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(<t, v>) on T^dim, with F the filled digit vectors, and its level product hat F_level(t) = Prod_(j<level) hat F(base^j t), the normalised transform of S_level; the norms are I_level = Int_(T^dim) |hat F_level|, the shifted grid sum Sigma_N(x) = Sum_(i in (Z/base^N)^dim) Prod_(j<N) |hat F(base^j (x + i/base^N))|, and the exponents alpha_1 = dim + lim (1/level) log_base I_level and alpha_1*, the exponent of the sup-over-box sum a large sieve consumes.
• The threshold throughout is dim/2: below it a Farey-point large sieve saves a power of the level, at or above it the same route saves nothing.
• Conventions: Y = base^level is the side and X = base^(dim level) the box, every exponent printed here is in Y units, and a one-dimensional exponent 27/77 in X units is the same statement as dim x 27/77 in Y units.

WHAT IT COMPUTES

• The one-digit sum. For a design that is the full residue cube less one vector, Sum_(v in F) e(<t, v>/base) = base^dim [t = 0] - e(<u, t>/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(<t, v>); 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<base^N) F_N(x + i/base^N)^s, so m_s >= (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 <d, t>) 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.