# 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.