# Digit Transform Norms - The `l^1` norms of a design's digit transform in `dim` dimensions: the two numbers a missing-digit sieve reads out of a set, computed for the gasket and the carpet on the 2D grid and for their Morton codes at base `base^dim`, and calibrated against the published one-dimensional constants. - The object is `hat F(t) = (1/fill) Sum_(v in F) e()` on `T^dim`, with `F` the filled digit vectors, and its level product `hat F_level(t) = Prod_(j/base) = base^dim [t = 0] - e(/base)`, so `|hat F(t/base)|` is `1` at `t = 0` and `1/fill` at the other `base^dim - 1` points and `Sum_(t in (Z/base)^dim) |hat F(t/base)| = 2` exactly, for every base and every dim; the row prints the exact `2`, the scan that confirms it, and `log_base 2`, which is `1` for the gasket and `0.630930` for the carpet against the base-10 exponent, certified here inside `[0.3505775, 0.3505797]`, above it in one case and below it in the other, so the per-digit number is arithmetic and not a bound. - The direct grid sums `Sigma_level(0)` and their exponents at `level 1..10`. A finite-level grid sum is neither a floor nor a ceiling on `alpha_1`; the table is a trend, and the base-10 rows are its control. - The sandwich. Substituting `t = (i + y)/base^N` in `I_(N+M)` gives `base^(-dim N) min_x Sigma_N(x) I_M <= I_(N+M) <= base^(-dim N) max_x Sigma_N(x) I_M`, hence `(1/N) log_base min_x Sigma_N <= alpha_1 <= (1/N) log_base max_x Sigma_N` for every `N`, and `Sigma_N` has period `base^(-N)` in each coordinate, so the scan runs over one cell. - The slack. `|hat F(t) - hat F(t')| <= Lip |t - t'|_inf` with `Lip = (2 pi/fill) Sum_(v in F) |v|_1`, since the derivative in coordinate `c` is `(2 pi i/fill) Sum_v v_c e()`; the factor at digit `j` moves at `base^j Lip`, the product of factors bounded by `1` moves at their sum, and there are `base^(dim N)` terms, so `Lip(Sigma_N) <= Lip (base^N - 1)/(base - 1) base^(dim N)`, and a grid of `m^dim` shifts on the cell leaves half a step `h/2 = 1/(2 base^N m)`. `Lip` is `4 pi/3` for the gasket and `4 pi` for the carpet; the slack is derived, never fitted. - The window bound. `G(w)` is the supremum of `|hat F|` over the box of side `2 base^(-n_d)` behind a window `w` of `n_d` digit-vectors, bounded above by a sub-scan of `m^dim` cell centres plus `Lip h/2`; the transfer matrix carries a window to its `base^dim` successors with weight `G`, and `Sum_a sup_box |hat F_level| << lambda^level` with `lambda` its Perron root, so `alpha_1* <= log_base lambda`. The moment version raises `G` to a power `s` and reads `g(s) = log_base lambda(s, n_d)`. - The criterion. A pair `(base, a_0)` carries the one-dimensional sieve when `g(s) < (1/5)(1 + c/2)(2 - s)` for some `s` in `[3/2, 2)`, `c = log(base-1)/log base`; the rows print `g`, the criterion and the margin at `s = 1, 3/2, 235/154, 1.6, 1.7, 1.8`. - The contraction rate. For `gcd(d, base) = 1` the per-digit rate `max_t Prod_(l < ord_d(base)) |hat F(base^l t/d)|^(1/ord_d(base))` over every nonzero `t` in `(Z/d)^dim`, against `c(base,fill)^(1/m_d)` with `m_d = floor(log_base(d/2)) + 1` computed by an integer loop, and against the weaker `c(base,fill)^(1/ord_d(base))`. - The corollary band. Exact integer digit counts by residue give `max_a |#{x in S_level : x = a mod d}/fill^level - 1/d^dim|` as a rational, against the window bound `c(base,fill)^floor(level/m_d)` and the old orbit bound `c(base,fill)^floor(level/ord_d(base))`, over every modulus coprime to the base and every level in the band; the base-peel row does the same for `T_(e m)(level)` against `(fill_e/fill)/m^dim`, checking the exact identity `T_(e m)(level) = T_m(level-1)` on the way. - The order band. For a one-missing-digit pair the shift sandwich runs at a power: `base^(-N) min_x Sigma_N^(s)(x) I_M^(s) <= I_(N+M)^(s)` with `I_level^(s) = Int_0^1 F_level^s` and `Sigma_N^(s)(x) = Sum_(i= (1/N) log_base min_x Sigma_N^(s)`, a lower bound on the true exponent where the window matrix gives