# 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`.