# The slice ladder: rate and dead routes

- 2026-08-28 [Conjecture] The roots-of-unity circulant correction `Q_dim = fill/3 + (2/3)(-1)^(dim-1)(dim-1) cos(2 pi dim / 3)` is worse than bare `fill/3`: mean absolute error `11.05299392007777488084818` against `0.1511524922667271359757626` over `dim = 2..50`, a factor `73.1248`, with `|rho - Q_dim| / |rho - fill/3|` reaching `1067922.7` at `dim = 50`; its sign matches `(-1)^(dim+1)` only when `dim = 0 mod 3`, 16 of 49 cases; the order-`dim` root-of-unity term is cancelled by finite-boundary effects, the empirical correction factor collapsing to `-9.363982332e-7` at `dim = 50`, so any model of the excess must derive the boundary cancellation.
- 2026-08-28 [Conjecture] The three-block DFT decomposition of the carry matrix does not exist: the span of `1`, `omega^c`, `omega^(2c)` is not invariant for any `dim >= 5` (relative Frobenius residual `0.318` to `0.523`), carry residues mod 3 are coupled for every `dim >= 3` so `M` does not commute with `diag(omega^c)`, and the three-root average `(P(1) + P(omega) + P(omega^2))/3` misses the Perron root by `-9.191` to `+11.263` while the true excess is `-0.0435` at `dim = 20`; the mod-3 block version has off-diagonal Frobenius mass of order one, ratio `0.609` to `4.111` over `dim = 3..30` with slope `-0.00162 +- 0.00773` per `dim`, `p = 0.836`, and a Schur correction at `fill/3` positive for every `dim`, `0.343` of `fill/3` at `dim = 30`; one exact row-sum identity survives.
- 2026-08-28 [Conjecture] The saddle-point route is closed: the transfer operator is coefficient decimation, `(Mv)(c) = [t^(c+dim)] P_dim(t) V(t^3)` on a finite carry window, not multiplication by a scalar symbol; geometric vectors `z^c` are not eigenvectors, the all-ones vector is the only reflection-even one and is not an eigenvector either; on the unit circle `max|P_dim(e^(i theta))| = P_dim(1) = fill`, whose cube root is exponentially smaller than `rho_dim`, while `max|P_dim|/3` is exactly `fill/3` and misses the whole effect; no non-tautological `f_dim(theta)` with `rho_dim = max|f_dim|` was found.
- 2026-08-28 [Conjecture] Induction on `dim` is closed from both ends: the same-size correction between `M_dim` and `M_(dim+1)` at odd `dim` has full rank at every `dim = 3..19`, determinants from `-54` at `dim = 3` to `-441065669103434214513656226772598887664331096` at `dim = 19`, so the matrix determinant lemma has no low-rank update to consume; the threshold determinant sequence `d_dim` satisfies no recurrence surviving holdout - constant-coefficient orders 1 to 6 with degrees 0 to 5 on the full sequence and each parity subsequence, all 62 identifiable holonomic pairs with `r s <= 40`, normalisation by `fill^n` and by `dim^beta` for `beta = -4..4`, Berlekamp-Massey over three primes at maximal linear complexity (20 for 39 terms, 10 per parity), the one determined fit (order 4, degree 2, odd subsequence) failing at `dim = 35, 37, 39`; the 2-adic valuation of `d_dim` fits none of the tested elementary forms.
- 2026-08-28 [Conjecture] Cauchy interlacing is closed: over all 27 pairs `2 <= dim <= 28`, `M_even(dim)` is not the upper-left block of `M_even(dim+2)` and none of the `(n+1)^2` row-column deletions of the larger matrix is the smaller, so no bordering `u`, `v`, `alpha` exist; the eigenvalue chain holds for every even start and fails for every odd one, witness `lambda_1(3) = 7.372281323269` against `lambda_2(5) = 16.965208741322`; the threshold count it was meant to prove is nevertheless exact on `dim = 2..30` - no eigenvalue above `fill/3` at even `dim`, exactly one at odd `dim`.
