# Conjecture S: even half

- 2026-08-28 [Proved] The reduction chain for Conjecture S runs at every odd `base`, middle-digit design, `dim >= 2`: the digit polynomial is palindromic with strictly positive support; the transfer recursion `hat u_(level+1)(psi) = (1/base) sum_r Phi(y_r) hat u_level(y_r)` holds with exact phase cancellation (the palindromic centre is the carry offset `dim m`); the step identity is `W_k = (-1)^(dim-1)(dim-1) base^(k-1) V(k-1)` with `V(level) = base m0(level) - b(level)`; the carry core has reachable set exactly `{|c| <= floor((dim-1)/2)}`, is irreducible and aperiodic, and its Perron root is `rho_dim`; eventual contraction (`V(level) >= 0` for all `level >= level_0`, even `dim`) implies `rho_dim <= fill/base`; and `det(fill I - base M_even) == fill^n mod p` for any `p | base`, primality unused, gives strictness whenever `p nmid fill`; the Perron-Frobenius asymptotic is unnecessary (`b(level) >= (M^level)[0,0]` suffices) and the base-3 contraction hypothesis weakens to its eventual form since the odd-`level` dip is a finite transient. Witness: slice-sign-even-half.
- 2026-08-28 [Proved] The exact two-step reduction `9 b(2j+2) - fill^2 b(2j) = -(dim-1)[fill V(2j) + 3 V(2j+1)]`, the two-step weight identity `C_2(y) = g_5(2y)` (the base-3 two-step symbol is the base-5 symbol, the comb constant having minimal polynomial `x^3 - 9x - 9`), and the orbit identities `G(3t) = cos(t) G(t)` and `Ntilde(3^a pi) = ((dim-2)/(dim+2))^a Ntilde(pi)`. Witness: slice-sign-even-half.
- 2026-08-28 [Proved] The Collatz-Wielandt certificate closes the even half per dimension at any odd base, strictness included: `B >= 0`, `x > 0`, `Bx < theta x` componentwise imply `rho(B) < theta` with no irreducibility needed; with `beta_K = (M^T)^K 1 > 0` (positive by column-sum positivity alone, `colsum(c) = (fill + (-1)^(dim-1)(dim-1)(base[base|c]-1))/base > 0`), if `base beta_(K+1)(c) < fill beta_K(c)` for every `|c| <= (dim-1)//2` then `rho_dim < fill/base` strictly, bypassing the mod-`p` determinant lemma and every exceptional class; the base-5 mass identity on the right Perron vector of the full core is `5 rho_dim = fill - (dim-1)(5 p_dim - 1)`, so the even half is `p_dim > 1/5`, and the left-vector reading is false at `dim = 8` (`p_LEFT = 0.1428 < 1/5`); `K = 2` certificates are exact at `dim = 16, 30, 44, 60`. Witness: slice-sign-even-half.
- 2026-08-28 [Verified] The row certificate `v^T M^t >= 0` is sound - `V(level) = sum_j (v^T M^t)_j u_(level-t)(j)` with both factors nonnegative, so one integer `t` with `v^T M^t >= 0` entrywise plus the exact prefix `V(0..t-1) >= 0` proves `V(level) >= 0` for all `level`, monotone in `t` - and at base 5 its minimal depth is `t(dim) = max(1, ceil(log_5(2 dim - 3)) - 1)`, breakpoints exactly at `R = (5^(k+1)-1)/4`, checked to even `dim = 400` with fresh rows at 150, 250 and the boundary `314|316`; the `782` at `k = 4` is an extrapolation unobservable below `dim = 1566`. Witness: slice-sign-even-half.
- 2026-08-28 [Proved] The 2-adic strictness lemma, complementary to the mod-`p` lemma: `det(fill I - base M_even) == (-base)^n det(M_even) (mod fill)` by principal-minor expansion (every `k < n` term killed by `fill^(n-k)`), so a prime `p | fill`, `p nmid base` with `v_p(det M_even) < v_p(fill)` forces `det(fill I - base M_even) != 0`, that is `rho_dim != fill/base`, which upgrades a certificate's `<=` to `<`; the two lemmas' silent classes (`p | base` against `p | fill`) are complementary; at base 5 `v_2(fill) = 2(dim-1) + v_2(dim+4)` while `v_2(det M_even) <= 26` out to `dim = 156`, so the test holds everywhere including the exceptional class `dim == 6 (mod 10)` to `dim = 156`, and at base 3 the class `dim == 4 (mod 6)` to `dim = 118`; the 5-adic side is large and erratic and the mod-25 angle is dead; open: a uniform bound on `v_2(det M_even)`, `<= n` sufficing for all even `dim >= 4`. Witness: slice-sign-even-half.
