The repunit layer

• 2026-09-06 [Proved] The binary-weight floor on the repunits w = R_k = (3^k - 1)/2 in closed form: Phi_k = sum_{q | rad R_k} mu(q) q^(-1) sum_{t mod q} P_{q,t}^(k/ord_q 3) with P_{q,t} = prod_{r < d} (1 + e(t 3^r/q)), C-finite in k prime by prime, N_k(2) = 2^(k-1), N_k(p) = (2^k + p - 1)/p whenever 2 is a power of 3 mod p (p = 5, 7, 23), and Phi_k = 2^k - 2 whenever R_k is prime; values 2, 6, 8, 30, 24, 126, 112, 450, 460, 1958, 1344, 8190, 8064, 27360 at k = 2..15 by three methods. Witness: lab/py/ratio-set-saving (ratio.py repunit), coprime.md THE REPUNIT FLOOR EXACTLY.
• 2026-09-06 [Proved] The repunit excess is a lift family and a deep tail: a binary K is a multiple of R_k exactly when its column counts satisfy sum_r c_r 3^r = 0 mod R_k; below 3^(2k) the multiples are K_T = a_(T^c) + 3^k a_T with multiplier 1 + 2 a_T and R_(2k); each lift set Occ_T is occupied and stable in k, so Z(R_k) >= |union_T Occ_T|, and at prime R_k (k >= 15, first at k = 71) Z(R_k) - Phi_k >= 2^k/7 - 4 F(k+1) - 126. Witness: lab/py/ratio-set-saving (ratio.py repunit), coprime.md THE REPUNIT EXCESS.
• 2026-09-06 [Verified] The repunit excess Z(R_k) - Phi_k reads 0, 0, 0, 0, 0, 6, 6, 50, 70, 402, 290, 2198, 2376, 8830 at k = 2..15; the lift union equals Z(R_k) at k <= 10 and falls short by 18, 16, 108, 162, 624 at k = 11..15; minimal witnesses reach 436 digits with a column used 27 times at k = 13; the drift factorises exactly as Z(R_k)/R_k^(log 2/log 3) = 2^(log 2/log 3) (1 - 3^(-k))^(-log 2/log 3) delta_k (1 + X_k) with delta_k = Phi_k/2^k and X_k rising 0.0476 to 0.3227 over k = 7..15. Witness: lab/py/ratio-set-saving (ratio.py repunit).
• 2026-09-06 [Conjecture] The repunit drift is unbounded: X_k = (Z(R_k) - Phi_k)/Phi_k rises at every step from k = 8 and beats the random-lift limit sum_T 1/m_T = 1.41 at k = 13, so no pointwise Z(w) <= C w^(log 2/log 3) holds on the repunits and every exponent above log 2/log 3 survives; the blocking lemma is whether |union_T Occ_T| plus the deep tail is O(2^k). Witness: lab/py/ratio-set-saving (ratio.py repunit), coprime.md THE REPUNIT DRIFT.
• 2026-09-06 [Refuted] That 1.5975 (the maximum of Z(w)/w^(log 2/log 3) below 8192, at w = 1093) bounds the layer: the repunits read 1.7845 and 1.963681 at k = 11, 13, so any pointwise C w^(log 2/log 3) needs C >= 1.9636. Witness: lab/py/ratio-set-saving (ratio.py repunit).
• 2026-09-07 [Verified] The deepest first return of the critical band automaton grows below the critical sqrt(w) as a sign and not as an exclusion: log d_max on log w over 27 weights gives 0.4055, 95 percent [0.3293, 0.4818], but one deletion moves the slope to 0.4261 and the interval to 0.5034, covering 1/2, so the leave-one-out range [0.3829, 0.4261] is what stands; the median has no single exponent, 0.1802 on every weight against 0.2798 at Z >= 8 with 4 of 27 weights having Z <= 2; and a two-predictor fit puts 0.3385 on log w and 0.1313 on log Z, so controlling for sample size lowers the exponent and the drift below 1/2 is understated. Witness: lab/py/band-return-times critical.
• 2026-09-07 [Proved] The submasks of a binary K divisible by a divisor m of it are closed under complement in supp K, under disjoint union and under nested difference, so N_K(m) is even and every solution is a disjoint union of irreducible ones; the decomposition is not unique, so N_K(m) is the number of distinct unions of pairwise disjoint irreducibles and satisfies N_K(m) <= #packings <= 2^iota for the irreducible count iota, with N_K(m) = 2^iota if and only if the irreducibles are pairwise disjoint, and then they partition supp K. Depth-free. The antipodal and run families of the repunit lift are the equality case, which is why their counts are exact powers of two, and the converse fails, k = 7 with multiplier 19 having a power-of-two count and overlapping irreducibles. Witness: lab/py/band-return-times lift and check, the three closures asserted over every one of the 2^(k-1) sets T at every k = 2..9, and both witnesses pinned.
• 2026-09-07 [Verified] The depth-2 lift census of the repunit reads M_k = 2, 6, 14, 36, 68, 172, 306, 728, 1338, 2814, 5224, 11852, 20888, 43364, 84124, 172516, 327092 at k = 1..17 with M_k/2^k inside [1, 2.89356], and its Hankel matrix is 9 by 9 of full rank on all seventeen terms, so no linear recurrence of order at most 8; the irreducible supply Sum_T iota_T / 2^k sits inside [0.738281, 0.890625] at k = 2..12, the even readings falling from k = 6, while max_T iota_T grows 2, 3, 4, 5, 6, 10, 14, 24, 31, 50, 68, and the equality case holds for 1970 of the 2048 multipliers at k = 12 against 1986 whose count is a power of two. The sweep is exhaustive over every T inside [1, k-1]; the multipliers meet the residue classes 1 and 7 mod 9 and never 4, which is forced by a_T = Sum 3^i with i >= 1 and not a reading. Witness: lab/py/band-return-times lift.
• 2026-09-07 [Verified] A column transfer for the submask count of the lift half needs at least 253 states where the return half needs b/2: a machine reading the k columns with a state set free of k is a linear representation of the count as a series over the column word, so its state count is at least that series' Hankel rank, and the rank reads 3, 7, 14, 31, 62, 126, 253 at word length 1..7 on each side against the full 3, 7, 15, 31, 63, 127, 255, deficiency 0, 0, 1, 0, 1, 1, 2. The words reach length 14, so the floor holds at k <= 15 and is already worse than the 2^(k/2) meet in the middle; whether the rank is unbounded is observed and not proved, and a rank levelling off would be a poly-time machine, so the route is blocked and not closed. The irreducible count has the same full-rank Hankel to word length 5. Witness: lab/py/band-return-times lift.