an upper one. `Sigma_N^(s)` falls in `s` because every factor is at most `1`, and the criterion falls in `s` because it is linear, so a finite chain of orders covers a whole interval; the chain's anchor at `s = 2` is exact, `Sigma_N^(2)(x) = (base/(base-1))^N` for every `x` by Parseval, since two `N`-digit integers congruent mod `base^N` are equal. - The least base. For a one-missing-digit set the window bound runs in one dimension at any base: `G(w)` is the supremum of `|hat F|` over the cell `[w/base^(n_d), (w+1)/base^(n_d))`, the transfer matrix collapses to `(M y)(v) = Sum_(c < base) G(v base + c) y((v base + c) mod base^(n_d - 1))` on `base^(n_d - 1)` states, and its Perron root gives `alpha_1 <= log_base lambda`; the same scan taking the infimum over the cell gives `alpha_1 >= log_base lambda_inf`, so one sweep brackets the exponent from both sides and the bracket is read against the threshold `1/4`. - The digit symmetry. `|hat F|` is unchanged by `a0 -> base - 1 - a0`, so a base carries `floor((base+1)/2)` distinct sets, and for even `base` the naive range `0 <= a0 <= base/2` scans the mirror pair `base/2 -> base/2 - 1` twice; at `a0 = (base-1)/2` the phase `e((a0 - (base-1)/2) t)` is constant and that set is the cheapest in its base, not the dearest, so the extreme digit is never assumed and always scanned. - The one-digit closed form. For `base_missing(base, a0)` the factor is `|sin(base Pi th)/sin(Pi th) - e((a0 - (base-1)/2) th)|/(base - 1)`, three trigonometric calls instead of `base - 1` complex exponentials, checked against the character sum on 2000 arguments. - The entropy floor. `Int_(T^dim) log|hat F(base^j t)| dt` is independent of `j` and Jensen gives `I_level >= exp(level Int log|hat F|)`, hence `alpha_1 >= dim + (1/log base) Int_(T^dim) log|hat F|`; for the gasket the inner integral is closed by Jensen's formula, `Int_0^1 log|a + e(u)| du = log max(|a|, 1)`, leaving `2 Int_0^(1/3) log(2 cos(pi u)) du = 0.323065947219`, so the soft bound reaches `0.881123` only and a certificate is not decoration. ## THE CERTIFICATE - Certified rows are interval arithmetic with directed rounding, not floats with a margin. Every argument of the scan is an integer multiple of `1/Q`, so a table of `Q` enclosures of `cos(2 pi j/Q)` is built once with `mpmath.iv` at 96 bits and each endpoint pushed one ulp outward. - `fill^2 |hat F(t)|^2 = Sum_(d in F - F) mult(d) cos(2 pi )` is a table lookup per difference; every add, multiply, square root and divide rounds outward through `nextafter`, so the printed pair encloses the true value. The kernel is checked against 40-digit `mpmath` on 1200 sampled arguments, no point outside its enclosure and the widest enclosure `1.5e-08`. - The gasket rows scan `min_x Sigma_N` and `max_x Sigma_N` over `m^dim` shifts, subtract and add the slack, and print `alpha_1 >` truncated down and `alpha_1 <` rounded up, each digit checked against `base^(N e)` before it prints. - The carpet row bounds the Perron root by the Collatz-Wielandt test: any positive `w` with `A^T w <= mu w` componentwise gives `rho(A) <= mu`, so a float power iteration supplies `w` and one certified pass supplies `mu`; the unweighted row sum, which is the same bound at `w = 1`, prints beside it and does not reach the threshold at any window length reached here. - The one-dimensional rows evaluate the closed form under directed rounding: all three arguments are integer multiples of `2 pi/Q` with `Q = 8 m base^(n_d)`, and a twiddle split `j = i_1 B + i_0`, `B = isqrt(Q) + 1`, builds their enclosures from `2 sqrt(Q)` interval cosines through `cos(x + y) = cos x cos y - sin x sin y` instead of tabulating `Q/2` of them, every product and difference rounded outward. - A Perron root is certified on both sides by Collatz-Wielandt: `M y <= mu y` componentwise gives `rho <= mu` and `M y >= mu y` gives `rho >= mu`, the test vector being the float Perron vector of the matrix being certified. A cell whose