Digit designs and the Euler product

Claims · 36 dated claims on Digit designs and the Euler product, each with its tag and its witness; newest 2026-09-21.

2026-09-07 Proved The indicator of S_F, the integers whose base digits all lie in the digit set F, is multiplicative exactly at the full digit set. 1 in F is forced by f(1) = 1; if a digit c >= 2 is missing take the least, and R_c R_(c+1) has no carry because its base^m coefficient is min(m+1, c, 2c-m) <= base-1, so its digit set is exactly {1..c} while gcd(R_c, R_(c+1)) = R_1 = 1; if only 0 is missing then odd base gives the coprime pair (2, (base^2+1)/2) with product base^2 + 1 = 101, and even base gives (base^2-1, base^2+1), coprime and odd, whose product base^4 - 1 has every digit base-1 while base^2+1 does not lie in the set. Over all 8177 sets with 2 <= base <= 12 the constructed witness is asserted at each of the 4083 sets that pass f(1) = 1 and are not full, and an independent search finds a minimal witness for every one, hardest base = 12, F = {1}, pair (5, 377). No design outside the full set carries an Euler product over primes; 0 excluded and a single digit both fail. Witness: lab/py/mrly-euler verb wall.
2026-09-07 Proved For every F strictly inside {0..base-1} the design zeta and the design Mobius series obey a disjunction and not a universal: if 1 is outside F the constant coefficient of zeta_F M_F is 0; if a prime p of S_F has p^2 outside S_F the coefficient at p^2 is -1, since (p,p) is the only admissible factorisation; and otherwise zeta_F M_F = 1 forces the least element g > 1 of S_F to be prime with every power g^j in S_F, a necessary condition on an escapee and not a contradiction. At the full digit set the two are inverse, zeta_F = zeta and M_F = 1/zeta. Over 257 sets the least n > 1 with a nonzero coefficient is at most 50, first at n = 4 for base 3 {0,1} and n = 9 for base 10 missing 9, while the eight full sets have none below 4000. Witness: lab/py/mrly-euler verb pair.
2026-09-07 Proved The position product. With G_level(t) = prod_(i<level) sum_(d in F) e(d base^i t) = fill^level hat F_level(t), uniqueness of the digit expansion gives int_0^1 G_level(t) e(-nt) dt = 1_(D_level)(n) for every integer n, hence sum_(n in D_level, n >= 1) a(n) n^(-s) = int_0^1 G_level(t) A(s,t) dt for every absolutely convergent Dirichlet series, with A(s,t) = sum_(n >= 1) a(n) e(-nt) n^(-s); a = 1 is the periodic zeta of DLMF 25.13.1 and a = mu the Lerch-Mobius series, so zeta_F and M_F are pairings of one set-only product against one arithmetic-only kernel. The set enters through the digit positions and never through the primes. Checked to 1.95e-16 and 2.04e-16 at base 10 missing 9, level = 3 and level = 4, and 2.9e-16 at base 3 {0,1}, level = 3, 4, 5. Witness: lab/py/mrly-euler verb position.
2026-09-07 Proved The tree's pair route is Holder on the position identity. When 0 is in F, at x = base^level the identity is finite on both sides, M_F(base^level) = int_0^1 G_level(t) S_level(t) dt with S_level(t) = sum_(n < base^level) mu(n) e(-nt), so abs(M_F(base^level)) <= (int_0^1 abs(G_level)) max_t abs(S_level) is at most fill^level base^(level(alpha_1 - 1)) x^b = x^(alpha + alpha_1 - 1 + b); when 0 is outside F the same upper bound holds after summing the levels, a geometric sum of ratio base^(alpha + alpha_1 - 1 + b) > 1 by the floor alpha + alpha_1 >= 1 of mobius.md. It sits under the trivial x^alpha exactly when alpha_1 < 1 - b, which is the bar of coprime.md and mobius.md derived rather than posited, with b = 3/4 + eps under GRH from Baker and Harman 1991. Witness: lab/py/mrly-euler verb position.
2026-09-07 Proved The fibres of the Lerch-Mobius series are inverse Dirichlet L-functions. Splitting n by g = gcd(n,Q) and expanding on the characters of (Z/(Q/g))^* gives M(s, a/Q) = sum_(g divides Q) mu(g) g^(-s) phi(Q/g)^(-1) sum_(chi mod Q/g) tau_a(chi) L(s,chi)^(-1) prod_(p divides Q not Q/g) (1 - chi(p) p^(-s))^(-1), so M(s, a/Q) continues to C with singularities in Re s > 0 only at zeros of L(s,chi) of modulus dividing Q, and M(s,0) = 1/zeta(s). Since G_level(a/base^j) = fill^(level-j) G_j(a/base^j) are the largest values the position product takes, the design's major arcs are the base-power rationals, and on that family holomorphy in Re s > 1/2 is exactly GRH for base-power modulus. Coefficient identity checked to 2.6e-12 at eleven pairs (Q,a) including Q = 3, 9, 27, 100, the Euler-factor step to 7.4e-16. Witness: lab/py/mrly-euler verb fibre.
