Research audit: F4 alternating trace is not by itself a phase-compilation obstruction (no new kappa)

research/f4-alternating-interface/CLAIM.md

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+# Local structural-interface claim
+
+OS-random seed 42; structural decoder track (40-49). Date: 2026-10-13.
+Base: 7ee02b7c96d85f22b38d28bd7ed49bb8c1aeb261.
+Branch: penguin/scoped-math-audit. No pinned coordination issue was returned;
+the initial open-issue list contained only #650 (Discussions).
+
+Three candidates considered:
+1. Strengthen extension-ring copy floors: #770/#798/#801/#814 already cover it.
+2. Finish historical Hurwitz asymmetric search: newer #785 addresses that model.
+3. Audit F4 alternating-trace exclusion against the existing binary phase
+   normal form. Chosen: exact radical census, complete-square kernels and
+   an ordinary-dot-product embedded block, without constructing a supplier.
+
+Inspected current selected record, ledger, negative results and open PR titles.
+#41 supplies alternating pair-block predecessor context; no code imported.
+The completed enumeration includes all Hermitian F4 matrices through dimension
+3 and all nonsingular alternating binary matrices of dimensions 2 and 4 with
+all even refinements. No stochastic optimizer or solver timeout is involved.
+
+## Completed handoff
+
+Submitted research-note PR #831: https://github.com/CrocSwap/integer-mult-bounds/pull/831
+452 exact phase cases, 114752 kernel entries and four rejected controls.
+No new kappa. Normal and optimized receipts agree; source pins use Git blobs.

research/f4-alternating-interface/NOTICE.md

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+# Attribution
+
+Prepared for Maximus Putnam / @Maximooch with substantial OpenAI assistant
+(Penguin) assistance in mathematics, implementation, verification and writing.
+Apache-2.0. No predecessor implementation is copied.
+
+Joel Pulikkan / @GamingPuzzled (#620): historical extension-ring report.
+Repository endpoint-gauge and community-final audit authors: the existing
+alternating residual compiler, pinned in SOURCES.json; not claimed as new.
+Alternating pair-block predecessor #41: hipotures at 03bde70c0c2a453f1ceae20c619b602841405467.
+Copy-floor proposals #770/#785/#798/#801/#814 inspected for overlap; no
+supersession claimed. Finite Python checking is not a formal theorem proof.

