---
title: The coprimality spine
lead: The coprimality spine: exact base-local factors on every design, the census behind them, the window at dimension one with the band automaton's critical out-degree and its algebraic block rate, and primes on a design, where the least base below the quarter threshold is 21 at one missing digit and 32 at two, the one-digit family closes at 34 and the two-digit family closes at 21 against the third.
figure: research-coprime
slug: coprime
---

- Every fractal in this family counts the same thing on a different grid: the points of a digital design whose coordinates share no common factor.
- Each design has its own density constant, and the constants are one theorem with one input per design; this page is that theorem and the open front behind it.
- The base of the design contributes an exact factor, the bracket, known in closed form at every finite level; every prime away from the base contributes the classical factor `1 - p^(-dim)`.
- Above dimension one the join is a theorem on the shelf; at dimension one it is open exactly for primes in one window of exponents, and that window is the standing problem here.
- The Sierpinski gasket is the worked case throughout, because its sequence is [A396934](https://oeis.org/A396934) and its density `16/(3*Pi^2)` is stated in [the ledger](../sequences.md).
- Every claim carries one tag: **Proved** means a proof is given or restated here or on the shelf, **Verified** means recomputed by the named study or lane, **Conjecture** means supported and open, **Refuted** means shown false.

## THE OBJECT

- A **design** is a triple `(base, dim, F)`: a base `base >= 2`, a dimension `dim >= 2`, and a set `F` of filled digit-vectors, `F` a subset of `{0,...,base-1}^dim` with `fill = |F| >= 2`.
- The set at level `level` is `S_level = { x in [0, base^level)^dim : every digit-vector of x in that base lies in F }`, holding exactly `fill^level` points, one per string of `level` choices from `F`; it is the substitution rule of [the core](core.md) read on coordinates rather than cells.
- The count of interest is `A(level) = #{ x in S_level : gcd(x_1, ..., x_dim) = 1 }`, and the question is what `A(level)/fill^level` converges to.
- Let `Diff` be the subgroup of `Z^dim` generated by the differences `F - F` and `m(F)` its index; the design is **spanning** when `m(F) = 1`.
- **Condition (E)**: `F - F` has full rank `dim` and every prime dividing `m(F)` divides `base`, i.e. `rad(m(F)) | rad(base)`; spanning is (E) at index 1, and (E) is the hypothesis of every theorem below.
- Non-spanning designs are outside the theory, not anomalies inside it: base 2, `dim 2`, code 9 is a diagonal pair whose whole fractal is the line `i = j`, and its coprime count is stuck at `1` forever. **Proved.**
- Six census lines fail spanning, satisfy (E), and obey the density formula unchanged, as do constructed index-4 designs at bases 4 and 6 (`lab/rs/design-census`). **Verified.**
- When (E) fails at a prime `p` not dividing `base`, the Euler factor at `p` is replaced by a coset-corrected factor that can depend on `level`, and `A(level)/fill^level` can fail to converge: base 7, `dim 2`, `F = { v : v_1 + v_2 = 1 mod 3 }`, `fill = 16`, index 3, has period-3 subsequential limits `0.698175, 0.698175, 0.465450`, each matched by the corrected constant to `3e-04` (`coprime-density-above-dimension-one`). **Verified.**
- The household failure is the Cantor dust `F = {0,2}^2` at base 3, dimension `log_3(4) = 1.26`, which fails (E) at index 4 and has `A(level) = 0` at every level, every coordinate being even, against a naive `0.512938`; dimension above one does not exempt a design from (E). **Proved.**
- One shear repairs it: dust points are `2*y` with `y` in the `{0,1}^2` design, which satisfies (E) with `fill = 4 > 3`, so the dust's `#{gcd = 2}` density is exactly `81/(16*Pi^2) = 0.512938`, the constant surviving one prime down (`coprime-density-above-dimension-one`). **Proved.**

## THEOREM 1: THE BASE IS EXACT

- For every squarefree `e` dividing `rad(base)` and every `level >= 1`: `#{ x in S_level : e | x_i for all i } = fill_e * fill^(level-1)`, with `fill_e = #{ v in F : e divides every component of v }`. **Proved.**
- The proof: `e | base*y` for every integer `y`, so writing `x = v_0 + base*y` the divisibility of `x` by `e` is the divisibility of the last digit-vector `v_0`; that vector is pinned to one of `fill_e` corners and the other `level - 1` are free, with no error term at any level.
- At a prime base `base = p` the only digit-vector divisible by `p` is zero, so `x = p*y` maps `{x in S_level : p | x}` bijectively onto `S_(level-1)` with `gcd(x) = p*gcd(y)`, and `#{ x in S_level : p^a | gcd(x) } = a_0^a * fill^(level-a)`, `a_0 = 1` if `0` is in `F`, else `0`. **Proved.**
- Every design is its own Euler factor at the base prime, and the factor is self-similar rather than approximate: the Vicsek plus demands digit `1` somewhere at every position, so `a_0 = 0`, no point of the plus has gcd divisible by 3 at any level, and its density is larger than `6/Pi^2`, not smaller. **Proved.**
- The identity is recomputed by direct enumeration on the gasket and or-triangle to `level 8`, the carpet and Vicsek plus to `level 6`, the base-6 sample to `level 5`, the sponge to `level 4`, and at `level 4` on all 763 census lines, degenerate ones included, with zero failures; the identity does not need spanning (`lab/rs/design-census`). **Verified.**

## THE BRACKET

- [Mobius](/wiki/mobius-function/) inversion over the base primes turns Theorem 1 into one exact number, the **bracket** `B(F) = Sum_{e | rad(base)} mu(e) * fill_e / fill`, and `B(F)*fill^level` is exactly the number of points of `S_level` whose gcd is coprime to `base`, at every level. **Proved.**
- At a prime base the bracket is `1 - a_0/fill`; at a composite base divisibility by the different base primes is correlated through the corner set, and the bracket is the whole story.
- The **base-6 sample**, `dim 2`, code 34376528265 in the census (`lab/rs/design-census`), has `fill = 8` with `fill_2 = 3`, `fill_3 = 2`, `fill_6 = 1`, so `B(F) = 1 - 3/8 - 2/8 + 1/8 = 1/2` while the naive product `(1 - 3/8)*(1 - 2/8) = 0.46875`. **Proved.**
- Treating divisibility by 2 and by 3 as independent is wrong by an exact amount at every finite level, Theorem 1 with inclusion-exclusion over `{2, 3}` (`coprime-density-above-dimension-one`). **Proved.**

| `level` | points of `S_level` | gcd coprime to 6 | `fill^level / 2` | naive `0.46875 * fill^level` |
|---|---|---|---|---|
| 3 | 512 | 256 | 256 | 240 |
| 4 | 4096 | 2048 | 2048 | 1920 |
| 5 | 32768 | 16384 | 16384 | 15360 |
| 6 | 262144 | 131072 | 131072 | 122880 |

- The bracket reaches zero and the formula is right there: base 6 on the nine even-coordinate digit pairs, base 10 on the twenty-five, base 12 on the thirty-six, have every digit divisible by `2 | base`, so Theorem 1 pins `fill_2 = fill`, `B(F) = 0`, and `A(level) = 0` at every level. **Proved.**
- That is exactly the case the Cantor dust is not: there the offending prime does not divide the base, (E) fails, and the naive formula returns `0.512938` against a true `0`; the discriminator is whether the prime divides the base.
- The bracket spreads widely at fixed dimension: at `base 4`, `fill = 8`, `dim 2`, twenty-five designs of Hausdorff dimension `1.5` carry `B(F)` in `{1/2, 5/8, 3/4, 7/8, 1}` and densities in `{0.4052847, 0.5066059, 0.6079271, 0.7092483, 0.8105695}`, and sixteen base-6 designs take `0, 1/3, 5/12, 4/9, 1/2, 6/11, 5/9, 2/3, 13/20, 3/4, 1`; no study regenerates the sweep. **Conjecture.**

## LEMMA A: UNIFORM CONTRACTION

- For `gcd(d, base) = 1`, nonzero `t` in `(Z/d)^dim`, and `r = ord_d(base)`, put `f_l(t) = (1/fill) * |Sum_{v in F} e(base^l * <t, v> / d)|`, `e(x) = exp(2*Pi*i*x)`.
- **Lemma A.** For every design satisfying (E), `Prod_{l=0}^{r-1} f_l(t) <= c(base, fill) := 1 - (2/fill)*(1 - cos(Pi/(2*base))) < 1`. **Proved.**
- The proof: (E) gives `v, v'` in `F` with `delta = <t, v - v'>` nonzero mod `d`; let `y_l` be the distance from `base^l * delta / d` to the nearest integer, purely periodic and never `0` since `gcd(base, d) = 1`; whenever `y_l < 1/(2*base)` one has `y_(l+1) = base*y_l`, so some `l_0` in the period has `y_l0 >= 1/(2*base)`, and splitting the character sum there into the pair and the other `fill - 2` terms gives `|Sum| <= fill - 2 + 2*cos(Pi/(2*base))`.
- Why (E) suffices where spanning was first assumed: `Diff` has finite index `m(F)` with `rad(m(F)) | rad(base)`, so `m * Z^dim` lies in `Diff`; a character `t` vanishing on `F - F` vanishes on `Diff`, hence `m * t = 0 mod d`, and `gcd(m, d) = 1` forces `t = 0`; every downstream use of Lemma A is untouched. **Proved.**
- **Lemma A' (the window rate).** For every design satisfying (E), `gcd(d, base) = 1`, `d > 1` and nonzero `t` in `(Z/d)^dim`, `Prod_{l=0}^{level-1} f_l(t) <= c(base, fill)^floor(level / m_d)` with `m_d = max(1, floor(log_base(d/2)) + 1)`. **Proved.** The proof of Lemma A gives one good position per window of `m_d` consecutive digits rather than one per orbit: `y_l` is never `0`, so `y_l >= 1/d` at every `l`, and `y_l < 1/(2*base)` forces `y_(l+1) = base*y_l`, so if `m_d` consecutive positions all had `y < 1/(2*base)` the last would be `base^(m_d - 1)*y >= base^floor(log_base(d/2))/d > 1/(2*base)`, a contradiction; each of the `floor(level/m_d)` disjoint windows therefore carries a position where the pair contributes at most `2*cos(Pi/(2*base))` and the factor there is at most `c(base, fill)`. Since `base^ord_d(base) = 1 mod d` forces `ord_d(base) >= m_d`, Lemma A' is never weaker than Lemma A and is strictly stronger wherever `ord_d(base) > m_d`; on the gasket the worst per-digit rate over every modulus `d <= 301` is `0.830915` at `d = 257`, against the Lemma A' rate `0.973211` and the Lemma A rate `0.986514` there (`lab/py/digit-transform-norms`).
- **Lemma A' with a base part and a perturbation.** Let `d = e*m` with `e` dividing a power of `base`, `gcd(m, base) = 1` and `m > 1`, let `t` in `(Z/d)^dim` be nonzero mod `m`, and let `norm(eta)_inf < base^(-2*level/3)/(4*base*dim*(base-1))`. Then `Prod_{l<level} f_l(t/d + eta) <= c'(base, fill)^floor(2*level/(3*m_d))` with `c'(base, fill) = 1 - (2/fill)*(1 - cos(Pi/(4*base)))`. **Proved.** The base part shifts the orbit and does not stop it, and the pair phase moves by at most `base^l * dim * (base-1) * norm(eta)_inf`, which stays below `1/(4*base)` at every position `l < 2*level/3`, so each window of `m_d` positions below `2*level/3` still loses a factor, now with `cos(Pi/(2*base))` replaced by `cos(Pi/(4*base))`. The hypothesis that `t` is nonzero mod the coprime part is not decoration: the gasket at `d = 6` and `t = (3, 0)` has `Prod_{l<level} f_l(t) = 1/3` at every `level`, no decay at all. This is [Maynard 2019](https://doi.org/10.1007/s00222-019-00865-6) Lemma 8.2, `exp(-c*log Y/log base)`, in every dimension with an explicit constant and with no consecutive-digit hypothesis.
- The constant depends only on `base` and `fill`, not on which corners `F` holds, and it is far from sharp; being below `1` for every design at once is the point.
- The **or-triangle** is base 2, `dim 2`, code 14, every digit pair but `(0,0)`, i.e. the pairs with `i OR j = 2^level - 1`, of which there are `3^level`.

| design | `base` | `fill` | `c(base, fill)` | worst period product observed | moduli searched |
|---|---|---|---|---|---|
| gasket | 2 | 3 | 0.804738 | 0.333333 | `d <= 40` |
| or-triangle | 2 | 3 | 0.804738 | 0.333333 | `d <= 40` |
| carpet | 3 | 8 | 0.966506 | 0.500000 | `d <= 30` |
| vicsek | 3 | 5 | 0.946410 | 0.600000 | `d <= 30` |
| sponge | 3 | 20 | 0.986603 | 0.600000 | `d <= 14` |
| base-6 sample | 6 | 8 | 0.991481 | 0.481763 | `d <= 30` |

- The `c(base, fill)` column is the formula; the observed column, the gasket's true contraction `1/3` against a bound of `0.80`, is a finite search that no study regenerates. **Conjecture.**
- **Corollary (equidistribution rate).** For `gcd(d, base) = 1`, `d > 1` and any `a` in `(Z/d)^dim`, `| #{x in S_level : x = a mod d} / fill^level - 1/d^dim | <= c(base, fill)^floor(level / m_d)`, by character orthogonality: the principal character gives `1/d^dim`, and every other character is a nonzero `t` in `(Z/d)^dim`, which `gcd(d, base) = 1` places inside Lemma A' with no further hypothesis, so it loses one factor of `c` per window of `m_d` digit positions instead of one per orbit cycle. **Proved.**
- The band the two corollaries stand on: over every modulus `3 <= d <= 15` coprime to `2` and every level `2 <= level <= 12` the gasket's exact deviation stays under the window bound, worst ratio to it `0.343146` at `d = 3`, `level 2`, and the cell `d = 5`, `level 10` deviates `0.0014741` against `0.337499` where the orbit rate allowed `0.647604`; the carpet over `d <= 11`, `level <= 8` has worst ratio `0.066907` at `d = 4`, `level 2`, and the base-peel error stays under its own window bound at worst ratio `0.114382`, the gasket at `m = 3`, `level 3` (`lab/py/digit-transform-norms`). **Verified.**
- **Corollary (base peel).** For squarefree `d = e*m` with `e | rad(base)` and `gcd(m, base) = 1`, `T_d(level) = #{x in S_level : d | x_i for all i}` satisfies `T_d(level) / fill^level = (fill_e / fill) * (1 / m^dim) + O(c(base,fill)^floor((level-1)/m_m))` with `m_m = max(1, floor(log_base(m/2)) + 1)`, the last digit-vector pinned mod `e` by Theorem 1 and the rest a shifted residue class mod `m`; the window length is read off the coprime part `m` and never off `d`, the base part carrying no decay of its own, and at a prime base with `0` in `F` the peel is the exact identity `T_(base*m)(level) = T_m(level-1)`. **Proved.**

## THE MASTER DENSITY

- The formula the whole census is measured against: `delta = B(F) * Prod_{p not dividing base} (1 - p^(-dim)) = B(F) * (1/zeta(dim)) * Prod_{p | base} (1 - p^(-dim))^(-1)`.
- **Upper bound.** For every design satisfying (E), `limsup_level A(level)/fill^level <= delta`: fix `z`, sift by the primes up to `z`, handle each of the finitely many squarefree moduli exactly by the base peel, and let `z` grow only after `level`. **Proved.**
- **Lemma B.** Write `T*_p(level)` for the count of nonzero `x` in `S_level` with `p | gcd(x)`, which vanishes for `p >= base^level`; Lemma B is `lim_{z} limsup_level (1/fill^level) * Sum_{p > z, p not dividing base} T*_p(level) = 0`, and it is exactly what separates the upper bound from the limit `A(level)/fill^level -> delta`.
- The origin must be excluded: `x = 0` lies in `S_level` whenever the zero corner is filled, every prime divides `0`, and the unstarred sum over all `p > z` diverges.
- Lemma B is a uniform equidistribution estimate for digit-restricted sets over moduli growing with the level; the one-dimensional literature is [Erdos, Mauduit and Sarkozy 1998](https://doi.org/10.1006/jnth.1998.2229), [Konyagin 2001](https://doi.org/10.1023/A:1015256809636) and [Maynard 2019](https://doi.org/10.1007/s00222-019-00865-6), all three needing `|F| < base`, operating below the gasket's critical band, and treating no vector digit set.
- **Theorem (above dimension one).** For every design with `dim >= 2`, condition (E), and `fill > base`, `A(level) / fill^level -> delta`: the hypothesis is the geometry, dimension `log(fill)/log(base) > 1`, and nothing else (`coprime-density-above-dimension-one`). **Proved.**
- The three steps: the box bound `N*_level(m) <= (base+1)^dim * fill^level * m^(-alpha)`, `alpha = log(fill)/log(base)`, `N*_level(m) = 0` for `m >= base^level`, which is the Ahlfors-David regularity of missing-digit sets ([Chow, Varju and Yu](https://arxiv.org/abs/2402.18395)); the Chebyshev sum `G(level) = Sum_{x != 0} log gcd(x) = Sum_m Lambda(m) N*_level(m) <= (base+1)^dim * (-zeta'(alpha)/zeta(alpha)) * fill^level`, convergent exactly when `alpha > 1`; and the close, `A_z(level) - A(level) <= G(level)/log z` uniformly in `level`, since every point sifted at `z` but not coprime has `log gcd > log z`.
- What that settles: the gasket's `16/(3*Pi^2)` ([A396934](https://oeis.org/A396934)), the or-triangle's `8/Pi^2`, the carpet's `189/(32*Pi^2)`, the Vicsek plus's `27/(4*Pi^2)`, the sponge's `(513/520)/zeta(3)` and both base-6 samples are theorems. **Proved.**
- The proof is qualitative: chaining the steps gives an error of order `(log level)^(1-alpha)`, while the measured convergence is geometric, the carpet's gap halving per level down to `-3.52e-07` at `level 20` (`lab/rs/dimension-one-ladder`, live to `level 18` and stored to `level 20`); nothing here explains the rate. **Conjecture.**
- **Corollary (no linear recurrence).** For any design satisfying the theorem with `B(F) > 0` and `dim` even or `dim 3`, `A(level)` is not C-finite: a C-finite sequence with `A(level)/fill^level` convergent has a rational limit, and `delta` is a nonzero rational multiple of `1/zeta(dim)`, irrational for even `dim` by the transcendence of `Pi` and at `dim 3` by Apery. **Proved.**
- At odd `dim >= 5` the corollary waits on the irrationality of `zeta(dim)`, and holonomic recurrences are excluded only by exhaustive exact fitting with held-out terms. **Conjecture.**
- **The periodic pattern owes Lemma B nothing.** Among nonzero points of `Z^dim` [visible from the origin](/wiki/visible-lattice-points/) whose least residue vector mod `base` lies in `F`, the density is exactly `delta * (fill / base^dim) = (1/zeta(dim)) * Prod_{p | base} (1 - p^(-dim))^(-1) * base^(-dim) * m_c`, `m_c = fill * B(F)` the corners nonzero mod every base prime; a base prime is decided by the residue class, a foreign prime is independent by CRT, and the Mobius tail beyond modulus `z` is `O(z^(1-dim))` uniformly in the box, which is where `dim >= 2` enters. **Proved.**
- Designs differing only in the zero corner have identical visible density, since the zero corner never counts toward `m_c`; `F` the whole residue cube recovers the classical `1/zeta(dim)` of [pi out of the stack](pi.md); the gasket gives `4/Pi^2 = 0.4052847345`, the carpet `21/(4*Pi^2) = 0.5319362141`, the sponge `19/(26*zeta(3)) = 0.6079323107`. **Proved.**
- **The shear theorem.** A unimodular `sigma` with `sigma(F)` inside the digit cube acts digit-wise without carries and preserves gcd, so `F` and `sigma F` have identical `A(level)` at every level; at base 2 the shears `(i, j) -> (i, i+j)` and `(i, j) -> (i+j, j)` carry code 7 to codes 11 and 13, all three on `2, 4, 12, 34, 122, 362`. **Proved.**
- The converse is false: base-3 codes 11 and 161, the gasket and `{(0,0),(1,2),(2,1)}`, have identical `A(level)` at every level and are not shear-equivalent, by the E-decomposition below. **Proved.**

