smith-window

• Builds the even carry core at base 3, at the odd carry parameter D = 2R + 1, through its symbol rather than its entries: with P = (1 + t^2)^(D-1)(1 + D t + t^2) and H_x = x_0 t^R + sum_(j >= 1) x_j (t^(R+j) + t^(R-j)), the row at c' of M_even x is the coefficient of t^(3 nu + 1) in H_x P at nu = R - c', and H_x P is palindromic about 3R + 1, so the R + 1 rows are every exponent == 1 mod 3 in [0, 6R + 2].
• Reads the second 2-adic Smith layer off the same symbol: a mod-2 kernel vector's mod-4 obstruction is ((H_x P)[3 nu + 1] mod 4)/2, it lifts exactly when that vector lies in the image of the mod-2 symbol on the same coefficient box, and V_2 is the set of family coordinates c(z) that pass.
• Builds the kernel family H_i = x^s (1 + x^3)^i (1 + x)^(2^b) with b = ceil(log_2(3R - 1)) and i = i_0 + 2j, j = 0..K, and extracts from V_2 its minimal-degree element g_D and its top degree C_D.
• Checks eight statements per row. V_2 = g_D F_2[z]_(<= C_D - deg g_D), a divisor-plus-ceiling window. The generator g_D = z^m c_t(z^(2^e)) with c_t the F_2 Fibonacci polynomials c_0 = 1, c_1 = 1 + y, c_t = y c_(t-1) + c_(t-2) and 2t + 1 = J(k), k + e = b - 1. The ceiling C_D = K - 2J(e-1) when k is even and C_D = K when k is odd, with L_2 = C_D - deg g_D + 1. The family shift law H^(j+1) = psi H^(j) - 2 Z_j with psi = t^3 + 2 + t^(-3) and Z_j = H^(j) + (t^3 H^(j) AND t^(-3) H^(j)), together with Z_j palindromic and inside the box, A(Z_j) in the image, and ob(X_(j+1)) = Lambda ob(X_j) + A(Z_j) for Lambda = S + S^(-1) folded at the centre. The slot tent identity. The reach law. The floor identity off the escaping rows. And the corrector law C_D - deg g_D = min(K - deg g_D, floor(reach/3)) with reach = R - jmax, jmax the least index whose columns of the mod-2 symbol already span the generator's obstruction, whose statement is now read from the closed form rather than from the span test.
• The slot tent. Write g = |2R - 2^(b-1) - 1|, e = min{e >= 1 : J(e) >= (g+1)/2}, k = b - 1 - e, and give the slot its length N = J(e) - J(e-1), its offset u = (g+1)/2 - J(e-1) - 1 and its position p = u above the octave centre R = 2^(b-2), p = N - 1 - u below it. The tent identity is min(p, N - 1 - p) = C_D - deg g_D: Law E's window length is the distance to the nearer end of the slot, in the slot's own coordinate.
• The reach law. With w = min(p, N - 1 - p), reach = 3w + 2 [e even] + [k odd] (1 + (p mod 2)), and p == R mod 2 whenever e >= 3, so the parity term is the parity of R. One row per odd octave escapes it, the e = 1 row above centre, D = 4^m + 3, where reach = 5 for m >= 2 and reach = 3 at D = 7.
• What the reach law buys. Off the escaping rows floor(reach/3) = w + [k odd and e even]; on them it reads 1 against w = 0, and there the cap K - deg g_D is 0 too, so min(K - deg g_D, floor(reach/3)) = w = C_D - deg g_D at every row and the branch of the min is a slot statistic and not a measurement: the floor binds strictly iff k is even and e >= 2, the cap binds strictly iff k is odd and e is even or D = 4^m + 3, and they tie otherwise.
• What it does not buy. The psi-orbit bound is capped by the family and reads C_D - deg g_D >= min(K - deg g_D, floor(reach/3)); the uncapped C_D - deg g_D >= floor(reach/3) is false wherever the cap binds strictly, at D = 25 (K = C_D = deg g_D = 0, jmax = 9, reach = 3, so 0 >= 1) and at 29 of the 115 rows D = 23..251. So only the >= half of the ceiling law becomes a consequence, on the strength of two swept laws and not of a proof, and one half stays open: that z^(w+1) g_D does not lift. Only the corrector law's statement is span-test-free; the deduction is not, since reach is defined by the span test and the psi-orbit needs its corrector valuation maximal.
• Censuses which branch of that min binds, and asserts that the floor-strict rows are exactly the rows with C_D < K. Prints the residue classes of R mod 4 and mod 8 in the swept range.
• Runs the same obstruction with the corrector taken out of the coefficient box, where the image has corank exactly 1 and every family obstruction meets it, so the unboxed layer-2 window is the whole mod-2 kernel.

