rho-decoupling

• Computes N_F(level; d), the number of length-level digit strings over a digit set F in base base whose value sum f_j base^j is divisible by d, exactly, by dynamic programming on residues mod d with u128 counts; the value map is injective on fixed-length strings, so this counts integers below base^level with padded digits in F.
• For each family and depth the sweep over divisors 2 <= d <= D with (d, base) = 1 prints the worst normalized error d |N_F(level;d) - fill^level/d| / fill^level among d coprime to the digit-difference gcd, the multiplicative order of base at the worst d, the per-digit rate worst^(1/level), the per-factor ceiling gamma = max_a |sum_{f in F} e(a f / d)| / fill at that d, and the slack against the proved bound d (1 - 8/(fill^2 d^2))^level.
• The proved bound is asserted exactly at every census cell where its hypotheses hold: d |N - fill^level/d| * (fill^2 d^2)^level <= d * fill^level * (fill^2 d^2 - 8)^level in big integers, no floats in the claim path.
• Families whose digits share a factor also print the unrestricted worst error, the wall where the count does not equidistribute.
• The Type I line accumulates sum mu(d) (N_F(level;d) - fill^level/d) and its absolute-value trivial bound over squarefree d <= D coprime to base as exact fractions, printing both relative to fill^level and their ratio.
• The decouple line probes the fixed divisor d = 7 at the deepest level of each family: the per-digit error rate against the per-factor ceiling, across bases 3, 4, 5, 10, 100, which is where the large-base decoupling is visible.
• Four pinned probes read single divisors of base^t - 1 at base = 100 (d = 101, 9999, 3367, 999999, the first inside the sweep range, the rest beyond it): each line prints the rate (|N - fill^level/d|/fill^level)^(1/level) and the orbit-mean damping factor mean = rate * d^(1/level), reading 0.106 at d = base^2 - 1 against fill^(-1/2) = 0.1005 and 0.237 at d = base^3 - 1 against fill^(-1/3) = 0.216.
• The carry DP counts N_F(level; e) for e | base^t - 1 without a state per residue: the value of a length-level string is congruent mod base^t - 1 to V = sum_{i<t} c_i base^i, where c_i sums the digits in the positions congruent to i mod t, so e | value exactly when V = j e for one of the (s+1) g + 1 targets with s = floor(level/t) and g = (base^t - 1)/e, and each target is counted by a carry DP across the t base-base columns of V, the column holding s_i positions weighted by the coefficients of P_F(z)^{s_i} for P_F(z) = sum_{f in F} z^f, with carries bounded by s+1.
• The cost is O(t s^3 g) per (level, t, g), polynomial in level at every t; the target is carried as a base-base digit vector and never as an integer, so t = 40 at base = 10 runs although base^t - 1 exceeds u128, and the counts themselves stay in u128 for fill^level <= 9^40.
• The carry sweep runs base = 3, F = {0,1} and base = 10, F = {0..9} minus {7} over every modulus e = (base^t - 1)/g with g | base - 1, e >= 2 and t <= level <= 40, and prints per depth the exact ratio Sigma_level/Abs_level of the signed sum Sigma_level = sum mu(e) T_level(e) to the absolute sum Abs_level = sum |T_level(e)| over the squarefree moduli of that set, together with Abs_level/fill^level, the count of terms within a factor of ten of the largest, the four largest terms at level = 40 as exact integers, and the ratio at every level = 3..40.
• Here T_level(e) = N_F(level; e) - fill^level/e is an exact rational and e T_level(e) = e N_F(level; e) - fill^level an exact integer; the sums are formed over a common denominator, so only the printed ratios are floats.
• mu(e) is read off a complete factorisation, never a table: base^t - 1 = prod_{d | t} Phi_d(base) splits first, then trial division to 10^5, Pollard-Brent, and Miller-Rabin on the thirteen bases 2 to 41, deterministic below 3.317 * 10^24; a cofactor called prime above that bound is not certified, so its modulus leaves the sums and counts instead in the unknown column of the depth row, and at t <= 40 both bases certify every prime factor and every unknown count reads zero.