infimum falls to zero can empty a whole row, and such a row prints no lower bound rather than a false one. - The cheapest-pair row checks itself against the other matrix: the infimum matrix of that same set at that same window is run once more and its certified `alpha_1 >` is asserted to sit below the printed `alpha_1 <`. The expectation is Perron monotonicity, `Glo <= Ghi` entrywise forcing `rho(M_inf) <= rho(M_sup)`, so it is a theorem about the two matrices and owes nothing to the sweep it checks, which never builds the infimum matrix at all. - The test vector is only a test vector, so a bad one weakens the certificate and never breaks it; but the power iteration must still clear the transient near-zero cluster, whose row sums are `base`, or the certificate it hands back is worthless. The cluster's plateau flattens with the window length, its first relative move falling `1.1e-03`, `1.8e-06`, `0.0e+00` at `n_d = 4, 5, 6` in base 21, so any break on a single small relative move stops inside it and returns `lambda = base`. The break here needs fifty consecutive relative moves under `1e-13` and at least 300 steps. - Safe rounding everywhere: lower bounds truncate down, upper bounds round up, and no digit is printed past what the enclosure establishes. ## THE PAIR FAMILY - The object. For a base and an excluded pair `{a, c}` the set is `F = {0..base-1}` less `{a, c}`, `fill = base - 2`, and the window machine runs unchanged: only the one-digit factor `G(w)` changes, and the transfer matrix, the Perron root and the safe rounding are the same code. - The closed form. `|hat F(t)| = |K(t) - e((a - (base-1)/2) t) - e((c - (base-1)/2) t)|/(base - 2)` with `K(t) = sin(base pi t)/sin(pi t)`, whose square expands to `K^2 + 2 + 2 cos(2 pi D t) - 4 K cos(pi S t) cos(pi D t)` with `D = a - c` and `S = a + c - (base - 1)`: five trigonometric calls instead of `base - 2` complex exponentials, checked against the character sum on 2000 arguments at `9.7e-14` and `3.9e-15`. - The census. The pair enters only through `|D|` and `|S|`, which is one implication and not an equivalence class: `{0,2}` and `{0,8}` at `base 10` read `(2,7)` and `(8,1)` and still share a transform, and grouping by `(|D|, |S|)` alone overcounts at every base tested, 25 against 21 at `base 10`. The two moves that do 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 then the one-missing-digit sets of a `(base-1)`-digit interval read at that base, and the count of distinct transforms is `(C(base-2, 2) + floor((base-2)/2))/2 + floor(base/2)`, `7, 16, 21, 31` of the `15, 36, 45, 66` excluded pairs at `base 6, 9, 10, 12`. The scan list is built from that law and asserted to meet every class. - The least base against `1/4`. The interval class `{0, 1}` 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 `2.2 x 10^(-5)`, while `base 31` certifies `[0.2518967, 0.2519717]` and fails. Every base `4 <= base <= 31` is swept at every distinct set, shortest window in `2, 3, 4, 5` that decides. - The unclear cells. Ten of the 2373 sets of `4 <= base <= 31` get no positive lower certificate at five window digits, three at `base 26`, one at `base 29`, five at `base 30` and one at `base 31`, and every one of them has `S = 0` or `D = base/2` with `base/2` odd. That is a correlation and not a mechanism: a real zero of `|hat F|` empties a cell and lowers the Perron root of the infimum matrix without zeroing it, and the headline set is itself a witness, since `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 still certifies `alpha_1 > 0.2499087`; 13 of the 14 `S = 0` classes at `base 31` certify above `1/4` as well. What the machine reports is which cells it cannot bracket, never why. - The `1/3` rung. The bar `1/3` is the same criterion with a Mobius exponential-sum exponent `2/3` in place of the `3/4` a zero-free half plane buys, so it is printed beside `1/4` and never in place of it. Some pair first clears `1/3` at `base 13`, the interval class at `alpha_1 < 0.3318819` on three window digits, with `base 12` above `1/3` at all 31 of its sets even at five; every pair clears `1/3` from `base 21` on, and `base 20` is refuted by the witness `{3, 11}` at `alpha_1` in `[0.3356579, 0.3356674]`. ## RUN - `uv run python research/lab/py/digit-transform-norms/norms.py check` in 3 seconds: the exact one-digit sums, the exact anchor, the entropy floor, the gasket and carpet grid sums, the contraction rate to `d = 129`, the base-10 calibration, the gasket certificate at `N = 2` and the carpet certificate at four digit-vectors. - `certify` in 6 seconds: both gasket certificates and both carpet certificates. - `corollary` in under a second: the equidistribution band and the base-peel band for both designs. - `criterion` in 36 seconds: the closed-form check, the order band for base 9 missing `4` at `N = 4` with both covers, the sharper single orders at `N = 5`, and the 2D window moments against the 1D ones in the same units. - `grid` in 4 seconds, `moments` in 6, `windows` in 8, `lemma` in 14, `sandwich` in 52, of which the carpet at `N = 3` and `m = 400` is 40. - `least` in 93 seconds: the closed-form checks, four published calibrations, the least base carrying one clearing set, the two family sweeps, the witness base, the two ladders, the grid-machine cross-checks and the exact grid sums. - `six` in 200 seconds and under 1.8 GB resident: the headline certificate at six window digits in base 21 and the base-10 calibration at seven. The window tables are two arrays of `base^(n_d)` doubles, 686 MB each at base 21 and six digits, and the transfer step reads them in place as `(base, base^(n_d - 2), base)` rather than gathering a successor index per state, so the run never holds a third array of that size. - `family` in 151 seconds: every excluded digit of every base `35 <= base <= 125` against the threshold, each base at the shortest window in `2, 3, 4` that clears. - `pairs` in 80 seconds: the two closed-form checks, the census law at four bases, the least base against `1/4` with its witness, the `1/3` witness at base 20, the pinned first-clearing rows at `base 13` and `base 12`, and the certified interval-class ladder. - `pairfail lo hi [third]` sweeps every distinct pair of every base in the range for a certified `alpha_1 >= 1/4` (or `1/3`), `pairclear` for a certified `alpha_1 <` the bar, `pairsome` for one failing witness per base, `pairfirst` for the cheapest pair per base and `pairone base a c nd m [wide] [third]` for one set; every pair sweep takes the bar as a trailing word and prints its verdict against the bar it was given, never against an assumed one. `pairfail 4 31` is 9 minutes without the five-digit escalation and 25 with it. - Prints only, writes nothing; every printed row that has a target asserts it, and the run ends by raising if any row is off. ## WITNESSES - coprime.md LEMMA A: UNIFORM CONTRACTION, Lemma A': the worst per-digit rates `0.830915` at `d = 257`, `0.830253` at `255`, `0.809637` at `129`, `0.808166` at `127` over every modulus `d <= 301` coprime to `2`, against `c(2,3) = 0.804738`, the Lemma A' rate `0.973211` and the Lemma A rate `0.986514` at `d = 257`, with zero violations. - coprime.md LEMMA A: UNIFORM CONTRACTION, the two corollaries: over every modulus `3 <= d <= 15` coprime to `2` and every level `2 <= level <= 12` the gasket's exact deviation is at most `c^floor(level/m_d)`, worst ratio to it `0.343146` at `d = 3`, `level 2`, and the cell `d = 5`, `level 10` reads `0.0014741` against `0.337499` where the orbit rate gave `0.647604`; the carpet over `d <= 11`, `level <= 8` has worst ratio `0.066907` at `d = 4`, `level 2`. The base peel holds `T_(e m)(level) = T_m(level-1)` exactly and its error stays under `c^floor((level-1)/m_m)`, worst ratio `0.114382` at the gasket `m = 3`, `level 3` and `0.032333` at the carpet `m = 2`, `level 2`. - coprime.md PRIMES ON A DESIGN, the order band: for base 9 missing `4` the shift sandwich