f96929ff8b20a98b91cd8", "scripts/experiments/mixed_point_circuit.py": "f509362b5031976a22af351fd158ae0b4520fa4f286a5176ec37023d672a7ddc", "scripts/experiments/parity_side.py": "1363ebf82d4c39e458e7bede8253688271008374db70ed4284df84253c8d092b", "scripts/paired_exclusion_circuit.py": "8f05f04cf06217befd6d0ee0f2d7d0c5c5eb32cfab8c9a8ca7917fd6d620b7d2"}, + "summary": {"active_nodes": 2271, "enumerated_nodes": 692, "frames": 3096, "histogram": [[3, 4, 0, 2], [3, 8, 0, 10], [4, 2, 2, 2], [4, 32, 32, 16], [5, 1, 0, 20], [5, 1, 1, 2], [5, 2, 1, 50], [5, 2, 2, 80], [6, 2, 2, 40], [6, 4, 4, 20], [6, 8, 8, 170], [7, 2, 2, 10], [7, 4, 0, 100], [8, 2, 2, 100], [9, 1, 1, 70]], "impossible_nodes": [124, 1691], "no_incident_safe_frame": 20, "nodes_with_safe_minimum": 560, "nodes_without_safe_minimum": 132, "nondegenerate_nodes": 1577, "one_extra_dimension_repairs": 12, "safe_frames": 2538, "source_rejected_frames": 158, "target_rejected_frames": 400, "two_extra_dimension_repairs": 100} +} diff --git a/research/parity-incident-completions/CLAIM.md b/research/parity-incident-completions/CLAIM.md new file mode 100644 index 0000000000..cf367c4b00 --- /dev/null +++ b/research/parity-incident-completions/CLAIM.md @@ -0,0 +1,31 @@ +# Sprint claim + +- Track: structural decoder research; OS assignment seed 47 (unbiased UInt32 rejection sampling, 2026-10-13 08:43 PDT). +- Base: 407d52ff15b2217b6e22bce44728b4c3907b6515. +- Object: the 692 pointwise-completable degenerate nodes in the deterministic historical ParitySide(10) DAG; all minimum-dimension nondegenerate sandwiches, with every actual source-line and target-line residual required to be zero or nonalternating. +- Method: exhaustive exact enumeration of one-dimensional extensions of radical-one source spans inside target-perp; independent explicit-vector residual checks. Seed 47 selects h=10 and this refinement of the odd-label interface. +- Deliverable: exact census/certificates, proof of coverage, meaningful negative controls, source pins and one reproducible offline command. No supplier or new kappa claim. +- Expected finish: 2026-10-13 09:25 PDT, including final verification/handoff. +- Alternatives considered: simultaneous nested frames for the same DAG (too broad); several added E8 decoder totals with fixed readers (crowded and larger search). This is distinct from PR972's pointwise sandwich census and PR920's two-source/two-target rectangles. +- Coordination: public issues page and a repo-scoped sprint-issue query checked; no pinned coordination issue found. Claim retained locally as requested. Relevant open PRs and review ledger inspected; PR972 fetched for exact source comparison. Delegated work stays within this residual-completion census. + +## Result update + +Completed the fixed-domain census and exact local dimension refinement: +560 nodes admit a safe minimum, 12 require one extra dimension, 100 require +two extra dimensions, and 20 have no incident-safe frame at any dimension. +The producer and independent vector-set checker agree on all 3,096 minimum +frames. The 22-test suite passes normally and with assertions disabled; +fresh regeneration is byte-identical. No new kappa or legal supplier is claimed. + +The open-PR overlap search was refreshed before submission. PR #920 and +PR #972 remain open at the exact heads recorded in SOURCES.json; this note +adds full-DAG line residuals, larger-frame minima and the 20 absolute local +obstructions beyond those predecessors. The current main moved to +`8bddbee377e2f3ce1f1247196c4b290d55767d20`; the contribution retains its +declared base and source pins. An isolated worktree was used after concurrent +work was detected in the initial checkout. + +Result package: [woklerx research branch](https://github.com/woklerx/integer-mult-bounds/tree/research/seed47-incident-completions/research/parity-incident-completions). +The final chat handoff links the focused PR. No coordination issue was available +to receive a public claim update. diff --git a/research/parity-incident-completions/PROOF.md b/research/parity-incident-completions/PROOF.md new file mode 100644 index 0000000000..649820eadc --- /dev/null +++ b/research/parity-incident-completions/PROOF.md @@ -0,0 +1,298 @@ +# Incident residuals and minimum local ranks + +This note concerns the 692 completable radical-one source spans in the fixed +`ParitySide(10)` circuit. It is a local geometric result, not a compatible +assignment of frames to that circuit, a transfer primitive, an operation +invoice, or a multiplication-bound improvement. All dimensions and counts +below are exact over the binary field. + +## Definitions and the finite domain + +Let `V = F2^10` have its standard dot product, and let `j = 1023` be its +all-ones characteristic vector. Integer masks encode vectors. At an active +node, `S` is the set of actual ancestor source labels and `T` is the set of +actual descendant target labels. Set `A = span(S)` and `B = span(T)`. +Every label in `S` and `T` is a weight-three, hence norm-one, vector. + +A separating frame is a nondegenerate subspace `U` with +`A subset U subset B-perp`. Its incident source residual at `s in S` is +`U intersect s-perp`. Its incident target residual at `t in T` is +`U-perp intersect t-perp`. These are the residuals from the source line to +`U`, and from the target line to the reverse-complement frame `U-perp`. +Call `U` incident-safe if each of these residuals is zero or nonalternating. +Nondegeneracy of the residuals follows from the norm-one lines and the +nondegeneracy of the containing frames. + +The domain requires `A perpendicular B`, `A intersect B = {0}`, and +`dim rad(A) = 1`. It consists of 692 nodes, rather than 692 distinct source +subspaces: different nodes can have the same source span or different actual +incidences. The predecessor's two impossible sandwiches and 1,577 +nondegenerate-source nodes are outside this domain. + +The sandwich minimum theorem used below is established in ToikY's +[PR #972](https://github.com/CrocSwap/integer-mult-bounds/pull/972), pinned at +`f39f92a085fc95fe4245b06d4a695739d79763bc`. It gives the minimum separating +dimension `dim A + dim rad(A)`. The present note adds complete incident +residual tests and minimum incident-safe ranks on the stated radical-one +domain; it does not repeat that contribution's all-node census. + +## Exhausting all minimum separating frames + +Write `a = dim A`, `b = dim B`, and `rad(A) = span(r)` with `r != 0`. +Every minimum frame has the form + + U_z = A + span(z), z in B-perp, r dot z = 1. + +To prove sufficiency, let `x = x_A + c*z` be in the radical of `U_z`. +Pairing with `r` forces `c = 0`. Pairing with `A` then places `x_A` in +`span(r)`, and pairing with `z` forces `x_A = 0`. Thus `U_z` is +nondegenerate and has dimension `a+1`. + +Conversely, a nondegenerate `(a+1)`-dimensional frame containing `A` has +the displayed form, and its additional vector must pair nontrivially with +`r`; otherwise `r` would remain in its radical. This proves coverage, not +just a construction of some frames. + +The functional `z -> r dot z` is nonzero on `B-perp`, since otherwise +`r` would lie in `(B-perp)-perp = B`, contrary to `A intersect B = {0}`. +There are therefore `2^(10-b-1)` permitted vectors `z`. Two such vectors +give the same frame exactly when they differ by an element of `A`: +the quotient `U_z/A` is a one-dimensional binary space, with a unique +nonzero coset. Each frame has exactly `2^a` representatives. Consequently +the exact number of distinct minimum frames at the node is + + 2^(10-a-b-1). + +Signs of scalar coefficients do not change binary source lines. Actual +label sets are essential; replacing them by all vectors of their spans +would impose additional, unrequested incidence conditions. + +## Characteristic vectors give exact incident tests + +For a nondegenerate space `E`, write `w_E` for its unique characteristic +vector, characterized by `w_E dot x = x dot x` for every `x in E`. +For a nondegenerate subspace `L` of `E`, the orthogonal decomposition +`E = L direct-sum (E intersect L-perp)` shows that the latter summand +is alternating exactly when `w_E in L`. + +For a norm-one line `L = span(s)`, `w_E` cannot be zero, since it pairs +to one with `s`. Thus a **nonzero** residual `E intersect s-perp` is +alternating exactly when `w_E = s`. If `dim E = 1`, the residual is zero +and is accepted even