2026-09-07 Proved The reflection moves the kernel and not the design. Solving Hurwitz's formula DLMF 25.13.3 at x = t and x = 1-t gives Z(s,t) = ((2 pi)^s Gamma(1-s)/(2 pi i))(e^(pi i s/2) zeta(1-s,t) - e^(-pi i s/2) zeta(1-s,1-t)) for s not a positive integer, the derivation dividing by 2i sin(pi s); this is DLMF 25.13.2 recovered, the gain being the range Re s > 0 in place of Re s > 1. The position identity turns it into a dual integral of the same G_level against Hurwitz zetas at 1-s, never a relation between zeta_F(s) and zeta_F(1-s); the design's own symmetry is the base-adic scaling G_level(t) = g(t) G_(level-1)(qt), whose transfer eigenvalue fill base^(-s) is what makes the vertical pole lattice. Formula checked to 2.1e-30 at s = 3.3, 2.7 + 1.9i and 0.6 + 4.1i. Witness: lab/py/mrly-euler verb dual.
2026-09-07 Proved The design's multiplicative shadow is a Lyndon Euler product with no RH content. On the free monoid over F with norm N(w) = base^(abs(w)), sum_w N(w)^(-s) = 1/(1 - fill base^(-s)) = prod_(level>=1) (1 - base^(-level s))^(-c_fill(level)) with c_fill(level) the Lyndon count, by Chen-Fox-Lyndon: every word factors uniquely as a non-increasing product of Lyndon words, so the free monoid on F is equinumerous by norm with the free abelian monoid on Lyndon words and is not equal to it. The primes are the Lyndon words, the zeta is zero-free, its Mobius is supported on the empty word and the letters so its Mertens is 1 - fill beyond norm 1, and its poles are exactly s = alpha + 2 pi i m / log base, the design pole lattice. All RH content of zeta_F therefore sits in the cofactor zeta_F(s)(1 - fill base^(-s)). Expansion verified through u^16 at fill = 2, 3, 4, 9, 10, c_2(level) being A001037. Witness: lab/py/mrly-euler verb word, A001037.
2026-09-07 Proved The Beurling system of a design with non-unit digit gcd is finitely generated, on two branches. If gcd(F) = a > 1 every element of S_F is a multiple of a; when a is prime the primes of the design are {a}, N_F is the powers of a and M_B(x) = 0 for x >= a, and when a is composite S_F holds no prime at all, N_F = {1} and M_B is identically 1, witness base = 10, F = {0,4,8}. Either way the eight scaled census families of mobius.md are exactly the columns the Beurling route cannot see, while the scaling transfer reads them exactly. Witness: lab/py/mrly-euler verb beurling.
2026-09-07 Verified The Beurling census on the primes of a design, to x = 10^6. Base 3 {0,2} has the single prime 2 and M_B identically zero past 2; base 3 {0,1} has 525 primes, N_F(920483) = 2198, running max abs(M_B) = 98 and exponent 0.3339 against alpha/2 = 0.3155; base 10 missing 9 has 35139 primes, N_F(10^6) = 488864 against x^alpha = 531441, M_B(10^6) = 1860, running max 1866, and exponent log(running max)/log x reading 0.4203, 0.4882, 0.5452 at 10^4, 10^5, 10^6 against alpha/2 = 0.4771, where full base 10 as control reads 0.4084, 0.4241, 0.4276 at the same points against its own alpha/2 = 0.5. What the census reads is the level and not a trend: +0.068 over alpha/2 for the design against -0.072 for the control, a running maximum climbing in both. There is cancellation, 0.545 against the trivial alpha = 0.954, and it is above alpha/2, so the census supports cancellation and does not support the square-root conjecture on N_F; N_F is not S_F. Full base 10 reproduces -23, -48, 212 at 10^4, 10^5, 10^6, A084237. Witness: lab/py/mrly-euler verb beurling, A084237.
2026-09-07 Proved The identity that replaces zeta M = 1 on a design. For every (base,F) with 1 in F the indicator 1_(S_F) has a Dirichlet inverse nu_F, given by nu_F(1) = 1 and nu_F(n) = -sum_(d divides n, d > 1, d in S_F) nu_F(n/d), so zeta_F(s) N_F(s) = 1 with N_F(s) = sum nu_F(n) n^(-s); the support of nu_F lies inside the multiplicative semigroup generated by S_F and strictly inside it, since 9, 27 and 36 lie in the semigroup with nu_F = 0 while 16, 48 and 52 lie in the semigroup and outside S_F, so the semigroup is a third set beside S_F and the Beurling integers on the primes of the design and the support is a fourth, and nu_F is mu exactly at the full digit set, where the classical identity is the special case. If rho is a zero of zeta_F with Re rho > alpha then sigma_c(N_F) >= Re rho, by the identity theorem on the connected pole-free half plane Re s > max(sigma_c(N_F), alpha), so sum_(n <= x) nu_F(n) is not O(x^(Re rho - eps)) for any eps > 0; the converse bound sigma_c(N_F) <= sup Re rho is not claimed. Checked against mu term for term on the full digit set to n = 131072 at base = 2 and n = 177147 at base = 3, and the partial sums of N_F(sigma) meet 1/zeta_F(sigma) to 1.60e-3 at sigma = Re rho + 0.08 = 0.8008 and 1.96e-4 at sigma = Re rho + 0.20 = 0.9208 at base 3 {0,1}, and to 1.72e-2 at sigma = 1.0816 and 2.39e-3 at sigma = 1.2016 at base 10 missing 9, both offsets sitting above Re rho. Witness: lab/py/mrly-pairing verb inverse, lab/py/design-zeta.
