Digit Uniform Bound

• One bound on the digit transform of every one-missing-digit set in a base at once: it names the base and forgets the digit, so what it proves is a statement about base and not a list of bases.
• The target is the l^1 exponent alpha_1 of a one-missing-digit set against the threshold 1/4, where the pair route turns a Mobius asymptotic on.

THE DOMINATION

• The set is F = {0..base-1}, the digits of the code, less {a0}, fill = base - 1, and its normalised transform is hat F(t) = (1/(base-1)) Sum_(a != a0) e(a t).
• The full residue sum is the Dirichlet kernel up to a phase: Sum_(a<base) e(a t) = e(((base-1)/2) t) D_base(t) with D_base(t) = sin(base pi t)/sin(pi t).
• Subtracting the excluded term and taking absolute values, |hat F(t)| = |e(((base-1)/2) t) D_base(t) - e(a0 t)|/(base-1) <= (|D_base(t)| + 1)/(base - 1) =: u_base(t), for every t and every a0.
• The right side names no digit, so G(w) = sup_(cell w) |hat F| <= sup_(cell w) u_base dominates all base excluded digits at once, and |hat F| <= 1 gives the free sharpening u_base -> min(1, u_base).
• The cost is exact: the triangle inequality throws away the phase e((a0 - (base-1)/2) t), so u_base is attained only where that phase is antipodal to the kernel and the bound is strict everywhere else. The uniform bound is weaker than the per-digit one at every base, by construction.

THE RUN DECOMPOSITION

• The shift sandwich reads alpha_1 <= (1/N) log_base max_x Sigma_N(x) with Sigma_N(x) = Sum_(i < base^N) Prod_(j<N) |hat F(base^j (x + i/base^N))|, and Sigma_N <= Sigma_N^u, the same sum with u_base in place of |hat F|.
• Expanding Prod_j (|D_base(base^j t)| + 1) over subsets E of {0..N-1} and telescoping each maximal run, D_base(t) D_base(base t) ... D_base(base^(l-1) t) = D_(base^l)(t), gives the exact identity (base-1)^N Sigma_N^u(x) = Sum_E Sum_(i<base^N) Prod_(runs (s,l) of E) |D_(base^l)(base^s (x + i/base^N))|.
• Every term is a Lebesgue sum of a Dirichlet kernel over an arithmetic progression of points. Peeling the run of largest s first, whose factor depends on the fewest low digits of i, bounds each term by base^N Prod_r lambda_(l_r) with lambda_l = L_(base^l)/base^l and L_M = max_x Sum_(j<M) |D_M(x + j/M)|.
• The peel is checked term by term: every one of the 2^N - 1 run terms sits under base^N Prod_r lambda_(l_r) and max_x Sigma_N^u sits under a_N (base/(base-1))^N, at base 3, 4 to N = 4 and base 5, 7, 10 to N = 3, slack about 12%.
• Summing over E by first letter gives a_N = a_(N-1) + Sum_(l<N) lambda_l a_(N-1-l) + lambda_N, whose growth root solves z = 1 + Sum_(l>=1) lambda_l z^(-l), and alpha_1 <= log_base(z base/(base-1)).

THE LEBESGUE CONSTANT

• Writing the M points as distances d_j to the nearest integer and |sin(M pi x)| = |sin(M pi d_j)|, the sum is s(theta) Sum_j 1/sin(pi d_j) at offset theta in [0, 1/2], with pairs at (m + theta)/M and (m + 1 - theta)/M.
• With mu = pi(m + 1/2)/M and delta = pi(1/2 - theta)/M, the pair identity s = cos(M delta) and csc(mu - delta) + csc(mu + delta) = 2 sin mu cos delta/(sin^2 mu - sin^2 delta) turn the pair into 2 csc mu times X = cos(M delta) cos delta sin^2 mu/(sin^2 mu - sin^2 delta).
• For m >= 1, sin delta/sin mu <= delta/mu = r/(2m+1) <= r/3 with r = 2 M delta/pi in [0,1], and cos(pi r/2) <= 1 - r^2 because 2 sin^2(pi r/4) >= r^2, so X <= (1 - r^2)/(1 - r^2/9) <= 1 and the pair is largest at theta = 1/2.
• For m = 0 the pair is at most (M/pi)(1 + pi^2/(24 M^2)) max_theta sin(pi theta)(1/theta + 1/(1 - theta)) = (4M/pi)(1 + o(1)), which is the theta = 1/2 value again.
• At theta = 1/2 the sum is 2 Sum_(j < M/2) csc(x_j), x_j = (2j+1) pi/(2M); splitting csc = 1/x + g with g convex increasing, the midpoint rule gives Sum g(x_j) <= (M/pi) Int_0^(pi/2) g = (M/pi) log(4/pi), and H_(2n) - H_n/2 bounds the reciprocal-odd sum.
• Hence L_M <= M((2/pi) log M + gamma' ) + 2/pi with gamma' = (2/pi)(gamma + log(8/pi)) = 0.96252282676, printed and used rounded up at 0.9625229, so lambda_l <= (2/pi) l log base + gamma' + (2/pi) base^(-l) at every l, and the coarser lambda_l <= (2/pi) l log base + c_0 with c_0 = 0.97 needs base^l >= 86, since 0.97 - gamma' = 0.0074771 has to cover (2/pi)/base^l. The scan confirms the shape: L_M/M - (2/pi) log M falls from 0.965217 at M = 4 to 0.962523, and the maximising offset is 1/2 at every M tested.