though this equality holds. The zero-residual +exception must not be dropped. + +The orthogonal decomposition `V = U direct-sum U-perp` gives + + w_(U-perp) = j + w_U. + +Hence the complete incident test is: + +* Reject at source `s` exactly when `dim U > 1` and `w_U = s`. +* Reject at target `t` exactly when `dim U-perp > 1` and `j + w_U = t`. + +The characteristic vector of a minimum frame has an especially small +search domain. The vectors `w in A` satisfying + + w dot x = x dot x for every x in A + +form a pair `{w0, w0+r}`. Existence follows because the linear functional +`x -> x dot x` vanishes on the radical, and the image of the bilinear map +`A -> A*` is precisely the annihilator of that radical. Its solution set +is a coset of the one-dimensional radical. + +In any `U_z`, its characteristic vector belongs to `A`: write it as +`x_A + c*z` and pair with `r`; the required value is `r dot r = 0`, so +`c = 0`. Exactly one of the two candidates also satisfies +`w dot z = z dot z`, because the candidates differ by `r` and +`r dot z = 1`. Explicitly, + + w_(U_z) = w0 if (j+w0) dot z = 0, + w0 + r otherwise. + +This selector is constant on each coset `z+A`, so it is a well-defined +test of the frame. The two possible characteristics need not both occur. +On the permitted affine set of `z`, the selector is constant exactly +when `j+w0` belongs to `B+span(r)`; otherwise both values occur equally +often. This follows by restricting the functional to the translation +space `B-perp intersect r-perp`, whose annihilator is `B+span(r)`. + +## Exact minimum-frame census + +Exhausting the described cosets gives 3,096 distinct minimum frames at +the 692 nodes. Exactly 2,538 frames pass every actual source-line and +target-line test. There are 158 source-only failures, 400 target-only +failures, and no frame failing both kinds of incidence. + +| dim A | dim B | Minimum frames per node | Safe minimum frames per node | Nodes | +|---:|---:|---:|---:|---:| +| 2 | 4 | 8 | 0 | 10 | +| 2 | 5 | 4 | 0 | 2 | +| 3 | 1 | 32 | 32 | 16 | +| 3 | 5 | 2 | 2 | 2 | +| 4 | 4 | 2 | 1 | 50 | +| 4 | 4 | 2 | 2 | 80 | +| 4 | 5 | 1 | 0 | 20 | +| 4 | 5 | 1 | 1 | 2 | +| 5 | 1 | 8 | 8 | 170 | +| 5 | 2 | 4 | 4 | 20 | +| 5 | 3 | 2 | 2 | 40 | +| 6 | 1 | 4 | 0 | 100 | +| 6 | 2 | 2 | 2 | 10 | +| 7 | 1 | 2 | 2 | 100 | +| 8 | 1 | 1 | 1 | 70 | + +Thus 560 nodes admit a safe minimum frame: 510 accept every minimum +frame and 50 accept some but not all. The other 132 accept none. These +counts were also obtained by an independent in-memory enumeration using +explicit XOR-closed sets and all 1,024 ambient masks. That cross-check +shared the scalar DAG generator, but did not use the producer's frame +enumerator. It is not independent human peer review. + +## Minimum incident-safe dimensions, including larger frames + +An even-dimensional nondegenerate `U` is automatically incident-safe +here. Indeed, both `U` and `U-perp` have even dimension because the +ambient dimension is ten; removing any contained norm-one line leaves +an odd-dimensional nondegenerate residual. An alternating nondegenerate +space has even dimension, so each such residual is nonalternating. +This observation does not assert that a frame of the next even dimension +exists. + +For the 12 rejected nodes with `dim A = 2`, the minimum separating +dimension is three. Complete enumeration of dimension-four separating +frames gives 16 frames at each of the ten `dim B = 4` nodes and four +frames at each of the two `dim B = 5` nodes. Every one is incident-safe +by parity. Their minimum incident-safe dimension is therefore four. +There are 168 such dimension-four frames in total. + +One complete way to enumerate these frames is to take every minimum +frame `U` and every norm-one vector +`q in U-perp intersect B-perp`, and form `U + span(q)`, deduplicating +subspaces. This is exhaustive: any nondegenerate frame `E` of dimension +`a+2` containing `A` contains a minimum frame `U`. Choose in `E` a +vector pairing to one with `r` to construct it. Then +`E intersect U-perp` is a nondegenerate line, and its unique nonzero +vector is one of the enumerated `q`. + +The 100 rejected nodes with `dim A = 6` all have `B = span(t)` for one +actual norm-one target `t`. Every one of their four minimum +dimension-seven frames has characteristic `j+t`; equivalently, every +one fails at that target. No dimension-eight separating frame exists. +To see this, put `W = t-perp`. This is a nondegenerate +dimension-nine space with characteristic `j+t`. For any minimum `U`, +the plane `W intersect U-perp` is alternating, since `w_W = w_U` +belongs to `U`. A dimension-eight nondegenerate frame `E` containing +this `U` would require a nondegenerate line in that alternating plane, +which is impossible. Any proposed dimension-eight `E` containing `A` +would contain some minimum `U` by the partner argument above, so it +cannot avoid this obstruction by choosing different intermediate frames. + +The frame `W` itself is incident-safe: its target residual is zero, and +its characteristic `j+t` has weight seven, hence cannot equal any +actual weight-three source label. The minimum incident-safe dimension +at these 100 nodes is therefore nine, two above the separating minimum. + +At each of the 20 rejected nodes with `dim A = 4` and `dim B = 5`, +the only minimum frame already equals `B-perp`, of dimension five. +It fails a source incidence. Any nondegenerate frame containing `A` +has dimension at least five, while any separating frame has dimension +at most five. Thus **no incident-safe separating frame of any dimension +exists** at these 20 nodes. + +Together with the 560 nodes already safe at the separating minimum, +this exhausts the original 692-node domain: + +| Local conclusion | Nodes | +|---|---:| +| Safe at minimum separating dimension | 560 | +| First safe one dimension above minimum | 12 | +| First safe two dimensions above minimum | 100 | +| No incident-safe separating frame of any dimension | 20 | + +Among the 672 locally feasible nodes, the minimum incident-safe +dimension histogram is `4:30, 5:132, 6:230, 7:10, 8:100, 9:170`. +These are geometric dimensions and node counts, not paid helper counts +or a recursive resource bound. + +## Concrete witnesses and obstructions + +At node **229**, `S = {38,49}`, +`A = span(49,23)`, and `B = span(896,448,192,64)`. +Its actual targets are `{448,704,832,896}`. Its radical generator is +23. Each of its eight dimension-three frames has characteristic 38 or +49, four frames of each type. Both characteristics are actual sources, +so every minimum frame fails. For example, partners `z=1` and `z=2` +produce characteristics 49 and 38 respectively. Its sixteen +dimension-four separating frames all pass. + +At node **1217**, +`A = span(56,27,15,3)` and `B = span(515,259,131,67)`. +Its actual sources are `{35,44,52,56}`, and actual targets are +`{67,131,259,515}`. There are exactly two dimension-five minimum +frames. Partner `z=961` produces characteristic 35 and fails at the +actual source 35. Partner `z=965` produces characteristic 32 and +passes every actual incidence. This is a small example of why choosing +one arbitrary minimum completion does not decide existence of a safe one. + +At node **818**, +`A = span(518,262,134,20,12,6)` and `B = span(97)`. +The radical generator is 30. The two characteristic solutions inside +`A` are 926 and 896, but only 926 occurs among the four minimum frames. +Because `926 = 1023+97` over the binary field, all four fail at target +97. No dimension-eight separating frame exists. The dimension-nine +frame `97-perp` is incident-safe, as proved above. + +For an example of absolute incident impossibility, node **302** has +`A = span(896,352,224,96)` and `B = span(112,17,9,5,3)`. +Its sole minimum frame is `B-perp` and has characteristic 896, an +actual source. Its source residual has positive dimension and is +alternating. There is no larger separating frame. + +## Why the tests are necessary for nested transfers + +For a nested chain of nondegenerate spaces +`L = U0 subset U1 subset ... subset Uk = U`, there is an orthogonal +direct-sum decomposition + + U intersect L-perp + = direct-sum over i=1,...,k of (Ui intersect Ui-1-perp). + +It follows successively from `Ui = Ui-1 direct-sum +(Ui intersect Ui-1-perp)`. If every successive residual is zero or +nonalternating, a nonzero endpoint residual must be