- 2026-08-28 [Proved] The even half of Conjecture S at base 5 is a theorem entire: on the base-5 middle-digit solid `rho_dim < fill/5` for every even `dim >= 2`, hence `slice dimension < solid dimension - 1` at every even `dim` - the Fourier form `beta_K(c) = 5^(-K) sum_n F_K(n) e^(2 pi i n c/5^K)` (the product-formula phase cancellation iterated), the exact telescoping `fill beta_K - 5 beta_(K+1) = 5^(-K)(dim-1) Sigma_K(c)` with the `fill`-power leading terms cancelling identically, the frequency-separation lemma `Q_K(n)/Q_K(1) <= 0.768` for every `n != +-1` (three-branch residue analysis in rigorous intervals, maximum `0.7679580`, read `0.7679541` in an earlier check, the `j >= 3` branch exhaustive at `j = 3, 4, 5`), and the criterion at depth `K(dim) = Theta(log dim)`, analytic for even `dim >= 18` (`K(dim) >= 2` for `dim >= 34` since `4(dim-2) >= 128 > 25`) with exact integer certificates below; the log depth is necessary, every fixed `K` dying at `dim = 16, 66, 316` for `K = 1, 2, 3`; the criterion holds directly at every even `dim = 34..600` and at every depth transition to `dim = 10^6`; the death law `dim = 2 ceil(5^(K+1)/4) + 2` is known at three depths only; with the odd-`dim` theorem, Conjecture S at base 5 is settled everywhere except strictness at odd `dim == 1 mod 5`, exact through `dim = 80`. Witness: slice-sign-even-half.
- 2026-08-28 [Verified] The even half at base 3 holds per dimension for every even `dim = 2..102` and on the grid `106, 110, ..., 178` - `rho_dim < fill/3` with strictness at each, by the exact-integer Collatz-Wielandt certificate `3 beta_(K+1)(c) < fill beta_K(c)`, no determinant lemma and no exceptional class `dim == 4 mod 6` needed; `beta_K(0) = b(K)` exactly, so `K_min >= level*(dim) + 1` with `level*` the last level with `V(level) < 0`, and `K_min = level* + 1` or `+ 2` at every tested `dim`; spot depths `K_min = 8, 50, 140, 291` at `dim = 12, 30, 50, 72`; two implementations sharing no code reproduce all 36 rows to every digit, including the two non-monotone slack rows. Witness: slice-sign-even-half.
- 2026-08-28 [Verified] The odd half's last gap narrows to the same 2-adic bound: at base 3 and odd `dim == 1 mod 3`, where the mod-3 strictness lemma is silent, `v_2(det M_even) < v_2(fill) = (dim-1) + v_2(dim+2)` at every `dim = 13, 19, ..., 241`, silent only at `dim = 7` (`v_2(det) = 7 >= 6`, closed by the direct computation `det(fill I - 3M) != 0`), so `rho_dim != fill/3` on `13 <= dim <= 241` and, with `rho_dim >= fill/3` at every odd `dim`, `rho_dim > fill/3` strictly at every odd `dim <= 241`; the reference `v_2` rows on the two other classes read `1, 2, 3, 1, 4` (base 3, `dim == 4 mod 6`) and `2, 1, 2, 3, 2` (base 5, even); both exceptional classes of Conjecture S reduce to one uniform statement, an upper bound on `v_2(det M_even)`, and `v_2(det M_even) <= n = (dim+1)/2` at every `dim = 13..241` in the class, failing only at `dim = 7`, is exactly strong enough. Witness: slice-sign-even-half.