THE THRESHOLD

• With the exact hypothesis lambda_l <= c_1 l log base + gamma' + c_1 base^(-l), c_1 = 2/pi, the root equation is (z-1)^3 = c_1 (log base) z + gamma'(z - 1) + c_1 (z-1)^2/(base z - 1), and alpha_1 < 1/4 follows from z < base^(1/4)(1 - 1/base). The coarser c_0 = 0.97 form drops the last term and is the same statement wherever base^l >= 86.
• Certified at 120 bits at w = base^(1/4)(1 - 1/base) for every base in [125, 3000), and the chain fails at base 124; the certificate first checks that the constant it uses sits above the true gamma', which the earlier printed 0.9625153 did not.
• Above base 211 the closed-form cap z <= 1 + sqrt(2 c_1 log base + c_0), which follows from z >= 2 and z <= 2(z-1), is below base^(1/4)(1 - 1/base) and both sides are monotone, so the tail needs no evaluation at all.
• The threshold is base_u = 125: alpha_1 < 0.249980 at base 125, and it falls from there at every excluded digit, with no per-base spectral computation anywhere in the chain. The c_0 = 0.97 form of the same chain gives 126, one base worse, so the sharper constant strengthens that statement and does not contradict it.

ANY NUMBER OF EXCLUDED DIGITS

• Nothing in the chain used m = 1. With E any set of m excluded digits, |Sum_(e in E) e(e t)| <= m gives |hat F(t)| <= min(1, (|D_base(t)| + m)/(base - m)) =: u_(base,m)(t), again naming no digit, and the free cap |hat F| <= 1 still applies.
• The run decomposition gains one weight. Expanding Prod_j (|D_base(base^j t)| + m) over subsets E of {0..N-1} carries m^(N - |E|) on the positions outside E, and each maximal run of E telescopes to the same Dirichlet kernel, so (base-m)^N Sigma_N^u(x) = Sum_E m^(N - |E|) Sum_(i<base^N) Prod_(runs (s,l) of E) |D_(base^l)(base^s(x + i/base^N))|, the identity checked term for term against the direct product.
• The peel is unchanged, since it bounds one run term at a time; summing over E by first letter gives a_N = m a_(N-1) + m Sum_(l<N) lambda_l a_(N-1-l) + lambda_N and max_x Sigma_N^u <= a_N (base/(base-m))^N.
• The growth root solves z = m + m Sum_(l>=1) lambda_l z^(-l), so with lambda_l <= c_1 l log base + gamma' + c_1 base^(-l) the certificate is (z - m)(z - 1)^2 <= m(c_1 (log base) z + gamma'(z - 1) + c_1 (z-1)^2/(base z - 1)) and alpha_1 < e follows from z < base^e (1 - m/base). At m = 1 this is the cubic of the section above, so the general form is a check on the special one and not a rewrite of it.
• The thresholds, certified at 120 bits on [base_u, 3000) with the chain failing at base_u - 1: against 1/4 it gives base_u(1) = 125, base_u(2) = 649, base_u(3) = 1873; against the weaker bar 1/3 it gives 32, 105, 230. So every base >= 649 clears alpha_1 < 1/4 at every excluded pair, with no per-pair spectral computation anywhere. The 1/3 rung at m = 1 lands at base 32, where base^l = 32 is below 86, so it is exactly the rung the coarser c_0 form may not be used on.
• The cost of uniformity grows with m because the phase thrown away is m characters rather than one: the majorant loses m/(base-m) where the truth loses the interference between them, and the two-digit numeric floor is far below 650 (lab/py/digit-transform-norms, verb pairfail).