- 2026-08-28 [Refuted] The decay rate of the slice-dimension excess is `3/4`, `4/3` or `8/3` - at 320 digits over `dim = 2..100` the eigenvalue-scale one-step ratio extrapolates to `0.742874554813847413`, residual `0.00712544518615` from `3/4`, and `r_inf = 1.34612251727283689`, residual `0.0127891839395` from `4/3`, both far outside the `4.5643e-8` parity split and the fit-order spread; on the dimension scale `2 r_inf = 2.6922450` against `8/3 = 2.666667`; the coarse `dim <= 50` reading `0.373`, inverse `2.68`, and the sentence "the per-dimension factor approaches `3/4` from above" conflate the two scales; the constant is identified as `prod_{k>=2} cos(2 pi/3^k) = 0.7428747134`, within `1e-8` at `dim = 61`. Witness: slice-recurrence-order.
- 2026-08-28 [Refuted] The excess has the clean shape `slice dimension - (solid dimension - 1) = (-1)^(dim+1) C r^(-dim) + o(r^(-dim))` with a constant `C` - `|delta_dim| r_inf^dim` climbs from `52.4976468882` at `dim = 60` to `88.3872395676` at `dim = 100`, a log-linear fit puts the prefactor at `dim^1`, and the form is `|delta_dim| ~ A dim r_inf^(-dim)` with `A ~ 0.897520192686`; the linear factor is the parity factor `dim - 1`. Witness: slice-recurrence-order.
- 2026-08-28 [Refuted] Conjecture S stated against the threshold `base^(dim-1)` - at base 3, `sgn(rho_dim - 3^(dim-1))` matches the hypothesised sign in 25 of 49 cases over `dim = 2..50`, is wrong already at `dim = 3` where the difference is `-1.627718676730986`, and exact real-root counting finds no eigenvalue above `3^(dim-1)` at any `dim = 2..20`; at base 5, `rho_dim < 5^(dim-1)` at every tested `dim`, `rho_3 - 25 = -1.5341439003`; the only threshold that carries the statement is `fill/3`, the eigenvalue-scale form of `d - 1`. Witness: slice-recurrence-order.
- 2026-08-28 [Refuted] The even carry block has exploitable matrix structure - over `dim = 2..20` the banded-plus-low-rank form does not exist (Toeplitz displacement rank equal to the full dimension from `dim >= 5`, tridiagonal remainder of rank `n-1` at odd and `n` at even `dim`), total nonnegativity fails for every `dim >= 4` with an exact negative minor per row, only `dim = 2` is symmetric and only `dim = 3, 4` are positively diagonally symmetrizable (weights `(1,6)`), and `M_even - (fill/3) I` is Metzler rather than a Z-matrix, so the M-matrix route is circular; the matrices have `n` distinct real roots at every tested `dim`, which is spectrally useless. Witness: slice-recurrence-order.
- 2026-08-28 [Refuted] A simple positive test vector certifies the Collatz-Wielandt bound - the all-ones vector has ratios mixed around `fill/3` for every `dim = 3..50` (`dim = 2` excepted, that matrix being one by one), one-parameter cosine, alternating and centred-quadratic corrections succeed only at `dim = 2, 3, 4`, and the Gaussian `exp(-3 i^2 / dim)` and binomial-centre profiles only at `dim = 2`; the Perron vector certifies at every `dim <= 50` (worst discrepancy `9.15e-46`), is peaked at index 0 and monotone non-increasing rather than bell-shaped, and has no closed form. Witness: slice-sign-even-half.
- 2026-08-28 [Refuted] A cheap route proves the spectral separation `rho_dim/|lambda_2| -> 1` - the common-amplitude Gaussian kernel predicts a limiting ratio 9 where the truth is `(dim+2)/(dim-2) -> 1`, discarding an order-`dim` parity modulation; the zero-shift `2x2` Schur complement has median relative error 0.625 over `dim = 2..40` and worst 0.9998, deteriorating with `dim`; the Perron profile peaks at index 0 for all 39 tested `dim`, a boundary-centred half-Gaussian at median `R^2 = 0.99999`, not near `dim/6`; the second eigenvector has one sign change at every even `dim` and at `dim = 3, 5` but several at every odd `dim` from 7 to 39; separation holds numerically to `dim = 60`, and a proof must be uniform in a margin of order `4/dim`. Witness: slice-recurrence-order.