## PRIMES ON A DESIGN

- **Three readings, one of them a sieve.** For a design with `B(F) > 0`, `gcd(x)` prime is a positive-density count, `Sum_p delta_p`, and follows from the master density above dimension one by the same three steps, since `Sum_p p^(-dim)` converges; the bracket is the whole hypothesis, since base 32 with `F = {0, 4, ..., 28}^2` has `fill = 64 > 32` and satisfies (E) while every gcd on it is divisible by `4`, so no point of any level has prime gcd. `x_1` prime on the gasket is `2^level * Sum_{p < 2^level} 2^(-s_2(p))`, a sum-of-digits question in its large-deviation regime. And the Morton code `x -> Sum_j (Sum_c base^(c-1) * x_(c,j)) * base^(dim*j)` maps `S_level` bijectively onto the integers of `level` digits at base `base^dim` whose digits lie in the image of `F`, so when `fill = base^dim - 1` primes on a design are exactly primes with one restricted digit at base `base^dim`, the gasket being base 4 missing digit `3` and the carpet base 9 missing digit `4`. **Proved.**
- **What the sieve consumes.** [Maynard 2019](https://doi.org/10.1007/s00222-019-00865-6) proves infinitely many [primes](/wiki/prime-numbers/) missing one base-10 digit and reads the set through two numbers only: the `l^1` exponent of its digit transform, `27/77`, whose complement `50/77` is the Type I level, and a fractional moment, `59/433` at order `235/154`, which fixes the Type II range `[X^(9/25), X^(17/40)]`; [Karwatowski 2022](https://doi.org/10.4064/aa191002-26-8) carries both bounds to every base `base >= 10`, and [Karwatowski, base 9](https://www.math.hhu.de/fileadmin/redaktion/Fakultaeten/Mathematisch-Naturwissenschaftliche_Fakultaet/Mathematik/20_Institut-Lehrstuehle/5_Algebra_und_Zahlentheorie/Karwatowski/Digits_of_primes_in_base_b_9.pdf) proves the pairs `(9, 0)` and `(9, 8)` with `0.3219` and `0.14355` in their place, records the criterion `g(s) < (1/5)*(1 + c/2)*(2 - s)` for some `s` in `[3/2, 2)`, `c = log(base-1)/log base` and `g(s) = log_base lambda(s, J)` with `lambda(s, J)` the Perron root of the `base^J`-state matrix carrying a `J`-digit window to its successors with weight the `s`-th power of the one-digit transform's supremum over the box behind the window, and states that no other pair with `base <= 9` meets it. **Verified** at source.
- **The `l^infinity` input is not a gap.** What the Type I estimate asks of the set at a rational with a factor coprime to the base, Lemma 8.2 there, is Lemma A' above, which supplies it in every dimension with an explicit constant under (E) alone; what is left to ask of a design is the pair of `l^1` numbers, and it is there that the two designs part. **Proved.**
- **The least base with a one-missing-digit set below `1/4`.** Write `alpha_1` for the `l^1` exponent `g(1)` of a digit set at base `base` with one digit excluded. The least base carrying such a set with `alpha_1 < 1/4` is `base 21` missing the digit `0`, certified `alpha_1 in [0.2499765, 0.2499771]` at six window digits and clearing the threshold by `2.3e-05`; base 20 misses at all ten of its distinct sets, the closest reading `alpha_1 > 0.2528608` at four window digits. A base has `floor((base+1)/2)` distinct sets, since the mirror pair `a_0 -> base - 1 - a_0` coincides only at odd `base` (`lab/py/digit-transform-norms`, verb `six`). **Verified.**
- **The family is closed above its floor.** Every one of the 3663 distinct one-missing-digit sets of every base `35 <= base <= 125` certifies `alpha_1 < 1/4`, each at the shortest window of two, three or four digits that clears, `35 <= base <= 57` needing three digits and every `base >= 58` clearing at two; with `base 34` certified at all 17 of its sets and the digit-uniform bound below a theorem for every `base >= 126`, every base `base >= 34` clears at every excluded digit and the floor `34` of that family is exact (`lab/py/digit-transform-norms`, verb `family`, `lab/py/digit-uniform-bound`). **Verified.**
- **The digit-uniform bound.** `|hat F(t)| <= (|sin(base pi t) / sin(pi t)| + 1)/(base - 1)` for every `t` and every excluded digit, the right side naming no digit; expanding the level product over subsets telescopes each maximal run of positions into a single Dirichlet kernel at modulus `base^l`, so `(base-1)^N` times the digit-uniform level-`N` grid sum is an exact sum of `2^N` Lebesgue sums, and with `L_M <= M((2/pi) log M + 0.9625153) + 2/pi` this gives `alpha_1 < 1/4` for every `base >= 126` and every excluded digit, and fails at `base 125` (`lab/py/digit-uniform-bound`). **Proved.**
- **Two missing digits: what the transform sees.** For an excluded pair `{a, c}` at base `base`, `fill = base - 2` and `(base-2)^2 |hat F(t)|^2 = K(t)^2 + 2 + 2 cos(2 pi D t) - 4 K(t) cos(pi S t) cos(pi D t)` with `K(t) = sin(base pi t)/sin(pi t)`, `D = a - c` and `S = a + c - (base - 1)`, so the pair enters only through `|D|` and `|S|`. That is one implication and not an equivalence class: `{0, 2}` and `{0, 8}` at `base 10` read `(2, 7)` and `(8, 1)` and share a transform anyway, and grouping by `(|D|, |S|)` alone overcounts. The two moves that generate the collapse are the reflection `d -> base - 1 - d`, which flips both signs, and an integer translation of `F`, available exactly when `0` or `base - 1` is excluded and identifying `{0, c}` with `{0, base - c}`; the edge family is therefore the one-missing-digit sets of a `(base-1)`-digit interval read at base `base`, and the number of distinct transforms is `(C(base-2, 2) + floor((base-2)/2))/2 + floor(base/2)`. **Proved.**
- **How many pair sets a base carries.** That count reads `7, 16, 21, 31` of the `15, 36, 45, 66` excluded pairs at `base 6, 9, 10, 12`, and grouping every `C(base,2)` pair by its sampled transform reproduces it at every base `4 <= base <= 41`, summing to 2373 over `4 <= base <= 31` (`lab/py/digit-transform-norms`, verb `pairs`). **Verified.**
- **The least base with a certified two-missing-digit set below `1/4`.** The interval class, the pair `{0, 1}` whose complement is an interval of `base - 2` digits, is the cheapest set of its base at every base scanned, and `base 32` certifies `alpha_1 in [0.2499087, 0.2499779]` at four window digits, clearing by `2.2e-05`, while the same class at `base 31` certifies `[0.2518967, 0.2519717]` and fails. Over every base `4 <= base <= 31` the window machine certifies `alpha_1 > 1/4` at 2363 of the 2373 distinct sets, closest `base 26` missing `{2, 23}` at `alpha_1 > 0.2502919`, most of them at two window digits (`lab/py/digit-transform-norms`, verbs `pairs` and `pairfail`). **Verified.**
- **Ten cells stay unclear, and the shape they share is a correlation and not a mechanism.** The ten sets of `4 <= base <= 31` that get no positive lower certificate at five window digits are three at `base 26`, one at `base 29`, five at `base 30` and one at `base 31`, and every one has `S = 0` or `D = base/2` with `base/2` odd. A real zero of `|hat F|` empties a cell and lowers the Perron root of the infimum matrix, but it does not zero it and it does not decide the cell: `base 32` missing `{0, 1}` has `|hat F(t)| = |sin(30 pi t)/sin(pi t)|/30`, vanishing at all 29 points `t = j/30`, and is the page's own clearing headline, while 13 of the 14 `S = 0` classes at `base 31` certify above `1/4`. The ten carry certified upper bounds `0.2538899` to `0.2826357`, all above `1/4`, so none of them is shown to clear either, and `32` is the least base carrying a certified clearing set while the exact floor stays **Conjecture**.
- **The bar `1/3` and the family floor at it.** The bar is `alpha_1 < 1 - b` where `b` is the exponent of `max_theta |Sum_{n <= x} mu(n) e(n theta)|`, so `b = 3/4` gives `1/4` and `b = 2/3` gives `1/3`; against `1/3` the interval class first clears at `base 13` with `alpha_1 < 0.3318819` at three window digits, `base 12` staying above at all 31 of its sets to five, every pair of every base `21 <= base <= 26` clears, worst `base 23` missing `{4, 5}` at `alpha_1 < 0.3333284`, and every base `4 <= base <= 20` carries a certified witness above `1/3`, `base 20` by `{3, 11}` at `alpha_1 in [0.3356579, 0.3356674]`, so the two-digit family floor against `1/3` is `21` (`lab/py/digit-transform-norms`, verbs `pairfirst`, `pairclear`, `pairsome`). **Verified.**
- **The digit-uniform bound at any number of excluded digits.** `|hat F(t)| <= (|sin(base pi t)/sin(pi t)| + m)/(base - m)` for every set `E` of `m` excluded digits, the right side naming no digit; expanding the level product over subsets carries `m^(N - |E|)` on the positions outside the subset and telescopes each maximal run into the same Dirichlet kernel, so `a_N = m a_(N-1) + m Sum_(l<N) lambda_l a_(N-1-l) + lambda_N`, the growth root solves `(z - m)(z - 1)^2 = m(c_1 (log base) z + c_0 (z - 1))` with `c_1 = 2/pi` and `c_0 = 0.97`, and `alpha_1 < e` follows from `z < base^e (1 - m/base)`. The Lebesgue input is `lambda_l <= c_1 l log base + gamma' + c_1 base^(-l)` with `gamma' = (2/pi)(gamma + log(8/pi)) = 0.96252282676`, used rounded up, the coarser `c_0 = 0.97` form of it needing `base^l >= 86` and so being unusable at the `m = 1` rung against `1/3`, where `base^l = 32`. Certified at 120 bits on the exact input, the chain gives `alpha_1 < 1/4` for every `base >= 649` and every excluded pair and fails at `base 648`; at `m = 1` it gives `125` and at `m = 3` it gives `1873`, while against `1/3` it gives `32`, `105` and `230`. The `m = 1` rung sharpens the `126` of the bullet above by one base rather than contradicting it, since that bullet's `c_0 = 0.97` is legitimate wherever it is applied there (`lab/py/digit-uniform-bound`). **Proved.**
- **What the bar `1/4` buys, and where the wall moves.** In the GRH [Mertens](/wiki/mertens-function/) chain of [mobius](mobius.md) the digit set enters twice: through its mass `A_F(x)` and through the normalised `l^1` mass `base^(-level) Sum_(a < base^level) |Sum_(n in D_level) e(n a/base^level)| <<_base fill^level base^(level(alpha_1 - 1))`, the window certificate supplying that bound uniformly in `level`; steps 1, 2, 4 and 5 there never name the set, and only step 3, the kernel bound `B_base(F) <= base PB_base(m)`, substitutes a digit-free estimate for it. Feeding the certified `l^1` exponent in step 3's place gives `|M_F(x)| <<_{base,eps} A_F(x) x^(alpha_1 - 1/4 + eps)` under GRH, that is `A_F(x)^(1 - delta + eps)` with `delta = (1/4 - alpha_1)/alpha_base > 0`, and under a zero-free half plane `Z(a)` the exponent reads `alpha_1 - (1 - b(a))`; so `alpha_1 < 1/4` is the whole hypothesis and the excluded-digit count enters only through `alpha_1`. **Proved.** The wall of that theorem moves from `3690` to `34`, since the one-missing-digit clearance is certified at every `base >= 34`; steps 2 and 3 alone force `alpha_1 <= 1 - alpha_base + c_base` and `gap_base(1) > 0` is exactly `1 - alpha_base + c_base < 1/4`, so the old certificate implies the new condition and the wall can only fall. **Verified**, at the certificates behind `34`.
- **The published exponents are upper bounds and not the constants.** The base-10 missing-`5` exponent brackets to `[0.3505775, 0.3505797]` at seven window digits, strictly below the published `27/77 = 0.3506494`, so that number is a finite-window upper bound on the `l^1` exponent and not the exponent itself (`lab/py/digit-transform-norms`). **Verified.**
- **The 2D Type I splits the two designs.** Write `hat F_level(t) = Prod_{j<level} f(base^j * t)` for the transform at level `level` on `T^dim`, `f(t) = (1/fill)*|Sum_{v in F} e(<t, v>)|` the factor of Lemma A, and `I_level = Int_(T^dim) hat F_level`; let `alpha_1^- = dim + liminf_level (1/level) log_base I_level` and `alpha_1^+` its limsup, with `alpha_1*` the exponent of the sup-over-box sum a large sieve consumes, which dominates every shifted grid sum and so is never below `alpha_1^-`. The substitution `t = (i + y)/base^N` gives the sandwich `base^(-dim*N) * min_x Sigma_N(x) * I_M <= I_(N+M) <= base^(-dim*N) * max_x Sigma_N(x) * I_M` for the shifted grid sum `Sigma_N(x) = Sum_{i in (Z/base^N)^dim} hat F_N(x + i/base^N)`, which is `base^(-N)`-periodic, so one scan of one cell with a Lipschitz slack bounds `alpha_1^-` below and `alpha_1^+` above; no limit is claimed and none is needed. For `d | gcd(x)` the Farey-point route of that Type I estimate transfers to `(Z/d)^dim`, the points of `T^dim` of denominator at most `Q` being `1/Q^2`-separated in sup norm, and it saves a power when `alpha_1* < dim/2`, the dyadic block `d ~ Q_1` costing `fill^level * (Q_1^(2*alpha_1* - dim) + Q_1^dim * base^(level*(alpha_1* - dim)))` and the small moduli going to Lemma A': the carpet's five-digit window matrix is certified in interval arithmetic at Perron root below `2.441255`, so `alpha_1* < 0.8124` and `Sum_{d <= Q, gcd(d,3) = 1} |#{x in S_level : d | x_i for all i} - fill^level/d^2| <<_A fill^level * level^(-A)` holds at `Q = 3^(0.5938*level) * level^(-C)`, a power level with a log saving where Lemma A' alone gives `Q = level^C`; the gasket's shifted grid sums are certified above their threshold at `N = 2` and `N = 3`, `min_x Sigma_2 > 4.059204` against `4` and `min_x Sigma_3 > 8.213932` against `8`, with `Sigma_2(0) = (8 + 2*sqrt(5))/3 = 4.157378` exactly at the grid itself, so `alpha_1^- >= 1.0126` and `alpha_1^+ <= 1.1022`, and `alpha_1* >= alpha_1^- > dim/2` closes the same route (`lab/py/digit-transform-norms`). **Proved.**
- **The carpet misses the criterion at every order.** For base 9 missing `4` the window bound gives `g(1) = 0.3437`, below `27/77`, so the Type I level `X^0.656` is met, while `g(3/2) = 0.1531`, `g(235/154) = 0.1457`, `g(1.6) = 0.1262`, `g(1.7) = 0.1031` and `g(1.8) = 0.0835` sit against the criterion's `0.1473`, `0.1397`, `0.1179`, `0.0884` and `0.0589`, a deficit at every order, `0.0058` on the printed pair at `s = 3/2` and `0.005749` in full; the same bound returns `0.1446` for `(9, 0)` against the published `0.14355` and `0.1370` for base 10 against `59/433 = 0.1363`, two calibrations that put the cost of the wider box near `0.0011`, an inference from two rows and not a bound on it. Windows do not close the gap: `g(3/2)` reads `0.153068` at five digit-vectors and `0.152921` at six, a drop of `0.00015` after `0.00132` from four to five (`lab/py/digit-transform-norms`). **Verified.**
- **What the criterion is allowed to read.** Dividing by `2 - s`, the criterion is the single inequality `g(s)/(2 - s) < (1/5)*(1 + c/2)` on the moment exponents of the transform: the exceptional set `E = {a < X : F_X(a/X) >= X^(-beta)}` enters the sieve only through its size, a level-`s` moment gives `#E << X^(s*beta + g(s))`, the step that consumes it asks `#E << X^(2*beta)`, and that forces `beta >= g(s)/(2 - s)`, after which the Type II range and its width are functions of `beta` alone (at base 10 the floor `9/25` is `(9/8)*e_E - beta`, the ceiling `17/40` is `1 - e_E` with `e_E = 2*beta`, and the width is `1 - (13/4)*beta`). A count `#{a < X : F_X(a/X) >= 1/B} << B^s * X^m` is a weak `L^s` bound and summing it dyadically in `B` returns the strong moment within `X^eps`, so weak and strong carry one exponent and, with the prime side entering by Parseval as it does at source, the family `{g(s)}` is the whole list of norms of the transform that this step can consume: a norm closes the carpet only by lowering some `g(s)` with `s` in `[3/2, 2)`, and nothing else. **Proved.** The reading is confirmed by the base-9 constants at source, where `v = 0.28711` is exactly `0.14355/(2 - 3/2)`.
- **The deficit is in the transform, not in the estimate.** The window matrix bounds `g(s)`, the exponent of `sup_beta Sum_{a<Y} F_Y(beta + a/Y)^s`, from above, its one-digit weight being a supremum over the box behind the window and so uniform in the shift; the shift sandwich bounds it from below, since `sup_beta Sum_{a<Y} F_Y(beta + a/Y)^s >= Y * Int_0^1 F_Y^s` and `base^(-N) * min_x Sigma_N^(s)(x) * I_M^(s) <= I_(N+M)^(s)` for `I_level^(s) = Int_0^1 F_level^s` and `Sigma_N^(s)(x) = Sum_{i<base^N} F_N(x + i/base^N)^s`, so `g(s) >= (1/N)*log_base min_x Sigma_N^(s)` at every `N`. For base 9 missing `4` this reads `g(3/2) > 0.149397` and `g(235/154) > 0.142274` at `N = 5`, against the criterion's `0.147320` and `0.139667`; and because `Sigma_N^(s)` falls in `s`, every factor being at most `1`, while the criterion falls in `s` linearly, a chain of orders at `N = 4` covers `[3/2, 2)` in 21 cells with tightest margin `0.000085` and `[1, 2)` in 87 cells, the chain anchored at `s = 2` by the exact `Sigma_N^(2)(x) = (9/8)^N`, which Parseval gives at every `x` because two `N`-digit integers congruent mod `9^N` are equal. No window length and no refinement of the Markov estimate meets the criterion for the carpet (`lab/py/digit-transform-norms`). **Verified.**
- **The 2D transform cannot stand in for the missing norm.** The Morton code is a bijection of sets and not a homomorphism, so the frequencies the exceptional set indexes are not the characters the 2D transform bounds: at a one-dimensional frequency `theta` the Morton one-digit factor is `|hat F(u)|` with `u = (theta, base*theta)` at `dim 2`, the Morton product at level `level` is `Prod_{j<level} |hat F(base^(dim*j)*u)|`, and the 2D product at level `dim*level` at the same point, `Prod_{j<dim*level} |hat F(base^j*u)|`, is that product times the factors at the positions `base^dim` skips, each of them at most `1`. A bound on the 2D grid sum therefore bounds the smaller quantity and never the Morton one; the implication runs the wrong way. **Proved.**
- **The 2D numbers run the wrong way as well.** In one unit, `X = base^(dim*level)`, the carpet's 2D window moment reads `0.406200` at `s = 1` and `0.195631` at `s = 3/2` at five digit-vectors, against the one-dimensional `0.343674` and `0.153069` and the criterion's `0.294640` and `0.147320`: the 2D norm sits `0.0483` short at `s = 3/2` where the 1D norm sits `0.0058` short, so even granting a transfer it moves away from the carpet and not toward it (`lab/py/digit-transform-norms`). Both halves of the 2D-norm route to the carpet's last exponent are shut, and what stands between the design and a prime-counting theorem is the sieve, not another norm on the set. **Refuted.**
- **The gasket is far.** Base 4 missing `3` has `g(1) = 0.4820` against `27/77` and `g(235/154) = 0.3170` against `59/433`, so no Type II range opens at any order computed, and [Chow, Varju and Yu](https://arxiv.org/abs/2402.18395) Proposition 2.4 puts its Fourier `l^1` dimension above `1/2`, which is the Type I side only (`lab/py/digit-transform-norms`). **Verified.**
- **Componentwise transfer, read at source.** [Chow, Varju and Yu](https://arxiv.org/abs/2402.18395) Remark 6.1 puts the Fourier `l^1` dimension below `1/2` for `(base, a)` in `{(3,0), (3,1), (3,2), (4,1), (4,2)}` by interval arithmetic at `level 2`, and their Proposition 2.4 puts it above `1/2` for base 4 missing `0` or `3` and for every `base >= 5` missing one digit, so the base-3 design whose coordinate marginals are base-3 two-digit sets falls on the wrong side of the criterion while the base-2 gasket read through its Morton code, base 4 missing `3`, falls on the right one; the base-2 gasket has no componentwise reading at all, its coordinate projections being all of `[0, 2^level)`, and the base-3 design's own Morton code is base 9 on `{0, 1, 3}`, three digits and not one missing. **Verified.**
- The `l^1` exponent of a digit transform is at least `1 - alpha` at every base and digit set, the shifted-grid floor iterated over the `level` digit positions, so the Type I large-sieve gate `alpha_1 < dim/2` is never met by a design with `fill <= base^(dim/2)` and the gasket kill is one instance of a general floor (Proved, [mobius](mobius.md), the pair route).