• The riesz module computes the even moments S_p(level) = sum_{a mod base^level} |hat F_level(a/base^level)|^p of the digit transform hat F_level(y) = sum_{n in D_level} e(ny) on the value set D_level of the length-level strings, exactly through the identity S_p(level) = base^level E_r(level), p = 2r, where E_r(level) counts the 2r-tuples of length-level strings with n_1 + .. + n_r = n_{r+1} + .. + n_{2r} mod base^level, together with the integer count E_r^int(level) (equality over the integers, which is int_0^1 |hat F_level|^p); both come from one carry DP whose state is the pair of carries of the two sides, reduced by the swap symmetry to r(r+1)/2 states, so each sequence is C-finite of that order and Lambda(p) = lim S_p(level)^{1/level} is base times the Perron root of the transfer matrix.
• For each family it prints the characteristic polynomial of that matrix in exact integers, its factorisation over Z (integer roots by a scan bounded by the maximal row sum, monic quadratic factors by a bounded search over the divisors of the constant term, a cofactor of degree at least four certified irreducible modulo a printed prime by the distinct-degree test), an exact two-sided bracket on the Perron root from the Collatz-Wielandt bounds at a positive rational vector on the states reachable from the zero carries, the factor holding the Perron root located by a sign change on the bracket with every other factor certified nonzero there, then Lambda(p), its exponent theta_p = log Lambda(p)/log base, the floors max(fill^p, base fill^{p/2}) and the ceiling base fill^{p-1} asserted at every depth to level = 60, the recurrence asserted at every depth, the ratio E_r/E_r^int, the share of the zero frequency in S_p, and the growth of the primitive part S_p(level) - fill^p S_p(level-1).
• The type II rows print per family the exponent theta_4/4 + 1/4 that Holder with the fourth moment and Parseval on the bilinear side yields for a Type II sum over the digit set, its miss against alpha = log fill/log base, the bilinear saving 1 - alpha/2 such a route would need on the minor arcs, and the large-values threshold eta_4 = (1 + 3 alpha - theta_4)/2 below which the fourth moment counts the frequencies near the maximum better than Parseval does; the arcs rows evaluate |hat F_level(a/base^level)|^2 at every grid point at one or two depths per family, mark the points within 1/(d x^{3/5}) of a fraction with denominator d <= x^{2/5}, and print the share of the Parseval mass off those points against (fill/base)^level, the share below which a moment route could still work.
• The riesz families are the census families with every reflection class of one excluded digit at base = 10; the higher moments run p = 6, 8, 10 at base = 3, F = {0,1} and p = 6 at four more families.
• The large module reads the same grid as a complex table, hat F_level(a/base^level) at every a for base^level <= 10^6, asserts Parseval and the exact S_4 on it to 10^{-10} relative, and for six thresholds eta per cell (0, eta_4/2, eta_4, 2 eta_4, 0.25, 0.5) forms the large set A_eta = {a : |hat F_level(a/base^level)| >= fill^level x^{-eta}}: it prints its size against the fourth-moment count S_4 fill^{-4 level} x^{4 eta} and the Parseval count base^level fill^{-level} x^{2 eta}, both asserted above the size, the least gap between two of its points in grid units, the number of maximal runs of consecutive grid points and the longest run, the floor sum_{a in A} |hat F_level|^2/fill^level of the set's large-sieve constant for polynomials supported on D_level, and that constant itself as the top eigenvalue of the Gram matrix hat F_level((a - a')/base^level) on the set, bracketed by the Rayleigh quotient of a power iteration below and by the least of the Gershgorin row bound, the Frobenius norm and base^level above, while the set has at most 1500 points, capped by fill^{level-j} base^j when the set lies on the level-j grid; every printed float is a two-sided band or rounded safe, bounds up and floors down.