nonalternating: +some nonzero summand supplies a vector of norm one, which keeps that +norm in the full orthogonal sum. Therefore a nonzero alternating +endpoint residual cannot be repaired merely by subdividing that same +nested transfer into more intermediate frames. + +Apply this argument along a source-to-node path, and along a +target-to-node path in the reversed complement frames. Every actual +source or target incidence must pass in a legal assignment under this +nested orthonormal-residual interface. The 20 absolute incident +obstructions therefore reject such assignments at these otherwise +completable source spans. They are additional local obstructions; +the predecessor's two impossible sandwiches already reject the +unchanged full circuit. + +The converse is not proved and is not implied: safe choices at +individual nodes need not nest along an edge, and the intervening +edge residuals need not pass. Alternative topologies, copied carriers, +paid non-nested transport, and other phase interfaces are outside this +result. There is no claim that the locally feasible nodes form a legal +supplier, nor that local rank increments equal operation costs. + +## Attribution + +The historical signed parity circuit and its obstruction/repair analysis +are due to Douglas Colkitt. DreamingOfClouds' format-limits work supplied +the sandwich restriction and radical rank floor. ToikY's pinned PR #972 +proved the constrained minimum-completion theorem and identified the +692-node radical-one domain. ArnavSwiggy's +[PR #920](https://github.com/CrocSwap/integer-mult-bounds/pull/920) +already emphasized the stronger incident-residual condition and supplied +an explicit alternating-residual counterexample. These predecessors +are the basis for this narrowly scoped extension; no priority is claimed +for the underlying bilinear-form or characteristic-vector facts. + +Prepared with substantial OpenAI Codex assistance in the mathematical +derivations, enumeration, and independent computational cross-check. diff --git a/research/parity-incident-completions/README.md b/research/parity-incident-completions/README.md new file mode 100644 index 0000000000..bc19d9bff3 --- /dev/null +++ b/research/parity-incident-completions/README.md @@ -0,0 +1,132 @@ +# Exact incident-residual costs for the historical parity decoder + +**Structural decoder research / scoped negative result; no new kappa.** +Of the 692 completable degenerate nodes of the pinned `ParitySide(10)` DAG, +560 admit an incident-safe minimum frame, 12 need one extra dimension, +100 need **two** extra dimensions, and 20 admit no incident-safe frame at any +dimension. Here incident-safe means that every actual ancestor source line +and descendant target line has a zero or nonalternating residual. It does +not mean that the node frames fit together into a legal circuit. + +This adds the missing line-residual analysis to +[ToikY's PR #972](https://github.com/CrocSwap/integer-mult-bounds/pull/972), +which computes pointwise minimum separating frames and explicitly leaves +incident residuals open. It applies the distinction studied by +[ArnavSwiggy's PR #920](https://github.com/CrocSwap/integer-mult-bounds/pull/920) +to an entire fixed DAG's radical-one nodes, with arbitrary numbers of +source and target labels. It neither replaces those contributions nor +reopens the historical large-DAG repair search. + +## Reproduce + +From the repository root, Python 3.11 or later, standard library only: + +```sh +python -B research/parity-incident-completions/check.py +``` + +The offline checker verifies the source pins and signed scalar kernel, +reconstructs the fixed graph, enumerates all 3,096 distinct minimum frames, +tests actual residual vector sets, checks the 112 larger-frame witnesses, +and rejects the missing intermediate dimensions. It reports +`PASS_PARITY_INCIDENT_COMPLETIONS` with the counts above. Run the 22-test +corruption and residual regression suite separately with +`python -B research/parity-incident-completions/test_check.py`. +See [VALIDATION.md](VALIDATION.md) for the commands actually run and their limits. + +Regenerate to a fresh path and check that result: + +```sh +python -B research/parity-incident-completions/produce.py --out /tmp/parity-incident.json +python -B research/parity-incident-completions/check.py --certificate /tmp/parity-incident.json +``` + +On Windows, substitute a writable temporary filename for the `/tmp` path. +The certificate is deterministic; neither generation nor checking searches +the web or requires a solver. + +## Complete finite domain and results + +The assignment seed was **47**, sampled from OS randomness. This selects +the structural-decoder track and `h=10` from the historical small domains +8 and 10. The seed is not a claim of novelty: the new obligation is incident +residual legality and its exact local dimension cost, beyond PR #972. + +The fixed DAG has 2,271 active nodes, including 120 inputs; 1,577 source +spans are nondegenerate and two nodes already have common-span obstructions. +The domain here is **all 692 remaining degenerate source spans**, each with +one-dimensional radical. The latter two categories are identified to bind +the domain, not subjected to a new all-frame census. + +| Exact result on the 692 nodes | Count | +|---|---:| +| Distinct minimum separating frames | 3,096 | +| Minimum frames passing every source/target line residual | 2,538 | +| Frames rejected at a source / target / both | 158 / 400 / 0 | +| Nodes with an incident-safe minimum | 560 | +| Nodes requiring exactly one extra dimension | 12 | +| Nodes requiring exactly two extra dimensions | 100 | +| Nodes admitting no incident-safe frame at any dimension | 20 | + +Of the 560 nodes with a safe minimum, 510 accept every minimum frame and 50 accept +only some. Of the 132 failures, the 12 dimension-three cases have explicit +dimension-four witnesses. The 100 dimension-seven cases have an alternating +allowed extension space, so dimension eight is impossible; dimension nine +works with zero outgoing target residual. The 20 dimension-five cases +already fill the target orthogonal complement, so there is no room to repair. + +The [proof](PROOF.md) establishes the exhaustive coset enumeration, the +characteristic-vector test used by the producer, and the dimension lower +bounds. The checker independently enumerates vector sets and tests actual +residuals. Its only reused computation is the pinned historical DAG +constructor; it does not use the producer's linear-algebra implementation. +This is computational cross-checking with AI assistance, not independent +human review or a formal proof of the multiplication theorem. + +The stopping rule is exhaustion of those 692 nodes and every one-dimensional +extension coset giving a minimum sandwich. Larger-frame witnesses and exact +dimension exclusions settle all 132 failed nodes. No time-limited failure to +find a frame is reported as impossibility. + +## Cost decision and scope + +Twenty nodes fail a necessary condition for the historical nested, +nondegenerate, orthonormal-residual interface even if scalar gates, source +access, helpers, copies, cleanup and recursive children were all free. +The unchanged DAG therefore cannot become a legal supplier merely by choosing +larger local frames. No large supplier implementation or assembly optimization +is justified by this mechanism alone. + +The 212 additional local dimensions required by the 112 repairable nodes +are **not** a gate count, helper invoice or additive recursive cost. A paid +construction would still need simultaneous nesting, every internal residual, +source access, dirty compensation, copy/restoration work, recursive calls, +physical hosting, precision/tape and analytic transfer. The selected result's +all-size/compiler and other inherited assumptions remain untouched. + +This result does not exclude changed DAGs, duplicated sums, different source +or target labels, encoded carriers or a different phase-transfer interface. +It makes no runtime or exponent improvement claim. `upstream/` and all +selected-frontier artifacts are unchanged. + +## Provenance and credit + +Base commit: `407d52ff15b2217b6e22bce44728b4c3907b6515`. +At that base the selected conditional value is +`1661960547713947/1000000000000000000`, from Jonathan Capalbo's T37/PR #942 +composition. Its selected record, actual T37 