RUN

uv run python research/lab/py/smith-window/smith_window.py

About seventy seconds: 799 rows of the default domain D = 5..1601 plus the unboxed sweep over odd D = 5..601, 299 rows. Two integer arguments set a different odd range; 1603 2401 is 400 rows and about eight minutes.

WITNESSES

• claims/2-adic-smith-cascade.md: the layer-2 window structure V_2 = g_D F_2[z]_(<= C_D - deg g_D), the generator z^m c_t(z^(2^e)) and the ceiling C_D, at 1199/1199 rows of odd D = 5..2401.
• The family shift law and the exact intertwiner ob(X_(j+1)) = Lambda ob(X_j) + A(Z_j), at the same 1199 rows.
• The slot tent identity and the reach law, at the same 1199 rows and at the far rows D = 4099 and D = 16387 of the escaping family; the floor identity floor(reach/3) = C_D - deg g_D + [k odd and e even] off the escaping rows, with those rows read as 1 against C_D - deg g_D = K - deg g_D = 0; and the corrector law read from the closed form, with the branch census in floor, cap, tie order: 448, 424, 327 over all 1199, 216, 258, 325 over the default 799, and 340, 284, 176 over the 800 out-of-sample rows; over odd D = 5..401 it is 48, 61, 90 and over 403..601 it is 60, 38, 2.
• The tent identity and the closed-form corrector law are pure slot arithmetic once Law E is granted, and hold at 999999/999999 rows of odd D = 5..2000001.
• spectra, THE TENT IDENTITY: the statement, its eight lemmas and the upper half free wherever C_D = K, with the free-against-open census the sweep asserts and prints, 583 against 216 over the default 799 rows and 751 against 448 over the 1199 rows of odd D = 5..2401, the open count equal to the floor-strict count on the branch line.
• The unboxed image of corank exactly 1 and the collapse L_2 = L_1, at 299/299 rows of odd D = 5..601.

NOTE

• The slot, the window bounds and the closed forms are the shelf lane's arithmetic line for line. What is independent here is the object side: the kernel family, the mod-4 symbol, the obstruction, the extraction of V_2, and the reach law.
• The reach law was read off the 399 rows D = 5..801 and never adjusted after. Its out-of-sample support is the 800 rows D = 803..2401, swept cold, plus the single rows D = 4099 and D = 16387, and the escaping family D = 4^m + 3 was visible inside the fit at D = 7, 19, 67, 259 and predicted at D = 1027, D = 4099 and D = 16387 before any sweep reached them.
• The min in the corrector law was chosen after the 5..401 overshoot, so 5..401 is in sample for it too; with the reach law that min is no longer a choice but a computation, and the branch census is predicted from (e, k) at every row swept.
• Dropping the 2 from psi breaks the shift law at D = 29, 31, 47, 115, 251; the pair (psi, Z_j) has to move together.
• The unboxed sweep bounds what an untruncated argument can see, and nothing more: corank 1 is the cokernel dimension the one-class window lemma already records.
• Nothing here is fitted. Every printed count is an equality test between two exactly computed objects, and a failure prints its D.