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.