2026-09-07 Proved The design's own Mobius has anti-cancellation, and that is what makes the decoupling a blessing. Winding boxes on zeta_F by the argument principle certify one zero each and pin Re rho to the box edges: winding 1 on Re in [0.72074, 0.72084], Im in [28.60563, 28.60573] at base 3 F = {0,1} with contour minimum abs(zeta_F) = 8.298e-4 against the engine bound 6.284e-30, and winding 1 on Re in [1.00150, 1.00168], Im in [2.73915, 2.73925] at base 10 missing 9 with contour minimum 6.865e-4 against 2.798e-23, while the control rectangle Re in [0.99900, 1.00050], Im in [2.73810, 2.74030] there returns winding 0. Both boxes lie strictly right of alpha = 0.6309297536 and 0.9542425094, so sum_(n <= x) nu_F(n) is not O(x^(0.72074 - eps)) and not O(x^(1.00150 - eps)) respectively: the limsup of the design's own Mertens function exceeds the design's own mass A_F(x), and at base 10 missing 9 exceeds x itself, the box lying right of Re s = 1. The square-root conjecture in the alpha/2 shape is therefore false for nu_F and can only be carried by mu restricted to S_F; the sibling's decoupling theorem is what protects it. Pointwise the census is far below both limsups, max/A_F = 0.0738 at base 3 level = 16 and max/x = 0.0847 at base 10 level = 7: the running maximum of sum nu_F(n) grows by 9.4474, 11.5000, 10.2220, 10.0354 per level at base 10 missing 9, level = 4..7, against base^(Re rho) = 10.036661 and the trivial fill = 9, only the last of the four landing on the predicted rate, with max/A_F(base^level) rising 0.1043, 0.1094, 0.1398, 0.1588, 0.1771; at base 3 {0,1} the geometric mean of the four steps level = 12..16 is 2.059 against 2.207512 and 2 while the arithmetic mean of the five printed level ratios is 1.9972, below the trivial 2, a census too short to separate them. Witness: lab/py/mrly-pairing verbs box and inverse, lab/py/design-zeta.
2026-09-07 Verified The pair zeta_F M_F = 1 + D_F gains nothing: D_F has abscissa exactly alpha. Absolute convergence of zeta_F^2 puts sigma_a(D_F) <= alpha and that half is proved; for the other half, if sigma_c(D_F) were below alpha then M_F(sigma) = (1 + D_F(sigma))/zeta_F(sigma) would tend to 0 as sigma -> alpha+, since zeta_F has nonnegative coefficients and is singular at its abscissa by Landau, so zeta_F(sigma) -> +infinity, and there is no circularity in the argument because M_F is dominated termwise by zeta_F and so converges absolutely at every sigma > alpha with no hypothesis on theta(F). That half rests on a measurement, unconditional in shape since sigma_c(D_F) < alpha would force P(x) = o(x^alpha): P(x) = sum_(n <= x) c_F(n) divided by x^alpha is bounded away from 0 and from infinity, reading 0.493767, 0.699235, 0.758519, 0.587055 at four sampling phases at base 3 {0,1}, the four phases being needed because P(x)/x^alpha is log-periodic and sampling only at x = base^level aliases every Fourier mode onto one number. The M_F(sigma) -> 0 limit test is not a witness here: the tail the generator prints beside it is base^(-level alpha/2), which assumes the square-root conjecture, and against the unconditional tail (fill-1) base^(-level eps)/(1 - base^(-eps)) from A_F(base^l) = fill^l no printed M_F value at base 10 missing 9 is distinguishable from 0. Since M_F = (1 + D_F) N_F and sigma_c(N_F) > alpha, the glue is not neutral but lossy. Witness: lab/py/mrly-pairing verb glue.