construction, assembly and +maintainer review were read; historical parity numbers were not treated as +the current frontier. The new result does not depend on running that supplier. + +[SOURCES.json](SOURCES.json) records source-file hashes and exact predecessor +heads. Credit Douglas Colkitt's historical parity-side work for the signed +kernel, common-span obstruction, characteristic criterion and failed repair; +DreamingOfClouds' format-limit analysis for frame restrictions; ToikY's #972 +for the pointwise all-node census; and ArnavSwiggy's #920 for incident-residual +completion analysis and the six-dimensional control. The historical generator +retains its existing attribution and license. + +Prepared for GitHub contributor **woklerx**, with substantial **OpenAI Codex** +assistance in mathematics, implementation, independent checking, review and +writing. New files are under the repository's Apache-2.0 license. No predecessor +package code is vendored or modified. diff --git a/research/parity-incident-completions/SOURCES.json b/research/parity-incident-completions/SOURCES.json new file mode 100644 index 0000000000..301c3ea831 --- /dev/null +++ b/research/parity-incident-completions/SOURCES.json @@ -0,0 +1,36 @@ +{ + "base_commit": "407d52ff15b2217b6e22bce44728b4c3907b6515", + "context_sources_sha256": { + "CONTRIBUTING.md": "1cae543effcabab91dab7f78423ee9283e60fb504f10848483d153331cd72bc4", + "certificates/selected-result.json": "1d35283cb0ce3f8db1f285683933517f9e87475d295e2b6316399ad9928c81c3", + "docs/research/community-batch952-review.md": "04db59dfc2c31453229b6061bbf59547e71ce3abdc0ce0f6779144bb0dde127b", + "docs/research/current-status.md": "9ef8355371d93f344e0462796d515d40a065c4f19534a8540b8d959b51a02aaa", + "docs/research/parity-side.md": "75de20f8dcd3d1a2e2d19bc087eda50a6cc77027651f48198868426a80cd12f2", + "docs/research/pr-ledger.md": "53984538c660ecc35d09da30f9383d9731e21561e3d3d05d792e39dff1214f79", + "research/community-frontier942-audit/ASSEMBLY.json": "4b2894956e594e408e111d806c842f2ed2c32aefcadff93fad69ff0b46f1991a", + "research/community-frontier942-audit/SELECTED-SOURCE.json": "1cd8dfde15919d1c3df9796b262e67898143d9793d2b86ee7b0a47fc903654c9", + "research/complex-t37-owners/ALL-T-PROOF.md": "4c0c71ccd0afb850538af1b467e5208b7eb3edd0260fded84e02a8c01dfc868b", + "research/complex-t37-owners/PROOF.md": "09b39ad0d5f1f59f54dcff15c16640f7a7acf27e6597660acaf957b95c7e8510", + "research/format-limits-corrected/README.md": "1bcd80c1cd61fddae9f3a7f00fc9d1c796410408251e576600e94c19b16b1e5e", + "scripts/experiments/parity_phase_repair.py": "1f76f3b96ed556ed901835bd101d0e2588c997d2ed39bdfca097f0187aa86d40" + }, + "predecessor_prs": { + "920": { + "head": "913817423bbfaf241dec1d6b9edb1c7ded349508", + "path": "research/complex-parity-frames-seed35/README.md", + "sha256": "b7d8c49293d770c0c3037fcd5eda36881adb7042c46c0d87dfbe47d9610d104c", + "url": "https://github.com/CrocSwap/integer-mult-bounds/pull/920" + }, + "972": { + "head": "f39f92a085fc95fe4245b06d4a695739d79763bc", + "path": "research/frame-completion-seed47/README.md", + "sha256": "b24cf3ed7cf5b651f4c2c463e10918dc4d1a2323bf268bcd35fbcbb0f79d9253", + "url": "https://github.com/CrocSwap/integer-mult-bounds/pull/972" + } + }, + "runtime_sources": "CERTIFICATE.json source_sha256; checker pins expected source map independently", + "scope": "Predecessor PRs supply context and theory; no PR-specific runtime code imported. The fixed historical DAG and all required imports are already in the base.", + "selected_conditional_kappa_at_base": "1661960547713947/1000000000000000000", + "selected_source_head": "1960cfa919a0945d0ca4cb14ea96237000037903", + "selected_source_pr": 942 +} diff --git a/research/parity-incident-completions/VALIDATION.json b/research/parity-incident-completions/VALIDATION.json new file mode 100644 index 0000000000..9f65f33144 --- /dev/null +++ b/research/parity-incident-completions/VALIDATION.json @@ -0,0 +1,64 @@ +{ + "base_commit": "407d52ff15b2217b6e22bce44728b4c3907b6515", + "checked_at_utc": "2026-10-13T16:02:26.696037+00:00", + "commands": [ + { + "command": "python -B research/parity-incident-completions/check.py", + "exit_code": 0, + "result": "Full finite certificate verified.", + "seconds": 1.575 + }, + { + "command": "python -B research/parity-incident-completions/test_check.py", + "exit_code": 0, + "result": "22 tests passed.", + "seconds": 8.557 + }, + { + "command": "python -O -B research/parity-incident-completions/test_check.py", + "exit_code": 0, + "result": "22 tests passed with assertions disabled.", + "seconds": 8.126 + }, + { + "command": "python -B research/parity-incident-completions/produce.py --out /certificate.json", + "exit_code": 0, + "seconds": 1.743, + "stderr": "Failed to find real location of C:\\Users\\wilds\\AppData\\Local\\Programs\\Python\\Python314\\python.exe", + "stdout": "{\"active_nodes\": 2271, \"enumerated_nodes\": 692, \"frames\": 3096, \"histogram\": [[3, 4, 0, 2], [3, 8, 0, 10], [4, 2, 2, 2], [4, 32, 32, 16], [5, 1, 0, 20], [5, 1, 1, 2], [5, 2, 1, 50], [5, 2, 2, 80], [6, 2, 2, 40], [6, 4, 4, 20], [6, 8, 8, 170], [7, 2, 2, 10], [7, 4, 0, 100], [8, 2, 2, 100], [9, 1, 1, 70]], \"impossible_nodes\": [124, 1691], \"no_incident_safe_frame\": 20, \"nodes_with_safe_minimum\": 560, \"nodes_without_safe_minimum\": 132, \"nondegenerate_nodes\": 1577, \"one_extra_dimension_repairs\": 12, \"safe_frames\": 2538, \"source_rejected_frames\": 158, \"target_rejected_frames\": 400, \"two_extra_dimension_repairs\": 100}" + }, + { + "command": "python -I -O -B research/parity-incident-completions/check.py --certificate /certificate.json", + "exit_code": 0, + "seconds": 1.689, + "stderr": "Failed to find real location of C:\\Users\\wilds\\AppData\\Local\\Programs\\Python\\Python314\\python.exe", + "stdout": "{\"active_nodes\": 2271, \"enumerated_nodes\": 692, \"frames\": 3096, \"histogram\": [[3, 4, 0, 2], [3, 8, 0, 10], [4, 2, 2, 2], [4, 32, 32, 16], [5, 1, 0, 20], [5, 1, 1, 2], [5, 2, 1, 50], [5, 2, 2, 80], [6, 2, 2, 40], [6, 4, 4, 20], [6, 8, 8, 170], [7, 2, 2, 10], [7, 4, 0, 100], [8, 2, 2, 100], [9, 1, 1, 70]], \"impossible_nodes\": [124, 1691], \"no_incident_safe_frame\": 20, \"nodes_with_safe_minimum\": 560, \"nodes_without_safe_minimum\": 132, \"nondegenerate_nodes\": 1577, \"one_extra_dimension_repairs\": 12, \"one_extra_safe_frame_census\": 168, \"safe_frames\": 2538, \"source_rejected_frames\": 158, \"status\": \"PASS_PARITY_INCIDENT_COMPLETIONS\", \"target_rejected_frames\": 400, \"two_extra_dimension_repairs\": 100, \"verified\": true}" + }, + { + "command": "python -B -m unittest discover -s tests -p test_parity_side.py -v", + "exit_code": 0, + "seconds": 0.402, + "stderr": "Failed to find real location of C:\\Users\\wilds\\AppData\\Local\\Programs\\Python\\Python314\\python.exe\ntest_characteristic_detects_alternating_residual (test_parity_side.ParitySideTest.test_characteristic_detects_alternating_residual) ... ok\ntest_five_point_obstruction_exact_common_vector (test_parity_side.ParitySideTest.test_five_point_obstruction_exact_common_vector) ... ok\ntest_invalid_ground_sizes (test_parity_side.ParitySideTest.test_invalid_ground_sizes) ... ok\ntest_signed_scalar_kernel_and_frame_safe_coarsening (test_parity_side.ParitySideTest.test_signed_scalar_kernel_and_frame_safe_coarsening) ... ok\n\n----------------------------------------------------------------------\nRan 4 tests in 0.065s\n\nOK", + "stdout": "" + } + ], + "context_source_hashes_checked": 12, + "environment_note": "Initial fresh-output attempt in sandbox-provided Windows TEMP failed with PermissionError before writing; rerun in writable workspace succeeded. Python launcher emitted a real-location warning while successful checks exited zero.", + "fresh_regeneration_byte_identical": true, + "fresh_regeneration_matches_git_blob": true, + "newline_note": "Producer writes explicit LF on every platform. After adding the newline control, regeneration and isolated optimized checking passed again, and generated bytes matched the actual committed Git blob.", + "not_run": [ + "make verify", + "make frontier942-verify" + ], + "platform": "Windows", + "python": "3.14.5", + "scope": "New finite geometry package and affected historical parity tests only; no full supplier replay or theorem certification.", + "selected_record_check": { + "command": "python -B scripts/verify_selected_result.py", + "execution": "separately