certifies `m_(3/2) > 0.149397` and `m_(235/154) > 0.142274` at `N = 5`, `m = 3000`, against the criterion `0.147320` and `0.139667`; at `N = 4`, `m = 4000` the chain of orders covers `[3/2, 2)` in 21 cells, tightest margin `+0.000085` at `s = 1.52`, and `[1, 2)` in 87 cells, tightest margin `+0.000023` at `s = 1.492`; the Parseval anchor reads `(9/8)^4 = 1.601807` against the scan `1.599035`. - coprime.md PRIMES ON A DESIGN, the 2D norm: in `X = base^(dim M)` units 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 1D `0.343674` and `0.153069` and the criterion `0.294640` and `0.147320`. - coprime.md PRIMES ON A DESIGN, the 2D Type I: the gasket at `N = 3` and `m = 256` certified `min_x Sigma_3 > 8.213932` against `2^3` and `max_x Sigma_3 < 9.893778`, so `alpha_1` in `(1.0126, 1.1022)`; the exact anchor `Sigma_2(0) = (8 + 2 sqrt(5))/3 = 4.157378651667` against `4`; the carpet at five digit-vectors certified `lambda < 2.441255`, so `alpha_1* < 0.8124` and the Type I level is `3^(0.5938 level)`. - coprime.md PRIMES ON A DESIGN, the carpet criterion: `g(1) = 0.343673`, `g(3/2) = 0.153068`, `g(235/154) = 0.145727`, `g(1.6) = 0.126215`, `g(1.7) = 0.103092`, `g(1.8) = 0.083503` at five digit-vectors against the criterion `0.294639`, `0.147320`, `0.139667`, `0.117856`, `0.088392`, `0.058928`, a deficit at every order, smallest `0.005749` at `s = 3/2`, `0.0058` on the pair rounded to four places; `g(3/2)` reads `0.154389` at four digit-vectors, `0.153068` at five and `0.152921` at six, a drop of `0.00015` after `0.00132`, so windows do not close it; the calibration rows `0.144609` for base 9 less `0` against the published `0.14355` and `0.137022` for base 10 less `5` against `59/433 = 0.136259`, so the box convention costs at most `0.0011`. - coprime.md PRIMES ON A DESIGN, the gasket is far: `g(1) = 0.482002` and `g(235/154) = 0.316958` at eight digit-vectors against `27/77` and `59/433`. - Printed here and carried by no page line yet: the exact one-digit sum `2` for all six sets and its exponent `log_base 2`; the direct grid exponents to `level 10`, the gasket rising to `1.055279` and base 4 falling to `0.483633`; the float sandwich rows for the interleaved sets, `alpha_1 < 0.486325` at base 4 and `< 0.370342` at base 9; the base-10 `l^1` calibration `lambda(1, 5) = 2.242123`, exponent `0.350659`, a five-digit float row the certified seven-digit bracket `[0.3505775, 0.3505797]` supersedes; and the entropy floor `0.881123` for the gasket. - THE LEAST BASE, printed by `six`: base 21 missing `0` certifies `alpha_1 in [0.2499765, 0.2499771]` at six window digits and sub-scan 8, a bracket `6 x 10^(-7)` wide clearing `1/4` by `2.3 x 10^(-5)`, the least base carrying a one-missing-digit set with `alpha_1 < 1/4`; the same cell reads `[0.2499715, 0.2499821]` at five digits and `[0.2498658, 0.2500871]` at four, so four window digits do not decide it and six do. - THE LEAST BASE, printed by `least`: base 20 certifies `alpha_1 > 0.2528608` at every one of its 10 distinct digits, best `a0 = 0` at `[0.2528608, 0.2531118]` and worst `a0 = 6` at `[0.2831944, 0.2834134]`. Base 34 certifies `alpha_1 < 0.2493701` at every one of its 17 distinct digits, worst `a0 = 16` at `[0.2493107, 0.2493701]` and best `a0 = 0` at `[0.2246400, 0.2247052]`, the least base whose whole one-missing-digit family clears; base 33 missing `15` certifies `alpha_1 > 0.2506145` at four digits, the witness that 34 is least for the family. - THE LEAST BASE, the family closed, printed by `family`: all 3663 distinct one-missing-digit sets of all 91 bases `35 <= base <= 125` certify `alpha_1 < 1/4`, sub-scan 8, each base at the shortest window in `2, 3, 4` that clears, no base failing; bases `35 <= base <= 57` need three window digits and `base >= 58` clear at two, and the ceiling of the printed band is `base 58` missing `28` at `alpha_1 < 0.2499305`, a margin `7 x 10^(-5)` that a longer window widens. With `base 34` certified here and the uniform