2026-09-07 Proved The position pairing is exact on the grid and its l^1 mass sits at the top level, which kills the per-denominator split. For 0 in F, M_F(base^level) = base^(-level) sum_(a mod base^level) G_level(a/base^level) S_level(a/base^level) exactly, both factors being trigonometric polynomials of degree below base^level; writing a = base^v a' with base not dividing a' and j = level - v gives G_level(a/base^level) = fill^(level-j) G_j(a'/base^j) and the exact level decomposition C_level = sum_(j=0)^level fill^(level-j) c_j of the l^1 mass, with C_j = fill C_(j-1) + c_j. The l^1 floor C_j >= base C_(j-1) forces the top-level share c_level/C_level >= 1 - fill/base = m/base at every base and digit set, measured 0.485846, 0.602606, 0.687994, 0.510055 against floors 0.333333, 0.500000, 0.600000, 0.100000, with levels j >= level/2 carrying 0.995116, 0.996061, 0.997043, 0.942350. Since the Baker-Harman Proposition beats the uniform x^(3/4) only below j = level/2, weighting the Mobius input per denominator saves exactly log(C_level/c_level)/(level log base), a constant factor capped by base/m: the numerator is 0.657068 at base 3 {0,1}, identical at every level = 6..14. The split exponents are 0.988106, 0.912502, 0.905006, 1.012881 uniform and 0.941173, 0.879287, 0.879188, 0.964150 per denominator against alpha = 0.630930, 0.500000, 0.430677, 0.954243, while the Cauchy-Schwarz split is (alpha+1)/2 exactly since int abs(G_level)^2 = fill^level; the base 2 and base 3 full-set controls return 0.500000, the classical RH exponent. Witness: lab/py/mrly-pairing verb split.
2026-09-07 Proved The principal fibre of the grid pairing has exponent alpha - 1/2 under RH, below the conjectured alpha/2, and that is an asymptotic statement only. The a = 0 term of the grid pairing is base^(-level) fill^level M(base^level), of exponent alpha - 1/2 under RH, and alpha - 1/2 < alpha/2 for every alpha < 1, so in the limit the classical Mertens function cannot carry the conjectured size of the design meter. At finite depth it carries a great deal: the a = 0 term reads -0.31857 of 11, 0.11133 of 6, -0.05924 of 9 and 112.66549 of 276 at base 3 {0,1} level = 14, base 4 {0,1} level = 11, base 5 {0,1} level = 9 and base 10 missing 9 level = 6, shares -0.028961, 0.018555, -0.006583, 0.408208, and exactly all of the meter on the two full-set controls. So at base 10 missing 9 the principal fibre carries 40.8 percent of the meter at the only measured level, which refutes any claim that the square-root conjecture lives entirely off the principal fibre at finite depth: the exponent gap there is 0.454243 against 0.477121, and a factor of 10 between them needs x = 10^44. Witness: lab/py/mrly-pairing verb split.
2026-09-07 Proved The one-step constant of the digit transform never exceeds the triangle-split bound, and is strictly below it at every family measured beyond level = 1. With H(t) = sum_(r mod base) abs(g_F((t+r)/base)) and B_base(F) = sup_t H(t), the identity C_level = sum_(a mod base^(level-1)) abs(G_(level-1)(a/base^(level-1))) H(a/base^level) gives C_level <= B_base(F) C_(level-1), so C_level/C_(level-1) <= B_base(F) at every level and every family with no computation at all; the inequality is not strict in general and equality is attained, C_1/C_0 = 4 = B_base(F) exactly at base 3 {0,1}, so strictness needs level >= 2. Verified there: C_level/C_(level-1) reads 3.889888518, 5.032783116, 6.410132461, 18.369402635 at base 3 {0,1}, base 4 {0,1}, base 5 {0,1} (all level = 9) and base 10 missing 9 (level = 6), against B_base(F) = 4.000000000, 5.226251860, 6.472135955, 19.888543820, and the ratio agrees between the two consecutive level the generator prints to 8.5, 7.4, 10, 5.0 digits by family, so the stability is family by family and two values of level are all that is measured. Witness: lab/py/mrly-pairing verb split.
2026-09-07 Proved The design Mobius of the two-digit design is base-free. Let S* be the nonzero 0/1 polynomials of Z[x], M* the monoid they generate, nu* the Dirichlet inverse of 1_(S*). For F = {0,1} at every base >= 2, nu_F(n) = sum over P in M* with P(base) = n of nu*(P). Evaluation is a bijection S* -> S_F, a monoid homomorphism, and of finite fibres, since an element of M* has nonnegative coefficients so P(base) = n caps every coefficient by n and deg P by log(n)/log(base); the pushforward g therefore exists, g(1) = 1 because 1 is the only element of M* of value 1, and grouping the pairs (D, Q) in S* x M* with D(base) Q(base) = n by P = DQ turns 1_(S*) * nu* = delta into 1_(S_F) * g = delta, where the Dirichlet inverse is unique. Hence sum_(n <= x) nu_F(n) = sum over P in M* with P(base) <= x of nu*(P) at every x: the base enters only as the order in which one base-free function is summed, and at base = 2 the classical mu is that pushforward. Checked term for term with 0 mismatches to n <= 2^15, 3^10, 4^8 and 5^7, where 108978 elements of M* collapse onto 32768 integers at base 2. Witness: lab/rs/carry-free-mobius verb lemma.
