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.