• Beside the set it prints the inequality chain of a large-values Type II with bounded coefficients: the small set's exponent min(alpha + 1/2 - eta, (1 + alpha)/2) from the threshold and Parseval on the bilinear side, the large set's exponent from the sup and the count, alpha + log(#A)/(2 log x), from the set's own mass, log(sum_{a in A} |hat F_level|^2)/(2 log x), and from the Parseval cap (1 + alpha)/2, all against Parseval on the bilinear side (the large-sieve constant of any grid subset for base^level consecutive frequencies is base^level exactly), the sup form with the fourth-moment count in place of the size, the count-based c sup = alpha - max(small, large sup) and the chain c = alpha - max(small, min of the three), together with the l^1 exponent c_1 = log(sum_a |hat F_level(a/base^level)|)/log x at the depth.
• A per-cell row prints nu_4(level), eta_4, c_1, the second-largest grid value over fill^level, which equals max_{b != 0 mod base} |g_F(b/base)|/fill by the last non-integer factor of the product formula, and the threshold eta_1(level) below which the large set is the zero frequency alone.
• A witness row per cell counts, on the box m, l in [M, 2M) with M = floor(x^{1/2}/2), the representations R = #{(m, l) : ml in D_level}, which is the Type II sum at a_m = b_l = 1, and the balanced sum at a_m = 1_{base | m}, b_l = 1, R_base - fill^level base^{-level} M #{m : base | m}, each against fill^level and as an exponent of x.
• The menergy module computes the multiplicative energy E_x(level) = #{(n_1, n_2, n_3, n_4) : n_i in D_level, n_i != 0, n_1 n_2 = n_3 n_4} of the same digit column exactly, by forming the K^2 products of the column with itself in u128, hashing them into as many partitions as the memory cap allows, sorting each partition and summing the squares of the run lengths, which is sum_m r(m)^2 for r(m) the number of ordered factorisations of m inside the column; it prints K, E_x(level), the exponent log E_x/(level log base) against 2 alpha, the ratio of E_x to the exact diagonal count 2K^2 - K, the largest r(m), the share of the excess over the diagonal that the shift solutions explain, and the same ratio and the direct quotient for one uniform random subset of [1, base^level) of the same size drawn by the module's own generator.
• Two bounds are asserted at every cell in exact integers: the band 2K^2 - K <= E_x(level) <= K^3 (the diagonal and the swap agree on the K quadruples with n_1 = n_2; fixing three entries determines the fourth) and E_x(level) <= K^2 max_m r(m), which with r(m) <= d(m) pins the exponent at 2 alpha for every digit set.
• When 0 is a digit the module also asserts a third floor E_x(level) >= 2K^2 - K + T(level), where T(level) counts the shift solutions (base^i u, base^j v, base^{i'} u, base^{j'} v) with i + j = i' + j', i != i', u != v and u, v of nonzero last digit, a closed form in fill and level alone; the printed share (2K^2 - K + T)/E_x says how much of the energy the diagonal and the shifts explain.
• The type II rows sweep the dyadic boxes m in [M, 2M), l in [N, 2N) with MN in [x/32, x/4] and M, N >= 8, and for each box count R = #{(m, l) : ml in D_level}, the restricted energy E_x(M, N) = sum_m (#{l : ml in D_level})^2, and the balanced energy E_x^bal(M, N) = sum_{l_1, l_2} |W(l_1, l_2)| for W(l_1, l_2) = sum_m psi(m l_1) psi(m l_2), psi = 1_{D_level} - K/base^level with K the column size, whose Cauchy-Schwarz bound (M E_x^bal)^{1/2} is what such a route returns for a bilinear sum with bounded coefficients; the pairs with W off its baseline are counted by sorted keys, the rest by the histogram of the column counts, and the identity sum_{l_1, l_2} W = sum_m (c_m - N fill^level/base^level)^2, derived independently, is printed as a self-check.
• Each cell prints the extreme boxes by the bound exponent among the boxes carrying a representation, the worst box inside the arc regime M, N >= x^{2/5}, the ratio of the bound to that box's trivial bound, and the margin alpha - log bound/log x, for the digit column and for a random column of the same density side by side.