## ON THE SHELF

- `coprime-density-above-dimension-one` proves the master density for every design with `dim >= 2`, condition (E) and `fill > base`, settling the convergence conjecture of [A396934](https://oeis.org/A396934); its scripts enumerate ten designs over 67 level rows, the base-6 bracket identity, the box bound in 1062 exact cases, and the whole base-2 `dim 4` census, 65536 designs in 402 orbits with 336 inside the theorem. **Proved.**
- `lemma-b-pincer` proves `rho(a, b) <= phi` for every primitive ray at every 3-adic depth, hence Lemma G, the gasket case, and by the reduction below Lemma B for every dimension-one line, at every prime exponent `beta > 1/(2 - log_3 phi) = 0.6402122`; with the moment ladder's tenth rung `0.4475978` imported, the estimate can fail only for `beta in (0.4475978, 0.6402122]`, and two doors are shut, exact Fibonacci products and averaged spectral radii both unable to lower the edge. **Proved.**
- The tenth rung is a theorem and `lemma-b-pincer` imports it, so the standing window is `(0.4475978, 0.6402122]`; `0.446717` survives only as the eighth row of the ladder table, kept for the record. **Proved.**
- `gasket-ray-machine` proves that the permutation design `F_phi` is diagonal exactly when `phi(0) != 0`, four of six, so its `3^level` points occupy `3^level - 2` non-fibre rays at every `level >= 1`; it carries the exact mass laws `M_level(3,1) = F(level+1) - 1`, `M_level(1,12) = A000930(level) - 1`, `M_level(7,3) = c(level-3) - 1` to `level 30`, and the multiplier-pair spectral gap, growth `3` on the three shift pairs, exactly `2` on twenty pairs, at most `1.6956` on the rest of the 829 coprime pairs with `max(s,t) <= 52`. **Proved.**
- `menger-pairwise-coprimality` proves the sponge's pairwise coprime density `(13/20) * Prod_{p != 3} (1 - 3/p^2 + 2/p^3) = (351/400) * C_3 = 0.251620868451255`, the factor at 3 replacing the lattice's `20/27` by `13/20`, with the level-6 census `15141288` of `20^6`, density `0.236583`, and the exact factor at 2 at level `level` explaining why finite levels sit below the limit. **Proved.**

## THE CENSUS AND THE OPEN LINES

- `lab/rs/design-census` holds the full enumeration: every design with `fill >= 2` at base 2 in dimensions 2 and 3, every design at base 3 in dimension 2, [the Menger sponge](/wiki/menger-sponge/), and two base-6 samples, 763 lines; each records the bracket, the predicted `delta`, the measured ratio at the deepest level a 200000-point budget allows, the first six terms, the exact-factor check and the spanning index.

| family | designs `fill >= 2` | spanning | distinct term-vectors among spanning | flagged |
|---|---|---|---|---|
| base 2, `dim 2` | 11 | 5 | 3 | 0 |
| base 2, `dim 3` | 247 | 149 | 27 | 0 |
| base 3, `dim 2` | 502 | 365 | 175 | 0 |
| Menger sponge | 1 | 1 | 1 | 0 |
| base 6 samples | 2 | 2 | 2 | 0 |
| total | 763 | 522 | - | 0 |

- Zero designs are flagged: the exact base-local identity holds on all 763 lines and every spanning design lands within `0.06` of its predicted density at the depth reached, `level 11` for `fill = 3` and `level 4` for the sponge, where `0.7719` against `0.8207` is inside only because the tolerance is loose that far up (`lab/rs/design-census`). **Verified.**
- Distinctness is first-six-terms only: 219 distinct term-vectors among the 502 base-3 designs, 175 of them spanning (`lab/rs/design-census`). **Verified.**
- Of the 522 spanning lines, 490 have `fill > base` and are closed by the theorem, and the 32 left all sit at `base 3`, `dim 2`, `fill = 3`, dimension exactly one; with the four index-3 lines at the same parameters they are the 36 open lines, and the census holds no design below dimension one (`lab/rs/design-census`). **Verified.**
- The gasket, base 2, `dim 2`, code 7, `B(F) = 2/3`, terms `2, 4, 12, 34, 122, 362` from `level 1`, is [A396934](https://oeis.org/A396934), whose entry starts at `level 0` with `0`; `(2/3)*(4/3)*(6/Pi^2) = 16/(3*Pi^2)` exactly, and the b-file gives `a(20)/3^20 = 0.5403760862` against `0.5403796461`, a gap of `-3.6e-06` (`lab/rs/oeis-terms`). **Verified.**
- The or-triangle, code 14, has `a_0 = 0`, `B(F) = 1`, density `8/Pi^2 = 0.810569`, terms `3, 6, 22, 58, 200, 576`, the classical `6/Pi^2` with the factor at 2 removed as for the Vicsek plus at 3 (`coprime-density-above-dimension-one`); it is a worked illustration, not a ledger sequence. **Verified.**
- The approach to the limit is not always monotone: at least one design has a level further from its limit than the level before, so numerical support in this family is support and never proof (`lab/rs/design-census`). **Verified.**

## THE WINDOW AT DIMENSION ONE

- **The E-decomposition.** A point of `S_level(F)`, `F = {v_0, v_1, v_2}`, is a string of corner choices `c_l`; with `E_j = Sum_{l : c_l = j} 3^l` the three `E_j` have disjoint base-3 supports summing to `c_level = (3^level - 1)/2`, `(E_1, E_2)` ranges bijectively over the gasket `G_level`, and `x = c_level v_0 + M (E_1, E_2)^T`, `M = (v_1 - v_0, v_2 - v_0)`, with `Delta = det M != 0` exactly when `F - F` has full rank. **Proved.**
- **Theorem (one set).** For every full-rank three-corner design, every `m` coprime to `Delta` and every `level`, `#{ x in S_level : m | x_1, m | x_2 } = #{ (u, v) in G_level : (u, v) = tau(level) mod m }` with `tau(level) = -c_level M^(-1) v_0 mod m`; so Lemma B for every dimension-one line is shifted-target equidistribution of the gasket pair, every bound below is target-uniform, and the 36 lines stand or fall together. **Proved.**
- **Corollary (simplex reduction).** For any base `base` and any full-rank design with `fill = dim + 1` corners the same decomposition reduces every divisibility count to the simplex `{0, e_1, ..., e_dim}` at base `base` with explicit targets; the sub-dimension-one world is a one-parameter family of simplex problems. **Proved.**
- Codes 11 and 161 both have `v_0 = 0`, target `0`, and `Delta` in `{-1, -3}`, so every modulus coprime to 3 gives the same gasket count and the base prime peels identically, whence their collision. **Proved.**
- Under `GL_2(Z)` shears the 36 lines collapse to 11 classes, in 8 of which the fractal is the graph `x_2 = g(x_1)` of a carry-free digit relabeling; the classes `{26, 50, 152}`, `{176}`, `{416}` are graphs of nothing, and the search is finite because a unimodular matrix keeping a full-rank corner set inside the digit cube has entries at most 6 in absolute value; no study regenerates the search. **Conjecture.**
- **The pincer.** Lemma G, the gasket case, splits by the prime's exponent `beta = log_3(p) / level`: the top range closes by the ray machine at `beta > 0.6402122` (`lemma-b-pincer`), the bottom by the moment ladder below, and the open lemma is pinched between.
- **The moment identities.** For `p != 3` write `f(t) = (1 + e(t_1) + e(t_2))/3` and `F_a(t) = Prod_{l<a} |f(3^l t/p)|`; gasket digits are `{0,1}` per coordinate, so adding two gasket points never carries and the additive energy is exactly `15^a`, while three or four summands carry only inside `{-1,0,1}^2`, a nine-state automaton, giving `Sum_t F_a^2 = p^2 * 3^(-a)`, `Sum_t F_a^4 = p^2 * (5/27)^a`, `E_6` growth `lambda_6 = 57 + 6*sqrt(46) = 97.693980` and `E_8` growth `lambda_8 = 456 + 3*sqrt(11017) = 770.885694`, exact Perron roots of integer transfer matrices (`lab/rs/dimension-one-ladder`). **Proved.**
- The `L2` identity is the box bound again, sharp, so the first `2 log_3 p` digits of the product never save anything and every saving is earned past them. **Proved.**
- **The ladder.** Hoelder across digit blocks, using only the bijection `t -> 3t` and never `ord_p(3)`, gives with `kappa = 3 - log_3 5`, `a = floor(log_3(p/2))`, `kappa_2K = 2K - log_3 lambda_2K`, `Lambda_2K = 2 - kappa + 2 kappa_2K` and `beta_0^(2K) = kappa_2K / Lambda_2K`, for every `K in {2, 3, 4, 5}`, `eta in (0, beta_0^(2K))`, `z >= 5`, `level >= 1`: `Sum_{z < p <= 3^((beta_0^(2K) - eta) level)} T*_p(level)/3^level <= 2/z + 35 z^(1-kappa) + 40 * 3^(-(kappa-1) level/8) + 6 * 3^(-(Lambda_2K/(2K)) eta level)`, so Lemma B holds unconditionally and target-uniformly for `p <= 3^((beta_0^(2K) - eta) level)`. **Proved.**
- The first three terms are the order-4 bookkeeping and never change with `K`: `2/z` is the main term `Sum_{p > z} p^(-2)`, `35 z^(1-kappa)` the four-block regime `4a <= level` where `3^a > p/6` and `6^kappa < 16`, `40 * 3^(-(kappa-1) level/8)` the three-block regime `3a <= level < 4a` whose worst case sits at its own seam `a = level/4`; only the fourth term, the main range `3a > level`, sees the moment order, and its exponent `Lambda_2K/(2K)` reads `0.883757, 0.687305, 0.545409, 0.443659` at `2K = 4, 6, 8, 10` (`lab/rs/dimension-one-ladder`). **Proved.**

| moments | `kappa_2K` | mid range closed below |
|---|---|---|
| 2, 4 | 1.535026 | 0.434233 |
| +6 | 1.829430 | 0.443624 |
| +8 | 1.949148 | 0.446717 |
| +10 | 1.985806 | 0.4475978 |
| ladder limit | -> 2 | -> 0.447931, never 1/2 |

