Audit common24 P112 with the complete colored completion matching

research/p112-common24-graph-audit/PROOF.md

@@ -0,0 +1,172 @@
+# Independent graph audit of the common24 application
+
+This audits PR3960 at `7e8d98e351ac153d81943218bf168dc8548b1cea`.
+It finds no mathematical defect in the stated theorem or the two transfers.
+The new finite result is stronger structure of the **complete colored
+completion graph**: it is a matching with 2,884 edges and 448 isolated vertices.
+This gives an alternative proof of the common-cut coverage step. It does not
+establish independent human review or formal-kernel verification.
+
+## 1. Exactly which finite objects are checked
+
+Write R for the 120 oriented doubled E8 root lines, F for the eight frame
+lines `2e_0 +/- 2e_1,...,2e_6 +/- 2e_7`, and P=R\F. The live table and simple
+system are canonical-main inputs. A fresh rational Gauss-Jordan calculation
+recovers integral simple coordinates and the original labels
+`ell(r)=(1,c(r) mod 2)` in F2^9. The verifier checks the full E8 shell, original
+row-label binding, distinct labels and live Gram matrix `112 I_8`. No
+predecessor executable is imported.
+
+A two-dimensional binary annihilator has exactly three nonzero elements
+`a,b,a+b`. Sort them. Its unique first two satisfy `a<b<a xor b`. Conversely
+every such pair is independent and generates this two-space. Consequently the
+nested nonzero-functional pair enumeration covers each annihilator exactly
+once, hence each seven-space exactly once by the nondegenerate standard
+pairing. It produces `511*510/6=43,435` seven-spaces. The verifier computes the
+full-root trace mask of each. Their largest trace has 36 lines. Lower-rank
+residual spans extend to seven-spaces.
+
+For each of the 120 root normals a and each seven-space V compute
+
+    C(a,V) = {r in P : r dot a != 0 and ell(r) not in V}.
+
+All 5,212,200 pairs are visited. Retain every record with cut size at most24.
+There are 6,216 such records, and exactly 6,216 distinct `(a,C)` vertices;
+no inclusion-minimality reduction or representative discard is used. The
+complete sorted record digest and the full-flat multiplicity digest match
+PR3960. `REVIEWED.json` binds that comparison to the exact published target
+and records hashes of the reviewed source/proof files. It is review provenance;
+the replay does not fetch or import pending PR3960 files.
+
+## 2. All-cut coverage by a matching
+
+Color a vertex `(a,C)` by its projective root normal a. Join two vertices
+of **different colors** exactly when their cuts have union size at most24.
+The checker compares every unordered pair of the 6,216 vertices. Its result:
+
+| Vertex degree | Number of vertices |
+|---:|---:|
+| 0 | 448 |
+| 1 | 5,768 |
+
+There are therefore 2,884 disjoint edges and no triangle. The checker also
+independently checks triangle absence through adjacency-bitset intersections.
+The matching property concerns the entire colored graph, including frame
+normals and all cut sizes, rather than just the size21 subgraph used in PR3960.
+
+Fix ONE S subset P with size at most24. Suppose three distinct root normals
+qualify: their residual supports have binary rank at most7. Extend each
+residual span to a seven-space V_i. Its complete cut C(a_i,V_i) is contained
+in S, so the corresponding retained vertices exist. Their three different
+colors and pairwise unions contained in S force a triangle. This contradicts
+the graph result. Thus at most two root-normal directions qualify.
+
+Only this implication is used. A triangle in the graph need not have total
+three-cut union at most24; pairwise compatibility is a relaxation. The controls
+include three singleton cuts with cutoff2: they form a triangle although
+their total union has size3. Triangle **absence** remains a sufficient proof.
+No assumption about unused sites of S, the relative cut sizes, or separate
+cuts for different directions enters the implication.
+
+An edge supplies an attaining pair: its union is an actual cut of at most24,
+and the two root normals are independent. RESULT.json lists an explicit pair
+and the actual cut labels. This corroborates attainment of two directions;
+it does not prove sharpness of the permitted cut size24.
+
+## 3. Audited characteristic-zero and distance implications
+
+Here is the analytic bridge checked against PR3960, stated explicitly because
+the graph enumerates rational root normals while the theorem quantifies over
+all characteristic-zero fields K.
+
+The verifier checks that the eight simple reflections permute the 240 signed
+roots, and that the simple roots span. Hence the generated reflection group
+is finite: its action on the roots is faithful. It also checks a rational
+rho with `rho dot alpha_i=1`. For any real v, maximize `rho dot g(v)` over this
+finite group. A negative simple pairing would be increased by its reflection,
+so a maximizing vector lies in the closed chamber. The checked uniform signs
+of each root's simple coordinates imply that its perpendicular roots are
