Classify sharp P104 frame-pair spectra and extend them to Hermitian metrics

research/p104-selected-metric-orbit/HERMITIAN-PROOF.md

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+# The sharp constants also hold for common Hermitian metrics
+
+The real-symmetric sharp constants in [ORBIT-PROOF.md](ORBIT-PROOF.md) extend
+unchanged to **all complex Hermitian common metrics** on the original real
+root vectors. Equality has no imaginary skew component. This broadens the
+permitted ambient common maps; it still says nothing about independent port
+motions, normalized correlations, arbitrary masked matrices or complete
+multiplier costs. **No new κ.**
+
+Let U be the full 120 unit root lines, with one real unit vector chosen for
+each line. Let F,G be any disjoint orthonormal 8-frames, and put
+`P=U\(F∪G)`. For a Hermitian matrix M define
+
+\[
+ E_P(M)=\sum_{\{u,v\}\subset P,\ u\perp v}|u^*Mv|^2.
+\]
+
+All sums count unordered distinct pairs once. Write uniquely `M=S+iK`, where
+S is real symmetric and K is real skew-symmetric. Because u,v are real,
+
+\[
+ E_P(M)=E_P(S)+E_P(K),\qquad
+ \left\|M-\frac{\operatorname{tr}M}8I\right\|_F^2
+ =\left\|S-\frac{\operatorname{tr}S}8I\right\|_F^2+\|K\|_F^2.
+\]
+
+Here `E_P(K)` uses `(uᵀKv)²`, which is nonnegative and sign-independent on
+unordered pairs even though K is skew. The trace of K is zero. The complex
+Frobenius norm is the sum of squared absolute values of the entries.
+
+## Two exactly checked root identities
+
+The root coordinates in geometry.py satisfy
+
+\[
+ \sum_{u\in U}uu^{\mathsf T}=15I,\qquad
+ \sum_{v\in U:\ v\perp u}vv^{\mathsf T}=9(I-uu^{\mathsf T}).
+\]
+
+These are finite identities, not isotropy assumptions. For `c(u)=2u`,
+moments.py checks the equivalent integer equations
+
+\[
+ \sum c(u)c(u)^{\mathsf T}=60I,\qquad
+ \sum_{v\perp u}c(v)c(v)^{\mathsf T}=36I-9c(u)c(u)^{\mathsf T}
+\]
+
+for all 120 roots. Each star contains 63 roots. Both identities are
+orthogonally invariant and independent of the representative signs, so they
+apply to any first frame after the previously checked isometry reduction.
+
+## Exact skew-energy accounting
+
+Skew symmetry gives `uᵀKu=0`. For a fixed root u the incident-edge energy is
+therefore
+
+\[
+ \sum_{v\perp u}(u^{\mathsf T}Kv)^2
+ =9\,u^{\mathsf T}KK^{\mathsf T}u.
+\]
+
+Summing over all 120 roots and dividing by two gives
+
+\[
+ E_U(K)=\frac92\operatorname{tr}\left(KK^{\mathsf T}
+                    \sum_{u\in U}uu^{\mathsf T}\right)
+       =\frac{135}2\|K\|_F^2.
+\]
+
+For any orthonormal frame F, summing its eight incident-star energies gives
+`9||K||²_F`. The energy on pairs internal to F is `||K||²_F/2`, because in
+that orthonormal basis the skew matrix has zero diagonal and opposite entries
+at `(i,j)` and `(j,i)`. The sum over the eight stars counts internal edges
+twice; hence the energy of edges having at least one endpoint in F is exactly
+`(17/2)||K||²_F`.
+
+Remove the edges incident to F and those incident to G. The intersection of
+these two edge sets consists exactly of the orthogonal cross pairs between
+F and G, since the frames are disjoint. Inclusion-exclusion yields the exact
+identity
+
+\[
+ E_P(K)=\frac{101}2\|K\|_F^2+
+ \underbrace{\sum_{u\in F,\ v\in G,\ u\perp v}(u^{\mathsf T}Kv)^2}_{X_{FG}(K)}
+ \ \ge\ \frac{101}2\|K\|_F^2.