• The sigma rows draw random sign vectors for both coefficient sequences on a fixed box and assert the Cauchy-Schwarz bound at every draw, printing the largest ratio of the measured |Sigma| to the bound; the identity sum_{l_1, l_2} W = sum_m (c_m - N delta)^2 printed beside them is an algebraic rearrangement of the sweep's own quantities and pins summation order, not correctness.
• The signed module computes the off-diagonal correlation Sigma_b = sum_{l_1 != l_2} b_{l_1} b_{l_2} W(l_1, l_2) of a coefficient vector b, the one part of the balanced energy that keeps the coefficients' signs, for nine coefficient vectors at once: the constant 1, the Mobius function, the Liouville function, three seeded random sign vectors, and three seeded random sign vectors supported on the squarefree integers only, which is the Mobius support with the Mobius arithmetic removed.
• The identity it evaluates is Sigma_b = sum_m V_m(b)^2 - sum_l b_l^2 W(l, l) for V_m(b) = sum_l b_l psi(ml), the full quadratic form being a sum of squares, and it is carried as the exact integer x^2 Sigma_b = x^2 (Q - P) - 2 K x (B C - P) + M K^2 (B^2 - S_2) in i128, where Q = sum_m u_m^2, u_m = sum_{l, ml in D_level} b_l, B = sum_l b_l, C = sum_l b_l c'_l, P = sum_l b_l^2 c'_l and S_2 = sum_l b_l^2; the printed Sigma_b is that numerator over x^2 rounded to the nearest integer and every ratio and exponent comes from the numerator itself.
• Mobius and Liouville come from one smallest-prime-factor sieve to twice the largest box side; the random vectors come from the module's own generator at fixed seeds, so the same vector is used at every family, depth and box.
• The signed sweep runs the same dyadic band MN in [x/32, x/4] at two depths for each of four families, selecting the box that maximises |Sigma_1| over the band and the box that maximises it inside the arc regime M, N >= x^{2/5}, and repeats the deeper depth on a random column of the same size, each on its own selected box; each row prints Sigma_1, Sigma_mu, Sigma_lambda, their exponents log |Sigma_b| / log x, the ratios to Sigma_1, the min-max band of each triple of random draws, and the ratios to the root mean square a random sign vector reaches.
• That root mean square, (2 sum_{l_1 != l_2} W(l_1, l_2)^2)^{1/2}, is the yardstick a signed sum is read against, and it is computed as sum_{l_1, l_2} W^2 = sum_{m, m'} (sum_l psi(ml) psi(m'l))^2 over the pairs of the shorter side, the only float in the module and never in a claim path.
• Each row also prints the min-max, over the eight signed coefficient vectors, of the full quadratic form divided by its own diagonal, and the ratio of the best possible bounded-coefficient sum sum_m |V_m(b)| to the Cauchy-Schwarz bound (M sum_m V_m(b)^2)^{1/2}, which says how much the inequality itself loses.
• A separate section drives the quadratic form down by greedy single-sign flips from the all-ones vector on a +-1 vector supported on the whole side, printing the flips used, the form, its diagonal, the resulting bound and the diagonal floor (1 - delta)(M R)^{1/2}; the engineered vector is built with full knowledge of the column, is available to no decomposition, and exists to show that the bound has no floor over all bounded coefficients.
• A summary row counts, over the digit cells, how often the Mobius value falls below, inside and above each band, how often it is smaller in absolute value than the Liouville value, the extreme ratios to the root mean square for the Mobius vector and for the support-matched draws, the extreme values of the form over its diagonal, the extreme Cauchy-Schwarz losses, and how often the signed sums fall below the box representation count R while the unsigned one stays above it.
• Two checks hold the signed rows: at b = 1 the quantities Q and C are asserted equal to the E_x(M, N) and R of the box sweep, computed along a different data path, and the tests re-evaluate the definition pair by pair over l_1 != l_2 with W by a direct triple loop in the same exact integers.