+exactly a simple face. The checked maximum face size on at most six nodes is
+36, and the seven-node face counts are `42,28,22,14,16,23,37,63`; the 63-line
+face has a root as its normal.
+
+For nonzero h in K^8 with live support at most60, let Z be the rational span
+of the live roots perpendicular to h. It cannot have rank8, because a nonzero
+rational 8x8 minor stays nonzero over K. If its rank were at most6, a generic
+real vector in Z-perp would avoid every root hyperplane not containing that
+space. Its perpendicular roots would then be exactly the full roots lying
+in Z, whose span is Z. Chamber transport would give a face of rank at most6,
+contradicting the at least52 perpendicular live roots. Thus Z has rational
+rank7. Its rational one-dimensional kernel extends to its kernel over K
+by Gaussian elimination, so h is a K-multiple of a rational vector. The
+corresponding real vector has at least52 perpendicular roots. Only the
+63-line seven-node face can occur, making its normal a root; inverse
+reflection transport preserves roots. The frame itself need not be preserved.
+No embedding of K in R or C is used.
+
+For a qualifying common24 h, the complete seven-space trace bound gives
+`|Off(h)| <= 24+36=60`, so this bridge applies. Every such direction is one
+of the 120 rational root lines already covered by the graph. They remain
+distinct projective lines over K, since rational rank is preserved. At most
+two projective lines imply span dimension at most2.
+
+The same short-support bridge explicitly supplies the distance premise in
+the caller transfer: if a nonzero codeword had support at most52, it would
+be a root-normal word. Direct evaluation gives 53 live sites for each of
+the112 live normals and56 for each of the eight frame normals, a contradiction.
+Live normals attain53. Evaluation is injective over K because the live Gram
+matrix is `112 I_8`, invertible in characteristic zero. Thus the code distance
+is exactly53. This is an implication of the audited argument, not an extra
+unproved distance assumption.
+
+## 4. Coefficients and the two quotient transfers
+
+Let eleven pieces p_1,...,p_11 be collectively K-linearly independent, T their
+span, C the evaluation code, and D=T intersect C with dim D>=5. Assume the
+binary ranks of their actual support labels are bounded respectively by
+
+    2,2,2,2,3,5,7,7,7,7,7
+    2,2,2,2,4,4,7,7,7,7,7.
+
+Ranks2,3,4 give support caps2,4,8 because the labels lie in a leading-one
+affine slice. The rank5 cap12 is **inherited from Nick Petalas's global
+capacity theorem**, independently checked in PR3951; this audit does not
+repeat its 3,309,747-space census. The first caller remains conditional on
+that explicitly named dependency. Its six small pieces have capacity sum
+`4*2+4+12=24`; the second has `4*2+8+8=24` and needs no rank5 capacity.
+The original TWOSIDE status and necessary-filter membership are inherited
+from PR3960 and PR3154/3282, rather than recounted as new inventory results.
+
+Let A=span(p_1,...,p_6) and S the ONE actual union of their supports. Then
+|S|<=24. Every element of A is supported in S, so distance53 gives A intersect
+C=0. Since A has dimension6, its quotient image in T/D is independent. On
+the other hand dim(T/D)<=11-5=6. Hence its dimension is exactly6, A maps
+isomorphically to T/D, and dim D=5.
+
+For each j=7,...,11 there is a unique a_j in A with c_j=p_j-a_j in D.
+If sum(lambda_j*c_j)=0, expansion in the original independent p_i basis has
+coefficient lambda_j on p_j for every j>=7. Thus all lambda_j vanish: the
+five c_j are independent, regardless of the arbitrary coefficients of a_j,
+intersections between supports, or cancellations inside S. Evaluation
+injectivity gives five independent normals h_j. Outside S, c_j equals p_j,
+so its residual label rank is at most7. The common24 bound limits their span
+to dimension2, contradiction. This proves exactly these two transfers.
+
+The executable receipt records graph facts and small caller arithmetic.
+It does not turn this written coefficient argument into formal verification.
+
+## 5. Scope, credit and remaining obligations
+
+This is a fresh computational and written audit in the same donor/assistant
+context as PR3960. It adds the complete matching certificate and a different
+finite-space enumerator. It does not claim independent human acceptance,
+a full C5 theorem, the full (5,6) cell, a decoder floor, supplier admission,
+or a change to the selected kappa. The field proof rechecks and explains
+PR3951/3960's argument; its underlying idea is credited there, not claimed
+as new here.
+
+Credit Nick Petalas for global capacities and the E8/root-shortening/shared-
+cut predecessors, DreamingOfClouds for PR3154/TWOSIDE/C5, David Arias for the
+quadratic model, joaqcoo and the retained E8/P112 lineage. The same donor's
+PR3931, PR3951 and PR3960 retain their respective earlier results. New audit:
+Chris R / Clear1nsight with substantial OpenAI Codex assistance, Apache-2.0.