research/f4-alternating-interface/PROOF.md

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+# Exact interface statement
+
+## 1. The trace obstruction is real, but narrower than compilation
+
+Put F4=F2[t]/(t^2+t+1), conjugation a -> a^2, and Tr(a)=a+a^2.
+For a Hermitian matrix H define
+
+    h(x,y) = conjugate(x)^T H y,
+    B(x,y) = Tr(h(x,y)).
+
+B is symmetric F2-bilinear. Hermitian symmetry forces h(x,x) to belong to
+F2, whose field trace is zero. Thus B(x,x)=0 for every x: B is alternating.
+No positive-dimensional restriction can have an orthonormal basis for B.
+
+The radical of B is exactly the radical of h, considered as an F2 space.
+One inclusion is immediate. For the other, suppose h(x,y)=a != 0. By
+scaling y by c, h(x,cy)=ca can take any field value, including t with
+Tr(t)=1. Then B(x,cy)!=0. Consequently a nonsingular Hermitian H of
+F4 dimension n has a nonsingular alternating trace form of F2 rank 2n.
+More generally its binary rank is twice its F4 rank. The checker independently
+counts H's nullspace by enumerating all field vectors, then compares it with
+packed binary elimination of B. It includes singular H rather than assuming
+all frames are nonsingular.
+
+## 2. An alternating block admits a valid fourth-root quadratic phase
+
+Let G be a symmetric nonsingular alternating binary r-by-r matrix. For
+any linear mask ell define
+
+    q(x) = 2 sum_{i<j} G_ij x_i x_j + 2 sum_i ell_i x_i (mod 4).
+
+A direct expansion gives
+
+    q(x+y) - q(x) - q(y) = 2 x^T G y (mod 4).
+
+All additions in the argument are XOR. This is a quadratic refinement;
+its values are even, but that does NOT make its polar form zero. In
+particular q(a,b)=2ab refines the hyperbolic matrix [[0,1],[1,0]].
+
+Let S=sum_z i^q(z) and let W(t)=sum_z i^q(z) (-1)^(z^T t).
+Shifting the summation variable by v gives the exact complete-square identity
+
+    W(Gv) = S i^(-q(v)).
+
+The Fourier multiplier kernel is therefore
+
+    K(x,y) = 2^-r W(G(x+y))
+           = 2^-r S i^(-q(x)) (-1)^(x^T G y) i^(-q(y)).
+
+This is the same normal-form identity as the pinned endpoint note in a
+different coordinate convention: that note starts with polarization G^-1
+and hence uses q(Gx). Here q has polarization G, so it uses q(x).
+There is no assumption of an orthonormal basis in either formula.
+
+Every nonsingular alternating form splits into hyperbolic pairs: choose
+nonzero u, choose v with B(u,v)=1, and split off their nondegenerate span
+by orthogonal projection. Repeat on its complement. On one pair a
+refinement is 2(ab+alpha a+beta b); direct summation gives
+
+    sum_{a,b} (-1)^(ab+alpha a+beta b) = 2 (-1)^(alpha beta).
+
+Hence r is even, |S|^2=2^r, and S=+/-2^(r/2).
+With b=(1-i)/2, S b^r is a fourth root of unity. The scaled Walsh kernel
+is a binary permutation of H0^tensor-r, where
+
+    H0 = (1/2) [[1,1],[1,-1]] = b diag(1,i) C diag(1,i),
+    C = (i I + X)/(1+i).
+
+Thus the normal form uses one rank-r C child, binary address changes,
+diagonal fourth-root phases and a scalar fourth-root phase, exactly as
+in the repository endpoint note. This inherits that compiler's physical
+and analytic conditions; it is not an independent all-size implementation.
+The child itself retains its ordinary precision bill; no free arithmetic,
+free wrappers, free copies or reduced recurrence cost is asserted.
+
+## 3. A concrete legal ambient block
+
+In F2^3 with its ordinary dot product, take u=110 and v=011. Both have
+even norm and u dot v=1. Their span is nondegenerate, with hyperbolic Gram
+matrix G. The ambient weight refinement is
+
+    q(a,b)=weight(a*u+b*v) mod 4 = 2a+2b+2ab mod 4.
+
+Its values are [0,2,2,2], Gauss sum -2, and S b^2=i. This supplies an
+actual even-weight ambient residual block, not just an abstract form.
+Its orthogonal complement is the nondegenerate line 111. The checker
+executes the complete kernel identity on this embedded block separately.
+
+## 4. Coverage and rejected controls
+
+The field census includes every Hermitian matrix in dimensions 1,2,3
+(2,16,512 matrices respectively), with all field-vector nullspaces. The
+phase census includes all symmetric alternating nonsingular matrices in
+binary dimensions 2 and 4 and every even-valued quadratic refinement
+(4 and 448 cases respectively). Every polarization pair and kernel entry
+is checked, as is the exact Gauss magnitude and fourth-root normalization.
+
+Controls reject: a changed phase that violates polarization; a singular
+Gram input; a perturbed Gauss scalar that spoils the kernel; and the
+replacement of the Hermitian product by an unconjugated product. For the
+last, x=t has Tr(conjugate(t)*t)=0 but Tr(t*t)=1. All checking conditions
+remain active under `python -O` (no implementation assertions).
+
+## What this does not settle
+
+The historical report was discussing a particular orthonormal phase
+calculus. Its statement remains an obstruction to that narrow method.
+The added finding is that it cannot be used as a general exclusion of
+alternating-compatible compilers already documented elsewhere in the
+repository. There is no contradiction in the trace computation itself.
+
+Degenerate trace restrictions remain a problem: nonsingular full H does
+not imply every support or intersection is nondegenerate. A supplier also
+requires fitting labels, source/read incidence, nested frames, paid adapters,
+restorable auxiliaries, physical owners and fresh composition. None is
+supplied here. Existing copy-floor results are not invalidated by this audit.