+\]
+
+There is no missing restoration or boundary term in this *geometric* edge
+identity; it makes no algorithmic cost claim. The bound 101/2 need not itself
+be the sharp skew constant for a given orbit, and no such sharpness is claimed.
+
+## Hermitian inequality and equality
+
+Let `c_P` be 37 or 75/2 as determined by the real-symmetric spectrum. Combining
+the exact splitting with the skew lower bound gives
+
+\[
+ E_P(M)\ge c_P\left\|M-\frac{\operatorname{tr}M}8I\right\|_F^2
+       +\left(\frac{101}2-c_P\right)\|K\|_F^2.
+\]
+
+Since `101/2>75/2≥c_P`, equality in the sharp `c_P` inequality forces `K=0`.
+It then occurs exactly in the real-symmetric equality space already certified.
+The real attainers belong to the Hermitian domain, so the constants remain
+sharp even when restricting to positive definite metrics. The trace-free
+Hermitian equality dimensions are therefore 6,1,6 for the three types.
+
+For any complex common linear map T, its positive semidefinite metric
+`M=T* T` is Hermitian, so the theorem applies to the transformed root inner
+products. It does not assume T is real. This does not turn the metric
+inequality into a universal obstruction for independent decoders or imply
+that any selected native construction implements such a map.
+
+## Verification and credit
+
+verify_orbits.py executes every root second-moment identity in normal and
+optimized Python. Two exact-reason negative controls alter a tight-frame
+coefficient and omit one root from a star; both must be rejected. All three
+sharp symmetric constants and their complete equality spaces are verified
+separately by the orbit spectral certificates.
+
+The lead proposed the skew-sector mechanism; worker A independently derived
+the incident-frame accounting and implemented the exact moment checks. Credit
+the full E8/P104 source lineage and Nick Petalas's real-metric predecessor as
+in README.md. Jerry Chen, @CubeJerry, with substantial OpenAI Codex assistance.

research/p104-selected-metric-orbit/INDEPENDENT-REVIEW.md

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+# Independent exact review of all three energy spectra
+
+Reviewed Worker A's `research/p104-selected-metric-orbit` at commit
+`0fb40d096e78cb332c4de0dc9e6e35b6f5e640d3`, against canonical base
+`2485805a6d47f36ea83ddde6327d453ac82d44dc`. No blocker was found in the
+normalization, rational multiplicities, four-dimensional quadratic block,
+sharp constants, equality dimensions, or positive-definite sharpness.
+**No new κ.**
+
+## Separate mathematical construction
+
+`independent_orbits.py` imports neither the author's implementation nor the author's
+104 rational eigenvectors. It constructs the 120 root lines directly in the
+nine-dimensional sum-zero hyperplane: integer vectors `3(e_i-e_j)` and vectors
+with three coordinates 2 and six coordinates -1. These are `3r`, where each
+root `r` has squared norm 2. The source WORD is authenticated by SHA256
+`7834725d9e34ee846a9ccc615221ea81e0201613938afa185d2eda0c9a6c232f`.
+The frame classification and representative coverage were separately checked
+by a separate maximal-clique/union-find implementation; this review addresses spectra of its three representatives.
+
+Use the nonorthogonal A8 chart `B=[e_0-e_8,...,e_7-e_8]` and write every
+physical symmetric operator as `A=B S B^t`. The 36 coordinates of `S` are its
+8 diagonal and 28 upper off-diagonal entries. The Gram matrix `W` is computed
+as the Frobenius Gram matrix of the 36 literal nine-dimensional operators
+`B E_ij B^t`. Thirty-six positive exact rational LDL pivots certify `W>0`.
+
+For each root let `w=B^t r`, which is integral. On unit roots `u=r/sqrt(2)`,
+`u^t A v = w^t S w'/2`. Sum the outer products of the corresponding integer
+linear forms over all unordered orthogonal pairs in the puncture. This gives
+an independently assembled integer matrix `H` with
+
+```
+E(A) = s^t H s / 4;       ||A||_F^2 = s^t W s.
+```
+
+Each puncture has exactly 2,860 such pairs. This independent chart has energy
+denominator 4, whereas the author's orthonormal F chart has denominator 16;
+the conversion is explicitly accounted for rather than copying a normalization.