• The signed families are F = {0,1} at base = 3 and depths 12, 14, {0,1,2} at base = 4 and depths 8, 9, {0,1,2,3} at base = 5 and depths 7, 8, and one excluded digit at base = 10 and depths 5, 6.
• The vaughan module censuses the real coefficient sequences a Vaughan decomposition hands the bilinear sum, on the same boxes the signed module selects: b^U_l = sum_{d | l, d <= U} mu(d), the l-side coefficient of the Type II term of Vaughan's identity for both mu and the von Mangoldt function, and the log-weighted variant sum_{d | l, d <= U} mu(d) log(l/d), each at the two cuts U = floor(x^(1/3)) and U = floor(x^(2/5)), together with the sign vector of the second.
• Each sequence is built by adding mu(d) and mu(d) log d along the multiples of every squarefree d <= U, so the two pieces cost one pass per divisor, and the tests check both against a direct loop over the divisors of every l to 1200 at three cuts.
• The Type II region of the identity is m > V and l > U, so the boxes that are Type II cells are those with both sides above the cut; the summary counts them and the wide boxes are printed beside them as the same sequence read where the identity does not put it.
• For each box it prints the two cuts, the ratio 2N/U that bounds the box maximum of the coefficient, the share of the box where the coefficient is nonzero, the box maximum, the full quadratic form over its own diagonal for all five sequences, the diagonal share w = diag/((1 - delta)^2 R) of the normalised coefficient, and its Cauchy-Schwarz bound over the diagonal floor both normalised by the box maximum and unnormalised.
• The split bound/floor = (form/diag * w)^(1/2) is an identity in the definitions and is asserted at every box, so a small bound is attributed to cancellation, to a sparse or small coefficient, or to a large box maximum, and never to all three at once.
• The summary row prints the extremes of every column over the sixteen boxes and the eighty coefficient cells, how many sit inside the band the signed module's summary prints, how many fall below one tenth, how many boxes carry a coefficient with box maximum one, and the closest any cell comes to either end of that band.
• The menergy census runs the census families to the depth the K^2 product count allows, level = 12 at the two-digit sets of base = 3, level = 8 at base = 4, level = 6 and level = 7 at base = 5, level = 4 and level = 3 at one excluded digit of base = 10, level = 9 for the pair and level = 2 for the lower half at base = 100, and the box sweep one to three levels deeper, to level = 14 at F = {0,1}, base = 3, where only membership in the column is needed.
• Census families: the three two-digit sets at base = 3; {0,1,2} at base = 4; {0,1,2,3} and {0,2,4} at base = 5; one excluded digit at base = 10 and base = 100; the lower half {0..49} and the pair {0,1} at base = 100. Depths to level = 96, divisor ranges to D = 500; floats appear only in printed readouts, truncated at forty decimal digits.

RUN

• bash scripts/cargo.sh cargo run --release -p rho-decoupling
• Under twenty seconds for the whole run, census, carry sweep, riesz moments, large values, multiplicative energy and signed correlation together; prints only, writes nothing.
• cargo test -p rho-decoupling runs thirty-nine tests: the residue DP against brute-force string enumeration at four bases, the residue vector total against fill^level, the exact splitting of N_F(level; d1 d2) across d1 | base, (d2, base) = 1 at base = 6, the Mobius sieve against hand values and M(100) = 1, and the exact bound inequality at base = 3, F = {0,1}, level = 16, d <= 60.
• The riesz tests pin the moment DP against a sumset histogram at p = 4 on every family and at p = 6, 8 on four, the exact S_4 against direct evaluation of the product formula on the grid and E_2^int against the 2 base^level-point quadrature at level <= 6, the factoriser and the irreducibility test on known polynomials, the scaling identities of {0,2} and {1,2} against {0,1} at base = 3, every fourth-moment bracket against the growth ratio at level = 40, the rendered algebra, value and type II rows of seven families, the minimal polynomials of seven higher moments, and the arcs counts at base = 3, level = 12.
