# 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.