Multilayers

• 2026-09-23 [Proved] For a digit-design multilayer (base b, digits D, equal optical thickness, normal incidence) the level-(k+1) characteristic matrix is the ordered product over d < b of M_k(delta) for d in D and the host slab M_B(b^k delta) otherwise, b - 1 multiplications a level. Witness: notes/multilayers.md "The level recursion".
• 2026-09-23 [Verified] The level recursion agrees with a 40-digit layer-by-layer transfer product to relative 8.4e-13 over 3^6, 4^5, 5^4, 7^4 and 9^3 cells at nA = 1.6, 2.3, 3.5. Witness: lab/py/design-multilayer check.
• 2026-09-23 [Proved] At zero contrast abs(d(r_k/t_k)/d nA) equals abs(d(r_0/t_0)/d nA) times abs(prod_(j<k) P_D(e^(2i b^j delta))), so its log averaged over uniform delta grows by exactly m(P_D) a level; this is a statement about the derivative at nA = nB only. Witness: notes/multilayers.md "The Born drift".
• 2026-09-23 [Verified] At nA = 1.455, nB = 1.45 the mean per-level drift of ln abs(r_k/t_k) over six to eight levels is 0.3832, 0.4425, 0.2818, within 0.001 of m(P_D) = 0.38225, 0.44214, 0.28120 for {0,1,3} base 4, {0,1,2,4} base 5, {0,1,5} base 7, and within 0.004 of 0 for three designs with cyclotomic P_D, standard errors below 0.002. Witness: lab/py/design-multilayer born.
• 2026-09-23 [Verified] Step law: where abs(r_k/t_k) < 1e-3 on 50 or more frequencies, ln abs(r_(k+1)/t_(k+1)) - ln abs(r_k/t_k) equals ln abs(Q_D(u^2, v^2)), u the phase of t_k and v the spacer phase, Q_D(x, y) = sum_i x^i y^(d_i - i), with median gap below 3e-6 on 25 of 27 design-contrast cells (nine designs at nA = 1.6, 2.3, 3.5); {0,1,3} base 4 and {0,1,2,4} base 5 at nA = 3.5 never reach 50 such frequencies. Witness: lab/py/design-multilayer deep.
• 2026-09-23 [Refuted] P_D cyclotomic implies a critical stack: {0,2,3,4,6} base 7, {0,1,4,7,8} base 9 and {0,2,3,4,5,7} base 8 have cyclotomic P_D, non-Boyd Q_D with m(Q_D) = 0.2513, 0.2513, 0.3181, and f_k sqrt(k) falling from 0.878, 0.883, 0.888 at level 2 to 0.548, 0.538, 0.406 at level 12 at nA = 2.3. Witness: lab/py/design-multilayer refute.
• 2026-09-23 [Refuted] At nA = 1.6 the deep drift of ln abs(r/t) (mean step where abs(r_k/t_k) < 0.02) tends to m(Q_D): over the last four levels it is 0.350 for {0,2,3,4,5,7} base 8 and 0.272 for {0,2,3,4,6} base 7 against m(Q_D) = 0.3181 and 0.2513, 6.3 and 4.3 standard errors above, flat over those levels. Witness: lab/py/design-multilayer deep.
• 2026-09-23 [Verified] The deep drift over the last four levels is at most 0.01 in absolute value, standard error at most 0.01, at nA = 1.6, 2.3, 3.5 on {0,2} base 3, {0,1,3,4} base 7 and {0,2,4,6} base 7, whose Q_D is Boyd-cyclotomic, and at least 0.26, standard error at most 0.02, on four designs whose Q_D is not. Witness: lab/py/design-multilayer deep.
• 2026-09-23 [Verified] m(1 + x + x^2 y), the lift of {0,1,3}, equals Smyth's constant 3 sqrt(3) L(chi_(-3), 2)/(4 pi) = 0.3230659472 to 1e-8. Witness: lab/py/design-multilayer smyth.
• 2026-09-23 [Refuted] At nA = 1.6 the {0,1,3} base 4 stack drifts at Smyth's constant 0.3231: its deep drift over levels 8 to 11 is 0.51 to 0.60, standard error at most 0.026. Witness: lab/py/design-multilayer deep.
• 2026-09-23 [Verified] All 39 designs at bases 3 to 6 with 0 in D and 2 <= abs(D) < b, up to shift and mirror, sort by the Boyd class of Q_D: flatness f_12 sqrt(12)/(f_6 sqrt(6)) at nA = 2.3 is 0.921 to 1.036 on the 26 Boyd-cyclotomic designs and 0.057 to 0.315 on the other 13, verdict critical above 0.9. Witness: lab/py/design-multilayer census.
• 2026-09-23 [Conjecture] A digit-design multilayer at fixed contrast has passband fraction f_k ~ C/sqrt(k) when m(Q_D) = 0 and geometric decay in k when m(Q_D) > 0; at nA = 2.3, f_12 sqrt(12) is 1.097 for {0,2,4,6} base 7, still falling slowly, and 0.901 for {0,1,3,4} base 7. Witness: lab/py/design-multilayer census and refute.