• The large tests pin the complex grid against a direct sum over every string at four families, the large set at five thresholds against the same direct values, the Gram eigenvalue at {0} (fill^level), at the base^2-grid (fill^{level-2} base^2, the witness set of the sparse large sieve) and at the full grid (base^level), the run statistics on hand sets, and the witness counts against an enumeration of the column.
• The menergy tests pin the partitioned energy against a four-fold brute-force enumeration of the solutions at six families, against itself at three partition caps, the band and the divisor ceiling at four families, the scaling identity E_x(cF) = E_x(F) at {0,2} against {0,1} and {0,2,4} against {0,1,2} together with the failure of translation invariance at {1,2}, the restricted and balanced energies of a box against a direct triple loop over the box, the shift count against an enumeration of the shift solutions themselves, and the Cauchy-Schwarz bound against sixty random sign draws on three boxes.
• The signed tests pin the Mobius and Liouville sieves against their first twelve values, the exact identity against a pair-by-pair evaluation of the definition at four families and all nine coefficient vectors, the unsigned quantities against the box sweep at three families, and the collapse of the unsigned correlation on a random column of the same size.
• The vaughan tests pin both pieces against a direct divisor sum at three cuts, the form and the diagonal against a pair-by-pair evaluation of the definition with the kernel by a direct triple loop, and the form of the integer piece against the exact integer sum_m (x u_m - K B)^2 over the box.
• The carry tests pin the carry DP against brute-force enumeration at base = 3, 5, 10 over every t <= level and every g | base - 1, against the residue DP at every reachable e <= 30000 at level = 8 and level = 12 in both carry families with zero mismatches, the certification of every cofactor of base^t - 1 at t <= 40 in both bases, and as rendered strings the depth rows, the four largest terms at level = 40, the ratio sequence level = 3..40, the mu tables and seven factorisations; one more test pins the digit-vector helpers.

WITNESSES

• mobius.md digit strings across divisors, the uniform equidistribution bound |N_F(level;d) - fill^level/d| <= fill^level (1 - 8/(fill^2 d^2))^level for (d, base) = 1 with d coprime to every digit difference: asserted exactly at every census cell.
• mobius.md digit strings across divisors, the worst divisors are the pinned ones: at every family's deepest level the sweep argmax d has ord_d(base) <= 8, so d | base^t - 1 with t <= 8, e.g. d = 164 at base = 3, d = 143 at base = 10, d = 101, 303 at base = 100; shallow levels can stray (d = 199, ord = 99, at base = 10, level = 6).
• mobius.md digit strings across divisors, the decoupling: at d = 7 the per-digit error rate falls 0.49, 0.33, 0.25, 0.11, 0.017 along base = 3, 4, 5, 10, 100 with one digit excluded, but stays 0.50 for F = {0,1} at base = 100: the gain is carried by the fill, not the base alone.
• mobius.md digit strings across divisors, a signed pinned sum against its absolute sum: over the moduli e = (base^t - 1)/g with g | base - 1, e >= 2, t <= level, squarefree, the ratio Sigma_level/Abs_level of the Mobius-signed sum of T_level(e) to the sum of |T_level(e)| is verified at -0.211, -0.123, +0.069, -0.498 for level = 10, 20, 30, 40 at base = 3, F = {0,1} and at +0.812, -0.495, -0.127, -0.192 for base = 10 with one digit excluded, while Abs_level/fill^level falls from 4.7e-2 to 2.1e-4 and from 1.1e-5 to 3.9e-12.
• mobius.md digit strings across divisors, no cancellation trend to depth 40: the ratio is printed at every level = 3..40 on both families and shows no decay toward zero with depth, swinging across the band and reading -0.50 at level = 40 in base three and -0.99 at level = 29 in base ten, so the sign of mu(e) buys no extra saving over the trivial absolute bound on this modulus set at these depths.
