spectra-from-cuts

• Composes the two-tile claim of notes/spectra.md with the carry automaton of notes/cuts.md run at an odd base b instead of base 3.
• The objects: P_b is the digit polynomial of the solid with at most one odd coordinate, P_b(t) = E^2 (E + 3 O) with E = 1 + t^2 + .. + t^(b-1) and O = t + t^3 + .. + t^(b-2); g = 3(b-1)/2 is the target digit; the automaton is M[c, c'] = P_b[c + g - b c'] on abs(c) <= 1.
• block recomputes the even block [[M[0,0], M[0,1]], [M[1,0] + M[-1,0], M[1,1] + M[-1,1]]] at every odd b = 3..25, checks that no carry leaves abs(c) <= 1, that the block equals the closed form spectra prints for the class of b mod 4, that (hexagons, triangles) = ((M^n)[0,0], (M^n)[1,0] + (M^n)[-1,0]) obeys the block at levels 0..8, and that a direct digit-sum census at heights h_n - 1, h_n, h_n + 1, h_n = 3(b^n - 1)/2, returns the same pairs at low levels.
• forms derives the five coefficients P_b[g], P_b[g+1], P_b[g+b], P_b[g+b-1], P_b[g+b+1] as polynomials in b within each class of b mod 4, by inclusion-exclusion on ((1 - x^n)/(1 - x))^3 and (1 - x^(n-1))(1 - x^n)^2/(1 - x)^3 with n = (b+1)/2, prints the base from which each extraction's active terms are stable, and checks every form against the digit polynomial at odd b = 3..101, which covers every base below the thresholds.
• split takes the trace and determinant of the block in each class, and proves rho_b > fill/b for b = 3 mod 4 and rho_b < fill/b for b = 1 mod 4 as polynomial inequalities in n: real roots isolated exactly, all below the first n of the class, and the sign read at that n.
• classes checks that the residue-class sums of P_b modulo b are the row sums of the automaton, prints them as polynomials in b, and checks P_b(w) (1 + w)^3 + 2 is divisible by 1 + w + .. + w^(b-1), so P_b(w) = -2/(1 + w)^3 at every b-th root of unity w != 1.
• Exact arithmetic throughout, integers and sympy; the whole run takes under one second.

RUN

uv run python research/lab/py/spectra-from-cuts/compose.py
uv run python research/lab/py/spectra-from-cuts/compose.py block
uv run python research/lab/py/spectra-from-cuts/compose.py forms
uv run python research/lab/py/spectra-from-cuts/compose.py split
uv run python research/lab/py/spectra-from-cuts/compose.py classes

WITNESSES

• spectra.md THE CLAIM, the matrices and the closed form: verb block, twelve odd bases, no mismatch.
• spectra.md THE COMPOSITION, the five coefficients and the block in each class: verb forms, run inside split.
• spectra.md THE COMPOSITION, the sign of rho_b - fill/b in each class: verb split.
• spectra.md THE COMPOSITION, the residue-class sums and the root-of-unity value: verb classes.
• cuts.md The ladder in every dimension, the base axis at dim = 3: verb block.