Diagonal slice ladder

• 2026-08-28 [Verified] The even half at base 7 carries exact Collatz-Wielandt certificates rho_dim < fill/7 at every even dim <= 26 and at dim = 172, 174, anchored by P(1) = 6^(dim-1)(dim+6) and a brute-force digit enumeration of P, with depth K_min = 0, 1, 2 stepping at dim = 4 and dim = 26 (and 3 at dim = 174), V(level) > 0 at every even dim = 2..40, level <= 25 with no dip, and c == 0 mod 7 immune at depth 0, so Sigma_K depends on c mod 7^(K+1). Witness: slice-sign-even-half.
• 2026-08-28 [Verified] An off-centre diagonal slice cannot break the alternation: a fixed target offset k changes only the initial vector, the transfer matrix commutes with carry reflection and e_0 is even, so e_k^T M^level e_0 = (1/2)(e_k + e_(-k))^T M^level e_0 and the odd component is annihilated, the off-centre slice seeing only the central even block; the boundary condition matches direct polynomial multiplication in 24 cases at dim = 2..7, level = 1..4, exact counts to level = 80 obey minimal rational recurrences on every term, and the dominant root equals the central one at every dim = 2..12 and offset |k| <= 2, the only movement being transient zero modes at dim = 4, |k| = 2 and dim = 2, |k| = 1, 2 that raise the order without touching the growth rate; offsets scaling with 3^level are untested. Witness: slice-recurrence-order.
• 2026-08-28 [Verified] The anti-diagonal slice profile factors across the Kronecker product, P_(A (x) B)(t) = P_A(t^(side_B)) P_B(t), since r + c = side_B (r_A + c_A) + (r_B + c_B), giving the stationary product prod_(j<level) P(t^(base^j)) and its mixed-radix form, so one identity covers fractal slices, mixed-product slices and the dimensional ladder; exact on all words of length 2 and 3 at base 2, dim = 2, zero mismatches. Witness: lab/py/slice-ladder-controls.
• 2026-08-28 [Verified] The level-1 central diagonal slice of the base-3 Menger analog is the vertex set of the cube's central cross-section: the hypersimplex vertex counts 2, 6, 6, 30, 20, 140, 70 at dim = 2..8 match C(dim, dim/2) for even dim and C(dim, (dim-1)/2)(dim+1)/2 for odd, so the ladder starts on a polytope rather than on an analogy. Witness: lab/py/slice-ladder-controls.
• 2026-08-28 [Conjecture] The 2-adic Smith form of M_even at base 3 and odd dim has elementary-divisor valuations 0^r 1^p 2^e A with exactly one divisor >= 2^3 and sum a_i = v_2(det), so v_2 = nullity + #{a_i >= 2} + max(a_max - 2, 0) reduces the uniform bound v_2 <= n to the tent rank law nullity_2(M_even) = min_(t in T) (|dim - t|/2 + 1), T = {2J(k)+1, 2J(k)+3 : k >= 2}, J(k) = (2^k - (-1)^k)/3 (exact 255/255 at odd dim = 3..511, peaks J(k-1) at dim = 2^k + 1, hence nullity <= ceil(n/3), troughs at the odd dim with 3 dim nearest a power of 2, so the rank deficiency measures the 2-versus-3 carry mixing) plus a small-excess bound whose constants are domain-limited: max a_i <= 9 and #{a_i >= 2} <= 5 hold at odd dim = 5..121, but max a_i <= 9 first fails at dim = 127 (12 by 511), #{a_i >= 3} = 1 at dim = 175 (reaches 5) and #{a_i >= 2} <= 5 at dim = 183 (reaches 21); the mod-2 form is P == (1+t)^(2D-3)(1+t^3) with 1 + Dt + t^2 == 1 + t + t^2 irreducible, dim = 7 is a tent trough with the whole valuation in the lone big divisor (a = {0,0,0,7}, max a_i = 7 > n = 4, a size effect), dim = 5 (a = {0,0,4}) is the only other v_2 > n at odd dim = 5..121, equality holds at dim = 9, 15, and the fold puts the 2-content in the even block because the palindromy row c' = 0 is 2 P[dim+c] entrywise and at dim = 7 the odd block is 2-adically unimodular. Witness: lab/py/smith-cascade; slice-sign-even-half.
