Part 0.0: full correlation matrix in cascade ratios; partial closure of OQ2

[dickie81]

Summary

The next forced implication of the Gram unification (PR #90): the Beta–Gamma reduction extends to general layer pairs, giving the entire correlation matrix in cascade slicing ratios and partially closing Supplement Open Question 2.

What's added

Theorem 14.X+1 (Full correlation matrix in cascade slicing ratios), thm:gram-matrix:
$$C_{ij} = \frac{R(d_i+d_j+1)}{\sqrt{R(2d_i+1),R(2d_j+1)}}$$
for any cascade layer pair $(d_i, d_j)$. Equivalently, $\log C^2_{ij} = -\Delta^2_{(k)}\log R|_{m}$ with step $k = |d_i - d_j|$ and centered doubled argument $m = d_i + d_j + 1$. Theorem thm:gram-R is the special case $k=1$.

Corollary (Eigenvalue deficit in cascade primitives), cor:eigenvalue-cascade:
The eigenvalue deficit $\epsilon(n, d_0)$ of Theorem 14.3 is the dominant-eigenvalue deficit of the explicit cascade-native $n \times n$ matrix. The first-order perturbation expression of Theorem 14.5 becomes a sum of cascade $R$-ratios.

Remark (Structural sign), rem:gram-sign-from-convexity:
The positivity of the Gram correction (cited as Cauchy–Schwarz in Part IVb) reduces to convexity of $\log\alpha$ on the cascade tower. Asymptotically $\alpha \sim 1/(2d)$, so $\log\alpha \sim -\log(2d)$ is convex and $\Delta^2 \log\alpha > 0$ structurally.

Why this is the "next forced implication"

The original Theorem 14.X (Gram correlation in cascade slicing ratios) used the Beta–Gamma reduction $G_{d,d+1} = \sqrt{\pi},R(2d+2)$ for adjacent layers. The same reduction $G_{ij} = \sqrt{\pi},R(d_i+d_j+1)$ holds for any pair $(d_i, d_j)$ — this is just the Beta function's argument structure, not a new mathematical fact. So the closed form generalises automatically.

What's gained:

• Full correlation matrix in cascade primitives (every entry an $R$-ratio).
• Eigenvalue deficit has a manifestly cascade-internal expression.
• OQ2 partially closes: the input matrix is no longer "a matrix of Beta function values" but "a matrix of cascade slicing ratios at doubled arguments."
• Sign rule derives from convexity of $\log\alpha$ rather than from $L^2$ Cauchy–Schwarz on integrands — a cleaner structural foundation.

Verification

tools/verifiers/gram_compliance_laplacian.py extended with:

• V4: full correlation matrix at non-adjacent pairs $(5,7), (5,12), (10,20), (14,21), (5,217)$. All match direct Beta-function computation to machine precision.
• V5: eigenvalue deficit $\epsilon$ from the cascade-native matrix matches direct Beta-function eigendecomposition for canonical paths $d=5..12, 6..13, 14..21$ at relative error $< 10^{-13}$.

All five verifications pass.

What remains open

OQ2 is partially closed, not fully:

• Closed: the matrix is in cascade primitives (not Beta values).
• Open: a closed-form expression for $\lambda_1$ in terms of $n$ and $d_0$ that eliminates numerical eigendecomposition.

Any closed-form $\lambda_1$ would derive from properties of $R$ at consecutive doubled arguments — a tractable problem in cascade primitives, no longer dependent on Beta function asymptotics.

Test plan

• LaTeX builds clean (no overfull hbox, no undefined references).
• Verifier passes all five checks (python tools/verifiers/gram_compliance_laplacian.py).
• Confirm the new Theorem and Corollary read coherently after Theorem thm:gram-R and Corollary cor:gram-laplacian.
• Confirm "What this section proves" item 2 reflects the full-matrix closed form.
• Confirm OQ2's "partially closed" framing is honest (the matrix is cascade-native; the eigenvalue closed form remains open).

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