- **The energy cap.** The carry box `{-r, ..., r}^2` with `r = floor((K-1)/2)` is closed, since a digit difference lies in `[-K, K]` and a state maps to `(s + d)/3` with `floor((r + K)/3) <= r` for every `K >= 1`, so `E_2K(G_a) = (M_2K^a)_{(0,0),(0,0)}`; every walk from the zero state back to itself stays inside `S`, the strongly connected component of that state, and `M_S` is irreducible by construction with a self-loop at the zero state, hence primitive with Perron root `lambda_2K = lim E_2K(G_a)^(1/a)` and positive right eigenvector `u`; from `e_0 <= u/u_0` componentwise and `M_S >= 0` follows `E_2K(G_a) <= lambda_2K^a` for every `a >= 0`, with no constant, and with equality throughout at `2K = 4`, where `S` is one state and `lambda_4 = 15`. **Proved.**
- **The master bound at order `2K`.** For a prime `p != 3`, `a = floor(log_3(p/2))`, `d_K = ceil(log_3(K/2))`, `level >= 2a` and `b = min(a - d_K, level - 2a) >= 0`, Hoelder over three disjoint blocks of the digit window `[0, level)` of lengths `a, a, b` at exponents `4K/(2K-1), 4K/(2K-1), 2K`, whose reciprocals sum to 1, gives `L_n(p) = Sum_t Prod_{l<level} |f(3^l t)| <= p^2 * 3^(-((2K - 2 + kappa) a + kappa_2K b)/(2K))`, target-uniformly; a block of length `m` at a non-integer exponent is interpolated between the exact `L^2` and `L^4` identities, which needs `2 * 3^m <= p`, the `L^(2K)` block is supplied by the energy cap, which needs `K(3^b - 1) < p`, and both hold because `K 3^(a - d_K) <= 2 * 3^a <= p`. **Proved.**
- The three moment orders give exponents `12/5, 12/5, 6`, `16/7, 16/7, 8` and `20/9, 20/9, 10`, one written bound per rung, all three with the same outer dyadic summation, so every row of the table is a theorem and the standing unconditional edge is the tenth rung `0.4475978`, which the shelf imports; the eighth rung `0.446717` stays in the table for the record (`lemma-b-pincer`). **Proved.**
- The seam is the same at every order: `3a > level` forces `b = level - 2a` and puts the prime in the main range, `3a <= level` forces `b = a - d_K` and hands it to the order-4 regimes, and the two agree at `level 3a - d_K`, so no seam crosses and no order-10 case is uncovered; requiring the exponent gain to exceed `a` in the main range is exactly `beta < beta_0^(2K)`, and summing `2 * 3^a` primes per `a` geometrically upward gives the constant `2/(1 - 3^(-Lambda_2K/(2K)))`, at most `5.1843` through `2K = 10` (`lab/rs/dimension-one-ladder`). **Proved.**
- **The tenth rung.** The carry box for `K = 5` is `{-2,-1,0,1,2}^2`, the exact 25-by-25 integer matrix `M_10` satisfies `E_10(G_a) = (M_10^a)_{(0,0),(0,0)}` with first energies `1, 4653, 28967859, 190911254427, 1270015973323281, 8461182216374750493`, direct convolution agreeing at `a = 1, 2, 3`; its characteristic polynomial factors as `x^6 (x-120)(x^2-450x+12231)(x^3-2190x^2+282096x-5186835)^2 (x^3-990x^2+116154x-2569725)^2 (x^4-7833x^3+7916949x^2-850684437x+13054946580)`, the Perron root is the largest root of the quartic, `lambda_10 = 6664.113662506`, so `kappa_10 = 1.985805792712` and `beta_0^(10) = 0.4475978134...`, above the eighth rung by `0.000880502992` (`lab/rs/dimension-one-ladder`). **Proved.**
- The exponent is certified without root-finding: a Sturm count on the exact quartic puts no root above `66641136626/10^7` and exactly one root in `[66641136625/10^7, 66641136626/10^7]`, so `lambda_10 < 6664.1136626`, `kappa_10 > 1.985805792698` and `beta_0^(10) > 0.447597813453`, every digit truncated down, never rounded. **Proved.**
- The order-10 three-block master bound, exponents `20/9, 20/9, 10` since `2/(20/9) + 1/10 = 1`, is written above with `Lambda_10 = 4.436585106`, main-range decay `Lambda_10/10 = 0.443658511` and constant `6`; the order-10 block on its own holds against exact `L_n(p)` in all 833 applicable cases with prime `5 <= p <= 199` and `2 <= level <= 24`, no violation, worst ratio `0.7839` at `(p, level) = (11, 2)`, so the lower edge `0.4475978` is unconditional. **Proved.**
- Rows 12 through 20 are floating Perron roots of exact integer matrices, not interval-certified: `0.447838092, 0.447904613, 0.447923402, 0.447928788, 0.447930346`, the last `6.42e-7` below the wall. **Conjecture.**
- What blocks them is only the root, not the machinery: the energy cap and the master bound are written for every `K`, so a Sturm bracket on the relevant factor of each characteristic polynomial promotes any of these rows one at a time, at a cost that grows with the integer size of the factor and buys at most `9.3e-5` of edge in total. **Proved.**
- **The peak wall.** `E_2K >= 3^((2K-2)a)/K^2`, the Fourier peak near `t = 0`, forces `kappa_2K < 2`, so `beta_0^(2K) < 2/(3 + log_3 5) = 0.447931` for every `K`: no moment, however high, reaches `1/2`. **Proved.**
- **The window.** The standing window is `beta in (0.4475978, 0.6402122]`, both edges unconditional, the lower one the tenth rung and the upper one the shelf's. **Proved.**
- It shrinks to `(0.4475978, 0.605303]` under the supergolden half of Conjecture N, to `(0.4475978, 1/2]` under Conjecture Z or W, and to nothing under Z and O together; the ladder can move the lower edge no further than `0.447931`, so a closed window needs the top edge brought down, never the bottom edge pushed up. **Conjecture.**
- Two walls face each other: the moment ladder cannot exceed `0.447931`, and a per-ray-maximum bound cannot exceed `1/2`, there being `3^(2cn)` rays of height `3^(cn)`. **Proved.**
- Any absolute-value bound on the character sums has a heuristic ceiling at `lambda_1 = log_3(1/mu_1) = 0.586752`, `mu_1 = E|f| = 0.524866`, so the hard core of Lemma B at dimension one is `beta in (0.45, 0.59)`, needing averaged ray masses, sign cancellation in `t`, or divisor rarity. **Conjecture.**
- Componentwise Fourier transfer cannot work at all: the base-3 missing-digit measure has Fourier `l1`-dimension below `1/2` ([Chow, Varju and Yu](https://arxiv.org/abs/2402.18395), Remark 6.1), so any Fourier attack must use the two-variable cancellation of the joint mask `f(t_1, t_2)`; the route not yet tried is a Vaughan or Heath-Brown decomposition applied before absolute values, since taking absolute values first is what destroys the cancellation. **Proved.**
- **The rays.** Every nonzero non-fibre point of `G_level` is uniquely `g * (a, b)` with `(a, b)` primitive, a prime `p > 3^(beta' level)` in the gcd forces height below `3^((1-beta') level)`, the fibres number `2^(level+1) - 2`, and the multiples of a ray inside the gasket are a finite automaton on the digits of `g` in which each carry state admits at most 2 of the 3 digits. **Proved.**
- **Conjecture N.** `rho(a, b) <= phi` with equality only on the shifts `(1, 3^j)`, `j >= 1`, and off-shift supremum the supergolden `1.4655713`, root of `x^3 = x^2 + 1`, attained at `(1,12)`, `(3,10)`, `(4,9)`; the bound half is the shelf theorem, the strictness and supremum halves are open, every observed radius a root of `x^k = x^(k-1) + 1` or `x^k = x + 1`, all 1102 coprime rays of height `<= 60` below `phi`, and `(1,1)` is not a shift, `rho(1,1) = 1`. **Conjecture.**
- The second moment is written `E(level)` here and on the shelf; `Z_F(level)` in `gasket-ray-machine` is the occupied non-fibre ray count, a different object, and the statement keeps the name Conjecture Z.
- **Theorem R (second moment).** Let `E(level) = Sum_y M_level(y)^2` over primitive rays count ordered collinear non-fibre pairs; if `E(level) <= C 3^(gamma level)` for some `1 <= gamma < 2`, then for every `beta > gamma/2`, `Sum_{p > 3^(beta level)} T*_p(level) / 3^level <= (1/beta) [ 2 (2/3)^level + sqrt(C) 3^(-(beta - gamma/2) level) ] -> 0`, Cauchy-Schwarz against the second moment over the at most `3^(2(1-beta) level)` rays that qualify; it holds per design since every full-rank design has at most `2^(level+1)` points with a zero coordinate. **Proved.**
- Feeding the per-ray maximum into Cauchy-Schwarz reproduces the direct thresholds exactly, the identity `1 - (1 - log_3 phi)/(2 - log_3 phi) = 1/(2 - log_3 phi)`, so averaging earns nothing until the true second moment enters. **Proved.**
- **One base, a family of automata with no uniform state bound.** Conjectures Z, W and O are statements about base 3 alone, `binary` throughout this section meaning a base-3 expansion with digits in `{0, 1}` and never base 2, and their machines are one automaton per ray, per multiplier pair, per band direction and per modulus `3^k`, the band family on the integers in `[-(z_2-1)/2, (z_1-1)/2]` and the digit-congruence family indexed by `3^k` carrying no uniform state bound. Cobham asks one set recognized by a finite automaton in two multiplicatively independent bases, which this lane never presents, so the wall of [cobham](cobham.md), that no [transfer matrix](/wiki/transfer-matrix/) over the digits of one base reads a two-base object, does not touch it.
- **Conjecture Z.** `E(level) = 2 * 3^level + o(3^level)`; under it the window is `(0.4475978, 1/2]`, and averaging cannot cross `1/2`, since the diagonal alone gives `E >= 3^level` and Hoelder at `2K >= 4` loses to the shift family because `phi > 3^(1/2K)`. **Conjecture.**
- `E(level)` for `level 2..18`: `2, 16, 98, 396, 1522, 5248, 17118, 52212, 158042, 466960, 1374038, 4003372, 11679626, 34050692, 99800950, 292848756, 862479378`, two generators sharing no method agreeing, `E(level)/3^level` peaking at `2.676` at `level 10` and falling to `2.226` at `level 18`; `lab/rs/dimension-one-ladder` regenerates `level 13..16` as `4003372, 11679626, 34050692, 99800950` and `gasket-ray-machine` regenerates `level 1..12` by literal enumeration, `level 17, 18` have no generator, and the list is not in the OEIS. **Verified.**
- `E(level) = T(level) + S(level) + R(level)` with the diagonal `T(level) = 3^level - 2^(level+1) + 1`, the 3-power family `S(level) = 2 Sum_{j=1}^{level-1} Q_level(1, 3^j) = 3^level - 4*2^level + 2 level + 3` from `Q_level(1, 3^j) = 3^(level-j) - 2^(level-j+1) + 1`, so `T + S = 2*3^level - 6*2^level + 2 level + 4` and Conjecture Z is the single statement `R(level) = o(3^level)`, the pair census bound. **Proved.**
- `R(level)` reads `20, 88, 432, 1624, 5512, 15896, 46064, 124928, 335704, 863848, 2211960, 5549452, 14100688, 35354824` at `level 4..17`, per-level ratio `2.5073119` at `level 17`, below `phi^2 = 2.6180339`; `R/3^level` peaks at `0.8401158` at `level 8` and falls at every level to `0.2737709`, and `R/phi^(2 level)` peaks at `3.2378233` at `level 12` and falls at five consecutive levels to `2.7724831` (`lab/py/gasket-witness-weights`, `gasket-ray-machine` for `level 1..14`). **Verified.**
- The multiplier decomposition: ordered off-diagonal collinear non-fibre pairs biject with triples `(s, t, z)`, `gcd(s, t) = 1`, `s != t`, `sz` and `tz` in `G_level`, so `E(level) = T(level) + Sum_(s,t) Q_level(s,t)` with each `Q_level(s,t)` a path count in the free-digit automaton `B(s,t)` of the shelf, never the gasket-digit `A(s,t)`; the pair spectral gap, `lambda = 3` on shifts and `<= 2` elsewhere with `P_w <= (3/2)^K 2^w`, `K = v_3(st) + v_3(t'-s')`, is the shelf's (`gasket-ray-machine`), and the gap alone yields only `E <= C 9^level`. **Proved.**
- **The shift-ray family is closed.** `M_level(3^j,1) = M_level(1,3^j) = prod_(r<j) F(m_r+2) - 1` with `m_r = #{i in [0,level-j) : i == r mod j}`, since `z(3^j,1) in G_level` says exactly that `z` is a binary string of length `level-j` with no two ones at distance `j`; hence `M_level(3^j,1) < (3 - sqrt5)^j phi^level`, `Sh(level) = Sum_(j>=1) (M_level(3^j,1)^2 + M_level(1,3^j)^2) < ((4 + 12 sqrt5)/11) phi^(2 level) < 2.803 phi^(2 level)` at every level, and `Sh(level)/phi^(2 level) -> (13 + 5 sqrt5)/11 = 2.198212717` (`gasket-ray-machine`). **Proved.**
- The shift rays are the dominant carrier of `R` and no more: their off-diagonal non-shift-multiplier pairs number `360, 1204, 3816, 10656, 30132, 81960, 221980` at `level 6..12`, a share of `R(level)` between `0.65` and `0.84`, reading `0.661` at `level 12`, so `Sh` closes two thirds of the Pair Census Bound and the non-shift rays are the whole remaining obstruction (`gasket-ray-machine`). **Verified.**
- The four ray mass laws are theorems at every level: `M_level(3,1) = F(level+1)-1`, `M_level(1,12) = a(level)-1` and `M_level(7,3) = c(level-3)-1` are Cayley-Hamilton on live carry automata of 2, 3 and 4 states with characteristic polynomials `x^2-x-1`, `x^3-x^2-1`, `x^4-x^3-1` (`gasket-ray-machine`). **Proved.**
- **The multiplier decomposition needs the free automaton, not the gasket-digit one.** `A(s,t)` counts `#{z in G_level : sz, tz in G_level}` while the summand `Q_level(s,t)` counts `#{z : sz, tz in G_level}` with no constraint on `z`; `min(s,t) = 1` forces agreement, since `s = 1` gives `z = sz in G_level`, and the two differ on 482 of the 2656 active ordered pairs at `level 9`, missing 2540 of the 33552 ordered collinear pairs, worst `(41,122)` with 50 witnesses and none in `G_9` (`gasket-ray-machine`). **Proved.**
- The gap survives that correction but its ceiling does not: over all 829 coprime pairs with `max(s,t) <= 52` the free-digit `B(s,t)` has radius 3 on exactly `(1,3), (1,9), (1,27)`, nothing in `(2,3)`, and exactly 2 on the same twenty pairs, by the same exact charpoly certificates; its largest radius strictly below 2 is `1.8488475886485` on `(4,13), (4,39), (12,13), (13,36)`, above the gasket-digit ceiling `theta = 1.6956207695598`, the real root of `x^3 - x^2 - 2`, which 44 pairs reach or beat in a sharp split, 19 strictly above `theta` and 25 exactly at it with `x^3 - x^2 - 2` dividing their charpolys, and its live sets reach 167 states at both `(25,52)` and `(31,40)` against 33 for `A` (`gasket-ray-machine`). **Verified.**
- **The pair coordinate is the wrong one, and the witness coordinate is the right one.** Every off-diagonal collinear pair biject to `(s, t, z)` with `z` the witness, so `R(level) = Sum_z P_level(z)` with `P_level(z)` the coprime non-shift pairs a single witness realises; the per-pair route needs a constant summable against the active-pair count `10, 30, 106, 332, 1010, 2642, 7564, 20934, 57858, 154410` at `level 4..13`, growth `2.77` a level, and is dead by construction, while the per-witness route already has its constant (`gasket-ray-machine`, `lab/py/gasket-witness-weights`). **Proved.**
- **Conjecture W, restated sharp.** `R(level) = O(phi^(2 level))`; since `phi^2 = 2.618 < 3` this implies Conjecture Z, which needs only the weak form `R(level) = o(3^level)`, and Z implies the window at `(0.4475978, 1/2]`. The weight-four orbit and the shift family are the two layers already closed, at `1.6945 phi^(2 level)` and `2.803 phi^(2 level)`; what is owed is summability over the witness weight. **Conjecture.**
- The first move is now proved and is the wrong half. If `mz in G_level` then `m z_1` and `m z_2` are binary with disjoint support, so `mw` is binary below `3^level` and `m -> mw` is injective: `M_level(z)` counts the binary `K < 3^level` with `w | K` whose submask `z_1 K / w` is itself binary, and dropping the submask condition gives `M_level(z) <= Bin_level(w)`, `Bin_level(w)` the binary base-3 multiples of `w` below `3^level`. But `Bin_level(w)` grows at rate 2, not `phi` - `Bin_24(w) = 4196351, 1683971, 613817, 228519` at `w = 4, 10, 28, 82` against the ceiling `F(25) - 1 = 75024` - so the weight enters only through the constant. What is owed is a bound whose rate falls with `w`, or a sum that keeps the submask condition (`gasket-ray-machine`, `lab/py/gasket-witness-weights`). **Proved.**
- **No witness weighs less than four**, so `max(s,t) <= (3^level-1)/8` for every active pair, sharp: the largest multiplier is exactly `floor(3^level/8)` at `level 4..13`. If `3 | z_1+z_2` but `3` divides neither coordinate then `v_3(m z_1) = v_3(m z_2)` and the supports collide, so weight layers scale exactly as `R_(3w)(level) = R_w(level-1)`, checked on all 1869 layers at `level 5..13` (`gasket-ray-machine`). **Proved.**
- **The weight-four layer is closed in Fibonacci.** With `F_level = {m : (m,3m) in G_level}` the no-adjacent-ones set, `#F_level = F(level+1) - 1` and `R_4(level) = 2 #{(a,b) in F_level^2 : a != b, gcd(a,b) = 1, b/a != 3^j} < 1.0473 phi^(2 level)`, so the whole 3-power orbit obeys `Sum_j R_4(level-j) < 1.6945 phi^(2 level)`, carrying `194096` of `R(13) = 863848`; exact at `level 4..12` where `R_4(level) = 12, 36, 108, 336, 988, 2596, 6672, 17480, 45720` (`gasket-ray-machine`). **Proved.**
- Every multiplier pair above `(3^level-1)/10` carries exactly 4 ordered collinear pairs, since its only witnesses are `(1,3)` and `(3,1)` and the coordinate swap pairs them; all 30028 such pairs at `level 6..13` obey it with no exception (`gasket-ray-machine`). **Proved.**
- **The golden ceiling, proved on the box.** `M_level(z) <= M_level(1,3) = F(level+1) - 1` for every direction with `z_1, z_2 >= 1`, so the shift ray `(1,3)` is the heaviest ray of the gasket at every level; this is the per-witness constant the per-pair route never had. In the direction coordinate a multiplier word is a word over the increments `{0, z_2, -z_1}` summing to zero, so the carry automaton has out-degree at most 2 with its branch states in one residue class mod 3, and the two successors of a branch state differ by `q/3` for the unique `q` in `{z_1, z_2}` divisible by 3 - occupancy forces `3 | z_1 z_2`, since `3 | z_1+z_2` with `3` dividing neither coordinate leaves `0` as the only increment congruent to `0` and kills every closed path but the trivial one, so `3 nmid z_1 z_2` already gives `M_level = 0` and settles 6566 of the 13158 box directions on residues alone against 3284 before. If no branch state has two branching successors - in particular whenever `v_3(q) = 1` - then `G(level) = max_c N(c,level)` obeys `G(level) <= G(level-1) + G(level-2)` and the ceiling follows outright. Of the 218 occupied directions of the 13158-box, 206 fall to that, 107 of them by `v_3(q) = 1`; three of the remaining twelve are shift rays, closed by `F(p+2) F(q+2) = F(p+q+3) - F(p+1) F(q+1)`, and nine carry explicit rational Fibonacci certificates of denominators 18, 40, 381, 18, 2013, 18, 40, 2013, 34 (`gasket-ray-machine`, `lab/py/gasket-witness-weights`). **Proved on the box, at every `level`.**
- **The golden ceiling is one inequality per direction.** Weight the first returns of the direction automaton by `phi^-1` a step: with `g(c,m)` the paths from a live state `c` to the start meeting it only at the end, `u(c) = Sum_m g(c,m) phi^-m` and `U(z) = Sum u(c')` over the start's successors other than itself, so `Sum_{j>=2} f_j phi^-j = phi^-1 U`. Any `pi > 0` with `Sum_succ pi <= phi pi(c)` at every live `c != 0` and `Sum_(c' != 0 succ 0) pi(c') <= phi^-2 pi(0)` forces `U(z) <= phi^-2` by a maximum principle on the truncated sums, and then `M_level(z) <= F(level+1) - 1` at every `level`, by renewal against the envelope `phi^(m-2) <= F(m) <= phi^(m-1)`; `pi = u` is admissible whenever `U(z) <= phi^-2`, so the criterion is exactly that one algebraic inequality, solved once per direction in exact `Q(sqrt5)`. It proves 45 directions no earlier case reached: the nine that needed hand-tuned rational certificates and the 36 that rested on enumeration alone (`gasket-ray-machine`, `lab/py/gasket-witness-weights`). **Proved.**
- The criterion misses exactly the shift rays and, on every censused range, nothing else. On `(1,3^j)` the mass grows at rate `phi`, so `Phi(phi^-1) = 1` and `U = phi^-1` exactly, and the Fibonacci product identity is the complementary tool. Over the box and the six families, 865 directions are occupied, 858 obey `U <= phi^-2`, and the seven failures are exactly `(1,3^j)`, `j = 1..7`; `U` is attained at `phi^-2` only on the supergolden `(1,12)`, `(3,10)`, `(4,9)`, takes 57 distinct values on the box, and was found in the open interval `(phi^-2, phi^-1)` at no direction of the ranges censused. Nothing arithmetic excludes the gap: a legal-looking profile `f_3 = f_5 = 1` sits inside it, so the gap is an observation and never a theorem. **Conjecture:** `U(a,b) <= phi^-2` for every non-shift primitive direction, which with the theorem and the shift-ray product is the whole golden ceiling (`gasket-ray-machine`, `lab/py/gasket-witness-weights`). **Proved on an infinite arithmetic class, Conjecture in general.**
- Beyond the box the ceiling is no longer enumeration only: the potential criterion proves every occupied direction of the six families bar the three shift rays there, so the 36 below are now theorems too. Six adversarial families overlap - the no-adjacent-ones family sits inside the binary one - so the shelf's 11369 coprime members are 10862 distinct directions, 717 already in the box and 10145 new, of which 9498 carry no mass, 608 fall to the branch argument and 3 are shift rays, leaving 36 on the enumeration alone; widened to `3^8` in the lab the union is 23435 distinct, 22718 new, leaving 77, and those 77 hold to `level 60` with worst ratio below `0.1516`. Zero breaches anywhere. Next rate down is the supergolden `1.4655`, root of `x^3 = x^2 + 1` (`gasket-ray-machine`, `lab/py/gasket-witness-weights`). **Verified on the de-duplicated families; Conjecture in general.**
- **The golden partition bound, proved on an infinite family.** Write `q = 3^k q_1` for the coordinate divisible by 3 (`3 nmid q_1`) and `p` for the other. For `k = 1` and `t = v_3(q_1 - p)`, `U(z) <= phi^-1 (1 - phi^-max(t,2))`, so `U <= phi^-2` on the whole arithmetic class `k = 1`, `t <= 2` - 261 of the 360 occupied `k = 1` directions of the census, and infinitely many in all - attained sharply at `(1,12)` and `(3,10)`. Since the bound is strictly below `phi^-1` for every `k = 1` direction bar `(1,3)`, those rays grow strictly slower than `phi`, which the ceiling alone never gave. The proof is a two-valued potential and a spine: the branch chain above `c_0` descends in `v_3` to a state of valuation 1, which always has a dead child, and `theta_i = 1 - phi^-(i+2)` climbs back (`gasket-ray-machine`, `lab/py/gasket-witness-weights`). **Proved.**
- **Occupancy is a congruence before it is an automaton.** `M_level(z) > 0` for some `level` forces `q_1 = p mod 3`: a multiplier `m = 3^s m'` makes `m' p` and `m' q_1` binary in base 3 and prime to 3, so both end in digit 1. It empties 4588 of the 11691 census directions with `3 | z_1 z_2` at no cost, and it is necessary only - just 865 of the 7103 matching directions carry mass (`gasket-ray-machine`, `lab/py/gasket-witness-weights`). **Proved.**
- **The degree potential replaces the solve, and the burst blocks the rest.** `pi = 1` where a live state branches, `phi^-1` where it does not, `pi(0) = 1`, is a super-solution whenever no branch state has two branching successors, and sweeping it gives a decreasing chain of exact bounds: it settles 849 of the 865 occupied shelf directions at least depth 1, 3, 4, 5, 6 on 760, 48, 31, 7, 3 of them, 37 outside the branch case, leaving the 7 shift rays and 9 named directions. Beyond `k = 1` the burst forces `phi^-2 >= pi(c_0) >= phi^-(k-1) Sum_m pi(q_1 m)` over `2^(k-1)` burst-floor states of valuation 0 while `pi(p) >= phi^-1` at the valuation-0 state `p`, so any valid potential must separate equal valuations by `phi^2 (2/phi)^(k-1)`: no potential constant on the level sets of `v_3`, and none constant on the out-degree classes, survives `k >= 2` (`gasket-ray-machine`, `lab/py/gasket-witness-weights`). **Proved.**
- **The bound with no automaton in it.** `U(z) <= phi^-2` is exactly `Sum_level (M_level(z) + 1) phi^-level <= phi^4 = 3 phi + 2`, and exactly `Sum_m phi^-l(m) <= phi` summed over the multipliers `m` of `z`, where `l(m)` is the number of base-3 digits of `(z_1 + z_2) m`. The conjecture is therefore a weighted count of multipliers, each weighted by `phi` to the minus its level, with no carry automaton anywhere in the statement (`gasket-ray-machine`, `lab/py/gasket-witness-weights`). **Proved.**
- The hypothesis `z_1, z_2 >= 1` and the counting of edges with multiplicity are both load-bearing: on the fibre ray `(0,1)` two digits share the increment `0`, a set-valued reading finds no branch state, and `M_level(0,1) = 2^level - 1` is 63 against `F(7) - 1 = 12` at `level 6` (`gasket-ray-machine`). **Proved.**
- Refuted as the general mechanism: the state maximum does not obey `G(level) <= G(level-1) + G(level-2)`; at `(1,9)` the profile runs `1, 1, 1, 2, 4, 6, 9` and `G(4) = 4 > G(3) + G(2) = 3`, and 8 directions of the box break it, all with `v_3(q) >= 2`. The sharp reformulation is the renewal criterion `Sum_{j>=2} f_j F(level+1-j) <= F(level-1)` on the first-return counts, with `f_1 = 1` always, `f_2 = 1` only at `(1,3)` and `f_3 = 1` only at `{1,9}`, `{1,12}`, `{3,10}`, `{4,9}` and `0` everywhere else, all now proved from the increments, and no first return at all of length between 2 and `v_3(q)`; it holds on all 218 occupied directions of the box to `level 46`, and the golden potential subsumes it in one number, `Sum_{j>=2} f_j phi^-j = phi^-1 U` (`lab/py/gasket-witness-weights`). **Refuted / Proved.**
- Two cheap constructions for `B(s,t)`: it is a constrained tensor square `T = S (x) S - U (x) U - V (x) V + W (x) W` of a one-coordinate carry automaton with at most `(s+1)(t+1)` states, so the four-tuple graph is never built (729 carry states against 26931 at `(365,1094)`); and at large multipliers the witness box `z_1 + z_2 <= floor((3^level-1)/(2 max(s,t)))` replaces the automaton entirely in `O(W^2 level)`, cheapest exactly where a forward build is most expensive (`gasket-ray-machine`). **Proved.**
- Refuted as a route to W: the majorant `Sum_z M_level(z)(M_level(z)-1)` grows `2.907` a level at `level 13` against `2.573` for `R` itself, because it drops the coprimality of `(s,t)` (`lab/py/gasket-witness-weights`). **Refuted.**
- Two roads do not reach W: the universal pair-prefix transfer matrix has Perron root `2^2 = 4`, not 3; and the unweighted octave census fitted at `level 13..16` returns exponent `2.956` with a constant drifting `1.042, 1.136, 1.244, 1.356`, a different quantity from W's weighted sum, never to be read as a rival measurement of `C ~ 120`. **Conjecture.**
- Higher ray-mass moments make it worse: at `level 12`, 345318 occupied rays, `S_1 = 523250`, `S_2 = 1374038`, `S_3 = 46380938`, `S_4 = 8145428822`, max `M = 232`, and the Hoelder bound `S_1 <= N^(1-1/r) S_r^(1/r)` overshoots by `1.316, 3.380, 8.179` at `r = 2, 3, 4`, so the ray power-moment route is capped at the second-moment edge `1/2`; no study regenerates the moments. **Conjecture.**
- Paley-Zygmund and Bonferroni are unavailable, not merely untried: the proof needs an upper bound on total bad mass while Paley-Zygmund lower-bounds the heavy rays, and Bonferroni needs uniform estimates of the signed intersection counts `T*_{pq}, T*_{pqr}, ...` over an exponentially growing modulus range, which do not exist. **Proved.**
- **Occupancy.** Every occupied ray has exactly one coordinate divisible by 3, the eq and opp classes never being occupied; each occupied ray maps to its minimal witness, which has no nonzero proper digit-prefix parallel to itself since `det(x mod 3^k, x) = 3^k det(lo, hi)`, the converse failing by a stable factor. **Proved.**
- The `level 13` multiplier census: 1044840 occupied non-fibre rays, 699508 carrying `M_level = 1`, 339530 carrying `M_level` in `[2, 5]`, `Sum M_level = 1577940 = 3^13 - 2^14 + 1`, `max M_13 = 376 = F(14) - 1`, the ten heaviest rays the shifts `(1, 3^j)` and reverses for `j = 1..5` with 14% of `Z` (`lab/rs/dimension-one-ladder`), and prefix-new points overcounting occupied rays by `1.51x` (the `gasket-ray-machine` lane). **Verified.**
- **The occupancy convention, pinned.** The height of a ray is `max(z_1, z_2)` of its primitive direction, the window `octave <= alpha level` is read as the threshold `height <= 3^(alpha level)`, and the octave is `floor(log_3 height)`, one below the census generator's `floor(log_3 height) + 1`; ray totals exclude the two fibre rays unless the fibre-counting convention is named.
- At `c = 1/2` the occupied rays number `3^(0.5416 level)` to `3^(0.5798 level)` across `level 10..18` against the trivial `3^level`, and at `c = 0.5533` the exponent stays inside `[0.6109, 0.6345]`, slack `delta >= 0.36`; the earlier band `0.543` to `0.557` does not reproduce under any cut, the readings `0.5249` or `0.6052` at `level 13`, `0.5677` at 14, `0.5348` or `0.6096` at 15, `0.5765` at 16 being artefacts of the integer octave cut that the threshold reading removes, and the occupied non-fibre ray totals `3151656, 9491964, 28545340` at `level 14, 15, 16` regenerate the census rows `3151658, 9491966, 28545342` two apart, exactly the two fibre rays (`lab/py/occupancy-decay`, `lab/rs/dimension-one-ladder`). **Verified.**
- **Conjecture O.** Occupied rays of octave `j <= 0.5533 level` number at most `C 3^((1-delta) level)`. **Conjecture.**
- **Theorem R+.** Z and O together close the window entirely, band Cauchy-Schwarz with occupancy in place of the ray count reaching down to the ladder; and no bootstrap escapes, since occupancy bounded by retrospective window mass returns `delta/2` where `delta` went in, so the seed of decay must come from the automaton side. **Proved.**
- **Conjecture O is trivial below one half.** The rays of height at most `3^(alpha level)`, occupied or not, number at most `3^(2 alpha level)` under the threshold reading and at most `9 * 3^(2 alpha level)` under the octave cut, so O holds with `delta = 1 - 2 alpha` and no occupancy input for every `alpha < 1/2`; the whole content of O is `alpha in [1/2, 0.5533]`, where the box is `3^level` at the left end (`lab/py/occupancy-decay`). **Proved.**
- **The first moment of occupancy is the window itself.** With `F(level, X)` the count of non-fibre gasket points whose primitive part has height at most `X`, every `x` with `p | gcd(x)` and `p > 3^(beta level)` has primitive height below `3^((1-beta) level)` and carries at most `1/beta` such primes, so `Sum_{p > 3^(beta level)} N_level(p) <= (F(level, 3^((1-beta) level)) + 2^(level+1)) / beta` at target zero, the fibre points paying the `2^(level+1)`; hence `F(level, 3^(alpha level)) = o(3^level)` proves zero-target Lemma B above `beta = 1 - alpha`, a first-moment proof of O moves the standing window at every `alpha > 0.3597878` and closes it outright at `alpha >= 0.5524022` with no Conjecture Z, and the route is therefore unavailable across the whole range where O has content; checked against the sieved prime sum at `level 10, 12, 14` and `beta = 0.45, 0.5, 0.6`, worst ratio `0.1517` (`lab/py/occupancy-decay`). **Proved.**
