Companion to Paper D · Spectral Energy Landscapes & Multiscale Window Entropies

Magic squares as spectral height fields

A normal magic square is a zero-mean permutation over a lattice in the (n²-2n-1)-dimensional nullspace of the rationalized magic-line Laplacian L̃_n = P_{1^⊥} A_n^T A_n P_{1^⊥}. Because its 3D spatial covariance is universal across all semi-magic squares, Paper D studies the exact integer spectrum of L̃_n, multiscale window entropies H̃_p(k), the Concentric Shell Quadrupole Identity, and cooperative (2,2) relaxation.

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Paper D Theorems & Identities

From spectral rationalization to multiscale window geometry

Thm 2.1 · Zero-Mean Spectrum L̃_n {0^[n²-2n-1], n-2, n^[2n-1]} (even n) · μ̄ = 2
Thm 3.1 · Even-Order Quartet Equivalence W_{2×2}^int(S)=0 ∧ H_line=0 ⇒ W_{2×2}^tor(S)=0
Prop 3.2 · Shell Quadrupole Identity M_xyz(S) = -8 Q_corner(S) - 3 Q_edge(S)
Sec 5.3 · Floppy-Mode Null Fraction f_null(n) = (n²-2n-1)/(n-1)² = 1 - 2/(n-1)²

Restricting the line-incidence Laplacian L_n = A_n^T A_n to the zero-mean hyperplane 1^⊥ collapses its irrational diagonal eigenvalues into exact integers with universal mean μ̄ = 2 (E[H_line] = 2/3 n²(n⁴-1)). At finer spatial scales, sliding k × k window entropies H̃_p(k) and concentric shell quadrupoles resolve all 12 Dudeney structural groups and explain why M_xyz = 0 on the 48 associative squares plus 8 coprime-resonance squares in Groups IX and X.

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2

Example square

Square

magic

Centered height field

3D render

Bars rise from the lattice by centered height Z. The camera is fixed so comparisons are stable across classes.

Line balance

Magic constraints as residuals

Spectral signature

Entropy by scale and exponent

Higher p concentrates attention on extreme centered values. Intermediate window scales are where valid and invalid squares usually separate.

Statistical summary

Current invariants

Local imbalance

Window residual distribution

Uses signed sums ΣZ over all k × k windows. Magic lines vanish, but local windows need not.

8x8 energy landscape

Ten heatmaps: valid magic fields and random controls

Each cell is colored by E_p(i,j)=|2a_ij-(8²+1)|^p. The first five panels are valid 8x8 magic squares from a doubly-even construction; the last five are random permutations.