• 2026-08-28 [Conjecture] The base-7 certificate extends with logarithmic depth and no transient to every even dim = 2..40, and the depth-death law is asymptotic rather than exact: both measured breakpoints land one even step early of dim = 2 ceil(7^(K+1)/4) + 2 because the depth-K positivity frontier f_K(dim) is still climbing when the window edge reaches it, so the corrected law reads K_min = min{K : (dim-2)/2 <= f_K(dim)}.
• 2026-08-28 [Conjecture] The central diagonal slice census of the base-3 dim-dimensional Menger analog obeys a linear recurrence of order exactly ceil(dim/2): the digit polynomial factors as P(t) = (1 + t^2)^(dim-1)(1 + dim t + t^2), the carry map c' = (c + dim - s)/3 contracts to {|c| <= floor((dim-1)/2)}, the symmetry v -> (2,...,2) - v gives P[s] = P[2 dim - s], so the Krylov subspace from e_0 sits in the reflection's +1 eigenspace of dimension floor((dim-1)/2) + 1 = ceil(dim/2) and the order bound holds at every dim; exactness is checked at dim = 2..14 by distinct eigenvalues of M_even with minimum gap above 6.9 and at dim = 2..24 by nonzero Hankel determinants, and is open for general dim (a square-free characteristic polynomial); controls: dim = 2 gives 2^level at order 1, dim = 3 gives 6, 42, 306, 2250, 16578, 122202 and A299916's 9a(n-1) - 12a(n-2) at order 2 by a route that never mentions a hexagon, dim = 4 gives 6, 132, 1848, 29040, 441408, 6772128 at order 2 with dominant root (11 + sqrt(385))/2, dim = 5 gives 30, 1000, 35700, 1321600, 49786200, dim = 6 gives 20, 4030, 242300, 24642700, and rational Hankel elimination on nine terms reads orders 1, 2, 2, 3, 3 at dim = 2..6. Witness: slice-recurrence-order; A299916.
• 2026-08-28 [Conjecture] Conjecture S, the sign law sgn(slice dimension - (solid dimension - 1)) = (-1)^(dim+1) at every dim: the even half at bases 3 and 5 and the odd classes dim != 1 mod 3 are settled on the shelf, the odd class dim == 1 mod 3 beyond dim = 80 is open, and the route through a named lemma, an explicit positive vector x_dim with sgn((M_even x)_i - (fill/3) x_i) = (-1)^(dim+1) at every index, has only the Perron vector, which supplies it numerically at every dim <= 50 with worst componentwise discrepancy 9.15e-46 at 90 digits, peaked at index 0 and non-increasing, with no closed form; its two silent hypotheses, a real spectrum (exact at dim <= 20, numerical at dim = 2..60) and no non-Perron eigenvalue crossing fill/3 (checked at dim = 2..60 against a 180-digit reference), are themselves unproved. Witness: slice-recurrence-order; slice-sign-even-half.
• 2026-08-28 [Conjecture] The second eigenvalue of the even carry block tracks the digit polynomial at -1: lambda_2 -> (-1)^(dim+1) 2^(dim-1)(dim-2)/3 = (-1)^dim P(-1)/3 with exponential convergence but never exactly (the characteristic polynomial is nonzero at that value in exact arithmetic at every dim = 2..40, so lambda_2 = -9007199254740992 = -2^53 to 22 digits at dim = 50, which is 2^49 * 48 / 3, is display rounding), hence rho/|lambda_2| -> (dim+2)/(dim-2) -> 1, measured 13/12 to 2.58e-22 at dim = 50 and 1.04081632653 at dim = 100, with |lambda_2|/rho = (dim-2)/(dim+2) to nine digits by dim = 36, so the spectral gap closes and no argument may assume a fixed one; the asymptote must not be quoted at small dim, where dim = 4 gives a true lambda_2 = -4.310708 against -16/3, a 19% gap consistent with an O(2^(-dim)) approach; measured at 420 to 650 digits. Witness: slice-recurrence-order.
