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.