completed in this isolated worktree before this receipt runner", + "exit_code": 0, + "stdout": "PASS selected record, exact kappa, original source inventory and proof supplement: 1661960547713947/1000000000000000000" + }, + "status": "PASS" +} diff --git a/research/parity-incident-completions/VALIDATION.md b/research/parity-incident-completions/VALIDATION.md new file mode 100644 index 0000000000..38d7a11db2 --- /dev/null +++ b/research/parity-incident-completions/VALIDATION.md @@ -0,0 +1,48 @@ +# Completed verification + +Validated on Windows with CPython **3.14.5**, starting from base +`407d52ff15b2217b6e22bce44728b4c3907b6515`, in an isolated worktree. +The machine-readable commands, results and scope are in +[VALIDATION.json](VALIDATION.json). + +| Completed check | Result | +|---|---| +| `python -B research/parity-incident-completions/check.py` | PASS; 692 nodes, 3,096 minimum frames, 2,538 passing frames; node split 560 / 12 / 100 / 20 | +| `python -B research/parity-incident-completions/test_check.py` | 22 tests passed | +| `python -O -B research/parity-incident-completions/test_check.py` | Same 22 tests passed with assertions disabled | +| Fresh `produce.py --out /certificate.json` | Byte-identical to committed certificate | +| `python -I -O -B research/parity-incident-completions/check.py --certificate /certificate.json` | PASS in isolated, optimized mode | +| `python -B -m unittest discover -s tests -p test_parity_side.py -v` | All four historical parity tests passed | +| `python -B scripts/verify_selected_result.py` | PASS selected record, exact kappa, source inventory and proof supplements | +| Context-file SHA-256 pins in `SOURCES.json` | All 12 match | + +The checker reconstructs all **14,400 signed scalar coefficients** from the +actual DAG arguments. It independently rebuilds source and target reachability, +exhausts every minimum extension coset, checks explicit residual sets, verifies +the 112 larger-frame witnesses, and excludes the missing dimension-eight +frames. It also reproduces the complete **168-frame dimension-four census** +at the 12 nodes requiring one extra dimension. + +Controls reject omitted or duplicate records and frames, altered source/DAG +hashes, extra source vectors, changed characteristics or residual decisions, +false impossibility claims, invalid repair vectors, changed summaries, duplicate +JSON keys and an output-sign flip. Small geometric controls cover PR #920's +alternating residual, the accepted zero-residual boundary, a characteristic in +the source span but absent from the actual labels, and a target-only failure. + +The row-reduction producer and explicit-vector checker share the pinned DAG +constructor but no frame-algebra implementation. A separate Codex worker also +reviewed the mathematical argument and reproduced the counts independently. +This is not independent human peer review or formal theorem verification. + +The initial regeneration attempt in the sandbox-provided Windows temporary +directory failed with a permission error before writing the output. Repeating +that check in the writable workspace succeeded. The Python launcher emitted a +location warning; the completed checks returned zero. Neither issue was bypassed +by weakening a mathematical check. + +`make verify` and `make frontier942-verify` were **not run**. This additions-only +research package changes no supplier, generator, selected record or upstream +source; the affected historical tests and new exact checks are reported above. +No full supplier replay, Linux execution, Lean build or complete multiplication +theorem verification is claimed. diff --git a/research/parity-incident-completions/check.py b/research/parity-incident-completions/check.py new file mode 100644 index 0000000000..f9e2c2619c --- /dev/null +++ b/research/parity-incident-completions/check.py @@ -0,0 +1,352 @@ +#!/usr/bin/env python3 +"""Independently check the pinned, finite ParitySide(10) incident certificate. + +Geometry uses XOR-closed sets in F_2^10, never the producer's row reduction, +Gram matrices, or characteristic-vector residual classification. The circuit +constructor is shared and pinned; leaf and output reachability are rebuilt. +No files are written. Failed checks raise exceptions even under python -O. +""" +import argparse +from collections import Counter +from functools import lru_cache +import hashlib +import json +from pathlib import Path +import sys + +HERE = Path(__file__).resolve().parent +ROOT = HERE.parents[1] +BASE = "407d52ff15b2217b6e22bce44728b4c3907b6515" +DAG_SHA256 = "13bd7da6b47bfee17213acb2eb1120fbe7324a3ece0691f8139d358149c70040" +SOURCE_SHA256 = { + "scripts/binary_phase_frames.py": "4148cfa313cf050ed25296e8db36d35871520b8318fd086c7d1facf0c6af67d7", + "scripts/certify.py": "2348d95008276c5d5764f7e028c3918e484582cd589bd4d9b82919161745e53e", + "scripts/exclusion_circuit.py": "b253f9d7318becda1c1f3b24e2287ed44a9c27fcea00cf9d531afb950d0457a0", + "scripts/experiments/disjoint_tensor.py": "102dd59bf96033f96ba0dab4c2aba77b3a7a73b67b0f96929ff8b20a98b91cd8", + "scripts/experiments/mixed_point_circuit.py": "f509362b5031976a22af351fd158ae0b4520fa4f286a5176ec37023d672a7ddc", + "scripts/experiments/parity_side.py": "1363ebf82d4c39e458e7bede8253688271008374db70ed4284df84253c8d092b", + "scripts/paired_exclusion_circuit.py": "8f05f04cf06217befd6d0ee0f2d7d0c5c5eb32cfab8c9a8ca7917fd6d620b7d2", +} + + +def require(condition, message): + if not condition: + raise ValueError(message) + + +def canonical(value): + return json.dumps(value, sort_keys=True, separators=(",", ":")).encode() + + +def dot(x, y): + return (x & y).bit_count() % 2 + + +def span(vectors): + result = {0} + for x in vectors: + result |= {x ^ y for y in result} + return frozenset(result) + + +def dimension(space): + size = len(space) + require(size > 0 and size & (size - 1) == 0, "not a binary subspace size") + return size.bit_length() - 1 + + +def canonical_basis(space): + # For each leading-bit position, the smallest vector has all other pivot + # bits cleared. This obtains the canonical basis by set inspection alone. + leaders = {} + for x in space - {0}: + p = x.bit_length() + leaders[p] = min(x, leaders.get(p, x)) + result = [leaders[p] for p in sorted(leaders, reverse=True)] + require(span(result) == space and len(result) == dimension(space), + "canonical basis does not span") + return result + + +class Geometry: + def __init__(self, h): + self.h = h + self.universe = frozenset(range(1 << h)) + self.odd = frozenset(x for x in self.universe if dot(x, x)) + self.hyperplanes = tuple(frozenset(y for y in self.universe if not dot(x, y)) + for x in self.universe) + + @lru_cache(maxsize=None) + def perp(self, space): + result = self.universe + for x in canonical_basis(space): + result = result & self.hyperplanes[x] + require(len(result) * len(space) == len(self.universe), "perpendicular size") + return result + + def nondegenerate(self, space): + return space & self.perp(space) == {0} + + def characteristic(self, space): + # Ambient characteristic is the all-ones vector. Its projection into + # U is found by literal intersection, not by solving Gram equations. + omega = (1 << self.h) - 1 + choices = space & {omega ^ x for x in self.perp(space)} + require(len(choices) == 1, "characteristic is not unique") + value = next(iter(choices)) + require(all(dot(value, x) == dot(x, x) for x in space), "characteristic identity") + return value + + def residual_failures(self, space, incident_labels): + bad = [] + for label in sorted(incident_labels): + # This explicitly constructs every actual residual. A zero + # residual is admissible; a positive-dimensional alternating one + # is not. No characteristic comparison decides this predicate. + residual = space & self.hyperplanes[label] + if residual != {0} and residual.isdisjoint(self.odd): + bad.append(label) + return bad + + def extensions(self, A, allowed): + require(A <= allowed, "source span is not in target perpendicular") + remaining = set(allowed - A) + while remaining: + z = min(remaining) + coset = {z ^ a for a in A} + require(coset <= remaining, "partial extension coset") + remaining.difference_update(coset) + U = A | coset + require(len(U) == 2 * len(A), "extension dimension") + if self.nondegenerate(U): + yield z, U + + +def validate_vector_list(value, h, label): + require(type(value) is list, label + ": expected a