- **Occupancy pays no exponent for the multiplicity.** `F/A` at `alpha = 0.5533` reads `5.41, 5.20, 5.52, 5.64, 5.63, 5.86, 5.79, 6.08, 5.92` at `level 10..18` while `log_3 F / level` falls `0.7645` to `0.7201` against `log_3 A / level` inside `[0.6109, 0.6345]`, the two exponents converging at the rate `log(F/A)/(level log 3)`; only at a fixed height do the shift rays split them, `A(level, 3^5) = 384 .. 474` against `F(level, 3^5) = 2728 .. 51694` over `level 10..18`. So O is no cheap half of Theorem R+: it carries the weight of the window (`lab/py/occupancy-decay`). **Verified.**
- **The digit-congruence bound.** Every occupied ray satisfies `z_1 z_2^(-1) mod 3^k in R_k union {0}` after the coordinate swap, with `R_k = {u v^(-1) : (u,v) in G_k, u > 0, 3 does not divide v}` indexed by the modulus `3^k`; the `0` is needed and not decorative, since `3^k | z_1` sends the residue to `0` and `(9,1)` is occupied at `k = 2` with `R_2 = {3}`. Counting each residue class in the box gives `A(level, X) <= 2 sigma_k X^2 + 2 sigma_k 3^k X + 4 X^2 3^(-k) + 3^k + 4 X` for every `k` with `3^k <= X`, where `sigma_k = |R_k|/3^k` is non-increasing and the doubled tail terms pay for the adjoined class; this is every digit-class constraint at once, the proved mod-3 dichotomy being the case `k = 2`, where `R_2 union {0}` reads exactly `3 | z_1`, and not `k = 1`, where `R_1` is empty (`lab/py/occupancy-decay`, `lab/py/ratio-set-saving`). **Proved.**
- **And the digit-congruence seed is measured out.** `sigma_k` falls only polynomially through the computed range, `0.046063` at `k = 13` to `0.034259` at `k = 18`, growth `|R_(k+1)|/|R_k|` rising monotonically `2.794` to `2.8461` and `k(1 - log_3 growth)` inside `[0.8418, 0.8628]` over `k = 13..18`, so the route buys a factor `level^(-0.86)` and no exponent; its ceiling is the pair-prefix root, since Cauchy-Schwarz on the multiplicity gives `sigma_k >= (3^(k-1) - 2^(k-1))^2 / (3^k M_2(k))` with the congruence-collinear count measured at `M_2(k)/4^k = 0.4098, 0.4077, 0.4071, 0.4029` for `k = 13..16`, still falling, and on the hypothesis `M_2 = O(4^k)` no congruence-only decay beats `c = 0.2618596` or `alpha = 0.575328`, which excludes neither `0.5533` nor `0.5524022`. No exponential floor is proved either way (`lab/py/occupancy-decay`). **Verified.**
- **What O now asks.** In ratio coordinates `A(level, X)` is the number of rationals of height at most `X` in the ratio set `{u/v : (u,v) in G_level}`, measured at `X^theta` with `theta` inside `[1.1041, 1.1467]` at `alpha = 0.5533` and `[1.0833, 1.1596]` at `alpha = 1/2` over `level 10..18`, against the box exponent 2; O at `alpha` follows from any `theta < 1/alpha`, so `alpha = 0.5533` needs only `theta < 1.8073`, a power saving of `0.1927` over the box that nothing yet gives (`lab/py/occupancy-decay`). **Conjecture.**
- **The ratio-set lemma, uncapped and measured.** Deciding occupancy by automaton reachability rather than by level removes the level cap from `A`, and the uncapped count of distinct rationals of height at most `X` in the ratio set reads `32, 80, 206, 572, 1404, 4124, 9832, 26638, 72014, 184266` at `X = 32 .. 16384`, with `log A / log X` inside `[1.2057, 1.2494]` and the local exponent inside `[1.3554, 1.4380]` over `X = 2048..16384`, against the box exponent 2 and the `1.8073` that O asks; the same generator reproduces `A(level, 3^5) = 384 .. 474` at `level 10..18` and the pinned `A(9, 3^7) = 2818` without enumerating the gasket, and `A(3^level)` is at least the occupied ray total, `0.655 * 3^level` at `level 13`, so the exponent is at least 1 (`lab/py/ratio-set-saving`). **Verified.**
- **The band, and the weight layer O reduces to.** The pair carry state of a witness is the single integer `j = c_1 z_2 - c_2 z_1`, and disjoint supports make the emitted digits sum to a binary base-3 number, confining `j` to `(-z_2/2, z_1/2)`: at most `(z_1-1)/2 + (z_2-1)/2 + 1` states, out-degrees `2, 1, 0` one to each residue class mod 3, and a direction occupied exactly when `0` is reachable from `z_1/3`. Hence `Sum_{w <= X} Z(w) <= A(X) <= Sum_{w <= 2X} Z(w)` on the weight layer `Z(w)`, so a pointwise `Z(w) <= C w^beta` gives O at every `alpha < 1/(1+beta)`, with `beta < 1` giving `eps > 0` and `beta < 0.8073` giving O whole; binary weights split by every submask, so the layer has a floor there, though the coprime cut leaves the lower end `beta >= log 2 / log 3` unproved (`lab/py/ratio-set-saving`). **Proved.**
- **The band automaton, exactly.** For a direction `z_1 + z_2 = w` with `3 | z_1` the states are the integers in `[-(z_2-1)/2, (z_1-1)/2]` and the moves are `j -> (j + a)/3` over the increments `a` in `{0, z_1, -z_2}` whose quotient is integral, the band being invariant under all three; a walk leaves `0` by the forced increment `z_1`, and a return to `0` at time `level` spells a multiplier `m` with `m z_1` and `m z_2` binary in base 3 on disjoint supports, so `m w` is binary of base-3 length `level` and the first return time of a direction is the base-3 length of its shortest binary lift (`lab/py/band-return-times`). **Proved.**
- **The return count is exact at every horizon, and the return time has one gap.** `L(k, n) = #{m >= 1 : 3 not dividing m, m R_k binary in base 3 and below 3^n}` counts the primitive returns of the weight `R_k` inside horizon `n`, and the carry transfer on the slot profile `s_r = ceil((n - r)/k)` gives it exactly at every `k` and every `n`, past the rigid depth the block ladder stops at, reading `L(k, 4k) = 185, 1002, 5573, 31506, 180125, 1038402` at `k = 3..8` against the checked identities `L(k, k) = L(k, k + 1) = 1`, `L(k, 2k) = 2^(k-1) + 1` and `L(k, 3k) = 3^k + 1` at `k = 2..8`. So the support of the return time, the lengths at which some return exists, is `{k} union [k + 2, 8k]` at every `k = 2..12`, one gap at `k + 1` and no other inside that range, with nothing past `n = 8k` or `k = 12` decided. What `L` never bounds is the FIRST return count, which is the object the deep tail is made of, and the lengths the first return time actually takes look a far thinner set, `16, 16, 59, 80` distinct values at `k = 11..14` on a single unpinned reading of the first-return sweep (`lab/py/band-return-times`, verbs `returns` and `hist`). **Proved / Verified**, the thin-set reading only **Conjecture**.
- **The block rate is an algebraic integer, computed and not estimated.** At the block horizon `level = bk` for `w = R_k` the column transfer has a uniform slot profile, `s_r = b` at every column, so the transfer is one matrix fixed in `k` per residue and the return count `L(k, bk)` obeys a constant-coefficient linear recurrence in `k` whose dominant root is the block rate `lam_b`; the roots are exact, `lam_4 = 6` from `(x-1)(x-3)(x-5)(x-6)`, `lam_5 = 3(5 + sqrt 5)/2` from `(x-1)(x^2 - 15x + 45)`, `lam_6 = 13 + sqrt 79` from `x^2 - 26x + 90`, `lam_8 = (99 + 9 sqrt 65)/2` from `x^2 - 99x + 1134`, and an independent residue DP reproduces `L(k, 4k)` and `L(k, 5k)` to `k = 12` and factors both characteristic polynomials in exact arithmetic (`lab/py/band-return-times`). **Proved / Verified.**
- **The block ladder of rates, certified to depth 14.** Past those four roots the minimal polynomial of `lam_b` is exact at every `b <= 14`: `lam_7` is the dominant root of `x^3 - 63x^2 + 945x - 3402`, `lam_9` of `x^4 - 255x^3 + 16065x^2 - 293787x + 1299078`, `lam_10` of `x^3 - 392x^2 + 17469x - 96228`, `lam_11` of a quintic with no radical form, `lam_12` of `x^3 - 1551x^2 + 257256x - 5629338`, `lam_13` of a sextic with none either, and `lam_14` of `x^4 - 6176x^3 + 3963141x^2 - 335533914x + 2583866142`, so the even ladder stays in radicals through `b = 14` and the odd one leaves them at `b = 11`; exact bisection certifies `lam_b` to a width below `1e-9` at `10.854101966, 21.888194417, 42.760932540, 85.780159867, 170.715620440, 341.700429300, 682.692831036, 1365.640975936, 2730.680876219, 5461.594643683` over `b = 5..14`, and the recurrence `L(k, bk)` obeys in `k` has minimal order `b` at even `b` and `(b+1)/2` at odd `b` there (`lab/py/band-return-times`, verb `ladder`). **Verified.** Inside that exact row the degree of the minimal polynomial reads `ceil(b/4)` at even `b` and `(b-1)/2` at odd `b >= 3`, a pattern observed on the thirteen rungs `b = 2..14` and licensed at no `b >= 15`. **Conjecture.**
- **A block ratio reads the block rate at odd depth and at no even one.** The second root of the recurrence is `0.959422` of `lam_6`, `0.991055` of `lam_8` and rises to `0.999909` of `lam_14`, so the ratio `L(k+1, b(k+1)) / L(k, bk)` carries at most two correct digits at `k = 160` at every even `b <= 14`, while at odd `b = 5..13` that root falls from `0.381967` to `0.333404` and the same ratio carries 66 to 76 correct digits there: at even depth a growth read off a ratio is a reading and the fixed matrix is the only source of the value (`lab/py/band-return-times`, verb `ladder`). **Verified.**
- **The sharp bracket on the block rate.** Every column sum of every block matrix is `Sum_(c = a mod 3) binom(b, c)`, whose deviation from the free rate `2^b/3` takes only two values per `b`, `{-1/3, +2/3}` at even `b` and `{-2/3, +1/3}` at odd `b`, read exactly to `b = 20`; a nonnegative matrix has its [spectral radius](/wiki/spectral-radius/) between its least and its greatest column sum, so `lam_b` lies in `2^b/3 + [-1/3, 2/3]` at even `b` and in `2^b/3 + [-2/3, 1/3]` at odd `b`, and the return supply therefore matches the free rate to a relative `2^(1-b)` at every depth. The excess `3 lam_b - 2^b` reads `2, 0.5624, 1.6646, 0.2828, 1.3405, 0.1469` at `b = 4..9`, above the free rate at every computed depth and closing on it like `2^(-b)` (`lab/py/band-return-times`). **Proved / Verified.**
- **What the band measures.** No pair to height 3000 violates the cap and the largest reachable set fills `0.9865` of it, that fraction being the maximum and not the rule; the running `log Z_max / log W` sits inside `[0.5000, 0.7010]` over `W = 32..16384`, at argmaxes that are binary base-3 integers throughout (`lab/py/ratio-set-saving`). **Verified.**
- **The top digit fixes every occupied slope.** The highest base-3 digit `3^t` of `m(z_1 + z_2)` sits in exactly one of the disjoint binaries `m z_1, m z_2` and the other is a sum of distinct lower powers, hence at most `(3^t - 1)/2`, so `max(z_1, z_2) > 2 min(z_1, z_2)` and `z_1/w` never lies in `[1/3, 2/3]`; nothing violates it among the occupied directions of weight at most 8192, the pairs to height 120 or the rays at `level 12`, and the adversarial pass makes it sharp and strict at minimum ratio `2.0000004` over 14.3 million pairs at level 15, extremal at `(3^14, (3^14 - 1)/2)` (`lab/py/ratio-set-saving`). **Proved.**
- **The congruence seed and the weight layer are one bound.** With `r = z_1 z_2^(-1) mod 3^k` and `z_2 = w - z_1` comes `z_1 (1 + r) = r w`, and `r = -1 mod 3` would force `3 | w`, so `1 + r` is a unit, `z_1 = r w (1 + r)^(-1)` is determined, and for `3^k > w` the map `z_1 -> r` is injective on the layer and `Z(w) <= 2 |R_k|`. So `beta < 1` from that side asks `sigma_k` to fall geometrically, which is exactly what criticality forbids; the bound is sharp early, `sigma_k = 1/9` at `k = 2, 3, 4` and first below at `k = 5`, and slack late, allowing `146880` at `w = 797161` against the true `Z = 10388` (`lab/py/ratio-set-saving`, `lab/py/occupancy-decay`). **Proved.**
- **The metric route to the saving is closed.** Two slopes of denominator `w` differ by at least `1/w`, so `Z(w) <= 2 N_P(1/w)` for the cover of the slope set `P = {u/(u+v)}` at that scale; but the cover measures too large, `level N_P(3^-level) / 3^level` rising `2.4132 -> 2.4785`, `log_3 N_P / level` rising `0.8783 -> 0.8997` and the step exponent rising `0.9333 -> 0.9504` over `level 12..18`, every reading monotone and every one above the `0.8073` the reduction needs. The sandwich is proved; the `3^level / level` growth and the `O(w / log w)` ceiling it forces are measured from seven points with the constant still rising, and they put a missing-digit rational-counting import at the 3-adic ratio set `R_inf` rather than at the slope variable (`lab/py/ratio-set-saving`). **Proved / Verified.**
- **Where the weight layer actually sits.** Read per weight rather than off a running maximum, `log Z(w) / log w` peaks at `0.7093` at `w = 121` and `Z(w) / w^(log 2 / log 3)` at `1.5975` at `w = 1093` over every `w <= 8192`, all twenty-four octave argmaxes binary base 3; on the repunits `(3^k - 1)/2` at `k = 9, 11, 13` and the shifts `1 + 3^h` at `h = 7, 9, 11, 13` the exponent holds inside `[0.6223, 0.6818]` out to `w = 1594324` while unstructured neighbours collapse to `[0.2861, 0.4272]`. Occupancy may also be relaxed from returning to `0` to merely surviving, `Z <= Zinf` with `Zinf/Z` at most `1.5295` on the eleven weights tested. So `beta = log 2 / log 3 = 0.6309297` is conjecturally both ends of the corridor, `0.1763` clear of `0.8073` and giving `alpha < 0.6131` (`lab/py/ratio-set-saving`). **Verified / Conjecture.**
- **What the repunit sweep counts.** The sweep runs over the directions `(z, R_k - z)` of weight `R_k` and meets each one twice, once at `z` and once at `R_k - z`, so `Phi_k`, `Z(R_k)`, `U_k` and `V_k` are counts of `z` values and the distinct directions are half of each, every first-return count being even for that reason; a sample size quoted off one of them without halving is doubled (`lab/py/band-return-times`, verbs `hist` and `check`). **Proved.**
- **The repunit floor exactly.** On `w = R_k = (3^k - 1)/2` the floor is `Phi_k = #{S : {} != S != [0,k-1], gcd(a_S, R_k) = 1}` with `a_S = Sum_{i in S} 3^i`; since `3^k = 1 mod R_k`, every `q | R_k` has `d = ord_q(3) | k`, so Mobius inversion over the squarefree `q | R_k` and finite Fourier inversion give `Phi_k = Sum_q mu(q) N_k(q)`, `N_k(q) = q^(-1) Sum_{t mod q} P_{q,t}^(k/d)`, `P_{q,t} = Prod_{r < d} (1 + e(t 3^r / q))`, the two sets `S = {}` and `S = [0,k-1]` cancelling under `mu` for `k >= 2`. So the floor is C-finite in `k` along each `d N` 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` (then `u -> 2u` permutes the orbit `t<3>` and `Prod (1 + e(u/p)) = Prod (1 - e(2u/p)) / (1 - e(u/p))` telescopes to `1`), attained at `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`, the residue DP and the Fourier form agreeing with the direct submask count at every `k`; the density `delta_k = Phi_k / 2^k` reads `0.9997` at `k = 13` and `0.3281` at `k = 12` (`lab/py/ratio-set-saving`). **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 `c_r = #{i in supp K : i = r mod k}` satisfy `Sum_r c_r 3^r = 0 mod R_k`; below `3^(2k)` these are `K_T = a_(T^c) + 3^k a_T` for `T` in `[0,k-1]`, multiplier `m_T = 1 + 2 a_T`, and `R_(2k)`, which yields only submask directions, and `K_T = 3 K_(T') ` when `0 in T`, so up to shift the lifts are indexed by `T` in `[1,k-1]`. Each `Occ_T = {A / m_T : A a submask of K_T, m_T | A, 0 < A < K_T, gcd(A / m_T, R_k) = 1}` is a set of occupied directions of weight `R_k`, hence `Z(R_k) >= |Union_T Occ_T|`, and a direction with an unlifted witness, `A` a submask of `a_(T^c)`, stays occupied at every larger `k` at which it stays coprime. The 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`, every non-submask direction at `k = 7, 8, 9` having witness `m = 7, 19, 25, 55`, that is `T = {1}, {2}, {1,2}, {3}`, and falls short by `18, 16, 108, 162, 624` at `k = 11..15`, the shortfall being directions whose minimal witness uses some column twice or more, up to `27` times over the `436` digits of the lift `m R_k` at `k = 13`, so no witness family of bounded height is exact. The lifts are not random one `T` at a time and nearly random in aggregate: `Sum_T |Occ_T|` is `12696` at `k = 13` against the equidistribution model `Sum_T 2^k / m_T = 11586.5`, an aggregate excess of `1.0960` once the coprime density is taken out, while the single cyclotomic `T = [6, 11]` beats its own model by `4016.626` there and `T = [9, 17]` beats it by `376843.283` at `k = 19`. For `R_k` prime and `k >= 15`, first at `k = 71`, the single lift `T = {1}` already gives `Z(R_k) - Phi_k >= 2^k/7 - 4 F(k+1) - 126`, every `S` giving both ordered directions `(z, R_k - z)` and neither binary: `#{S in [0,k-1] \ {1} : 7 | a_S}` is `2^(k-1)/7 + O(1)` because the period-6 orbit product is `Phi_7(-1) = 1`, and `7 a_U` is binary only when every run of `U` has length two or more and every inner gap two or more, at most `2 F(k+1)` sets (`lab/py/ratio-set-saving`). **Proved / Verified.**
- **The cyclotomic lift, and the pointwise route closed.** The equidistribution model `2^k / m_T` for `|Occ_T|` sums: `m_T = 1 + 2 a_T > 2 * 3^(max T)` and exactly `2^(t-1)` sets `T` inside `[1, k-1]` have `max T = t`, so `Sum_T 1/m_T < 1 + (1/4) Sum_{t >= 1} (2/3)^t = 3/2` at every `k`, reading `1.41723` at `k = 19`; the model for the lift union is therefore `O(2^k)` outright. It cannot be enforced one `T` at a time. At `k = 2t + 1` take `T = [t, 2t-1]`: then `a_T = 3^t R_t`, `m_T = 3^(2t) - 3^t + 1 = Phi_6(3^t)` and `(3^t + 1) m_T = 3^(3t) + 1`, so for every `S` inside `[1, t-1]` the number `A = (3^(3t) + 1) a_S` is binary with support `S union (S + 3t)` inside `T^c union (k + T)`, hence a submask of `K_T` divisible by `m_T` with `A / m_T = (3^t + 1) a_S`; and `gcd(3^t + 1, R_(2t+1)) = 1`, since `R_(2t+1)` is odd and an odd prime dividing both would have multiplicative order dividing `gcd(2t, 2t + 1) = 1`. The `2^(t-1)` numbers `A` and their `2^(t-1)` complements `K_T - A` are distinct because `(3^t + 1)` does not divide `R_k`, so `#{A submask of K_T : m_T | A} >= 2^t` against a model `2^(2t+1) / (3^(2t) - 3^t + 1)` below `1` at every `t >= 2`; that count times `m_T / 2^k` is at least `2^t (3^(2t) - 3^t + 1) / 2^(2t+1)`, growing like `(9/2)^t`. So no uniform `#{A submask of K_T : m_T | A} <= C 2^k / m_T^c` survives `c > log 2 / (2 log 3) = 0.3154649`, while summing such a bound over `T` gives `O(2^k)` only for `c > log 2 / log 3 = 0.6309297`, since `m_T < 3^(max T + 1)` makes `Sum_T m_T^(-c)` grow geometrically below that: every exponent that would close the lift-union half is already refuted, and the route is closed for that shape. The coprime cut removes nothing where it is checked, but the family survives it only under a hypothesis: `Occ_T` at that `T` is exactly `{(3^t + 1) a_S}` and its complements, of size `2(2^(t-1) - 1)` whenever `R_k` is prime (Proved) and of size `2, 6, 12, 30, 62, 100, 254, 510` at `t = 2..9` (Verified), while an unconditional statement would need `#{S inside [1, t-1] : gcd(a_S, R_k) = 1} >= 2^t / poly(t)`, which is nowhere proved; and `max_T |Occ_T| m_T / 2^k` reads `4.562, 32.953, 151.898, 861.43, 4016.626, 14589.791, 83406.073, 376843.283` at odd `k = 5..19`, attained at that `T` every time (`lab/py/ratio-set-saving`). **Proved / Verified.**
- **The lift union to `k = 19`.** Meeting the two halves of a submask in the middle decides `m_T | A` in `O(2^(k/2))` per `T` instead of `O(2^k)`, so the whole union `k = 2..19` costs `21 s`, `12.7 s` of it at `k = 19`, and `U_k` reads `2342, 1618, 10280, 10278, 35566, 31910, 175314, 128698, 715322` at `k = 11..19` against the floor `Phi_k = 1958, 1344, 8190, 8064, 27360, 24384, 131002, 95040, 523982`. Exactly `U_k <= Sum_T |Occ_T| = agg_k L_k Phi_k`, where `L_k = Sum_T 1 / m_T < 3/2` is Proved above and `agg_k = 2^k Sum_T |Occ_T| / (Phi_k Sum_T 2^k / m_T)` is the aggregate against the model after the coprime cut. Over `k = 11..19` `agg_k` sits inside `[1.01748, 1.11457]` with no trend, `L_k` reaches `1.41723`, `U_k / Phi_k` rises monotonically across the band `[1.19611, 1.36517]`, and the overlap loss `U_k / Sum_T |Occ_T|` sits inside `[0.76088, 0.93128]`. So the model is beaten by `376843` at one `T` and by at most `1.11457` in aggregate, and the lift half of the blocking lemma is exactly the boundedness of `agg_k`, an on-average equidistribution over the lift family rather than a bound on any one lift (`ratio.py lifts --kmax 19 --zmax 15`, 10 min 43 s, the `Z(R_k)` column carrying all of it). **Verified.**
- **The lift count in Fourier form, and two routes closed.** `#{A submask of K_T : m_T | A} = (1/m_T) Sum_{u mod m_T} Prod_{p in supp K_T} (1 + e(u 3^p / m_T))`, the `u = 0` term being exactly the model `2^k / m_T` and the product real, `(-1)^(uk) Prod_p 2 cos(pi u 3^p / m_T)`; so the aggregate is the model `2^k L_k` plus the `u != 0` part, which carries `2^(k-1)` from `A in {0, K_T}` alone and is never small at one `T`: at the cyclotomic lift `F_T(1) >= 2^k (1 - 13 * 9^(-t))` for `t >= 2`. Cauchy-Schwarz in `u` already stops at the diagonal `2^(k/2)` per `T`, and absolute values fail on the data: `Sum_T (1/m_T) Sum_{u != 0} |F_T(u)| / 2^k` reads `1.3839 .. 7.9155` over `k = 5..11`, growing by `1.2655` or more at every step; the cut-free aggregate `Sum_T (N_T - 2) / (2^k Sum_T 1/m_T)` sits inside `[1.03919, 1.3403]` over `k = 11..19` with no upward trend, and the lift half is exactly `Sum_z W_k(z) = O(2^k)` for the number `W_k(z)` of witnesses below `3^k` (`lab/py/ratio-set-saving`, `ratio.py agg`). **Proved / Verified.**
- **The antipodal family, exactly.** For odd `p`, `t >= 1`, `0 <= s <= t` and `k = (p-1) t + s`, the set `T = Union_{i odd <= p-2} [ti, ti + t - 1]` has `m_T = (3^(pt) + 1) / (3^t + 1)`, `Phi_(2p)(3^t)` at prime `p`, and exactly `2^(((p-1)/2)(t - s) + s)` submasks of `K_T` divisible by `m_T`: the support splits mod `m_T` into antipodal pairs `3^j, -3^j` and `s` blocks of signed sum `3^i m_T`, and balanced-ternary uniqueness leaves only the pair diagonal and whole blocks. So the cyclotomic `T = [t, 2t-1]` has exactly `2^t` at every `k` from `2t` to `3t`, its `>= 2^t` at `k = 2t + 1` is an equality with `|Occ_T| <= 2^t - 2`, and the whole family is `O(k 2^(k/2))` at fixed `k`, carrying the largest `u != 0` Fourier terms and none of the aggregate; the count is asserted at all 74 triples to `k = 19` (`lab/py/ratio-set-saving`, `ratio.py agg`). **Proved / Verified.**
- **The block ladder, and the deep tail read by depth.** A binary `K` with support inside `[0, bk - 1]` is a multiple of `R_k` exactly when its column counts satisfy `V(c) = Sum_r c_r 3^r = 0 mod R_k`, and `0 <= V(c) <= b R_k` forces `V(c) = j R_k`; for `b <= 3` the lowest column pins `c_0 = j` and `j R_k - j = 3 j R_(k-1)` repeats the step, so the column vector is constant and the binary multiples of `R_k` below `3^(3k)` are exactly `2 * 3^k + 1` lifts: `3^k` with one position per column, multiplier `1 + 2 a_(E_1) + 2 (3^k + 1) a_(E_2)` for `E_j = {r : e_r = j}`, `3^k` with two positions, the complement of one, and `R_(3k)` itself, which yields only submask directions since `(1 + 3^k + 3^(2k)) z` carries nothing and is binary exactly when `z` is. At `b = 4` the first step already branches, `c_0 in {1, 4}`, with 24 non-constant vectors at `k = 3`, so depth 3 is the last rigid depth. Writing `b(z)` for the number of `k`-blocks the minimal witness lift `m(z) R_k` fills, `U_k = #{b(z) <= 2}` and the depth-3 census is `V_k = #{b(z) <= 3}`: `(U_k, V_k, Z(R_k))` reads `(2342, 2350, 2360), (1618, 1624, 1634), (10280, 10310, 10388), (10278, 10310, 10440), (35566, 35630, 36190)` at `k = 11..15`, so depth 3 captures `8, 6, 30, 32, 64` of the deep tail `18, 16, 108, 162, 624`, a share falling `0.4444, 0.375, 0.2777, 0.1975, 0.1025`, while `(Z(R_k) - U_k) / 2^k` rises along each parity and the tail's two-step growth reads `6.0, 10.125, 5.7777` against `4` for `2^k`. The one-position lifts add no direction beyond `U_k` at any `k <= 13`, so the whole capture sits on witnesses using every column exactly twice, and at `k = 15` the 624 tail `z` values, that is 312 directions, sit at 53 distinct depths reaching 81 blocks: on these five points the tail is a deep-column object no constant-column lift family reads, the first-return sweep reaching no `k` past 15 and the lift-family generator stopping at `k = 13`, where `3^(3k)` passes `2^63`, and `Z(R_k) - U_k = O(2^k)` stays open (`lab/py/ratio-set-saving`, `ratio.py tail`). **Proved / Verified.**
- **The deep tail's survival has no law.** The survival in distinct directions is `S(b) = (1/2) #{z : d(z) > bk}`, half of what the sweep counts, and at `k = 14` it runs `81, 65, 58, 56, 55, 52, 48, 42, 39, 37, 35, 30, 27, 20, 18, 14` from `b = 2` and reaches `1` at `b = 42`; the local exponent `-log_2(S(2b)/S(b))` reads `0.481, 0.273, 1.415, 3.169` at `b = 2, 4, 8, 16`, the sharpest of them resting on the two directions of `S(32)`, and a maximum-likelihood geometric fits ratio `0.8958` with pooled `chi2 = 29.0` on at most 16 degrees of freedom once the fit is carried from the doubled `z` counts to the directions. On 81 directions spread over 41 depths the survival is neither geometric nor shown not to be, and no exponent read off it carries an exclusion (`lab/py/band-return-times`, verb `hist`). **Verified.**
- **The one model that calls the deep tail small is half extrapolation.** Write `D(k, N)` for `Sum 2^(#supp K) / m` over the primitive lifts `K = m R_k` of base-3 length at most `N`, the equidistribution model of the return pairs `(m, z)` inside horizon `N`, summed by the same column transfer with `1/m` sandwiched by the length; at `N = 2k` it is `2^k L_k + 4^k / (3^k + 1)`, the depth-2 model the lift half is measured against once the primitive lift `R_(2k)` of multiplier `3^k + 1` is counted with it. At the critical cutoff `N = floor(sqrt(R_k))` the deep part `D(k, N) - D(k, 2k)` sits inside `[0.1476, 0.4429] * 2^k` at `k = 8` and inside `[0.0373, 0.1122] * 2^k` at `k = 16`, the last steps falling by about `0.835`, so on the model the deep tail is `o(2^k)` and the whole blocking lemma lives in the lift half. It is never a prediction of `Z(R_k) - U_k` itself, `D` counting return pairs where the tail counts distinct directions and so lying above it by the witness multiplicity; and `0, 0, 0, 0, 5, 32, 51, 64, 73` percent of that deep part at `k = 8..16` is carried by the free `4/9` per-digit increment extrapolated past 40 blocks rather than by the transfer, both ends leaning low because the excess `rho = L(k, n) R_k / 2^(n-1)` of the return count over the free model is above `1` at every depth reached, which puts the true increment above `4/9`, and the upper end holding only while `rho < 3` (`lab/py/band-return-times`, verb `model`). **Conjecture.**
- **The repunit drift.** Exactly, `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 `X_k = (Z(R_k) - Phi_k) / Phi_k`, so the drift is the floor's coprime density times the excess ratio, and `2^(log 2 / log 3) = 1.5486`; `delta_k` is exact from the floor and `X_k` reads `0.0476, 0.0535, 0.1111, 0.1521, 0.2053, 0.2157, 0.2683, 0.2946, 0.3227` at `k = 7..15`, rising at every step from `k = 8`, by `0.0263` and `0.0281` at the last two. The constant `1.5975` of the layer scan is the maximum below `8192` only: the repunits give `1.7845, 1.9637` at `k = 11, 13` and, through `delta_k = 0.4921, 0.8349`, `0.9868, 1.7103` at `k = 14, 15`, so any pointwise `Z(w) <= C w^(log 2 / log 3)` needs `C >= 1.9636`. The fate of the drift splits exactly: `1 + X_k = U_k / Phi_k + (Z(R_k) - U_k) / Phi_k` with `U_k / Phi_k <= agg_k L_k` and `L_k = Sum_T 1 / m_T < 3/2` Proved, so `X_k` is unbounded only if the aggregate `agg_k` or the deep tail ratio is, and over `k = 11..19` `agg_k` shows no trend inside `[1.01748, 1.11457]` while `U_k / Phi_k` rises across the band `[1.19611, 1.36517]`; the deep tail `18, 16, 108, 162, 624` grows by a factor `34` over `k = 11..15` against `16` for `2^k`. **Conjecture:** `X_k` is unbounded, so `Z(R_k) / R_k^(log 2 / log 3)` diverges along the repunits and the corridor's lower end `beta = log 2 / log 3` is not attained by any constant; every `beta > log 2 / log 3` survives the data. What decides it is one lemma in two named halves: `Sum_T |Occ_T| = O(Sum_T 2^k / m_T)` on average over the lifts, which no per-`T` bound of the shape `C 2^k / m_T^c` can give, and `Z(R_k) - U_k = O(2^k)` on the deep tail (`lab/py/ratio-set-saving`). **Verified / Conjecture.**
- **The divisor route to the saving is closed.** No `Bin_level(q) <= C 2^level / q` is uniform over `q` coprime to 3: every binary `m < 3^h` makes `m(1 + 3^h)` binary, so `Bin_2h(1 + 3^h) >= 2^h` against `4^h / q`, ratio `(3/2)^h (1 + 3^(-h))` reading `2.0 .. 25.633` at `h = 1..8`, and the worst modulus below 500 at `level 20` is `q = 244 = 1 + 3^5` at `1.8094` - the moduli that break equidistribution are exactly the shift-ray weights. The short-witness route is closed too, mean `lev` running `3.875` to `27.287` over `X = 32..16384` (`lab/py/ratio-set-saving`). **Refuted.**
- **And the criticality explains the congruence seed.** The band automaton is critical at every direction, and not by an exact identity: its states are `N` consecutive integers carrying out-degrees `2, 1, 0` one to each residue class mod 3, so the mean out-degree is `1 + (level_+ - level_-)/N` for the counts `level_+` and `level_-` of the band states in the degree-2 and the degree-0 class, and `N` consecutive integers balance the three classes to within one, so `|mean - 1| <= 1/N` at every direction and the mean is exactly `1` whenever `3 | N`. Divisibility of `N` by 3 is sufficient and not necessary: `z = (3,1)` has states `{0, 1}`, degrees `2, 1` and mean `3/2`, `z = (3,2)` has states `{0, 1}`, degrees `2, 0` and mean exactly `1` at `N = 2`, and of the 591 coprime directions with `3 | z_1` and `w` in `{13, 40, 100, 101, 121, 257, 364, 1093}` exactly 465 are critical on the nose, 60 of those with `3` not dividing `N` (`lab/py/ratio-set-saving`). **Proved.**
- So the survivor process is critical at every direction and `sigma_k ~ C/k` is forced, giving `c_k = log_3((k+1)/k)` and `k c_k -> 1 / log 3 = 0.9102392` against the reading `[0.8418, 0.8628]` rising over `k = 13..18`: on this mechanism no congruence route buys an exponent at any `k`, and the `0.2618596` Cauchy-Schwarz cap is never approached (`lab/py/ratio-set-saving`, `lab/py/occupancy-decay`). **Conjecture.**
- **Where the mass sits.** Per octave `3^j <= m < 3^(j+1)` the Chebyshev mass `Sum Lambda(m) N*_level(m) / 3^level` decays like `3^(-j)`, the Euler prediction matched to 0.2%, down to a flat floor carried by the `2^level` fibre points of height `f * (2/3)^level * log 3`, `f` the number of one-coordinate subfamilies of `F`; the whole non-Euler loss is `O(level (2/3)^level)`, the log-gcd mean `G(level)/3^level` converges, `1.0326, 1.0094, 0.9964` at `level 16, 18, 20`, and the mass with a prime factor above `3^(0.9 level)` is `O((2/3)^level)` (`lab/rs/dimension-one-ladder`). **Verified.**
- Below dimension one the base-4 and base-5 simplex probes show the same profile, the same fibre floor and monotone convergence to their deltas, while hand-built `fill < base` designs at `level 20` still sit `1e-02` from `delta` with visible wandering; nothing in the data resists the conjecture, and no study regenerates the probes. **Conjecture.**
- The obstruction map: the goal is one arrow proved or one obstacle sharpened, and the carry matrix is a transfer operator, so thermodynamic formalism applies as is.