- 2026-08-28 [Proved] The even half of Conjecture S at base 3 is a theorem entire: `rho_dim < fill/3` for every even `dim >= 2`, hence `slice dimension < solid dimension - 1` at every even `dim`, base 3, middle-digit design, strictness included - by the Fourier/telescoping port to `base = 3` (the exact phase cancellation load-bearing, off-centre variants failing with integer witnesses), four nested frequency tracks (`+-1` at angle 0, the half-points `+-(3^(K+1)-1)/2` at the tripling fixed point `pi`; on-track prefixes nest, exits never return), exit-cost and window/subtree lemmas giving `E_K(dim) <= [4(K-1)(0.7528157^(dim-1) + 0.7052518^(dim-1)) + 2 * 0.2266816^(dim-1)] exp(2(K+1) 0.8900159^(dim-1))` for everything off the two leader pairs, and the criterion closing at `K_1(dim) = K*(dim) + O(1)`, analytic for even `dim >= 38` (182 interval-certified inequalities to `dim = 400`, monotone domination beyond, the margin term the true cosine deficit `delta(dim) = O(3^(-2 K_1))`, `< 4.1e-76` at `dim = 38`, since an absolute `1e-8` term fails at `dim = 399999998`), exact certificates below; checks: Fourier form to `9.4e-61`, telescoping to `2.6e-59`, tracks exhaustive over all `3 !| n < 3^10`, the subtree bound never exceeded (worst `sigma_5 = 1.023` against `8.97`), the E-bound dominating exact enumeration at all 35 `(dim, K)` points and at 28 fresh ones (`dim in {10, 14, 22, 26} x K in {3..9}`, worst ratio 19.74), class counts exact at `K = 9`, an exit-level sweep to `j = 60`, `dim = 4000` finding the caps asymptotically exact (worst attainment `0.999121`) but never breached, seven certified constants re-derived to 22 digits by exact interval arithmetic on a `10^-90` grid with outward rounding (a hand-rounded `0.6696` reads `0.66966`), and the theorem machine-checked in exact integers at `dim = 38, 40, 42` (the certificate holds at exactly `K_1`, fails at `0.8 K*`, `K_1 = K_min + 1` at all three); the chain's single global safety factor is 2 and the `h-exit(2)` attainment (two of four residues reach `C_H`) is load-bearing. Witness: slice-sign-even-half.
- 2026-08-28 [Proved] The base-3 transient is identified in closed form: `level*(dim)` is the greatest odd integer `<= K*(dim)`, `K*(dim) = [(dim-1) ln R + s_dim]/ln((dim+2)/(dim-2))`, `R = prod_(i>=2) cos(pi/3^i)/cos(2 pi/3^i) = 1.2553249438...` - the half-point frequency rides the `pi` fixed point with per-level magnitude advantage `cos(pi/3^i)/cos(2 pi/3^i) > 1` against per-level amplitude cost about `(dim-2)/(dim+2)`, its sign alternates as `(-1)^K` (that is the odd-`level` dip), and the crossing is the transient - so the certificate depth constant is `ln(R)/4 = 0.0568486146...` and `K_min in [level* + 1, ceil(K*) + 2]` for even `dim >= 38`; exact on 58 of 58 `level*` rows, every even `dim = 6..120`, each one exhausted by proof and not by margin - `M >= 0` and `u_level = M^level e_0 >= 0`, so a single `t` with `(M^T)^t v >= 0` entrywise forces `V(level) >= 0` at every `level >= t` and no census window can truncate the answer - with towers `level* = 79, 97, 107` at `dim = 38, 42, 44` and `level* = 811` at `dim = 120`; a census carried only to a `4 dim + c` window is unsound past `dim` about 70 since `level*` is quadratic, but no row of `dim = 6..120` is in fact false; the column-sum identity `fill - 3 colsum(c) = (dim-1) v_c`, which is `prop:mass` by root-of-unity filtering and needs no per-row check, makes that row certificate the even-half Collatz-Wielandt test itself, so the stopping level is `K_min` exactly, `level* + 1` on 36 rows and `level* + 2` on 22; scoped to `dim >= 6` since `dim = 4` has no dip; at `dim = 10, 20` the half-pair carries the largest magnitude in the spectrum, the dominant pair only third. Witness: slice-sign-even-half, lab/py/base3-transient-exhaustion.
- 2026-08-28 [Proved] Base 3 is the unique hard base: the half-point track exists at strength `|A_base(-1)|/A_base(1)` per level with `A_base(-1) = 1 - (-1)^((base-1)/2)`, so the ratio is `1` exactly at `base = 3`, `0` at every `base == 1 mod 4` (the symbol dies at `pi`, the base-5 case) and `2/(base-1) < 1` at every `base == 3 mod 4`, `base >= 7`. Witness: slice-sign-even-half.
- 2026-08-28 [Verified] Exact Collatz-Wielandt certificates give `rho_dim < fill/9` at every even `dim = 2..56` and `rho_dim < fill/11` at every even `dim = 2..74` (machine-pinned to `dim <= 42` and `dim <= 60`), `K_min <= 2`, `V(level) > 0` everywhere, no transient, and base `9 = 3^2` inherits nothing from base 3; at `base = 7` the `K = 2 -> 3` step lands at exactly `dim = 174` as the frontier-race law predicts, the frontier `f_2 = 85` converged from `dim = 160`, the asymptotic death law landing there too; earliness (asymptotic death minus true death) is monotone down in `K` and up in `base` - the depth-0 death is `dim = 4` at every base, so base 5 is one even step early at `K = 0` (4 against 6) and exact at `K = 1, 2`, `K_0(5) = 1`; `base = 7`: 1, 1, 0 steps; `base = 9`: 2, 1; `base = 11`: 2, 2 - the asymptotic law being exact for all `K >= K_0(base)`; at `base = 9` the window edge is immune when `h == 0 mod base` (`dim = 20, 38`), so tightness must be stated mod `base`; the 12 printed constants of the base-3 chain are asserted against interval endpoints and printed by ceiling. Witness: slice-sign-even-half.