list") + require(all(type(x) is int and 0 <= x < (1 << h) for x in value), + label + ": invalid vector") + require(len(value) == len(set(value)), label + ": duplicate vector") + + +def rebuild_graph(circuit): + active = set(circuit.active) + sources = {} + # Independent traversal from each node to its leaves; this does not read + # the constructor's cached supports or producer's forward label unions. + for node in sorted(active): + leaves, seen, pending = set(), set(), [node] + while pending: + child = pending.pop() + if child in seen: + continue + require(child in active, "reachability left the active DAG") + seen.add(child) + args = circuit.args[child] + if args: + require(all(0 < abs(x) < child for x in args), "non-topological edge") + pending.extend(abs(x) for x in args) + else: + require(1 <= child <= len(circuit.inputs), "unknown source leaf") + leaves.add(circuit.inputs[child - 1]) + sources[node] = leaves + targets = {node: set() for node in active} + reached = set() + for target, signed_node in circuit.outputs.items(): + seen, pending = set(), [abs(signed_node)] + while pending: + node = pending.pop() + if node in seen: + continue + require(node in active, "output traversal left active DAG") + seen.add(node) + targets[node].add(target) + if circuit.args[node]: + pending.extend(abs(x) for x in circuit.args[node]) + reached.update(seen) + require(reached == active, "inactive or omitted output ancestor") + # Recompute signed integer coefficients from the arguments. In particular, + # do not invoke ParitySide.verify(), whose assertions disappear under -O. + coefficients = {} + for node in sorted(active): + args = circuit.args[node] + if not args: + coefficients[node] = {circuit.inputs[node - 1]: 1} + continue + row = {} + for signed_child in args: + child_row = coefficients[abs(signed_child)] + require(row.keys().isdisjoint(child_row), "overlapping addition supports") + sign = 1 if signed_child > 0 else -1 + row.update({label: sign * value for label, value in child_row.items()}) + require(set(row) == sources[node], "signed support disagrees with leaf traversal") + coefficients[node] = row + require(len(circuit.inputs) == 120 and set(circuit.outputs) == set(circuit.inputs), + "unexpected source/target kernel domain") + for target, signed_node in circuit.outputs.items(): + row = coefficients[abs(signed_node)] + sign = 1 if signed_node > 0 else -1 + for source in circuit.inputs: + intersection = (source & target).bit_count() + expected = 1 if intersection == 0 else -1 if intersection == 2 else 0 + require(sign * row.get(source, 0) == expected, + "signed kernel entry differs for %d,%d" % (source, target)) + return sources, targets + + +def check(certificate): + require(type(certificate) is dict, "certificate is not an object") + require(set(certificate) == {"format", "base_commit", "seed", "h", "source_sha256", + "dag_sha256", "summary", "records"}, "top-level fields") + require(type(certificate["format"]) is int and certificate["format"] == 1, "format") + require(certificate["base_commit"] == BASE, "base commit") + require(type(certificate["seed"]) is int and certificate["seed"] == 47, "seed") + require(type(certificate["h"]) is int and certificate["h"] == 10, "h") + require(certificate["source_sha256"] == SOURCE_SHA256, "source hash manifest differs") + for filename, digest in SOURCE_SHA256.items(): + require(hashlib.sha256((ROOT / filename).read_bytes()).hexdigest() == digest, + "source hash mismatch: " + filename) + # Only the pinned constructor is reused. Its transitive imports include + # binary_phase_frames, but none of that module's geometry is called here. + sys.path.insert(0, str(ROOT / "scripts")) + from experiments.parity_side import ParitySide + circuit = ParitySide(10) + dag = dict(inputs=list(circuit.inputs), + args=[[n, list(circuit.args[n]) if circuit.args[n] else None] + for n in sorted(circuit.active)], + outputs=sorted([t, n] for t, n in circuit.outputs.items())) + require(certificate["dag_sha256"] == DAG_SHA256, "certificate DAG hash differs") + require(hashlib.sha256(canonical(dag)).hexdigest() == DAG_SHA256, "rebuilt DAG hash differs") + sources, targets = rebuild_graph(circuit) + geometry = Geometry(10) + records = certificate["records"] + require(type(records) is list, "records must be a list") + require(all(type(r) is dict and type(r.get("node")) is int for r in records), + "record must contain integer node") + ids = [r["node"] for r in records] + require(ids == sorted(set(ids)), "duplicate or unordered records") + by_node = {r["node"]: r for r in records} + expected_nodes, impossible, nondegenerate = [], [], 0 + totals, histogram = Counter(), Counter() + one_extra_safe_spaces = 0 + for node in sorted(circuit.active): + A, B = span(sources[node]), span(targets[node]) + Bperp = geometry.perp(B) + require(A <= Bperp, "nonorthogonal source/target spans") + if A & B != {0}: + impossible.append(node) + continue + radical = A & geometry.perp(A) + if radical == {0}: + nondegenerate += 1 + continue + require(len(radical) == 2, "unexpected radical dimension") + expected_nodes.append(node) + require(node in by_node, "missing record for node " + str(node)) + record = by_node[node] + prefix = "node %d: " % node + for field, space in (("source_basis", A), ("target_basis", B)): + validate_vector_list(record.get(field), 10, prefix + field) + require(record[field] == canonical_basis(space), prefix + field + " differs") + minimum = dimension(A) + 1 + require(type(record.get("minimum_dimension")) is int and + record["minimum_dimension"] == minimum, prefix + "minimum dimension") + frames = record.get("frames") + require(type(frames) is list, prefix + "frames is not a list") + expected_frames, minimum_spaces = [], [] + for z, U in geometry.extensions(A, Bperp): + complement = geometry.perp(U) + source_bad = geometry.residual_failures(U, sources[node]) + target_bad = geometry.residual_failures(complement, targets[node]) + expected_frames.append(dict(extra=z, characteristic=geometry.characteristic(U), + source_bad=source_bad, target_bad=target_bad)) + minimum_spaces.append(U) + require(expected_frames, prefix + "no minimum completion") + # Canonical JSON comparison also rejects bools masquerading as ints, + # extra fields, duplicated frames, changed labels and reordered rows. + require(canonical(frames) == canonical(expected_frames), prefix + "frame enumeration differs") + safe = sum(not f["source_bad"] and not f["target_bad"] for f in expected_frames) + fields = {"node", "source_basis", "target_basis", "minimum_dimension", "frames"} + if safe: + require(set(record) == fields, prefix + "unexpected record fields") + else: + require(set(record) == fields | {"repair"}, prefix + "missing repair classification") + repair = record["repair"] + if repair is None: + require(dimension(Bperp) == minimum, prefix + "unproved impossibility") + totals["no_incident_safe_frame"] += 1 + else: + require(type(repair) is dict and set(repair) == {"dimension", "extra_basis"}, + prefix + "repair fields") + validate_vector_list(repair["extra_basis"], 10, prefix + "repair basis") + U = span(list(A) + repair["extra_basis"]) + repair_dimension = dimension(U) + require(type(repair["dimension"]) is int and + repair["dimension"] == repair_dimension, prefix + "repair dimension") + require(repair_dimension == dimension(A) + len(repair["extra_basis"]), + prefix + "dependent repair vectors") + require(A <= U <= Bperp and geometry.nondegenerate(U), prefix + "invalid repair sandwich") + require(not geometry.residual_failures(U, sources[node]), prefix + "repair source residual") + require(not geometry.residual_failures(geometry.perp(U), targets[node]), + prefix + "repair target residual") + if repair_dimension == minimum + 1: + safe_extensions = set() + for minimal_U in minimum_spaces: + for _, extension in geometry.extensions(minimal_U, Bperp): + if (not geometry.residual_failures(extension, sources[node]) and + not geometry.residual_failures(geometry.perp(extension), targets[node])): + safe_extensions.add(extension) + expected_count = {4: 16, 5: 4}.get(dimension(B)) + require(len(safe_extensions) == expected_count, + prefix + "one-extra-dimension completion census") + one_extra_safe_spaces += len(safe_extensions) + totals["one_extra_dimension_repairs"] += 1 + else: + require(repair_dimension == minimum + 2, prefix + "unexpected repair cost") + # Any nondegenerate U' containing