2026-09-07 Proved The degree-graded mass of the base-free design Mobius is 1 - 2t exactly. Degree is a monoid homomorphism M* -> N with finite fibres because 1 is the only constant in M*, which holds for F = {0,1} and for no design carrying a digit at least 2, where a constant c >= 2 makes the degree-zero fibre {c^k} infinite; pushing 1_(S*) * nu* = delta along it with 2^d polynomials of degree d gives A(t)/(1 - 2t) = 1, so the graded sums are 1, -2, 0, 0, ... and sum over deg P < level of nu*(P) = -1 for every level >= 2. The design Mertens function is therefore pinned to -1 at every level boundary level >= 2 inside the carry-free window, and reads 0 at level = 1. Witness: lab/rs/carry-free-mobius verb sequence.
2026-09-07 Proved The design zeta has an explicit zero free half plane, and it closes the census right of the abscissa. Let a_min be the least nonzero digit of F, hence the least element of S_F, every element of two digits or more exceeding base. If a real sigma > alpha satisfies a_min^sigma zeta_F(sigma) < 2 then zeta_F has no zero in Re s >= sigma: the coefficients are nonnegative and the series converges for sigma > alpha, so for Re s = sigma' >= sigma one has abs(a_min^s zeta_F(s) - 1) = abs(sum_(n in S_F, n > a_min) (n/a_min)^(-s)) <= sum_(n > a_min) (n/a_min)^(-sigma) = a_min^sigma zeta_F(sigma) - 1 < 1. The hypothesis sigma > alpha is load bearing and the test is one real evaluation carrying the ladder's own error bound. On the grid alpha + 0.05 n the edge sigma_1 reads 0.5 at base 4 {1} to 1.75 at the three full digit sets over twenty-four designs, with a_min^sigma zeta_F(sigma) in [1.8635, 1.9995] and largest sigma_1 - alpha equal to 0.95, so a census of the zeros right of alpha needs no hand chosen right edge and the alpha + 3.02 strip of the locus sweep is three times wider than the zeros need. Witness: lab/py/transport-census verb census.
2026-09-07 Verified The transport census: every proper design censused carries zeros right of its abscissa, the full digit set alone carries none, and each rightmost is certified by a winding box. On the box alpha + 1e-6 < Re s < sigma_1, where the cofactor Z = zeta_F(s)(1 - fill base^(-s)) is analytic and its zeros right of alpha are exactly those of zeta_F, the transfer failing only at residue null poles which sit on the line Re s = alpha, the argument principle counts 157 zeros right of alpha below Im s = 40 over twenty-three designs, all 157 located, plus 2 at base 50 missing one digit below Im s = 4. The count is exact on the box and a lower bound for the half plane, since the sliver alpha < Re s <= alpha + 1e-6, the band 0 < Im s < 0.02, everything above the census height and the conjugate half plane are uncounted. Twenty-one of the twenty-four designs carry such a zero; the three that do not are the base 2, 3 and 4 full digit sets, whose windings read -1.97e-33, 1.73e-33 and 1.53e-33. Every rightmost carries the height it is read below, because the teeth of the level zero comb drift right with the pole index: base 20 missing one digit reads 1.000285484146 at Im s = 2.0988, 1.000549674321 at 4.1971 and 1.002685494779 at 14.6920. Below Im s = 40 the rightmost real parts run 0.441505537191 at base 5 {0,1} to 1.002685494780 at base 20 missing one digit, each certified by a winding 1 box on zeta_F of half width 5e-5 in Re s and in Im s whose sampled contour minimum, 1.2e-4 to 6.1e-3, beats the engine's error bound by at least eight orders of magnitude and whose distance to the pole lattice s_(i,j) = alpha - i + 2 pi i j/log base is at least 0.00517845, four hundred box half widths. The two published boxes of lab/py/mrly-pairing reproduce at their own edges, winding 1 and 1 with contour minima 8.298e-4 and 6.865e-4, and its control rectangle returns winding 0. Witness: lab/py/transport-census verb census.
2026-09-07 Proved A certified zero right of the abscissa refutes every square-root-shaped bound for the design's own Mobius, and the digit 1 is the hypothesis that bites. Let 1 in F, let rho be a zero of zeta_F certified by a winding 1 box with left edge x_0 > alpha containing no pole, and let nu_F be the Dirichlet inverse of 1_(S_F). The transport theorem gives sigma_c(N_F) >= Re rho >= x_0 > alpha, so sum_(n <= x) nu_F(n) is not O(x^(x_0 - eps)) for any eps > 0; since A_F(x) has exponent alpha the square-root exponent is alpha/2 <= alpha < x_0, so the design's own Mobius satisfies no square-root-shaped bound and misses even the trivial O(x^(alpha - eps)), the first inequality failing to be strict only at the two designs with alpha = 0, where A_F(x) grows like log x. Nineteen of the twenty-four designs censused meet all three hypotheses and get a bound, and seventeen of the twenty-two the locus and family sweeps censused, the bounds running theta(nu_F) >= 0.4414555 at base 5 {0,1} to theta(nu_F) >= 1.0026354 at base 20 missing one digit. Four designs have Re rho > 1, so their own Mobius outruns the count of all integers below x: base 10 missing two digits, base 10 missing the digit 9, base 20 missing one digit and base 50 missing one digit, at fill/base = 0.8, 0.9, 0.95, 0.98 and alpha = 0.9030900, 0.9542425, 0.9828779, 0.9948357; only the SIGN of Re rho - 1 is read and never its size, three of the four being censused to Im s = 40 and base 50 to Im s = 4. The fill/base reading dies on its control, base 5 {0,1,2,3} at the same fill/base = 0.8 with rightmost 0.989748105861. Two designs carry a zero right of alpha and no bound: base 4 {2,3} and base 4 {0,2,3} omit the digit 1, so 1 is outside S_F, the indicator vanishes there and nu_F does not exist. Witness: lab/py/transport-census verb law.
