# Dilated receptive fields

- 2026-09-23 [Proved] For a linear dilated stack at levels `j < L` with dilation `b^j`, independent random `C x C` taps, each zero-mean with iid entries of variance `sigma_k^2` or, at most one per level, the identity plus such a matrix, `E norm(dy_t/dx_(t-n))^2 = norm(u)^2 r_Q(n)`, where `q_k = [k is the residual] + C sigma_k^2` and `r_Q(n) = [z^n] prod_(j < L) Q(z^(b^j))` is the weighted base-`b` digit count; the undilated single-channel zero-mean case is Luo et al., section 2.2. Witness: notes/dilations.md, The field is a digit count.
- 2026-09-23 [Verified] Monte Carlo over `4000` random `8`-channel `8`-level stacks reproduces `r_Q` with median relative error `0.020` and correlation `0.9989` over `511` lags for `Q = 1 + z + z^2`, and `0.022` and `0.9999` over `256` lags for `Q = 1 + c(1 + z)` at `c = 1/2`; the pure stack's z-scores run heavy-tailed, `1.96%` of its lags past `3`. Witness: lab/py/dilated-receptive-field gradient.
- 2026-09-23 [Proved] For `Q = 1 + z + z^2` and `b = 2` the field is Stern's diatomic sequence, `r(n) = s(n + 1)` for `0 <= n < 2^L` (A002487), mirrored as `r(n) = r(2^(L+1) - 2 - n)` above, recomputed exactly for `L = 1..20`. Witness: notes/dilations.md, Stern in the plain stack; lab/py/dilated-receptive-field count.
- 2026-09-23 [Proved] The maximum of the `Q = 1 + z + z^2`, `b = 2` field at level `L` is the Fibonacci number `F_(L+1)`, attained at `n = m - 1` with `m = (2^(L+1) + (-1)^L)/3`. Witness: notes/dilations.md, Stern in the plain stack.
- 2026-09-23 [Verified] The maximum `F_(L+1)` sits at exactly the four lags `m - 1`, `3 . 2^(L-1) - m - 1` and their mirrors for `L = 3..20`, at `682, 852, 1194, 1364` for `L = 10`. Witness: lab/py/dilated-receptive-field count.
- 2026-09-23 [Proved] Exactly `2L + 1` lags of the `Q = 1 + z + z^2`, `b = 2` field carry a single path, `2^k - 1` and `2^(L+1) - 1 - 2^k` for `k = 0..L`, `21` at `L = 10`. Witness: notes/dilations.md, Stern in the plain stack; lab/py/dilated-receptive-field count.
- 2026-09-23 [Refuted] The single-path lag count `2(L + 1) + 1` for the `Q = 1 + z + z^2`, `b = 2` field: it counts the centre `2^L - 1` three times, and the exact census reads `2L + 1` at every `L = 1..20`. Witness: lab/py/dilated-receptive-field count.
- 2026-09-23 [Proved] Peak over mean of the `Q = 1 + z + z^2`, `b = 2` field is `F_(L+1)(2^(L+1) - 1)/3^L`, and its ratio from level `L` to `L + 1` tends to `2 phi/3 = 1.07869`, rational at every level, `1.07143` at `L = 3` and `1.14815` at `L = 4`; peak over mean is `3.0853` at `L = 10` and `6.5835` at `L = 20`. Witness: notes/dilations.md, Stern in the plain stack; lab/py/dilated-receptive-field count.
- 2026-09-23 [Proved] The depth limit `mu_Q` of a dilated field with nonnegative block polynomial `Q` at base `b` is absolutely continuous if and only if for every integer `k` not divisible by `b` some `Q(e^(2 pi i k/b^i))` vanishes, `i >= 1`, and is purely singular otherwise; for prime `b` and rational gains the test is that some cyclotomic `Phi_(b^i)` divides `Q`. Witness: notes/dilations.md, When the limit is smooth.
- 2026-09-23 [Refuted] For prime `b` and real nonnegative gains, an absolutely continuous depth limit forces some `Phi_(b^i)` to divide `Q`: `Q = (z^2 - sqrt2 z + 1)(z^4 + sqrt2 z^2 + 1)(1 + z + z^2)^2` at `b = 2` has least coefficient `2 - sqrt2`, covers every odd `k` at `i = 3` or `i = 4`, so its limit is absolutely continuous, and is nonzero at `-1`, `i`, `e^(3 pi i/4)` and `e^(pi i/8)`, so no `Phi_(2^i)` divides it. Witness: notes/dilations.md, When the limit is smooth; lab/py/dilated-receptive-field limit.
- 2026-09-23 [Proved] For `Q = 1 + z + z^2` and `b = 2` the fold modulo `1` of the depth limit carries the Stern rows, `s(2^L + j) = r(j - 1) + r(j - 1 + 2^L)` for `0 <= j < 2^L`, checked for `L = 1..20`. Witness: notes/dilations.md, When the limit is smooth; lab/py/dilated-receptive-field count.
- 2026-09-23 [Verified] The exact cyclotomic check agrees with the Fourier product `prod_i Q(e^(2 pi i t/b^i))/Q(1)` on all `35` block polynomials, `21` uniform, `6` single-convolution residual, `7` double-convolution residual and `(1 + z + z^2)^2`, the product below `1e-15` on every absolutely continuous row and at least `1.8e-3` on every singular one, a check of the generator's arithmetic since both sides read the same roots. Witness: lab/py/dilated-receptive-field limit.
- 2026-09-23 [Proved] A dilated stack of `K >= 2` uniform taps at base `b` has an absolutely continuous depth limit if and only if `b` divides `K`, and for `L >= 2` a constant field exactly when `K = b`; at `L = 1` every uniform field is constant. Witness: notes/dilations.md, Four stacks.
- 2026-09-23 [Proved] The residual block `Q = 1 + c(1 + z)` at `b = 2` has a singular depth limit for every gain `c > 0`, while the pure `K = 2` stack is flat. Witness: notes/dilations.md, Four stacks.
- 2026-09-23 [Proved] The double-convolution residual block `Q = 1 + c(1 + z + z^2)^2` at `b = 2` has an absolutely continuous depth limit exactly at `c = 1`, where `Phi_4` divides `Q`; with equal gains `q_1`, `q_2` across each convolution's taps the digit count covers the block and `c = q_1 q_2`. Witness: notes/dilations.md, Four stacks.
- 2026-09-23 [Proved] The square stack of `3 x 3` convolutions at dilations `2^j` in both axes, its nine taps of equal gain, has field `s(m + 1) s(n + 1)` on its lower quadrant, peak `F_(L+1)^2`, with `(2L + 1)^2` single-path cells. Witness: notes/dilations.md, Four stacks.
- 2026-09-23 [Refuted] Steps to learn a lag-`n` copy task scale like `1/r(n)`: a linear `K = 3`, `b = 2`, `L = 8`, `8`-channel stack under full-batch gradient descent takes median `615` steps at `r = 1` and `94` at `r = 34`, a factor `6.5` against `34`, log-log slope `0.55`. Witness: lab/py/dilated-receptive-field copy.
- 2026-09-23 [Conjecture] Steps to learn a lag-`n` copy task in a linear `K = 3`, `b = 2` stack fall monotonically with `r(n)`, roughly as `r(n)^(-1/2)`: at the default initial scale eight medians, slope `0.55`, correlation `0.984` with `log 1/r`; at half the scale slope `0.40`, correlation `0.983` over the same eight lags, one run capped and read as above the cap; the exponent moves with the scale. Witness: lab/py/dilated-receptive-field copy.