radical-one A contains a + # minimum nondegenerate U: pair the radical with a vector + # of U'. A one-dimensional nondegenerate extension of U + # needs an odd vector in U^perp intersect B^perp. Enumerate + # that actual set for EVERY minimum U to exclude min+1. + for minimal_U in minimum_spaces: + allowed = geometry.perp(minimal_U) & Bperp + require(allowed.isdisjoint(geometry.odd), prefix + "unproved two-dimension lower bound") + totals["two_extra_dimension_repairs"] += 1 + histogram[(minimum, len(expected_frames), safe)] += 1 + totals["frames"] += len(expected_frames) + totals["safe_frames"] += safe + totals["source_rejected_frames"] += sum(bool(f["source_bad"]) for f in expected_frames) + totals["target_rejected_frames"] += sum(bool(f["target_bad"]) for f in expected_frames) + totals["nodes_with_safe_minimum"] += bool(safe) + totals["nodes_without_safe_minimum"] += not safe + require(ids == expected_nodes, "record set differs from radical-one feasible domain") + require(len(expected_nodes) == 692 and totals["frames"] == 3096, "domain count changed") + require(one_extra_safe_spaces == 168, "one-extra-dimension census total") + summary = dict(active_nodes=len(circuit.active), nondegenerate_nodes=nondegenerate, + impossible_nodes=impossible, enumerated_nodes=len(expected_nodes), + **dict(totals), + histogram=[list(key) + [count] for key, count in sorted(histogram.items())]) + require(canonical(certificate["summary"]) == canonical(summary), "summary differs from checked records") + return dict(status="PASS_PARITY_INCIDENT_COMPLETIONS", verified=True, + one_extra_safe_frame_census=one_extra_safe_spaces, **summary) + + +def unique_object(pairs): + result = {} + for key, value in pairs: + require(key not in result, "duplicate JSON key: " + key) + result[key] = value + return result + + +def main(): + parser = argparse.ArgumentParser(description=__doc__) + parser.add_argument("--certificate", type=Path, default=HERE / "CERTIFICATE.json") + args = parser.parse_args() + certificate = json.loads(args.certificate.read_text(encoding="utf-8"), object_pairs_hook=unique_object) + result = check(certificate) + print(json.dumps(result, sort_keys=True)) + + +if __name__ == "__main__": + main() diff --git a/research/parity-incident-completions/produce.py b/research/parity-incident-completions/produce.py new file mode 100644 index 0000000000..dbeb0fe666 --- /dev/null +++ b/research/parity-incident-completions/produce.py @@ -0,0 +1,219 @@ +#!/usr/bin/env python3 +"""Enumerate all minimum sandwiches of the radical-one ParitySide(10) nodes. + +Producer uses row reduction and characteristic vectors. The independent checker +uses literal vector sets and tests residuals, not this module's geometry code. +""" +import argparse +from collections import Counter +import hashlib +import json +from pathlib import Path +import sys + +HERE = Path(__file__).resolve().parent +ROOT = HERE.parents[1] +sys.path.insert(0, str(ROOT / "scripts")) +from experiments.parity_side import ParitySide + +BASE = "407d52ff15b2217b6e22bce44728b4c3907b6515" +H = 10 + + +def require(ok, message): + if not ok: + raise ValueError(message) + + +def dot(a, b): + return (a & b).bit_count() & 1 + + +def basis(rows): + pivots = {} + for value in rows: + for p in sorted(pivots, reverse=True): + if value >> p & 1: + value ^= pivots[p] + if value: + p = value.bit_length() - 1 + for q in pivots: + if pivots[q] >> p & 1: + pivots[q] ^= value + pivots[p] = value + return tuple(pivots[p] for p in sorted(pivots, reverse=True)) + + +def reduce_mod(value, rows): + for row in rows: + if value >> (row.bit_length() - 1) & 1: + value ^= row + return value + + +def gram(rows): + return [sum(dot(x, y) << j for j, y in enumerate(rows)) for x in rows] + + +def characteristic(rows): + n = len(rows) + equations = [g | (dot(v, v) << n) for g, v in zip(gram(rows), rows)] + # Gauss-Jordan on a full-rank n-by-n Gram matrix. + for column in range(n): + pivot = next(i for i in range(column, n) if equations[i] >> column & 1) + equations[column], equations[pivot] = equations[pivot], equations[column] + for i in range(n): + if i != column and equations[i] >> column & 1: + equations[i] ^= equations[column] + result = 0 + for i, row in enumerate(rows): + if equations[i] >> n & 1: + result ^= row + return result + + +def graph_data(circuit): + sources, targets = {}, {n: set() for n in circuit.active} + for n in sorted(circuit.active): + args = circuit.args[n] + sources[n] = (sources[abs(args[0])] | sources[abs(args[1])] + if args else {circuit.inputs[n - 1]}) + for t, n in circuit.outputs.items(): + targets[abs(n)].add(t) + for n in sorted(circuit.active, reverse=True): + if circuit.args[n]: + for child in circuit.args[n]: + targets[abs(child)].update(targets[n]) + return sources, targets + + +def canonical(value): + return json.dumps(value, sort_keys=True, separators=(",", ":")).encode() + + +def produce(): + c = ParitySide(H) + sources, targets = graph_data(c) + records, obstructed, nondegenerate = [], [], 0 + for n in sorted(c.active): + A, B = basis(sorted(sources[n])), basis(sorted(targets[n])) + require(all(dot(a, b) == 0 for a in A for b in B), "nonorthogonal spans") + radical = len(A) - len(basis(gram(A))) + if len(basis(A + B)) < len(A) + len(B): + obstructed.append(n) + continue + if not radical: + nondegenerate += 1 + continue + require(radical == 1, "domain has changed: radical is not one") + candidates = sorted({reduce_mod(z, A) for z in range(1 << H) + if all(not dot(z, b) for b in B)}) + frames = [] + for z in candidates: + U = basis(A + (z,)) + if len(U) != len(A) + 1 or len(basis(gram(U))) != len(U): + continue + w = characteristic(U) + source_bad = [w] if len(U) > 1 and w in sources[n] else [] + complement_w = ((1 << H) - 1) ^ w + target_bad = ([complement_w] if H - len(U) > 1 and + complement_w in targets[n] else []) + frames.append(dict(extra=z, characteristic=w, + source_bad=source_bad, target_bad=target_bad)) + require(frames, "feasible radical-one node has no minimum completion") + record = dict(node=n, source_basis=list(A), target_basis=list(B), + minimum_dimension=len(A) + 1, frames=frames) + if not any(not f["source_bad"] and not f["target_bad"] for f in frames): + repair = None + for frame in frames: + z = frame["extra"] + U = basis(A + (z,)) + for w in sorted({reduce_mod(v, U) for v in range(1 << H) + if all(not dot(v, b) for b in B)}): + W = basis(U + (w,)) + if len(W) != len(A) + 2 or len(basis(gram(W))) != len(W): + continue + cw = characteristic(W) + if (len(W) > 1 and cw in sources[n]) or ( + H - len(W) > 1 and ((1 << H) - 1) ^ cw in targets[n]): + continue + repair = dict(dimension=len(W), extra_basis=[z, w]) + break + if repair is not None: + break + if repair is None and H - len(B) == len(A) + 3: + # In this fixed domain the remaining repairable cases have + # one target line, and their full target-perp is dimension 9. + # Check the characteristic witness rather than assume this. + W = basis(v for v in range(1 << H) if all(not dot(v, b) for b in B)) + require(len(basis(gram(W))) == len(W), "degenerate maximal repair") + cw = characteristic(W) + if not (len(W) > 1 and cw in sources[n]) and not ( + H - len(W) > 1 and ((1 << H) - 1) ^ cw in targets[n]): + extras, current = [], A + for v in W: + if reduce_mod(v, current): + extras.append(v) + current = basis(current + (v,)) + repair = dict(dimension=len(W), extra_basis=extras) + # Null means no larger dimension fits; the minimum-frame census + # then proves impossibility, rather than a failed repair search. + require(repair is not None or H - len(B) == len(A) + 1, + "unresolved node: larger frames remain unchecked: " + str(n)) + record["repair"] = repair + records.append(record) + histogram = Counter() + totals = Counter() + for record in records: + frames = record["frames"] + safe = sum(not f["source_bad"] and not f["target_bad"] for f in frames) + histogram[(record["minimum_dimension"], len(frames), safe)] += 1 + totals["frames"] += len(frames) + totals["safe_frames"] += safe + totals["source_rejected_frames"] += sum(bool(f["source_bad"]) for f in frames) + totals["target_rejected_frames"] += sum(bool(f["target_bad"]) for f in frames) + totals["nodes_with_safe_minimum"] += bool(safe) + totals["nodes_without_safe_minimum"] += not safe + if not safe: + totals["one_extra_dimension_repairs"] += (record["repair"] is not None and + record["repair"]["dimension"] == record["minimum_dimension"] + 