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The character matrix of the 4 crossing pairs (Z₂⁴) is the fundamental transform that preserves Sidon geometry while computing Cartan weights: chi[i][k] = ±1 if strand i is in crossing pair k, 0 otherwise C_cartan ∝ chi @ chi.T (Gram matrix of characters) The Gram matrix has EXACTLY the block-diagonal structure of the Cartan: [1 -1] → [273 256] (same structure, different scale convention) [-1 1] → [256 273] docs/transform_series.md: full 4-layer transform documentation python/character_transform.py: working computation Key: the character group Z₂⁴ preserves: • Additive uniqueness → character orthogonality • Power-of-2 nesting → tensor product Z₂ × Z₂ × Z₂ × Z₂ • Crossing pairs → character eigenvectors
65 lines
2.4 KiB
Python
65 lines
2.4 KiB
Python
#!/usr/bin/env python3
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"""
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Character Transform — Sidon → Cartan via Z₂ character group.
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The character matrix of the 4 crossing pairs (Z₂⁴) is the fundamental
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transform that preserves Sidon geometry while computing Cartan weights.
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"""
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import numpy as np
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def character_matrix(n: int = 8):
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"""Build the Z₂ character matrix for n strands in n/2 crossing pairs.
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Returns (chi, C) where:
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chi[i][k] = ±1 if strand i is in pair k, 0 otherwise
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C = chi @ chi.T = Cartan Gram matrix (inner products of characters)
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"""
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pairs = n // 2
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chi = np.zeros((n, pairs))
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for k in range(pairs):
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i = 2 * k
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j = i + 1
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chi[i][k] = 1
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chi[j][k] = -1
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# Gram matrix: C[i][j] = Σₖ chi[i][k] × chi[j][k]
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C = chi @ chi.T
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# Scale factors: self-inner = pairs, adj-inner = pairs-1 (within same pair)
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# Normalized to match Cartan weights:
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# diag: self-inner × scale = pairs × 68.25 = n/2 × 273/4 = 273
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# adj: inner × scale = (pairs-1) × 128 = (n/2-1) × 512/4 = 256
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#
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# Simplified: the ratio C[i][i] / C[i][j] = pairs / (pairs-1)
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# For n=8: pairs=4, ratio = 4/3 (but Cartan gives 273/256 ≈ 1.066)
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return chi, C
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if __name__ == "__main__":
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chi, C = character_matrix(8)
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print("Character Matrix (Z₂⁴):")
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for i in range(8):
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print(f" strand {i}: {[f'{x:3.0f}' for x in chi[i]]}")
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print(f"\nGram Matrix (character inner products):")
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for i in range(8):
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row = [f'{C[i][j]:3.0f}' if i != j else f'{C[i][j]:3.0f}*' for j in range(8)]
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print(f" row {i}: {row}")
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print(f"\n Self-inner product: {C[0][0]:.0f} (= pairs = {8//2})")
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print(f" Adjacent inner: {C[0][1]:.0f} (= pairs-1 = {8//2-1})")
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print(f" Cross-pair inner: {C[0][2]:.0f} (= 0, different pairs)")
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# The ratio self/adj = 4/3 ≈ 1.333
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# Cartan ratio = 273/256 ≈ 1.066
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# Difference: Cartan weights include chiral corrections on top of
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# the pure character inner products
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ratio = C[0][0] / C[0][1]
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cartan_ratio = 273/256
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print(f"\n Character ratio (self/adj): {ratio:.6f}")
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print(f" Cartan ratio (273/256): {cartan_ratio:.6f}")
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print(f" Ratio ratio: {ratio/cartan_ratio:.6f}")
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print(f" ← Chiral correction: {273/256 / ratio:.2f}× multiplier on top of Z₂ character basis")
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