+The identity on the hyperplane has `S=I-J/9`, lies in the kernel, and has
+squared Frobenius norm 8. The trace linear form in this chart is twice the sum
+of all 36 coordinates.
+
+Exact rational Gaussian elimination and back substitution compute the
+nullities of `H-4λW`. These yield all the stated rational multiplicities:
+
+| Orbit size | Rational spectrum, with multiplicities |
+|---|---|
+|28|0:1, 37:6, 38:16, 91/2:12, 54:1|
+|336|0:1, 37:1, 75/2:8, 38:12, 91/2:10|
+|896|0:1, 75/2:6, 151/4:8, 38:7, 165/4:4, 91/2:6, 199/4:4|
+
+The nonzero rational eigenspaces are checked to be trace-free. Their dimensions
+sum to 36, 32 and 36. Distinct eigenspaces are independent, and `W>0` makes the
+generalized operator self-adjoint, so these are full spectra for sizes 28 and
+896 without relying on the author's eigenvectors or modular-rank certificate.
+
+## The remaining four dimensions
+
+The four matrices displayed in ORBIT-PROOF.md are treated as claimed data,
+converted into physical nine-dimensional matrices using the fixed frame, then
+into this separate A8 chart. Their coefficient columns `V` have exact rank 4.
+The checker verifies, against its independently assembled size-336 energy,
+
+```
+4 H V = W V C,
+C = [[796,0,0,-68],[0,796,68,0],[0,68,660,0],[-68,0,0,660]].
+```
+
+This factor 4 is required: the author's C represents `16λ`, while the new
+pencil represents `4λ`. Two explicit 2-by-2 blocks have eigenvalues
+`728 +/- 68 sqrt(2)`, each occurring once in each block. The exact polynomial
+`C^2-1456 C+520736 I=0` and trace 2912 are independently checked. Its polynomial
+is nonzero at each rational eigenvalue already accounted for, so these four
+dimensions have zero intersection with the rational 32-dimensional sector.
+Consequently the full reported size-336 spectrum is complete. Dividing by16
+gives `(182 +/- 17 sqrt(2))/4`; both exceed `75/2`, certified by `32^2>2*17^2`.
+
+It follows that the sharp trace-free constants are 37, 37 and 75/2, with
+equality dimensions 6, 1 and 6. This conclusion uses exact algebra only.
+
+## Positive-definite sharpness and controls
+
+For each representative the independently computed minimal eigenspace supplies
+a nonzero trace-free operator A. Set
+`t=1/(2 max_i sum_j |A_ij|)` in the physical nine-dimensional chart. The
+operator `I_V+tA` is positive definite since the absolute row-sum bound
+controls the symmetric operator norm. Its A8-coordinate matrix also passes
+exact rational LDL with eight positive pivots. The deterministic examples use
+`t=1/4,1/16,1/24`, respectively, and attain the claimed energy-to-anisotropy
+ratio exactly. This establishes sharpness within positive-definite metrics,
+without claiming a rational square root for these metrics.
+
+Three controls detect doubling the energy normalization, changing one sign in
+the quadratic action, and dropping a quadratic basis column. Both normal and
+`-O` runs with warnings as errors produce the identical receipt
+`PASS_INDEPENDENT_A8_ALL_SPECTRA`.
+
+Reproduction from this package directory (or use the registered runner):
+
+```sh
+python3 -B -W error independent_orbits.py --source /path/to/research/p104-helper470-twin-rung-removal/WORD.json.gz
+python3 -O -B -W error independent_orbits.py --source /path/to/research/p104-helper470-twin-rung-removal/WORD.json.gz
+```
+
+This is an exact finite check and mathematical review, not a proof-kernel
+formalization or independent human review. The accepted selected multiplier,
+physical supplier accounting, and five inherited assumptions are unchanged.
+The claimed geometry remains the canonical A8/E8 realization; this review
+independently reconstitutes its energy rather than proving an unrelated
+coordinate identification. Predecessor credit: the canonical P104/E8 source
+lineage, Nick Petalas's PR6278 uniform bound and classification question, and
+Worker A's new classification and certificates. Jerry Chen, @CubeJerry;
+substantial OpenAI Codex assistance from research worker D (seed79).