THE WINDOW CEILING

• The same majorant run through the window machine measures how far the uniform idea can reach at all: U(w) = min(1, (min(base, S_1(w) S_2(w)) + 1)/(base-1)) on the cell [w/base^n, (w+1)/base^n), the transfer matrix (M y)(v) = Sum_(c<base) U(v base + c) y((v base + c) mod base^(n-1)), and alpha_1 <= log_base rho.
• The supremum is enclosed in closed form, never sampled. |sin(pi t)| is concave on [0,1], so sup_cell 1/|sin(pi t)| = 1/sin(pi delta(w)) exactly with delta(w) = min(w, base^n - 1 - w)/base^n; and sup_cell |sin(base pi t)| is 1 when the image interval [w, w+1]/base^(n-1) contains a half-integer and max of the two endpoint values otherwise. Both are monotone closed forms, so there is no sub-scan and no Lipschitz slack.
• The ceiling: the uniform window bound clears 1/4 first at base 75 and at every base above it, and the bound moves by less than 0.001 from three window digits to four (0.340029 -> 0.339085 at base 21, 0.297650 -> 0.297296 at base 34), so no refinement of the window rescues a base near 74.
• So base 75 is the floor of the digit-uniform route as a numeric fact, base_u = 126 is the floor of what the route proves uniformly in base, and the difference is slack in the run decomposition, not in the majorant.

THE CONSISTENCY CHECK

• Where both apply the uniform bound must be weaker than the per-digit ladder, and it is at every base tested, never stronger: 0.461532 against 0.323432 at base 9 missing 0, 0.440960 against 0.350684 at base 10 missing 5, 0.344129 against 0.283414 at base 20 missing 6, 0.339085 against 0.250088 at base 21 missing 0, 0.299529 against 0.253000 at base 33 missing 15, 0.297296 against 0.249371 at base 34 missing 16.
• The gap closes as the base grows, 0.1381 at base 9 down to 0.0479 at base 34, which is the discarded phase costing 1/(base-1).

THE CERTIFICATE

• The threshold inequality is mpmath.iv at 120 bits: base^(1/4), log base and every product and difference are intervals, and a row passes only when the lower endpoint of the margin is positive.
• The window rows are floats with every operation pushed outward by nextafter and every sine inflated by 2^-45 in the unsafe direction, the Perron root bounded above by Collatz-Wielandt, M y <= mu y componentwise giving rho <= mu, with the float Perron vector as the test vector and a positive floor on it.
• The power iteration accepts a root only after fifty consecutive relative moves under 1e-13 and at least three hundred steps; a shorter test admits the transient cluster of near-zero cells, whose row sums are base, and returns a root too large by a factor of sixteen.
• Every printed exponent is log_base mu rounded up and asserted against mu < base^e before it prints, and the certified L_M rows are the exact maximum over the offset, not a fit.

RUN

• uv run python research/lab/py/digit-uniform-bound/ubound.py check in 5 seconds: the domination on nine bases and every distinct digit, the run identity against the direct product at five base-level pairs, the Lebesgue scan, the threshold and its certificate, and the consistency rows.
• domination, identity, peel, lebesgue, threshold, window, least, compare run the pieces; least is 6 seconds and window is 5.
• Prints only, writes nothing; every row that has a target asserts it, and the run ends by raising if any row is off.

WITNESSES

• The domination |hat F| <= (|D_base| + 1)/(base-1) for every excluded digit, with the digit-slack column 1.000000 at base 3 falling to 0.010045 at base 200.
• The exact run decomposition of Sigma_N^u into 2^N Lebesgue sums of Dirichlet kernels at moduli base^l, and the peel bounding each of them.
• L_M <= M((2/pi) log M + 0.9625153) + 2/pi, with the maximum at the half-offset and the scanned constant falling from 0.965217 to 0.962523.
• The threshold base_u = 125, certified at 120 bits on [125, 3000), the chain failing at 124; the c_0 = 0.97 form of the same chain gives 126.
• The window ceiling base 75 at three digits, and the consistency rows against the per-digit ladder.