independent verification: r420 lattice parity and 480 added incidences

research/r420-lattice-parity/CLAIM.md

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+# sprint claim and single evidence-driven pivot
+
+- track: independent verification/formalization; os-random seed **76**, drawn
+  with `System.Security.Cryptography.RandomNumberGenerator.GetInt32(100)`.
+- base: `66849e1475c930fd7732b1a0067ffee7e1e95f60`.
+- start: 2026-10-13 04:54 utc; first claim: 04:58:34 utc.
+- initial candidates: an independent bit moment enclosure (overlap with #678),
+  a complex wrapper identity (overlap with #702/#712), and selected r420 root
+  geometry by construction-a and exterior products (chosen).
+- selected state inspected: `CONTRIBUTING.md`, the selected certificate, actual
+  #626 construction and maintainer review, current status, review ledger and
+  research/negative-result integration index. the selected conditional saving
+  was #626 with #543, `51110230636731/31250000000000000`.
+- no pinned sprint issue existed at the initial or pre-submission check, so
+  the claim was recorded locally. no reservation issue or comment was posted.
+- expected finish: 2026-10-13 05:54 utc; the final 15 minutes of the hour are
+  reserved for verification and handoff.
+
+## pivot at the overlap refresh
+
+the first package checked all 672 rational frames and odd-characteristic
+incidences. the 05:12 utc refresh found 581 open prs through #762, including
+sagar thakrar's #737 at `b41e73169430641be18c988c659d2855a6a6218a`, which now
+covered that object and method. the unsent initial package was retired; no pr
+was opened for it. this is the one evidence-driven pivot.
+
+final scope: the intrinsic quotient `L/2L` of the selected #543 lattice, the
+120 root-class bijection, and all 672 frames' binary-span root memberships.
+method: construction-a code enumeration, verified integral lattice coordinates,
+all 256 quadratic classes, exact binary spans with integral lift witnesses,
+and a separate pairing-coordinate implementation. root candidates are ordered
+by `(i-76) mod 120`; this seed-selected basis is checked primitive, not assumed.
+
+this is distinct from #737's odd-prime statements. the existing p111 code
+already uses lattice classes for affine rank; that idea retains attribution.
+the new finite coverage is the entire selected frame-incidence census in
+characteristic two, with original-coordinate witnesses. no search for a
+stronger supplier, retired family bound or new kappa was reopened.
+
+## bounded coverage and result
+
+all 120 roots, all 256 quotient classes, all 65,536 polarization pairs, all
+14,400 root pairings, all 672 frames and all 80,640 frame/root memberships are
+checked. stopping rule: complete those fixed domains, with no adaptive search
+or early success. the source-bound primitive-basis nomination is checked before
+any quotient claim is made.
+
+result: the 120 root lines give the 120 quadratic-value-one classes. all frame
+dimensions agree, while 180 dimension-six frames gain two roots apiece and
+120 dimension-seven frames gain one. the 480 extra memberships carry integral
+lifts and rational nonmembership witnesses. this is a scoped incidence
+distinction, not a production defect or a multiplication theorem.
+
+## final local handoff
+
+the 05:26 utc refresh covered 594 open prs through #775. the main head remained
+the recorded base and #737 remained at its inspected head. no matching
+intrinsic-quotient/frame-closure census was found. no pinned coordination issue
+was available, so this completes the local claim instead of posting an update.
+
+the finite certificate and six test groups pass, including optimized relocated
+execution, all 480 original-coordinate lifts, an independent pairing-coordinate
+census, and failure controls. the predecessor 660-column replay and selected
+source binding pass separately. see `VALIDATION.md` for limits. the final
+submission is one focused independent-verification pr; its link is retained
+in the accompanying handoff rather than a self-referential source edit.
+
+contributor: gojo (@6ojo), with substantial openai codex assistance. no
+delegated or independent human review is claimed.