• 2026-08-28 [Conjecture] The base-5 middle-digit analog (keep a cell when at most one coordinate is the middle digit 2) has P_5(t) = A(t)^(dim-1)(A(t) + dim t^2) with A(t) = 1 + t + t^3 + t^4, fill = 4^(dim-1)(dim+4) and carry rule c' = (c + 2 dim - s)/5, and sgn(slice dimension - (solid dimension - 1)) = (-1)^(dim+1) holds at dim = 2..15 down to a smallest excess of 1.055e-9 at dim = 15 (exact-integer sign sweep with 80-digit root refinement, agreeing with substitution-product convolution in all 16 cases at dim = 2..5, level = 1..4, the dim = 3 fill 112 of 125 matching the middle-digit count), yet A(-1) = 0 makes P_5(-1) = 0 for every dim, so the alternating mass that carries the base-3 explanation is absent while the alternation survives, and no mechanism yet survives that. Witness: slice-sign-even-half.
• 2026-08-28 [Conjecture] The alternation is universal across base-3 designs and its phase is not: over four families at dim = 2..7, Menger with at most one middle digit and P(-1) < 0 for dim > 2 gives -+-+-+-, at most two middle digits with P(-1) > 0 gives --+-+-+, Cantor with no middle digit and P(-1) = P(1) gives +-+-+-+, exactly one middle digit with P(-1) < 0 gives +-+-+ from dim = 3, so the phase tracks the sign of P(-1); the range stops at dim = 7 and the four sign patterns rest on a prose table alone.
• 2026-08-28 [Conjecture] The excess rho_dim - fill/3 decays at the rate r_inf = 1/prod_(k>=2) cos(2pi/3^k) = 1.3461220067642173 per dimension on the eigenvalue scale, 2 r_inf = 2.6922450 on the dimension scale, with a linear prefactor |delta_dim| ~ A (dim-1) r_inf^(-dim), A -> 2/(3 prod cos) = 0.89741, from the 3-adic angle-tower product formula, matched within 1e-8 by exact rational bisection at dim = 61; a third-order Richardson fit in 1/dim over dim >= 60 at 320 digits gave the one-step ratio 0.742874554813847413, r_inf = 1.34612251727283689 (seven true digits, the rest fit residue), even and odd extrapolations 4.5643e-8 apart and A ~ 0.897520192686, and the shape check (6A/ln 3)(dim/(dim+2))(2 r_inf)^(-dim) = 1.47e-21 at dim = 50 against the measured 1.42672e-21 tests the form and not the constant. Witness: slice-recurrence-order.
• 2026-08-28 [Conjecture] Conjecture S reduces to one separation lemma along an explicit chain: with M_even the reflection-even block of the carry automaton M[c,c'] = P[c + dim - 3c'], P(t) = (1+t^2)^(dim-1)(1 + dim t + t^2) and fill = P(1), if every non-Perron eigenvalue of M_even has modulus below fill/3 then sgn det(fill/3 I - M_even) = sgn(fill/3 - rho), and that determinant sign is (-1)^dim, exact in integer arithmetic at dim = 2..20 and to dim = 40, which is Conjecture S; the separation hypothesis is checked at dim = 2..60 and not proved, the row-sum lemma feeding it holds for the full carry matrix on states c = 0..dim and is false in the recurrent even basis (dim = 3 row sums (12,4) against the formula's (8,6)), and the product formula settles the odd half without separation, so this chain is a route to the even half only. Witness: slice-recurrence-order; slice-sign-even-half.
• 2026-08-28 [Conjecture] The balanced-mask homotopy reduces Conjecture S to a one-variable determinant inequality and owes two lemmas: with a_dim = (-1)^(dim-1)(dim-1) and Q_dim = P_dim - a_dim t^dim, the root-of-unity identity forces 1 + t + t^2 | Q_dim, and fill/3 I - M_dim = L_dim + a_dim K_dim exactly with K_dim = I/3 - E_dim, E_dim the dilation 1_(j=3i); then f_dim(z) = det(L_dim + z K_dim) = z h_dim(z) and S becomes h_dim(a_dim) < 0, exact at dim = 2..30; det L_dim = 0 is exact to dim = 30 but does not follow from residue balance, because N_dim lacks constant column sums in the unnormalised even basis after truncation and folding, and coefficient negativity of h_dim, which settles every odd dim since a_dim > 0 there, is useless at even dim where a_dim = -(dim-1) is negative, so a uniform root bound is still missing.