research/f4-alternating-interface/README.md

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+# F4 Hermitian trace: alternating is not automatically uncompileable
+
+## Finding
+
+The alternating-trace observation in Joel Pulikkan's historical
+`research/e8-decoder-floors-and-cold14-notes/s1-extension-rings/REPORT.md`
+is correct. Its inference that this alone rules out the complex-side phase
+calculus needs an interface qualification: alternating nondegenerate binary
+blocks do not have an orthonormal basis, but the repository's later
+`notes/endpoint-gauge-complex.tex` explicitly compiles them by quadratic
+phases, binary address changes and a rank-r child.
+
+This package provides an elementary proof and a fresh exact arithmetic audit
+of that distinction. **It does not construct an Eisenstein supplier or claim
+any new kappa.** The source report is historical and is left unchanged.
+
+## Reproduce from the repository root
+
+```sh
+python -B research/f4-alternating-interface/verify.py
+python -O -B research/f4-alternating-interface/verify.py
+```
+
+Both runs print the same JSON receipt (including the source digest):
+452 quadratic-phase cases, 114,752 kernel entries, four rejected negative
+controls, an explicit ordinary-dot-product embedding of a hyperbolic block,
+and this complete Hermitian trace-rank census:
+
+| F4 dimension | number of matrices by binary trace rank |
+| --- | --- |
+| 1 | rank 0: 1; rank 2: 1 |
+| 2 | rank 0: 1; rank 2: 5; rank 4: 10 |
+| 3 | rank 0: 1; rank 2: 21; rank 4: 210; rank 6: 280 |
+
+The checker uses only Python's standard library, integer Gaussian pairs,
+packed F2 vectors, and explicit F4 polynomial multiplication. It imports no
+repository implementation. `--output NEW.json` writes an exclusive-create
+receipt and never replaces existing evidence. Default runs write nothing.
+
+## Scope and handoff
+
+`PROOF.md` gives the general trace-radical argument and complete-square
+identity; finite enumeration checks the specified dimensions, not all sizes.
+No proof assistant was used. In particular, this does not establish that
+all supplier frames are nondegenerate, that their intersections remain so,
+that source-label containment is preserved, or that adapters and physical
+copies have an affordable bill. These obligations must precede any claimed
+Eisenstein construction. Supporting alternating blocks is necessary here,
+not sufficient for a useful decoder.
+
+The selected certificate, suppliers, upstream sources and historical reports
+are unchanged. Full `make verify` and the selected cold replay were not run:
+this isolated note modifies no production code or selected construction.
+See `CLAIM.md`, `SOURCES.json`, `RESULT.json` and `VALIDATION.md` for exact pins,
+selection history and executed checks.
+
+Prepared for Maximus Putnam / @Maximooch with substantial OpenAI assistant
+(Penguin) assistance in mathematics, implementation, checking and writing.
+Apache-2.0, following the repository contribution license.

research/f4-alternating-interface/RESULT.json

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+{
+  "embedded_hyperbolic_block": true,
+  "hermitian_trace_rank_census": {
+    "1": {
+      "0": 1,
+      "2": 1
+    },
+    "2": {
+      "0": 1,
+      "2": 5,
+      "4": 10
+    },
+    "3": {
+      "0": 1,
+      "2": 21,
+      "4": 210,
+      "6": 280
+    }
+  },
+  "kernel_entries": 114752,
+  "negative_controls": 4,
+  "phase_cases": 452,
+  "scope": "finite exact interface audit, not a multiplication theorem",
+  "source_sha256": "7d5d7350aa3e41806aff55165c23e779bfbfff4513f61cd98047695f0e86b28d"
+}

research/f4-alternating-interface/SOURCES.json

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+{
+  "base_commit": "7ee02b7c96d85f22b38d28bd7ed49bb8c1aeb261",
+  "sources": {
+    "research/e8-decoder-floors-and-cold14-notes/s1-extension-rings/REPORT.md": "e8de809976a1253087e9058d346dfcc7b098f6481c64f1569aa3072d27345e02",
+    "notes/endpoint-gauge-complex.tex": "dfee33b0281e5551bb6927f757ff0240abade60ae1800f715afd88853ae0191e",
+    "docs/research/community-final-audit.md": "8233d348873fe5168014fa49888a6a1da69ade6fa4fc2541f85a1264c1d6fc18",
+    "certificates/selected-result.json": "8f2bfbf11609194654d4da145866223d4c546c5418727cae056bfb9968cabcac"
+  },
+  "representation": "Committed Git blobs (LF), not platform checkout line endings"
+}

research/f4-alternating-interface/VALIDATION.md

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+# Executed validation (2026-10-13)
+
+CPython 3.11 on Windows; standard library only.
+
+- Normal `python -B verify.py` completed: 452 phase cases, 114,752 entries.
+- Optimized `python -O -B verify.py` completed with identical JSON.
+- All four internal rejection controls passed in both modes.
+- Embedded ambient hyperbolic block passed separately.
+- Independent enumeration of Hermitian nullspaces matches trace-form ranks.
+- Copied verifier alone to a transient directory within math: identical receipt.
+- Existing `--output RESULT.json` rejected; retained receipt unchanged.
+- All four consulted-source SHA-256 pins recomputed and matched.
+- `git diff --check` passed. Selected certificate and production files unchanged.
+
+No full make verify, selected cold replay, formal proof assistant or remote
+CI is claimed. Repository-wide validation would not strengthen this isolated
+finite interface result. Source-code and mathematical review remain welcome.