research/p112-common24-graph-audit/README.md

@@ -0,0 +1,77 @@
+# Complete colored-graph audit of the common24 P112 theorem
+
+An independent implementation audit of [PR3960](https://github.com/CrocSwap/integer-mult-bounds/pull/3960)
+at `7e8d98e351ac153d81943218bf168dc8548b1cea` finds no defect in its common24
+theorem or two literal caller transfers. The additional finite result is:
+**the entire different-normal completion graph is a matching**, with2,884
+edges and448 isolated vertices among6,216 vertices. A hypothetical three
+qualifying directions under one24-site cut would force a triangle. This
+replaces the target's specialized small-cut/candidate split with one complete
+graph certificate.
+
+The fresh verifier generates each two-dimensional annihilator by its unique
+sorted pair `a<b<a xor b`, rather than either target implementation's RREF
+or affine-space traversal. It checks all43,435 binary seven-spaces and all
+5,212,200 root-normal/space pairs. Both complete target inventory digests
+agree. Root labels are independently solved over rational numbers from the
+canonical table and simple system. No predecessor executable is imported.
+[PROOF.md](PROOF.md) gives the coverage proof, audits the characteristic-zero
+bridge, explicitly derives distance53, and checks coefficient independence
+in the quotient transfer. [RESULT.json](RESULT.json) contains the graph receipt
+and a two-direction witness. [REVIEWED.json](REVIEWED.json) identifies exact
+target bytes; it is provenance, with no pending-PR runtime dependency.
+
+The transfer concerns exactly `2^4,3,5,7^5` and `2^4,4^2,7^5`: eleven independent
+pieces, required code intersection dimension5, quotient dimension6, and one
+common support union of size at most24. The first transfer uses Nick Petalas's
+inherited rank5 support cap12, previously audited in PR3951. This audit does
+not repeat that capacity census. Original filter membership is inherited
+from PR3960/PR3154/3282. Caller arithmetic is executable; the general field
+and coefficient implications are written mathematics.
+
+## Reproduce
+
+From the repository root, using Python3.12 and its standard library:
+
+```sh
+python3.12 -I -Werror -B scripts/research_package.py --package research/p112-common24-graph-audit
+```
+
+The two registered optimized commands take about20seconds total on the
+contributor's workstation. They verify the graph and meaningful adversarial
+controls, including a triangle that does not correspond to a common cut,
+repeated-color suppression, complete small annihilator orbits, and actual
+CLI rejection of changed result, missing edges and changed input bytes.
+Direct commands also work in a Git-free declared-input export.
+
+## Scope and status
+
+This is a fresh computational and written audit within the same donor/AI
+assistant context as the target. It is not independent human acceptance or
+formal verification. No full C5 result, closure of the whole(5,6) cell,
+cut-threshold sharpness, decoder floor, supplier admission, full multiplication
+theorem or changed selected kappa is claimed. Frozen predecessors remain
+unchanged. The exact assigned acceptance gap and current overlap coverage
+are recorded in [TASK.json](TASK.json).
+
+## Open next questions
+
+- Review the chamber, rational-kernel and quotient proof independently; start
+  with PROOF.md sections3-4. Finite certificate success does not itself accept
+  those written implications.
+- A different literal caller needs a common30 bound or additional coefficient
+  structure. Follow the separate current investigation after overlap checking;
+  the common24 matching is not permission to extend its cutoff.
+
+Structured first steps, relevance and blockers are in TASK.json.
+
+## Credit
+
+Nick Petalas retains credit for global capacities, root geometry, shortening
+and shared-cut predecessors. DreamingOfClouds supplied PR3154/TWOSIDE/C5;
+David Arias supplied the quadratic model; joaqcoo and the retained E8/P112
+lineage retain their credit. This donor's PR3931, PR3951 and PR3960 supply
+the earlier common-cut and dependency results. The new contribution is the
+fresh complete-graph certificate and explicit application audit.
+
+Chris R / Clear1nsight, with substantial OpenAI Codex assistance. Apache-2.0.

