design-multilayer

• Light at normal incidence through a digit-design multilayer: layer A (nA) on the level-k design cells (base b, digits D), host B (nB = 1.45) elsewhere and on both sides, every cell of equal optical thickness delta.
• Builds the level-k characteristic matrix by the level recursion M_(k+1) = W_D(M_k, M_B(b^k delta)), b - 1 multiplications a level; a stack is dark once an entry passes 1e100, and a dark stack's matrix is rescaled by a positive number each level so the phase of t_k stays exact.
• Mahler measures: m(P_D) by Jensen on the roots, m(Q_D) by Jensen in x on 2^15 midpoints in y; the Boyd class of Q_D (monomial times cyclotomic polynomials in monomials) by exact factoring in sympy.
• One generator for every number in notes/multilayers.md.

RUN

uv run python research/lab/py/design-multilayer/multilayer.py
uv run python research/lab/py/design-multilayer/multilayer.py check born

• No argument runs every verb; about 46 seconds on one core. Needs numpy, sympy, mpmath and PARI (gp).
• check (2.4 s): the recursion against a 40-digit layer-by-layer product on five designs at nA = 1.6, 2.3, 3.5, asserting a relative gap under 1e-12; the zero-contrast derivative against prod P_D(e^(2i b^j delta)) at nA = nB(1 + 1e-8) in abs(gap)/max(1, abs(prod)), asserting 1e-5.
• born (2.1 s): per-level mean drift of ln abs(r/t) at nA = 1.455 against m(P_D) on six designs, with its standard error and abs(mean - m(P_D)), and the largest gap over the six.
• smyth (0.3 s): m(Q_(0,1,3)) against 3 sqrt(3) L(chi_(-3), 2)/(4 pi) from PARI, asserting 1e-8.
• deep (27 s): on nine designs at nA = 1.6, 2.3, 3.5, the deep drift (mean step where abs(r_k/t_k) < 0.02) with per-level standard errors, the tail mean of the last four levels with se the mean of their standard errors (an upper bound for the tail mean's standard error) and its distance from m(Q_D) in se, the deep lift and its largest tail gap from the drift, the step-law median gap where abs(r_k/t_k) < 1e-3 on 50 or more frequencies (n/a otherwise), the coupling abs(mean e^(2i(alpha - beta))) of block and spacer phases, the torus mean of ln abs(Q_D(u^2, v^2)) over every frequency, dark or open, and the dark fraction; then the step-law cell count and the zero-set band over the seven table designs.
• refute (3.8 s): the three designs with cyclotomic P_D and non-Boyd Q_D, asserting both, with f_k sqrt(k) to level 12 at nA = 2.3, beside two Boyd-cyclotomic controls.
• census (10.9 s): every design at bases 3 to 6 with 0 in D and 2 <= abs(D) < b, up to shift and mirror (39), its Boyd class and the flatness f_12 sqrt(12)/(f_6 sqrt(6)) at nA = 2.3, verdict critical above 0.9, and the flatness band of each class.

WITNESSES

• notes/multilayers.md "The level recursion": the 1e-12 agreement (check).
• "The Born drift": the derivative identity (check) and the weak-contrast drifts 0.3832, 0.4425, 0.2818, within 0.001 of m(P_D) (born).
• "The block-spacer lift": the step law below 3e-6 on 25 of 27 cells (deep).
• "Cyclotomic is not enough": the three counterexamples and the controls 1.097, 0.901 (refute).
• "The deep drift": the table, the refutation at nA = 1.6, the zero-set band and the {0,1,3} rows (deep, smyth).
• "Two regimes": the 39-design census, 26 critical and 13 decaying, matching the Boyd class on all (census).