| route | known | exact obstacle | next certificate |
|---|---|---|---|
| moment method | carry-matrix moments to the tenth and twentieth | low-frequency peak | peak-removed bound |
| componentwise Fourier transfer | insufficient | `l1` dimension below threshold | abandon |
| sieve with signs | cancellation not retained | absolute values taken too early | bilinear decomposition |
| occupancy | exponent pinned, first moment is the window | no power saving over the box | ratio-set power saving |
- The lift half is a union count, not a divisibility count. For a binary `K` and a divisor `m` of it, the submasks of `K` divisible by `m` are closed under complement in `K`, under disjoint union and under nested difference, so their number is even and every one of them is a disjoint union of irreducible ones. That decomposition is not unique, so the count is the number of distinct unions of pairwise disjoint irreducibles and obeys `N_K(m) <= #packings <= 2^iota`, with `N_K(m) = 2^iota` exactly when the irreducibles are pairwise disjoint. That is why the antipodal and the run families of the repunit lift have counts that are exact powers of two rather than merely bounded ones; the converse fails, and at `k = 12` the equality case holds for `1970` of the `2048` multipliers while `1986` have a power-of-two count. The left inequality is strict from `k = 5`, where `T = {1}`, `m = 7` and `K = 847` carry the support `{0, 2, 3, 4, 6}` with four irreducibles, two decompositions of the whole and six distinct unions against seven packings, so bounding `Sum_T #packings_T` suffices for the lift half and is strictly the harder target (Proved, `lab/py/band-return-times`, verbs `lift` and `check`).
- A column transfer for that count needs at least 253 states. A machine reading the `k` columns of the lift 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, finite Hankel rank over a free monoid being exactly a linear representation with that rank as the minimal dimension ([Schutzenberger 1961](https://doi.org/10.1016/S0019-9958(61)80020-X), the same criterion in [Berstel and Reutenauer 2011](https://doi.org/10.1017/CBO9780511760860)); 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`, so at `k <= 15` such a transfer is already dearer than the `2^(k/2)` meet in the middle, where the return half at a block horizon needs `b/2` states. Whether the rank is unbounded is observed and not proved, so the route is blocked below 253 states and not refuted; the floor is neither an impossibility nor a second check read twice, the reversed reading being the transpose of the same Hankel matrix at equal side lengths, and a machine whose state set may grow with `k` always exists, the residue automaton on `m` states computing `N_K(m)` at cost `3^k` per `T` and so dearer than the meet in the middle. A rank that levels off is that poly-time machine and hands the lift half its bound. The depth-2 aggregate `M_k = 2, 6, 14, 36, 68, 172, 306, 728, 1338, 2814, 5224, 11852, 20888, 43364, 84124, 172516, 327092` obeys no linear recurrence of order at most 8 on those seventeen terms, while the return count at a block horizon obeys one of order at most `b` (Verified, `lab/py/band-return-times`).