research/p112-common24-graph-audit/RESULT.json

@@ -0,0 +1,109 @@
+{
+  "seven_spaces": 43435,
+  "flat_multiset_sha256": "5009cb750c3a5c0a1f71b0331892b3b34a4fb8f1c025a69c109516428d22efb9",
+  "completion_records": 6216,
+  "completion_records_sha256": "9a2dd7d5d850c3f4194f5c10bbb19a94fea41d1dc192e99a4e7096425f028ae7",
+  "colored_vertices": 6216,
+  "vertices_sha256": "b7eeea54afce1b53d9489bcd6d9c0d6aa33609e5c0b3b7e2550acd61b5539c7d",
+  "different_color_edges": 2884,
+  "edge_sha256": "fbc252ef0b25bf31107a9d1d023164e15ad50cd04aecf358a40be25dcac121ea",
+  "degree_histogram": {
+    "0": 448,
+    "1": 5768
+  },
+  "triangles": 0,
+  "witness": {
+    "normal_indices": [
+      0,
+      20
+    ],
+    "normals": [
+      [
+        0,
+        0,
+        0,
+        0,
+        0,
+        0,
+        2,
+        -2
+      ],
+      [
+        0,
+        0,
+        2,
+        -2,
+        0,
+        0,
+        0,
+        0
+      ]
+    ],
+    "cut_labels": [
+      449,
+      361,
+      357,
+      461,
+      481,
+      329,
+      325,
+      493,
+      207,
+      103,
+      79,
+      231,
+      239,
+      71,
+      111,
+      199,
+      63,
+      151,
+      179,
+      27,
+      31,
+      183,
+      147,
+      59
+    ]
+  },
+  "caller_arithmetic": [
+    {
+      "weights": [
+        2,
+        2,
+        2,
+        2,
+        3,
+        5,
+        7,
+        7,
+        7,
+        7,
+        7
+      ],
+      "required_intersection": 5,
+      "quotient_bound": 6,
+      "basis_capacity_sum": 24,
+      "independent_residual_codewords": 5
+    },
+    {
+      "weights": [
+        2,
+        2,
+        2,
+        2,
+        4,
+        4,
+        7,
+        7,
+        7,
+        7,
+        7
+      ],
+      "required_intersection": 5,
+      "quotient_bound": 6,
+      "basis_capacity_sum": 24,
+      "independent_residual_codewords": 5
+    }
+  ]
+}

research/p112-common24-graph-audit/REVIEWED.json

@@ -0,0 +1,18 @@
+{
+  "pull_request": "https://github.com/CrocSwap/integer-mult-bounds/pull/3960",
+  "head": "7e8d98e351ac153d81943218bf168dc8548b1cea",
+  "scientific_commit": "fd3c969f4d69cfd53e6057451a4b287c6263fc7f",
+  "files": {
+    "PROOF.md": "e6a1f48219f9b3978838dfb6cd2102ce10ab14fd517d39b3c560dcf388db0e6c",
+    "RESULT.json": "9e885efa849ac1e3358ee8bc69eaae347caccf40fd9e4a515a79672c4ea6df84",
+    "verify.py": "20691cab23faddd3a33607190ba9d7ce4385ba5f6746354dee6497396b8a962e",
+    "independent.py": "a1235127f7d5e32958810123e57cb711272d38dd44583d72858117bdf1de5e1c",
+    "model.py": "f5c2041b552a65c1eb709d17abc6ea0447980984a08d452e049493728589fc15"
+  },
+  "compared_fields": {
+    "flat_inventory_digest": "5009cb750c3a5c0a1f71b0331892b3b34a4fb8f1c025a69c109516428d22efb9",
+    "critical_record_digest": "9a2dd7d5d850c3f4194f5c10bbb19a94fea41d1dc192e99a4e7096425f028ae7",
+    "critical_records": 6216
+  },
+  "scope": "Published result bytes are review provenance; executable audit uses only canonical-main table and simple-system inputs. No pending Git history is required."
+}