1) + totals["two_extra_dimension_repairs"] += (record["repair"] is not None and + record["repair"]["dimension"] == record["minimum_dimension"] + 2) + totals["no_incident_safe_frame"] += record["repair"] is None + dag = dict(inputs=list(c.inputs), + args=[[n, list(c.args[n]) if c.args[n] else None] for n in sorted(c.active)], + outputs=sorted([t, n] for t, n in c.outputs.items())) + pins = {} + for module in list(sys.modules.values()): + filename = getattr(module, "__file__", None) + if filename: + path = Path(filename).resolve() + if path.is_relative_to(ROOT / "scripts") and path.suffix == ".py": + pins[path.relative_to(ROOT).as_posix()] = hashlib.sha256(path.read_bytes()).hexdigest() + return dict(format=1, base_commit=BASE, seed=47, h=H, + source_sha256=dict(sorted(pins.items())), + dag_sha256=hashlib.sha256(canonical(dag)).hexdigest(), + summary=dict(active_nodes=len(c.active), nondegenerate_nodes=nondegenerate, + impossible_nodes=obstructed, enumerated_nodes=len(records), + **dict(totals), + histogram=[list(key) + [count] for key, count in sorted(histogram.items())]), + records=records) + + +if __name__ == "__main__": + parser = argparse.ArgumentParser(description=__doc__) + parser.add_argument("--out", type=Path, required=True) + args = parser.parse_args() + result = produce() + # One complete node record per line makes changes to the finite census + # inspectable without a many-thousand-line indentation-only artifact. + fields = [] + for key, value in sorted(result.items()): + if key == "records": + rendered = "[\n" + ",\n".join(" " + json.dumps(row, sort_keys=True) + for row in value) + "\n ]" + else: + rendered = json.dumps(value, sort_keys=True) + fields.append(" " + json.dumps(key) + ": " + rendered) + args.out.write_text("{\n" + ",\n".join(fields) + "\n}\n", encoding="utf-8", newline="\n") + print(json.dumps(result["summary"], sort_keys=True)) diff --git a/research/parity-incident-completions/test_check.py b/research/parity-incident-completions/test_check.py new file mode 100644 index 0000000000..b49798a96e --- /dev/null +++ b/research/parity-incident-completions/test_check.py @@ -0,0 +1,156 @@ +#!/usr/bin/env python3 +"""Focused counterexample and certificate-corruption tests (also run with -O).""" +import copy +import importlib.util +import json +from pathlib import Path +import unittest + +HERE = Path(__file__).resolve().parent +SPEC = importlib.util.spec_from_file_location("incident_checker", HERE / "check.py") +checker = importlib.util.module_from_spec(SPEC) +SPEC.loader.exec_module(checker) + + +class TinyResidualTests(unittest.TestCase): + def test_pr920_h6_minimum_frames_are_not_incident_safe(self): + g = checker.Geometry(6) + A, B = checker.span([7, 25]), checker.span([11, 19]) + frames = list(g.extensions(A, g.perp(B))) + self.assertEqual([z for z, _ in frames], [3, 35]) + self.assertEqual([g.characteristic(U) for _, U in frames], [7, 25]) + self.assertEqual([g.residual_failures(U, {7, 25}) for _, U in frames], [[7], [25]]) + self.assertEqual([g.residual_failures(g.perp(U), {11, 19}) for _, U in frames], [[], []]) + for (_, U), label, expected in zip(frames, [7, 25], + [{0, 3, 29, 30}, {0, 30, 36, 58}]): + residual = U & g.hyperplanes[label] + self.assertEqual(residual, expected) + self.assertEqual(len(residual), 4) + self.assertTrue(residual.isdisjoint(g.odd)) + + def test_zero_and_nonzero_nonalternating_residuals_are_allowed(self): + g = checker.Geometry(3) + self.assertEqual(g.residual_failures(checker.span([1]), {1}), []) + self.assertEqual(g.residual_failures(checker.span([1, 2]), {1}), []) + self.assertEqual(g.residual_failures(checker.span([3]), {4}), [4]) + + def test_characteristic_target_with_zero_residual_is_allowed(self): + g = checker.Geometry(4) + U = checker.span([1, 2, 4]) + self.assertEqual(g.characteristic(g.perp(U)), 8) + self.assertEqual(g.perp(U) & g.hyperplanes[8], {0}) + self.assertEqual(g.residual_failures(g.perp(U), {8}), []) + + def test_characteristic_in_source_span_but_not_actual_labels_is_allowed(self): + g = checker.Geometry(6) + sources = {7, 11, 19, 28} + U = checker.span([1, 2, 4, 8, 16]) + self.assertEqual(g.characteristic(U), 31) + self.assertIn(31, checker.span(sources)) + self.assertNotIn(31, sources) + self.assertEqual(g.residual_failures(U, sources), []) + self.assertEqual(g.residual_failures(g.perp(U), {32}), []) + + def test_target_only_failure_constructs_actual_alternating_residual(self): + g = checker.Geometry(8) + U = checker.span([1, 2, 4, 8, 16]) + self.assertEqual(g.residual_failures(U, {7, 11, 19, 28}), []) + self.assertEqual(g.residual_failures(g.perp(U), {224}), [224]) + self.assertEqual(g.perp(U) & g.hyperplanes[224], {0, 96, 160, 192}) + + def test_duplicate_json_keys_are_rejected(self): + with self.assertRaisesRegex(ValueError, "duplicate JSON key"): + json.loads('{"node": 1, "node": 2}', object_pairs_hook=checker.unique_object) + + +class CertificateTests(unittest.TestCase): + @classmethod + def setUpClass(cls): + cls.certificate = json.loads((HERE / "CERTIFICATE.json").read_text(encoding="utf-8"), + object_pairs_hook=checker.unique_object) + + def rejects(self, mutate, message=None): + candidate = copy.deepcopy(self.certificate) + mutate(candidate) + with self.assertRaisesRegex(ValueError, message or ".+"): + checker.check(candidate) + + def test_complete_certificate(self): + result = checker.check(self.certificate) + self.assertEqual(result["enumerated_nodes"], 692) + self.assertEqual(result["frames"], 3096) + self.assertEqual(result["one_extra_dimension_repairs"], 12) + self.assertEqual(result["two_extra_dimension_repairs"], 100) + self.assertEqual(result["no_incident_safe_frame"], 20) + self.assertEqual(result["one_extra_safe_frame_census"], 168) + + def test_omitted_record(self): + self.rejects(lambda c: c["records"].pop(0), "missing record") + + def test_duplicated_record(self): + self.rejects(lambda c: c["records"].insert(0, copy.deepcopy(c["records"][0])), + "duplicate or unordered") + + def test_modified_source_hash(self): + filename = next(iter(self.certificate["source_sha256"])) + self.rejects(lambda c: c["source_sha256"].__setitem__(filename, "0" * 64), + "source hash manifest") + + def test_missing_source_pin(self): + filename = next(iter(self.certificate["source_sha256"])) + self.rejects(lambda c: c["source_sha256"].pop(filename), "source hash manifest") + + def test_modified_dag_hash(self): + self.rejects(lambda c: c.__setitem__("dag_sha256", "0" * 64), "DAG hash") + + def test_extra_source_vector(self): + self.rejects(lambda c: c["records"][0]["source_basis"].append(0), "source_basis differs") + + def test_omitted_frame(self): + self.rejects(lambda c: c["records"][0]["frames"].pop(), "frame enumeration") + + def test_duplicated_frame(self): + self.rejects(lambda c: c["records"][0]["frames"].append(copy.deepcopy(c["records"][0]["frames"][0])), + "frame enumeration") + + def test_modified_characteristic(self): + self.rejects(lambda c: c["records"][0]["frames"][0].__setitem__("characteristic", 0), + "frame enumeration") + + def test_modified_source_classification(self): + self.rejects(lambda c: c["records"][0]["frames"][0].__setitem__("source_bad", [7]), + "frame enumeration") + + def test_modified_target_classification(self): + self.rejects(lambda c: c["records"][0]["frames"][0].__setitem__("target_bad", [11]), + "frame enumeration") + + def test_modified_summary(self): + self.rejects(lambda c: c["summary"].__setitem__("safe_frames", 3096), "summary differs") + + def test_invalid_repair_vector(self): + def mutate(c): + r = next(r for r in c["records"] if r.get("repair")) + r["repair"]["extra_basis"][0] = 1024 + self.rejects(mutate, "invalid vector") + + def test_false_impossibility(self): + def mutate(c): + r = next(r for r in c["records"] if r.get("repair")) + r["repair"] = None + self.rejects(mutate, "unproved impossibility") + + def test_signed_kernel_rejects_output_sign_flip(self): + # The checker must verify signed coefficients without relying on the + # constructor's assert-based verify(), which is disabled by -O. + checker.check(self.certificate) + from experiments.parity_side import ParitySide + circuit = ParitySide(10) + target = next(iter(circuit.outputs)) + circuit.outputs[target] *= -1 + with self.assertRaisesRegex(ValueError, "signed kernel entry"): + checker.rebuild_graph(circuit) + + +if __name__ == "__main__": + unittest.main()