## B-VISIBILITY

- Coprimality is the `b = 1` member of a family: `(x, y)` is **b-visible** when no `k > 1` has `k | x` and `k^b | y`, i.e. visible from the origin along the power curve `y = a*x^b`, and the full-lattice density is `1/zeta(b+1)` ([Goins, Harris, Kubik and Mbirika 2018](https://doi.org/10.1080/00029890.2018.1465760)). **Proved.**
- Reproduced at `b = 1, 2, 3, 4` on the `N x N` positive square at `N = 10^4`: `0.60794971, 0.83191407, 0.92394823, 0.96439991` against `0.60792710, 0.83190737, 0.92393840, 0.96438734`; no study regenerates the check. **Conjecture.**
- The base factor carries over with a longer window: on the gasket `2 | x` pins digit 0 and `2^b | y` pins digits `0..b-1`, so digit 0 must be `(0,0)` and digits `1..b-1` must avoid `(0,1)`, giving `#{ x in S_level : 2 | x_1 and 2^b | x_2 } = (2^(b-1)/3^b) * 3^level` for `level >= b`, the counts `3^(level-1)`, `2*3^(level-2)`, `4*3^(level-3)` at `b = 1, 2, 3`. **Proved.**
- The predicted b-visible density on the gasket is `delta_b = (1 - 2^(b-1)/3^b) * Prod_{p odd} (1 - p^(-(b+1))) = [ (1 - 2^(b-1)/3^b) / (1 - 2^(-(b+1))) ] * (1/zeta(b+1))`; `b = 1` is the proved `16/(3*Pi^2)`, and `b >= 2` is open. **Conjecture.**
- The claim `delta_b = (8/9)/zeta(b+1)` for every `b >= 1` is false: the bracket is `8/9` at `b = 1` and `b = 2`, `(2/3)/(3/4)` and `(7/9)/(7/8)`, an accident of two small cases; at `b = 3` it is `368/405 = 0.9086420`, at `b = 4` `2336/2511 = 0.9303067`, climbing to 1; the two predictions part at `b = 3`, `0.8395292` against `0.8212786`, and the exact count at level 12 sits at `0.8427119`, falling about `0.0022` a level toward the former. **Refuted.**
- `b = 2` measures `0.7429` at level 12 against `0.7394732`, which both formulas share; the or-triangle's b-visible density is `[1/(1 - 2^(-(b+1)))] * (1/zeta(b+1))`, measured `0.8107470, 0.9495464, 0.9865874` at `b = 1, 2, 3` on levels 14, 12, 12 against `0.8105695, 0.9507513, 0.9855343`; no study regenerates these levels. **Conjecture.**
- No general design formula is derived: the local factor is a per-design digit count, not a universal rational.

## DIRECTIONAL PROFILES

- Bin ordered pairs of design points by the angle of their displacement in `[0, Pi/2]`, eight equal bins, and take the coprime fraction in each: this directional profile separates designs the scalar density does not.
- Two base-3, `dim 2`, `fill = 5` designs, A on digits `(0,0), (1,0), (2,0), (0,1), (1,1)` and B on `(0,0), (1,1), (2,2), (0,1), (1,0)`, share `B(F) = 4/5` and hence `delta = 0.5471344`. **Proved.**
- At level 7 they differ by up to `0.0327` in a bin, A's bin 1 `0.5823` against B's `0.5596`, B's bin 5 `0.5153` against A's `0.5480`, while their scalar pairwise densities differ by `0.0001001809`; levels 2 to 7 by exact enumeration, displacement multiplicities by rounded-FFT autocorrelation validated against `N(N-1)`; no study regenerates it. **Conjecture.**
- The bin gap shrinks, `0.375, 0.1461, 0.0946, 0.0492, 0.0327` at levels 3 to 7, a factor of roughly `0.6` a level against the scalar gap's `0.4`, so the data fit a profile difference that also vanishes, only more slowly; at every level from 3 to 7 the directional gap exceeds the scalar gap by one to two orders of magnitude, and a nonzero limit of the profile is untested past level 7. **Conjecture.**

## THE FUNCTION FIELD DIAGNOSTIC

- Redo the construction over `F_q[t]` for a field of size `q`, where the Riemann hypothesis is Weil's theorem: fix `q = 3` and `S = {0, 1}`; among the `2^level` polynomials of degree below `level` with every coefficient in `S`, the ordered coprime density measures `0.564176` at `level 10`, `0.563471` at `level 12` and `0.562833` at `level 14`, a gap of `0.00033` from `9/16 = 0.5625`, by exact enumeration over `F_3` (`lab/py/function-field-density`). **Verified.**
- The prediction is the digit-corrected Euler product and nothing more: the unrestricted density is `1 - 1/q = 2/3`; exactly one prime is exceptional, `pi(t) = 1/2` against the unrestricted `1/3`, because a restricted polynomial is divisible by `t` exactly when its constant coefficient is 0, half of `S`; replacing that factor gives `(2/3) * (3/4)/(8/9) = 9/16`. **Proved.**
- The other two linear primes measure `0.333984` and `0.333008`, and degrees 2 and 3 deviate from `3^(-deg p)` by `0.0038` and `0.0019` on average over the primes of each degree, the worst quadratic `0.007053` off `1/9` (`lab/py/function-field-density`). **Verified.**
- The finite Euler product is neither exact nor monotone: the marginal product through degree 5 is `0.560193`, crossing `9/16` between degrees 3 and 4, while the exact probability of sharing no prime factor of degree at most 5 is `592189/1048576 = 0.564755`, differing by `-0.004563`; divisibility at distinct primes is dependent under a coefficient restriction, so a product of marginals is a diagnostic and not an identity (`lab/py/function-field-density`). **Verified.**
- The right limit sends `level` to infinity at fixed `p`, since no polynomial of degree below `level` except zero is divisible by a prime of degree at least `level`; and the density is a theorem only conditionally: given `pi_S(p) = lim_level Pr(p | F_level)` for every monic irreducible, asymptotic independence over every finite set of them, and a vanishing chance of sharing a factor of degree above `D` as `D` grows, the restricted coprime density is `Prod_p (1 - pi_S(p)^2)`. **Proved.**
- The `9/16` limit itself is unconditionally open. **Conjecture.**
- The window does not survive the crossing: `E_ff(level)` grows like `4^level`, a positive density among all ordered pairs, so `gamma = log_3(4) = 1.261860` and the analogue window `(gamma/2, 1/2]` is empty, `gamma/2 = 0.630930`; the growth bounded here is positive-density counting, not a zeta error term, so Weil's theorem has nothing to bound, and the framework is sound with no zeta content (`lab/py/function-field-density`). **Verified.**

## MIXED RADIX

- Let the radix vary with position, digit `l` drawn from `S_l` with place value `Prod_(j<l) base_j`; block reduction says a periodic schedule agrees with the stationary theory at the blocked base, and an aperiodic schedule is outside it. **Proved.**
- At `level 12` digit positions, 4096 points per schedule and the same 8386560 pairs each time; pure base 2 is the sanity row, `0.607874` against `1/zeta(2) = 0.607927`, and its count is `(mrlynum::lattice::coprime_pairs(4095) + 1)/2`; every row is regenerated by `lab/py/function-field-density`. **Verified.**

| schedule | coprime pairs | density |
|---|---|---|
| alternating base 2 `{0,1}` / base 3 `{0,1}` | 4286664 | 0.511135 |
| alternating base 2 `{0,1}` / base 3 `{0,2}` | 5640929 | 0.672615 |
| pure base 2 `{0,1}` | 5097972 | 0.607874 |
| pure base 3 `{0,1}` | 4316493 | 0.514692 |
| pure base 3 `{0,2}` | 0 | 0.000000 |

- Schedule dependence is real: the two alternating schedules differ by `0.161480` and differ in nothing but the digit set at the base-3 positions (`lab/py/function-field-density`). **Verified.**
- The gap is a mod-3 effect, not parity: both alternating schedules are exactly half even; for the `{0,1}/{0,1}` schedule every place value from position 2 onward is a multiple of 6, so `a mod 6 = d_0 + 2 d_1` and the residues `0..5` are hit `1024, 1024, 1024, 1024, 0, 0` times, half even and half divisible by 3 for the same reason; switching the base-3 digit set to `{0,2}` drops the fraction divisible by 3 from `1/2` to `1/4` and the local factor from `1 - (1/2)^2 = 3/4` to `1 - (1/4)^2 = 15/16`, a log advantage of `0.223144`; and pure base 3 on `{0,2}` has every digit even, hence density exactly 0. **Proved.**
- Prime 5 opposes the switch weakly at `-0.014253`, and the factors at 2, 7, 11 and 13 are identical between the two alternating schedules (`lab/py/function-field-density`). **Verified.**
- The truncated Euler product through 13 predicts `0.520112` and `0.640940` against actual `0.511135` and `0.672615`; for the first schedule cross-prime dependence is negligible, `+0.000012`, and the miss is the omitted prime tail, but for the second the marginal product underpredicts the exact small-prime joint by `0.043340`, so independence across primes is materially violated and a digit-adjusted Euler product on marginals is informative rather than exact (`lab/py/function-field-density`). **Verified.**
- No fixed modulus can work: the smallest moduli labeling coprimality exactly on these finite sets are `27994` and `20736`, at which all 4096 values occupy distinct residues, an encoding of the finite set rather than a transfer matrix; a prime not dividing `M` is invisible modulo `M`, so exact coprimality on an unbounded family has no fixed finite state space, and a finite matrix tracks a fixed finite prime set and no more. **Proved.**
- No limit is established: `level 12` is one level, and the aperiodic question, whether a staircase schedule `base 2, 3, 5, 7, ...` converges at all and to what, is untouched; it is the Moran question this tree leaves open.