2026-09-07 Refuted The gain of a design's rightmost zero over its abscissa is not a function of alpha and fill/base. The refuted functional is new: the locus row already refutes a law for the POSITION of the zeros, this refutes one for the single statistic the transport theorem reads, the rightmost real part less alpha. Four equal key families, one base and one digit count each so alpha and fill/base agree exactly and not to a rounding, read unequal gains: at alpha = 1/2, fill/base = 1/2 the four base 4 two digit designs give 0.0853043873, 0.4400124317, 0.2706238545, 0.3439264581, a spread of 0.35470804; base 5 at alpha = 0.4306766, fill/base = 0.4 spreads 0.37474232; base 3 two digit at alpha = 0.6309298 spreads 0.17605693; base 4 three digit at alpha = 0.7924813 spreads 0.060972003, which is still six hundred box widths. The two columns disagree in direction: the gain is largest at the sparsest designs, 0.5291214025 and 0.4485242462 at alpha = 0, while the rightmost real part itself is smallest there. What rises with alpha is the floor, the least rightmost real part at each alpha reading 0.4485242462, 0.4415055372, 0.5853043873, 0.7207876015, 0.9126562295, 0.9897481059, 1.0015143877, 1.0015892753, 1.0026854948, 1.0000614750 up ten rungs alpha = 0, 0.4307, 0.5, 0.6309, 0.7925, 0.8614, 0.9031, 0.9542, 0.9829, 0.9948, rising at every step but the first and the last, the last being where the census height drops from 40 to 4; one design per rung above alpha = 0.86 against six at alpha = 0.5, and no fit is taken. Witness: lab/py/transport-census verb law.
2026-09-11 Proved nu* vanishes at every polynomial divisible by x^2, and nu*(x b) = -nu*(b) at every b of nonzero constant term. The convolution runs over Z[x] divisors and M* is not divisor-closed, 1 + x^2 + x^4 = (1 + x + x^2)(1 - x + x^2), so the claim lives on A, the nonzero polynomials mod units, where nu* vanishes off M* by induction. A splits as N x R, R the classes of nonzero constant term and divisor-closed; x^k b lies in S* exactly when b lies in S*_odd, so 1_(S*) is the outer product of the all-ones function on N with 1_(S*_odd), inversion factors, and the inverse of the all-ones function on N is 1 - t. Witness: lab/rs/carry-free-mobius verb ladder.
2026-09-11 Proved Every element of M* of degree d has its coefficient of x^i at most binomial(d, i), so the maximum coefficient at degree d is exactly binomial(d, floor(d/2)), A001405, attained by (1+x)^d. A product of 0/1 polynomials of degrees summing to d is dominated coefficientwise by prod_k (1 + x + ... + x^(d_k)), each factor by (1 + x)^(d_k), and domination survives products of nonnegative polynomials, so the product is under (1 + x)^d, itself in M*. This retires the measured clause and the crude cap 2^(L-1) of the carry-bound row. Checked at every degree to 22, maximum 705432. Witness: lab/rs/carry-free-mobius verb ladder.
2026-09-11 Proved On R, the classes of nonzero constant term among the nonzero polynomials up to units, nu* is fixed by the reciprocal b -> x^(deg b) b(1/x). There the reciprocal is degree-preserving, multiplicative and involutive, hence a monoid automorphism, and it carries S*_odd onto itself by reversing the bitmask, so it preserves 1_(S*_odd) and its Dirichlet inverse. It is no invariance on all of Z[x]: the reciprocal drops the x power and nu*(x) = -1 against nu*(1) = 1. Checked with 0 mismatches over the 35121747 classes of degree 1 to 21. Witness: lab/rs/carry-free-mobius verb ladder.
2026-09-11 Proved The carry-free window of a two-digit design is (base+1)^(level-1) < base^level. A 0/1 polynomial of degree d has P(base) <= (1+base)^d and degrees add over a product, so the maximum of P(base) over M* at degree below level is exactly (base+1)^(level-1), attained by (1+x)^(level-1), and the least base holding every such element under base^level is the least base with (base+1)^(level-1) < base^level. Verified by enumeration at level = 3..14, reading 3, 3, 4, 4, 4, 5, 5, 6, 6, 6, 7, 7. The windows are 4, 7, 9, 12 and 15 at bases 3 to 7, and sum_(n <= base^level) nu_F(n) leaves -1 at level 5, 8 and 10, one level past the window each time. Witness: lab/rs/carry-free-mobius verb lemma.