criterion a theorem for `base >= 126`, every base `base >= 34` clears at every excluded digit and the family floor `34` is exact. - THE LEAST BASE, the ladder: at four window digits and sub-scan 8 the certified brackets at `a0 = 0` run `base 10 [0.3090500, 0.3106411]`, `base 14 [0.2780846, 0.2787259]`, `base 18 [0.2596685, 0.2599987]`, `base 20 [0.2528608, 0.2531118]`, `base 21 [0.2498658, 0.2500871]`, `base 22 [0.2470967, 0.2472930]`, `base 26 [0.2377912, 0.2379193]`, `base 30 [0.2305272, 0.2306165]`, `base 33 [0.2260074, 0.2260778]`, `base 34 [0.2246400, 0.2247052]`, and at the middle digit `a0 = floor(base/2)` they run `base 10 [0.3498936, 0.3512569]`, `base 14 [0.3129308, 0.3134889]`, `base 18 [0.2909812, 0.2912724]`, `base 20 [0.2828721, 0.2830946]`, `base 22 [0.2760085, 0.2761833]`, `base 26 [0.2649357, 0.2650507]`, `base 30 [0.2563012, 0.2563819]`, `base 34 [0.2493107, 0.2493700]`, with no lower bound at `base 21` and `base 33` where the constant-phase digit empties a row and the upper bounds read `0.2666524` and `0.2404222`. - THE PAIR FAMILY, the census: `15, 36, 45, 66` excluded pairs at `base 6, 9, 10, 12` fall into `7, 16, 21, 31` distinct transforms, matching `(C(base-2,2) + floor((base-2)/2))/2 + floor(base/2)` at every base, with the edge collapse `{0,c} = {0,base-c}` exhibited. - THE PAIR FAMILY, the least base against `1/4`, printed by `pairs` and `pairfail`: base 32 missing `{0,1}` certifies `alpha_1` in `[0.2499087, 0.2499779]` at four window digits and base 31 the same class `[0.2518967, 0.2519717]`; over `4 <= base <= 31`, 2363 of the 2373 distinct sets certify `alpha_1 > 1/4`, closest `base 26` missing `{2,23}` at `alpha_1 > 0.2502919`, and the ten undecided cells carry certified upper bounds `0.2538899` to `0.2826357`, all above `1/4`, so none of them is shown to clear either. - THE PAIR FAMILY, the interval-class ladder at four window digits and sub-scan 8: `base 20 [0.2830983,0.2833260]`, `base 24 [0.2688772,0.2690199]`, `base 28 [0.2582491,0.2583458]`, `base 30 [0.2538477,0.2539290]`, `base 31 [0.2518967,0.2519717]`, `base 32 [0.2499087,0.2499779]`, `base 33 [0.2481501,0.2482143]`, `base 36 [0.2431292,0.2431809]`, `base 40 [0.2374705,0.2375104]`. - THE PAIR FAMILY, the `1/3` rung, printed by `pairfirst`, `pairclear` and `pairsome`: the interval class first clears `1/3` at `base 13` at `alpha_1 < 0.3318819`, `base 12` staying above at all 31 sets to five window digits; every pair clears `1/3` at every base `21 <= base <= 26`, 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 `[0.3356579, 0.3356674]`. Each of those two rows brackets its own cheapest set from below by the infimum matrix, `base 13` reading `alpha_1 > 0.3227751` under the `alpha_1 < 0.3318819` it prints and `base 12` reading `alpha_1 > 0.3370434` under the `alpha_1 < 0.3371162` it prints, each lower certificate below the upper one on the same row as Perron monotonicity demands; the base 12 bracket lies above `1/3`, so that set is refuted at the bar and not merely undecided. - THE LEAST BASE, the calibrations and the cross-checks: base 10 missing `5` and box `2/10^(n_d)` certifies `[0.3504737, 0.3506832]` at five window digits, `[0.3505681, 0.3505891]` at six and `[0.3505775, 0.3505797]` at seven, so the exponent is strictly under the published `27/77 = 0.3506494`, which is a finite-window upper bound and not the constant; base 9 missing `0` at four digits certifies `[0.3201155, 0.3234318]`, containing the published `0.3219`; the closed form matches the character sum to `2.4e-13` on 2000 arguments at base 21 and `1.8e-13` at base 34; the grid machine agrees with the window machine, `[0.241060, 0.266434]` at base 21 missing `0` and `N = 4` and `[0.227429, 0.286502]` at base 34 missing `16` and `N = 3`, each containing the window bracket; and the exact grid sums for base 21 missing `0` read exponents `0.227670`, `0.240648`, `0.243576`, `0.245196` at `level 1..4`, rising toward the certified `0.2499768`.