- 2026-08-28 [Conjecture] The row certificate's sign law at base 5 is periodic, not one-sided: `(v^T M^k)_j >= 0` iff `dist(j, 5^(k+1) Z) <= (5^(k+1)-1)/4` (witness `dim = 40, k = 1, j = 19` positive), the threshold being the carry-drift radius around every multiple of `5^(k+1)`, not just around 0. Witness: slice-sign-even-half.
- 2026-08-28 [Conjecture] Three negatives on the base-5 even half: quintupling resummation is structurally empty, the two orbit relations of the base-5 tower summed over complete residue systems returning exactly the one-step identity `5 b(level+1) = fill b(level) - (dim-1) V(level)`, the nontrivial comb mapping into the trivial comb whose orbit factor is 1 and the correctly normalised scaling limit of `V` being `0 = 0`; any envelope bounding numerator and denominator independently dies at `psi = 0`, the cone having zero width at both census points `2 pi/5` and `4 pi/5` (both slack summands nonnegative with nonpositive sum), so only curvature-coupled envelopes remain; and the neutral Gaussian width of the transfer at `psi = 0` is exactly `a* = m2/(24 fill) = Var(digit sum)/24 = 5 dim (dim+3)/(48(dim+4))`, not `5 dim/48`, which explains the measured upward drift of `sig2/dim` toward `5/48`.
- 2026-08-28 [Conjecture] At every odd base `base >= 5` the even half falls to the base-5 template at depth `O(log dim)` with no transient: V-towers at `base = 5, 7, 9, 11, 13`, even `dim = 8, 12`, `level <= 25` show no dip anywhere off base 3. Witness: slice-sign-even-half.
- 2026-08-28 [Refuted] The base-3 resummation mechanism built on those identities - the scaling limit `3^n f_n(psi)/fill^n -> Sigma(psi)` is false, the exact tower falling geometrically to 0 as it must since a nonzero limit would contradict `rho_dim < fill/3`; the claimed absolute convergence is false, the per-decade absolute mass of `|G(m pi)|^(dim-1) Ntilde(m pi)` growing at `dim = 4` (block ratios 1.081 to 1.115 out to `m = 2e7`) and rising through 1 at `dim = 6, 8`; the tail-to-lead figures `-0.1812/-0.0464/-0.0121` are artifacts of the `m <= 199` cutoff, still moving at `m <= 2e5`; `Sigma(pi) > 0` is unproved at every `dim`; the leader bound `max_(m>1) |G(m pi)| = |G(7 pi)| = 0.2520527` holds to `m <= 20001`. Witness: slice-sign-even-half.
- 2026-08-28 [Refuted] `V_(2j+1) < 0` for every even `dim >= 4` - at `dim = 4`, `V_level > 0` for every `level <= 40` (`V_1 = +4` exactly) and at `dim = 6`, `V_3 = +135092 > 0`; the odd-`level` dip is a transient of length about `0.055 dim^2`, and `dim = 2, 4` never dip. Witness: slice-sign-even-half.
- 2026-08-28 [Refuted] Certificate depth `K = 2` closes every even `dim` at base 5 - `5 beta_3 < fill beta_2` holds for even `16 <= dim <= 64` and fails at every even `dim = 66..320`, first at `dim = 66` at the edge carry `|c| = 32 = (dim-2)/2`, relative deficit `-2.19e-43`; `beta_K` has Fourier support `5^(-K) Z`, the dominant frequency `n = +-1` gives `Sigma_K(c) ~ 2 T_K(1) cos(2 pi c/5^(K+1))`, so depth `K` sees only carries inside the quarter-period `|c| < 5^(K+1)/4` and dies at `dim = 2 ceil(5^(K+1)/4) + 2` - predicted deaths `16, 66, 316, 1566` at `K = 1, 2, 3, 4`, the first three exact - every fixed depth is finite, `Theta(log dim)` growth is necessary, and the minimal `K` equals the row certificate's `t(dim)` at every breakpoint tested (`14|16`, `64|66`, `314|316`). Witness: slice-sign-even-half.
- 2026-08-28 [Refuted] The base-3 certificate depth is exactly `(9/160) dim^2` - exact lower bounds put the residual at `+2.00` by `dim = 120` and `+9.78` at `dim = 178`; `9/160 = 0.05625` is the first two digits of the true constant `ln(R)/4 = 0.0568486146...`. Witness: slice-sign-even-half.