2026-09-11 Conjecture The base-free Mertens maximum of the two-digit design gains on the design's mass without reaching it. The running maximum of sum nu* over degree below level reads 1, 1, 2, 3, 4, 7, 15, 23, 45, 86, 162, 331, 741, 1665, 3173, 7508, 17753, 36147, 79645, 182432, 427806, 858703, 2026147 at level = 1..23, and the ratio max/2^level bottoms at 0.079102 at level = 11, falls for the last time at level = 15, and rises at every step from there to 0.241536 at level = 23. A turn-down at a deeper level kills the trend and none is seen to 23. Witness: lab/rs/carry-free-mobius verb ladder.
2026-09-11 Conjecture The rate of that maximum exceeds the design's mass rate 2, and its estimate is window-unstable. At depth 23 the geometric mean step reads 2.245836, 2.202419, 2.242075, 2.206405 over the last 4, 6, 8, 10 levels but 2.194975, 2.149760, 2.092480, 2.074545, 1.996559 over the last 12 to 20, a hull of [1.996559, 2.245836] straddling 2, the long windows opening inside the levels where the ratio still fell. Every short-window reading at depths 20 to 23 sits above 2.18 and above its depth-18 value. No constant is claimed; log_2(max)/level reaches 0.910883 at level = 23 unsettled. Witness: lab/rs/carry-free-mobius verb ladder.
2026-09-19 Proved Weil's theorem reaches a design zeta along no evaluation bridge: P -> P(base) on polynomials with coefficients in {0..base-1} is a bijection onto the nonnegative integers and additive only where no carry occurs, a coefficient of the polynomial product P R reaching (base-1)^2 (min(deg P, deg R) + 1) against the digit cap base - 1, so an integer product's digit string is the carry reduction of the polynomial product and R_c R_(c+1) is the shortest coprime pair whose carry-free product shows the missing digit c. Witness: lab/py/mrly-euler verb wall.
2026-09-19 Proved Under the hypothesis alpha < 1 the large sieve does not rescue the per-denominator split on the major arcs: writing a = base^v a' with base not dividing a' and j = level - v, the l^2 mass of the grid pairing over the levels j <= J is exactly fill^(2 level - J) base^J, and the spacing form of the large sieve on those base^(-J) spaced points gives (base^level + base^J) base^level, so the levels below J = u level cost x^(alpha + u(1-alpha)/2), strictly above alpha at every u > 0 and equal to alpha only at u = 0. Witness: lab/py/mrly-pairing verb split, Montgomery and Vaughan 1973.
2026-09-21 Proved The local floor of the Nyman-Beurling distance: for any set S of positive integers, with d_N^2(S) = inf over real c of norm(chi - sum_(n in S, n <= N) c_n rho_n)^2 in L^2(0, infinity), rho_n(t) = {1/(nt)}, chi = 1_(0,1], and delta_K(S) the least value over real sigma and c_n (n in S, n < K) of sigma^2 + sum_(k=1)^(K-1) int_k^(k+1) (sigma u - 1 - sum_(n in S, n <= k) c_n floor(k/n))^2 du/u^2, one has d_N^2(S) >= delta_K(S) for every N and K, because at u = 1/t < K the approximant is sigma u - sum_(n in S, n < K) c_n floor(u/n) with sigma = sum c_n/n; and delta_K(S) = 0 iff every squarefree integer below K lies in S, since vanishing forces sigma = 0 and (c * 1)(k) = -[k = 1] on [1, K), hence c_k = -mu(k) there by Mobius inversion. The floor is nondecreasing in K and identically zero on the full set. Witness: zeta.md, THE CLOSURE PROBLEM; lab/py/nyman-beurling verb floor
2026-09-21 Proved The closure problem sees exactly the squarefree numbers: for a set S of positive integers, chi lies in the L^2(0, infinity) closure of the span of rho_n, n in S, iff S contains every squarefree integer and the Riemann hypothesis holds. If S misses a squarefree q, the floor delta_(q+1)(S) > 0 keeps chi out; if S holds every squarefree integer, the approximants sum_(a <= n) mu(a) a^(-eps) rho_a of Baez-Duarte 2003, pages 3 to 5, lie in the span over S and converge to -chi under the hypothesis; and chi in the closure over any S is chi in the closure of the natural Beurling functions, which is the hypothesis by the easy half of Nyman-Beurling. Every proper digit design misses a squarefree integer, S_F counting x^alpha elements with alpha < 1, so liminf d_N^2(F) >= delta_(q_0 + 1)(S_F) > 0 unconditionally, q_0 the least squarefree integer outside S_F: 1 when 1 is not a digit, 2 when 1 is a digit and 2 is not, 19 at base 10 missing 9, 38 at base 10 missing 8; Burnol's bound reaches d_N through d_N >= D(1/N), the integer truncation sitting inside his continuous family at lambda = 1/N. Witness: zeta.md, THE CLOSURE PROBLEM; Baez-Duarte 2003; lab/py/nyman-beurling verb floor
