Part 0.0: k-step Gram-Laplacian — full correlation matrix as Jensen gap of log-compliance#94
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Part 0.0: k-step Gram-Laplacian — full correlation matrix as Jensen gap of log-compliance#94
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… gap of log-compliance
Extends Cor 14.4 (adjacent-pair Gram-Laplacian identity) to all layer
pairs. For any (d_i, d_j) with k = |d_j - d_i| >= 1:
log C^2_{ij} = -(1/2) Delta_k^2 log alpha |_{d_i+d_j+1}
= log alpha(midpoint) - (1/2)[log alpha(left) + log alpha(right)]
where left = 2 d_i + 1, right = 2 d_j + 1, midpoint = d_i + d_j + 1
(the arithmetic mean of left and right). k=1 reproduces Cor 14.4.
Reading: every entry of the full correlation matrix (Theorem on the
Gram matrix C in the supplement's setup) is a Jensen gap of the
log-compliance log(alpha). Cauchy-Schwarz C^2_{ij} <= 1 is then
equivalent to log-concavity of alpha on the cascade tower -- which
follows from R(d) ~ sqrt(2/(d+1)) so log alpha(d) = -log d + O(1/d) is
strictly concave.
Item 8 (cross-observable correlations from off-diagonal C_{ij}) was
also explored: shared-layer cancellation in observable ratios is real
and clean (Gram corrections to A and B sharing layers cancel
identically, leaving only the symmetric-difference path correction)
but is mechanical additivity rather than a new structural prediction.
Not landed in the paper.
Verifier tools/research/knockon_higher_order_laplacian.py confirms
both items at mpmath precision.
https://claude.ai/code/session_01KEi216CArEazmn38JExPwK
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Summary
Following the user's "anymore knock-ons?" prompt, this PR investigates two consequences of the recent Part 0.0 work:
Item 7 (LANDED): k-step Gram-Laplacian generalisation. For any cascade layer pair$(d_i, d_j)$ with $k = |d_j - d_i| \geq 1$ :
$$\log C^2_{ij} = -\tfrac{1}{2},\Delta_k^2 \log\alpha\big|_{d_i+d_j+1} = \log\alpha(\text{mid}) - \tfrac{1}{2}\bigl[\log\alpha(\text{left}) + \log\alpha(\text{right})\bigr]$$ $= 2d_i+1$ , right $= 2d_j+1$ , mid $= d_i+d_j+1 = (\text{left}+\text{right})/2$ .
where left
Reading: every entry of the full correlation matrix is a Jensen gap of the log-compliance$\log\alpha$ . Cauchy–Schwarz $C^2_{ij} \leq 1$ for $i \neq j$ is then equivalent to log-concavity of $\alpha$ on the cascade tower (which follows from $\log\alpha(d) = -\log d + O(1/d)$ being strictly concave).
The adjacent case$k=1$ recovers Cor 14.4. The general case extends the cascade-native Laplacian-of-compliance reading from the adjacent diagonal to the entire correlation matrix.
Verified to mpmath 50-digit precision across$(d, k)$ samples spanning cascade-physics scales:
tools/research/knockon_higher_order_laplacian.py.Item 8 (NOT LANDED): cross-observable correlations. Tested the shared-layer cancellation in observable ratios. The result is clean — Gram corrections to two observables sharing layers cancel identically in their ratio, leaving only the symmetric-difference path correction — but this is mechanical additivity of the Gram correction rather than a structurally new prediction. Verified in the script but not promoted to the paper.
What this adds
cor:gram-laplacian-k-step) in Part 0.0knockon_higher_order_laplacian.py)Test plan
python3 tools/research/knockon_higher_order_laplacian.pyruns to completion (verifies item 7 to mpmath precision)Honest framing
This is a modest result, not a major one. The structural identity is real and clean (and was previously undocumented for non-adjacent pairs), but it follows mechanically from the Beta-function form of$C^2_{ij}$ already in the supplement plus $\log\alpha = 2\log R - 2\log 2$ . Worth landing as a brief Corollary; not worth a full Theorem.
https://claude.ai/code/session_01KEi216CArEazmn38JExPwK
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