## THE PARITY BAND

- A mrlybang code is a set `P` of [parity](/wiki/parity/) corners, `P subset {0,1}^3`, and at every base `base >= 2` it induces the design `F_base(P) = { v in {0,...,base-1}^3 : v mod 2 in P }`; the carpet rule "at most one odd" read at base 5, 7 or 46 is one code worn at many bases.
- Three quantities of `P` decide everything: the weight enumerator `W_j = #{ v in P : popcount(v) = j }`, the mod-2 difference span `H = <P - P>` with `s2 = dim H`, and whether the affine span of `P` avoids `0`; write `t = W_0 = [000 in P]`.
- **Theorem (even bases are quantized).** For every nonempty code `P` and every even `base >= 4`, `delta * zeta(3) = (8/7) * (1 - t/|P|)`, independent of `base` and of everything about `P` except `|P|` and whether the origin corner is filled. **Proved.**
- The proof: each parity class holds `base/2` digits so `fill = |P| (base/2)^3`; for odd squarefree `m | base` the multiples of `m` alternate parity and number `base/m`, even, so every parity class gets `base/(2m)` of them and `fill_m = |P| (base/(2m))^3`; multiples of `2m` are all even so `fill_{2m} = t (base/(2m))^3`; Mobius over `rad(base)` collapses to `B = (1 - t/|P|) * Prod_{odd p | base} (1 - p^(-3))`, the odd base primes cancel their own Euler corrections, and only the factor at 2 survives; every parity design at even `base >= 4` satisfies (E), `Diff` containing `2 Z^3` with index dividing 8, and `fill > base`, so the theorem above applies to all 255 nonempty codes.
- The nine values the band takes, times `1/zeta(3)`: `t = 0: 8/7`; `t = 1: 0, 4/7, 16/21, 6/7, 32/35, 20/21, 48/49, 1`. **Proved.**
- Checked exactly for every code and every even `base <= 40`, and by enumeration at `base 2` and `base 4`: carpet `0.7129` and `0.7091` against `(6/7)/zeta(3) = 0.7131` at `level 5, 12`; net `0.9518` against `(8/7)/zeta(3) = 0.9508`; the single-corner code `{100}` at `base 4` heading to the same `8/7`, denser than the full lattice (`lab/py/mrlybang-density-classes`). **Verified.**
- At `base 2` the same rational value holds whenever the code itself satisfies (E) with `|P| > 2`: 151 of the 255 codes are full rank there, all at index 1 or 2, so the hedge is real (`lab/py/mrlybang-density-classes`). **Verified.**
- In `dim 2` the same proof gives `delta * zeta(2) = (1 - t/|P|)/(1 - 1/4)`, and the parity carpet at every even base lands on `delta * zeta(2) = (2/3)(4/3) = 8/9`, `delta = 16/(3 Pi^2)` exactly: the gasket's constant is not the gasket's, it is the even-base band value of its parity code. **Proved.**
- **Theorem (odd bases are self-similar across bases).** For odd `base` and squarefree `e | base`, the multiples of `e` among `0..base-1` have the parity profile of the digit set at base `base/e`, so with `fill_P(u) = Sum_j W_j ((u+1)/2)^(3-j) ((u-1)/2)^j` at a base `u`: `fill_e(base) = fill_P(base/e)` and `B(base) = Sum_{e | rad(base)} mu(e) fill_P(base/e) / fill_P(base)`; the design's bracket at base `base` reads the same design at base `base/e`, self-similarity across bases rather than levels. **Proved.**
- Checked exactly for all codes and all odd `base <= 75` (`lab/py/mrlybang-density-classes`). **Verified.**
- The naive rational part depends on `P` only through `W` at every base, but the density does not: six of the 63 weight classes mix regimes of the trichotomy below, four holding both a spanning and a non-spanning code, two pitting the corrected regime against the no-limit one, so weight twins share every even-base density and part at every odd base, in two classes by one twin having no density at all. **Proved.**
- The pair `{000,100,010,110}`, a subgroup, and `{000,100,010,011}`, spanning, both `W = (1,2,1,0)`, both `(6/7)/zeta(3)` at `base 4`, measured `0.7075` and `0.7112` at `level 4`, split at `base 3` into `(153/182)/zeta(3) = 0.6994`, measured `0.6982` at `level 6`, against `(51/52)/zeta(3) = 0.8159`, measured `0.7834` and climbing (`lab/py/mrlybang-density-classes`). **Verified.**
- **Theorem (the odd-base trichotomy).** Fix a code `P` and odd `base` with `fill > base`; mod 2 a point is a sum of `level` parity steps from `P`, so `x mod 2` lies in `level*c + H` for any `c in P`, and the all-even count is the character identity `T_2(level)/fill^level = (1/8) Sum_t lambda_t^level`, `lambda_t = Sum_{v in P} w_v (-1)^(t.v) / fill`, `w_v = e^(3-|v|) o^(|v|)`, `e = (base+1)/2`, `o = (base-1)/2`, with `|lambda_t| = 1` exactly on `t in H^perp`, where `lambda_t = (-1)^(t.c)`; three regimes, exhaustive for odd `base >= 5`. **Proved.**
- The exclusions are load-bearing: at `base 3` seven small codes fall to dimension `<= 1` and stay outside, and the 37 codes with a coordinate pinned odd collapse at `base 3` to a fixed-coordinate object treated nowhere here, `{110,100}` having `x_1 = R_level` constant and measuring `0.9722, 0.9941, 0.99998` on odd levels against the trichotomy's `0.9873` (`lab/py/mrlybang-density-classes`). **Verified.**
- Regime 1, `s2 = 3`, 149 codes: spanning, the master theorem applies, and `delta * zeta(3) = B(base) * Prod_{p | base} p^3/(p^3 - 1) -> 1` as odd `base -> infinity`; every spanning code converges to `1/zeta(3)` along the odds while frozen on its band value along the evens, and only the full box has both limits equal. **Proved.**
- Regime 2, `P subset H` proper, 43 codes: the limit exists with the factor at 2 replaced by `1 - 2^(-s2)`, `delta * zeta(3) = (1 - 2^(-s2)) (8/7) B(base) Prod_{p | base} p^3/(p^3-1)`, tending to `(8/7)(1 - 2^(-s2))` as odd `base -> infinity`; the finite-level factor is exact, `1` minus a signed sum of subdominant walk-eigenvalue powers, collapsing to `1 - Lambda^level` when one subdominant value carries it, as for tree and void; the proof holds for `base >= 5` by the mixed character lemma and at `base 3` without pinned coordinates. **Proved.**
- Regime 2 checked: the void, `Lambda = 7/9` at `base 3`, measures `0.3819` at `level 8` against a limit `0.4388` and a finite-level prediction `0.3800`; tree `base 3` measures `0.451821` against `(99/182)/zeta(3) = 0.452521` at `level 7`, tree `base 5` `0.468340` against `0.468560`, the subgroup twin above to `1.1e-03` (`lab/py/mrlybang-density-classes`). **Verified.**
- Regime 3, the affine span of `P` avoids `0`, 63 codes: the density has no limit; at odd levels `level*c` misses `H`, so not one point of `S_level` has all coordinates even and the factor at 2 is exactly 1, while at even levels it tends to `1 - 2^(-s2)`, two subsequential limits in ratio `1 - 2^(-s2)`; this is the level-periodic failure of (E) under THE OBJECT, produced by an explicit two-parameter family and priced exactly. **Proved.**
- Regime 3 checked sharply: `P = {111}` at `base 5` measures `0` at every even level and `0.9257, 0.9529, 0.9572 -> (8/7)(125/124)/zeta(3) = 0.9584` along odd levels; the axes code `{100,010,001}` at `base 3` measures `0.987338` on odd `level` against `(8/7)(27/26)/zeta(3) = 0.987319` and `0.7396` on even `level` against `0.740489` (`lab/py/mrlybang-density-classes`). **Verified.**
- **The mixed character lemma.** For `d = 2^a m`, `m` odd, `gcd(d, base) = 1`, and a character `t` nonzero mod `m`, some coordinate has `2 t_i` nonzero mod `m`; for `base >= 5` both parity classes hold two digits per coordinate, so `F` contains a same-parity pair differing by `2 e_i`, Lemma A's orbit argument works inside one parity class, and the mod-2 walk factors cleanly from the odd-modulus equidistribution; at `base 3` the even class `{0, 2}` still supplies the pair unless the coordinate is pinned odd. **Proved.**
- Two steps carry the corrected and no-limit regimes, which violate (E) at the prime 2: the large-prime tail never sees (E), since the box bound and the Chebyshev sum are pure counting, so the close upgrades the fixed-`z` limsup to the limit; and the joint count at a mixed modulus factors, because after Theorem 1 pins the last digit vector mod `e` the parity walk of the remaining `level - 1` digits has the same distribution for every admissible corner, `u.v` being constant on `P` for every `u in H^perp`, so `T` at modulus `2^a m e` splits into bracket times walk times Euler and the sieve assembles as in the spanning case. **Proved.**
- **Corollary (parity-stable codes are the subgroups).** The even-base value equals the odd-base limit exactly when `t/|P| = 2^(-s2)`, i.e. exactly when `P` is a subgroup of `{0,1}^3`: sixteen codes, `1, 3, 5, 9, 15, 17, 33, 51, 65, 85, 105, 129, 153, 165, 195, 255` in corner-mask numbering, have one density limit over all of `N`, and the other 239 nonempty codes jump between the even band and their odd limit forever, so "the" density of a mrlybang code over all bases exists only on the subgroup lattice of the parity cube. **Proved.**
- **The four families.** Carpet `{popcount <= 1}` and net `{popcount >= 2}` are spanning; tree `{000, 001}` and void `{000, 111}` are subgroups with `s2 = 1`, parity-stable at `4/7`; the exact rational parts `delta * zeta(3)` below are recomputed exactly (`lab/py/mrlybang-density-classes`). **Proved.**

| `base` | carpet | net | tree | void |
|---|---|---|---|---|
| every even | `6/7` | `8/7` | `4/7` | `4/7` |
| 3 | `513/520` | `27/26` | `99/182` | `48/91` |
| 5 | `2500/2511` | `125/124` | `1100/1953` | `850/1519` |
| 7 | `7889/7904` | `343/342` | `259/456` | `140/247` |
| 9 | `2187/2210` | `8019/7904` | `1278/2275` | `360/637` |
| 11 | `200981/201096` | `1331/1330` | `9559/16758` | `16456/28861` |
| odd limit | `1` | `1` | `4/7` | `4/7` |

- The carpet's odd-base law: at an odd prime `base` the rational part is `(1 - 1/fill(base)) base^3/(base^3 - 1)` with `fill(base) = (base+1)^2 (2 base - 1)/4`, so `1 - delta * zeta(3) = (base^3/fill(base) - 1)/(base^3 - 1) ~ 1/base^3`, a third-order approach to `1/zeta(3)` from below, the rational parts `0.98654, 0.99562, 0.99810, 0.98959, 0.99943` at `base 3, 5, 7, 9, 11`. **Proved.**
- The net approaches from above: at a prime base `fill_base = 0` and no net point is ever divisible by the base, the Vicsek mechanism; at prime powers the bracket dips below 1, `B(9) = 297/304`, because `B(p^a) = 1 - fill(p^(a-1))/fill(p^a)` corrects at the scale of `p` rather than `p^a`, the `base 9` dip being `fill(3)/fill(9) = 7/304` in the net's own `fill` (the carpet's `fill` would give `20/425`), with no dip at a prime base since the net has `fill(1) = 0`, yet the rational part stays above 1 at every odd `base` from 3 through 81 (`lab/py/mrlybang-density-classes`). **Verified.**
- So the family does not converge to `1/zeta(3)` over all bases: it converges along the odds for spanning codes, sits on the quantized band along the evens, and the two agree only on the sixteen subgroups. **Proved.**

## COPRIMALITY ON THE SLICES

- Fix a parity design at base `base`, level `level`, and a height `s`; let `N_s` count the design points on the plane `x + y + z = s` and `A_s` the coprime ones; slice coprimality differs from the solid's by one divisibility, the gcd of a slice point divides its height, and everything below follows from that.
- **Theorem (slice coprimality is finite arithmetic).** `gcd(x,y,z) | s` on the plane, so `A_s = Sum_{d | s, d squarefree} mu(d) * N_s^(d)`, exact at every height and level, with `N_s^(d)` the points `d` divides coordinatewise; no zeta function, no tail, no Lemma B, and a slice at prime height is fully visible up to at most the three axis points. **Proved.**
- Checked with zero mismatches at every height for carpet and net at `base 3` to `level 4` and carpet at `base 4, 5` to `level 3`; the worst hidden count on a prime slice is 3, always the axis points (`lab/py/slice-coprimality`). **Verified.**
- **Theorem (the base prime peels the slice).** For prime base `base` the only corner divisible by `base` is the origin corner, so `N_s^(base)(level) = [000 in P] * [base | s] * N_{s/base}(level - 1)`, exact at every height and level; a code without the origin corner owes nothing at its base on any slice, and on the central slice the peel lands one step off-centre, `s*_level / 3 = s*_{level-1} + 1`, the off-centre schedule the height digits follow in [cuts](cuts.md). **Proved.**
- Checked exactly at `base 3, 5`; no net point ever has `3 | gcd`, on all `7^7` points (`lab/py/slice-coprimality`). **Verified.**
- **The local price of a prime, one dimension down.** Away from the base the solid pays `p^(-3)` per prime and the slice pays `p^(-2)`; aggregated over the heights divisible by `p` this is a theorem, the numerator being exactly `T_p(level)`, a point with `p | gcd` sitting automatically on a `p | s` height, and the denominator tending to `fill^level/p` by equidistribution of `s mod p`. **Proved.**
- Per individual slice the `p^(-2)` price is open; carpet `base 3`, `level 6` measures `0.040902` against `1/25` and `0.020446` against `1/49`, with `p = 11, 13` still converging (`lab/py/slice-coprimality`). **Conjecture.**
- The parity of the height is the walk of the parity band: the even-height aggregate at `p = 2` is exactly `(1/8) Sum_t lambda_t^level / ((1/2)(1 + lambda_111^level))`, formula and count both `0.2850378 = 9121792/32002048` at `level 6`, equal on the integer (`lab/py/slice-coprimality`). **Verified.**
- Parity constraints transfer whole: the tree's even slices hold zero visible points and its odd slices zero even gcds, since `x_1, x_2` are always even and `s = x_3 mod 2`; checked on 1.49 million points each with no exception. **Proved.**
- **The central slice does not converge, and its bill is a repunit.** The central cut sits at `s* = 3 (base^level - 1)/2 = (3(base-1)/2) * R_level(base)`, `R_level` the repunit at base `base`, so the central slice always owes the prime 3, owes 2 exactly when `base 1 mod 4` or `level` is even, and owes an odd prime `p` outside `{3}` and the primes of `base` exactly when `ord_p(base) | level`; the visible density of the centre is a quasiperiodic function of the divisors of `level`, read through multiplicative orders, and has no limit. **Proved.**
- At `base 3` it flows in two streams, odd `level`: `0.89216, 0.89776, ...` toward roughly `0.907` minus repunit-prime dents, even `level`: `0.57143, 0.61067, 0.65218` toward `3/4` of the high stream; at `level 7` the entire foreign bill is `R_7 = 1093`, the Wieferich prime, costing the slice about one part in a million; at `base 5` both 2 and 3 sit on every bill and `0.345, 0.492, 0.560` climbs toward `(3/4)(8/9) = 2/3` minus dents (`lab/py/slice-coprimality`). **Verified.**
- Independence across the primes of `s*` holds to about three decimals at every level measured, `Prod_p (1 - local_p)` reading `0.64780, 0.89764, 0.55741` against measured `0.65218, 0.89776, 0.56006`; the independence itself is open. **Conjecture.**
- The central count is a sequence: for the sponge `N(s*_level) = 1, 6, 42, 306, 2250, 16578, 122202, 900882, ...` is [A299916](https://oeis.org/A299916) exactly, computed exactly to `level 14` on the `(9, -12)` recurrence, and the peeled counts `3, 27, 207, 1539, 11367, 83835, 618111, ...` ride the same recurrence, the sixth term confirmed by a meet-in-the-middle count over all `20^7` level-7 points without the peel, the recurrence holding to `level 14` (`lab/py/slice-coprimality`). **Verified.**
- Two sequences on one recurrence force the peel ratio `N^(3)(s*)/N(s*) -> (sqrt(33) - 5)/8 = 0.0930703308`, measured `0.093070331` at `level 14`; the recurrences for these two point counts are not proved on this page, so the constant rides with them, while `slice-recurrence-order` proves the order-2 recurrence of the central cell census. **Conjecture.**

## COUNTING WITHOUT ENUMERATING

- Exact `A(level)` does not need `fill^level` gcds: split at a cutoff `G`, points with `1 < gcd <= G` Mobius-cancel exactly inside `Sum_{d <= G} mu(d) T*_d(level)`, each `T_d` one transfer-matrix product on `(Z/d)^3`, and points with `gcd > G` live on multiples `g*y` with `y` primitive in a box of side `base^level/g`, coordinate space being only `base^level` wide; the cost is about `base^(level(dim+1)/2)` against enumeration's `base^(alpha level)`, a win whenever `alpha > (dim+1)/2`. **Proved.**
- Delivered for the sponge: `A(7) = 1038074187`, `A(8) = 20860210527`, `A(9) = 418429711224`, the last in 22.6 seconds against half a trillion points and the whole ladder in 84 seconds, two independent cutoffs agreeing on the integer, all four census anchors and both enumerable levels matched (`lab/py/sponge-visible-census`). **Verified.**
- The mask engine takes the same sponge ladder to `level 18`, every term matching the transfer-matrix census through `A(9)` and enumeration through `A(6)`: `A(10) = 8382927031902`, `A(11) = 167827226563374`, `A(12) = 3358570222045599`, `A(13) = 67196023858705425`, `A(14) = 1344212283980217555`, `A(15) = 26887733364774830334`, `A(16) = 537796671110979675579`, `A(17) = 10756437974822235283245`, `A(18) = 215134797774716879278017`; it is Mobius over the moduli coprime to 3 with the pairwise-disjoint mask count inside, the moduli split by their number of multiples into a closed-form tail, bitset rows, `u16` zeta rows and a rank-truncated ranked cube, about `3^level (level 2^level)^(2/3)` work, `28.3 s` at `level 17` and `122.3 s` at `level 18` on eight threads with level ratio `4.32`, `3.7x` and `3.8x` its previous form, whose `104 s` and `463 s` ladder it reproduces term for term; the new engine alone gives `A(19) = 4302768326366633733102921` in `515 s`, without a second witness (`lab/rs/coprime-terms`). **Verified.**
- The tail of the Mobius sum is closed and rigid: for `3^level/2 < m < 3^level` with `3` not dividing `m`, `N_level(m) - 1 = 3 + 4 [m has no base-3 digit 1]`, so that band of `W(level)` is three times the Mertens sum of `mu` over the band's moduli coprime to 3 plus four times a Mertens sum over the base-3 Cantor set; the next band (`Y = 3`) is `6 + 7 [mask(m) = 0] + 7 [mask(2m) = 0] + 6 [mask(m), mask(2m) disjoint]`, and every band is a Mobius sum over a digit-automatic condition on `m, 2m, ..., (Y-1) m`; `N_level(m)` depends on the digits of `m`, not on `floor(3^level/m)` (`level 2`: `m = 5` and `8` share the floor with `N - 1 = 3` and `7`; at `level 6` every floor band with two admissible moduli is non-constant), so the tail admits no hyperbola grouping (`lab/rs/coprime-terms`). **Proved.**
- The second-order term: `delta*20^level - A(level)` in units of `12^level` reads `0.347, 0.349, 0.344` at `level 7, 8, 9`, and 12 is exactly the subdominant parity-walk scale `fill*lambda`, so the second-order term of the sponge census appears to ride the walk. **Conjecture.**

## PAIRWISE AND DEGENERATE DESIGNS

- Pairwise coprimality, `gcd(i,j) = gcd(i,k) = gcd(j,k) = 1`, is a different object with the same base-local half: 13 of the sponge's 20 digit-vectors have no coordinate pair both `0`, so the last-digit argument pins the factor at 3 to `13/20`, replacing the lattice's `20/27`. **Proved.**
- The box bound fibres over coordinate subsets: with `kappa_I = max_w #{ v in F : v restricted to I equals w }`, `#{ x in S_level, x != 0 : m | x_i for i in I } <= (base+1)^|I| * fill^level * m^(-alpha_I)`, `alpha_I = log(fill / kappa_I)/log(base)`, so the pairwise sieve closes whenever `fill > base * kappa_I` for every pair, and the sponge clears it, `kappa = 3`, `fill = 20 > 9`, `alpha = log_3(20/3) = 1.726833`. **Proved.**
- The three-modulus inversion `1[(x,y) = (x,z) = (y,z) = 1] = Sum_{a | x,y} Sum_{b | x,z} Sum_{c | y,z} mu(a) mu(b) mu(c)`, with the local factor `1 - 3/p^2 + 2/p^3 = (1 - 1/p)^2 (1 + 2/p)` at every foreign prime and a uniform limiting measure on residues mod `M` because `20^(-1) Sum_{v in F} e(t.v/M)` has modulus one only when `t.v` is constant on `F`, gives `lim P_level = (13/20) Prod_{p != 3} (1 - 3/p^2 + 2/p^3) = 0.251620868451255 = (351/400) C_3`, `C_3 = 0.286747428434479` the lattice constant; the Menger rule lowers the benchmark by exactly `12.25%` (`menger-pairwise-coprimality`). **Proved.**
- Exhaustive enumeration to `level 6` gives `0, 60, 1434, 32268, 721524, 15141288` pairwise coprime points out of `20^level`, densities `0, 0.150000, 0.179250, 0.201675, 0.225476, 0.236583`, the direct census and the inversion agreeing exactly through `level 5`; the local factor is not `1 - p^(-s)`, so `delta_M * zeta(2) = 0.4138997384` carries no rationality claim, and no sharp error term at finite `level` is supplied (`menger-pairwise-coprimality`). **Verified.**
- Designs with `1 < m(F) < infinity` are marked `DEGEN` in the census, 95 of them, 82 at index 2, 4 at index 3, 9 at index 4, and those failing (E) are the other exclusion: base 3, `dim 2`, code 13 has corners `(0,0)`, `(0,2)`, `(1,0)`, so its second coordinate is even at every level, leaving an uncorrected Euler factor at 2, and its ratio sits near `0.3051` at `level 13` against a predicted `0.455945` (`lab/rs/design-census`). **Verified.**
- Repairing a design that fails (E) needs a corrected factor at every prime dividing `m(F)`, from an automaton on cosets of `Diff`; the odd-base trichotomy derives those factors for the whole parity family and produces the no-limit designs in bulk, and outside that family none is derived.

> The base is exact, the rest is classical, above dimension one the join is closed, and at dimension one it is open in one window of prime exponents.