2026-09-21 Verified Vasyunin's Gram formula as printed by Bettin and Conrey 2013, <rho_h, rho_k> = (log 2 pi - gamma)/2 (1/h + 1/k) + ((k - h)/(2hk)) log(h/k) - (pi/(2hk)) (V(h/k) + V(k/h)) with V(h/k) = sum_(m=1)^(k-1) {mh/k} cot(pi m/k) for coprime h, k, and <rho_(gh), rho_(gk)> = <rho_h, rho_k>/g, agrees with the exact quadrature (1/(hk)) int_0^1 {hw}{kw} zeta(2, w) dw to 1e-57 at eleven pairs from (1,1) to (40,81) and (6,10); with the target <chi, rho_n> = (log n + 1 - gamma)/n, the distance 1 - b^T G^(-1) b agrees with the quadrature of the residual itself at S = {1,2,3}, {1,3}, {1,2,3,4}, {1,3,4}, reading 0.0938372, 0.2328324, 0.0651289, 0.1256225 on both paths to forty digits. Witness: lab/py/nyman-beurling verb gram
2026-09-21 Verified The Nyman-Beurling table at N = 3^2..3^6: distances rounded up, on the full set d_N^2 = 0.024015, 0.015314, 0.010949, 0.008254, 0.006738 against Burnol's C/log N = 0.021023, 0.014015, 0.010511, 0.008409, 0.007008 with C = 2 + gamma - log(4 pi) = 0.046191, and d_N^2 log N = 0.052766, 0.050472, 0.048113, 0.045335, 0.044412 falls through C between N = 81 and N = 243, sitting below the constant at N = 243 and N = 729; Burnol's Theorem 1.3 says liminf d_N^2 log N >= C and nothing more, and no upper bound on d_N is known, d_N -> 0 being the hypothesis; Remark 1.1 of his estimate paper writes C/sqrt(log N) on d_N with C >= 2 + gamma - log(4 pi), which squares consistently only with C >= sqrt(2 + gamma - log 4 pi), and the squared reading is the one printed here; on base 3 {0,1} with 4, 8, 16, 32, 64 elements d_N^2(F) = 0.116043, 0.106008, 0.104951, 0.104689, 0.104610, and 0.232833 at N = 3, 0.104579 at N = 3^7 with 128 elements. The 1-norm condition numbers read 6.6e2 to 2.3e7 on the full set and 1.7e1 to 4.1e6 on the design; 60 and 90 digits agree on every printed digit. The L^2(0,1) variant reads 0.023064, 0.014950, 0.010763, 0.008148, 0.006668 on the full set and 0.158347, 0.097858, 0.091353, 0.090586, 0.090398, 0.090341, 0.090319 on the design at N = 3 to 3^7. Witness: lab/py/nyman-beurling verb table
2026-09-21 Verified The floor at base 3 {0,1} reads 0.070873 at K = 3 and 0.098689 at K = 40, largest steps at K = 3, 6, 8, 12, one past the missing squarefree 2, 5, 7, 11, and 0.100877 at K = 60, 0.101926 at K = 100, so lim d_N^2(F) lies in [0.101926, 0.104579]; the L^2(0,1) floor reads 0.085724 at K = 40 against 0.090319; the full set returns abs(delta_K) < 1e-75 at K = 2..12; at K = 40 the floors read 0.304563 at base 3 {0,2}, 0.217197 at base 5 {0,1}, 0.164256 at base 4 {0,1}, 0.064881 at base 3 {1,2}, 0.010772 at base 4 {1,2,3}, 0.002263 at base 10 missing 9, 0.000287 at base 10 missing 8. Witness: lab/py/nyman-beurling verb floor
2026-09-21 Verified Burnol's constant from the zeros: 2 sum_(0 < gamma < 1000) 1/(1/4 + gamma^2) over 649 zeros plus the smooth tail 2 (log(T/2 pi) + 1)/(2 pi T) reads 0.0461922 against 2 + gamma - log(4 pi) = 0.0461914, and 0.04619150 against 0.04619142 at T = 3000 over 2469 zeros. Witness: lab/py/nyman-beurling verb zeros
2026-09-21 Refuted A proper digit design does not carry the Nyman-Beurling criterion: d_N^2(F) does not fall like 1/log N beside the full set, since the floor delta_(q_0+1)(S_F) > 0 holds it above a constant at every N; the floor is zero on the squarefree numbers at every K, so it carries nothing about zeta, and a design is refused at u < q_0 + 1, before the tail u -> infinity where the hypothesis lives. Witness: zeta.md, THE CLOSURE PROBLEM; lab/py/nyman-beurling verbs table and floor