# The tent rank law

- 2026-08-28 [Proved] The palindromic module reformulation: mod 2 at odd `dim` (`R = (dim-1)/2`, `n = R+1`, `M = 6R+2`) the symbol is `G = (1+t)^(4R)(1+t+t^2)` with 4-block coefficients `C(R, a)`, and `nullity_2(M_even) = dim{H : no exponent == 1 mod 3, deg H <= M, G | H, t^M H(1/t) = H}` - palindromy folds the `R+1` kernel conditions onto one residue class mod 3; the explicit kernel vectors `H_(b,i) = t^s (1+t^3)^i (1+t)^(2^b)` with `s = (M - 3i - 2^b)/2`, `i` even, `i + 2^b >= 4R`, `3i + 2^b <= M` are independent since `3R - 1 <= 2^b <= 6R - 4` forces a unique `b`, and number `tent(dim)`, so `nullity_2 >= tent(dim)` with troughs exactly where `3R` is adjacent to a power of 2; the reversal involution on the `t^3`-chain of the single generator gives `nullity_even = ceil(nullity_full/2)` exactly; the staircase submatrix (rows `c' = R - j`, leftmost pivots at `R - 1 - 3j`) yields only `floor((R-1)/3) + 1` independent rows, the wrong third of the rank; without the `(1+t+t^2)` hypothesis the valuation lemma yields only `tent + 1` (witnesses `R = 2, 4, 7`); matrix nullity equals `tent` at 21 `dim` through 511 and at every odd `dim = 3..401`, the module identities to `R = 1024`. Witness: slice-sign-even-half.
- 2026-08-28 [Proved] Lemma M, sharp with equality: for every `d >= 2` and `r in {0, 1, 2}`, the maximal `(1+t)`-valuation over nonzero `H in F_2[t]` with no exponent `== r mod 3` and `deg H <= d` is `mu_r(d) = max_(2^b <= d) [floor((d - s_0 - 2^b)/3) + 2^b]`, `s_0 = (r + 2^b) mod 3`, attained by `t^(s_0)(1+t^3)^i(1+t)^(2^b)` - the Frobenius split `H = A^2 + t B^2` gives the exact case law `v(H) = 2v(A) / 2v(B) / 2 min / 2w + 1` (the equal-valuation case forced by `C' = B_1^2`, a unit at 1), classes move `r -> (2r, 2r + 1)`, the recursion `M_r(d) <= Phi(M_(2r)(floor(d/2)), M_(2r+1)(floor((d-1)/2)))` with `Phi(X, Y) = max(2X, 2Y, 2 min(X, Y) + 1)` has the closed form as supersolution by lifting the child's maximising Frobenius block `b -> b + 1` (six integer inequalities, `X = Y` forcing `b_e = b_o` by a numerator gap `>= 2^(min+1) - 3`, finite windows `d = 5..12` with 24 evaluations, 8 tight, and base cases `d = 2..4`); the corollary Lemma M' for `(1+t+t^2) | H` is `M'_r(d) = M_r(d - 3) + 1` for `d >= 5`, the cheapest purchase of valuation being one Frobenius block plus `(1+t^3)` padding at exchange rate 3:1, which is where `sup nullity/n = 1/3` comes from; brute-forced to `d <= 16000` (failure set exactly the four `d < 2` pairs), the upper bound certified independently by full rank of Lucas submask matrices at `d = 1023..8193`, the supersolution tight at 4926 points up to `2^60` with minimum slack 0, the case law exact on all `H < 2^17`, and the true minimum of the module `Y` computed at 204 `R` up to 1025. Witness: slice-sign-even-half.
- 2026-08-28 [Proved] The one-class window lemma, by the parity of an index: `Y = Z ∩ G F_2[t]` is `F_2[t^3]`-free of rank 2 with generator degrees `delta_1 < delta_2` in distinct classes mod 3 and `delta_1 + delta_2 = 12R + 5` exactly - truncation counting gives `dim_(F_2) Z/Y = (delta_1 + delta_2 - 2)/3`, the projection onto the missing exponent class identifies the cokernel of `Z -> F_2[t]/(G)` with `F_2[u]/gcd(A_0, A_1, u A_2)` of dimension exactly 1 (since `(1+t^3) | G` but `(1+t^3)^2` does not), so `dim Z/Y = deg G - 1 = 4R + 1`; the sum is odd, so `delta_1 <= 6R + 2 = M < delta_2` in two lines, margins 0 and 2 impossible and margin 1 iff `delta_1 = M`; explicitly `{delta_1, delta_2} = {12R - 2A + 2[a even], 2A + 3 + 2[a odd]}` with `a = floor(log_2(4R - 1))`, `A = 2^a`, from `A + 2 <= 4R <= 2A`; hence `nullity_2(M_full)(dim) = floor((M - delta_1)/3) + 1` and `nullity_2(M_even)(dim) = ceil(nullity_full/2)` in closed form for every odd `dim`; the identity generalises as `delta_1 + delta_2 = 3(deg G - deg_u gcd) + 2` at 400 random `G`, the closed form holds to `R = 200000`, margins below 5 lie in `{1, 3, 4}` exactly as parity predicts, `R = 683 = J(11)` has `delta_1 = M` (margin 1) and `R = 1365 = J(12)` margin 3. Witness: slice-sign-even-half.
- 2026-08-28 [Proved] The Jacobsthal tent rank law, entire: for every odd `dim = 2R + 1 >= 3`, base 3, middle-digit design, `nullity_2(M_even)(dim) = tent(dim) = 1 + dist(R, {J(a), J(a) + 1})` with `a = floor(log_2(4R - 1))`, and sharply `nullity_2(M_even) <= ceil(n/3)`, `n = (dim+1)/2`, with equality exactly at `dim in {3} ∪ {2^(2j) + 1}`; with `m = R - J(a)` all four (parity of `a`) x (branch) cells reduce to `nullity_full = 1 - 2m` (`m <= 0`) or `2m` (`m >= 1`), the parity of `a` cancelling completely, and halving gives `nullity_even = 1 + d_a(R)`; the nearest-trough index is `a` itself (an `a - 1` reading was rejected exhaustively), margins exactly 1 at both window endpoints propagate by 1-Lipschitzness, `R = 1` is the sole reason the cap is `ceil` rather than `floor`, and the odd-`a` peaks miss by exactly 1; closed form equals tent equals the real transfer-matrix nullity at every odd `dim = 3..1401`, closed-form checks to `R = 500000` with points to `2^60` (argmin strictly unique everywhere), no residue family past the cap (max excess 0); so `v_2(det M_even) = tent(dim) + X(dim)` with the tent capped at `ceil(n/3)`, and base-3 strictness rides on the cascade layers `X(dim)` alone. Witness: slice-sign-even-half.