• mobius.md digit strings across divisors, few terms carry the sum: at level = 40 the terms within a factor of ten of the largest count 6 at base = 3 out of 29 and 4 at base = 10 out of 60, the four largest sitting at t = 7, 9 in base three and t = 5, 7, 8, 10 in base ten, each printed as the exact integer e T_level(e).
• mobius.md the meter and its yardstick, the fourth moment: sum_{a mod base^level} |hat F_level(a/base^level)|^4 = base^level E(level) with E(level) the additive energy of the strings modulo base^level, counted by the carry DP; Lambda(4) = 18 at base = 3 for all three two-digit sets, 2(23 + sqrt 353) at base = 4, F = {0,1,2}, (275 + 5 sqrt 2369)/2 at base = 5, F = {0,1,2,3}, 95 at F = {0,2,4}, five quadratic irrationals across the excluded digit at base = 10, from 6566.412229970 (digit 5 excluded) to 6567.410368845 (digit 0 or 9 excluded), 8335000 for the lower half and 600 for {0,1} at base = 100, each inside [max(fill^4, base fill^2), base fill^3] and strictly above fill^4.
• mobius.md the meter and its yardstick, the large-values route to Type II: splitting the grid at |hat F_level| >= fill^level x^{-eta}, bounding the large set by the fourth-moment count and Parseval and the rest by the threshold and Parseval gives the exponent max(min(alpha + 1/2 - eta, (1 + alpha)/2), min((1 + alpha)/2, log(sum_{a in A_eta} |hat F_level|^2)/(2 log x))), read at 66 cells over six families and equal to (1 + alpha)/2, the l^2 route, at every one of them; at {0,1} base 3, level = 12, eta = eta_4, the large set holds 407 adjacent-or-isolated grid points in 331 runs against the fourth-moment count 4096, and its large-sieve constant 1.06009e5..2.13280e5 sits nearly at its own l^2 floor 1.05611e5.
• mobius.md the meter and its yardstick, the witnesses: at a_m = b_l = 1 the Type II sum on the box M = N = floor(x^{1/2}/2) is the representation count R, which reads 0.38 fill^level at {0,1} base 3 level = 12 and 0.26 fill^level at one excluded digit of base 10 level = 6, and the balanced sum at a_m = 1_{base | m}, b_l = 1 reads 0.10 fill^level and 0.0044 fill^level there, so no bound x^{alpha - c} uniform over bounded coefficients holds for either sum at the nine dense cells; the two sparse cells are void at that box.
• mobius.md the meter and its yardstick, the moment route to Type II: with Parseval on the bilinear side the Holder exponent theta_4/4 + 1/4 misses alpha by at least 0.250000 (one digit excluded at base = 100) and by 0.446753 at base = 100, F = {0,1}, and any pointwise bilinear input would have to beat x^{1 - alpha/2} on the minor arcs, below the root mean square of the bilinear sum, at every family.
• mobius.md the meter and its yardstick, the multiplicative energy of a digit column has no exponent of its own: 2K^2 - K <= E_x(level) <= K^2 max_m r(m) with r(m) <= d(m) forces E_x(level) = fill^{2 level} x^{o(1)} for every digit set, and the census reads the exponent falling 1.475642, 1.410978, 1.356938 at level = 4, 8, 12 for F = {0,1} at base = 3 toward 2 alpha = 1.261860, with max_m r(m) = 56 at level = 12.
• mobius.md the meter and its yardstick, the excess over the diagonal is shift structure: E_x/(2K^2 - K) reads 1.7523 for {0,1} at base = 3 and level = 12 against 1.0004 for a random column of the same size, and the closed-form shift count T(level) = 11144220 brings the explained share to 0.7603; at {1,2}, where no digit is zero and T = 0, the ratio is 1.0006 and the share 0.9994.
• mobius.md the meter and its yardstick, the type II route through the multiplicative energy: |Sigma|^2 <= M E_x(M, N) and E_x(M, N) <= 2N E_x(level)^{1/2} x^{o(1)} give |Sigma| <= (2MN)^{1/2} x^{alpha/2 + o(1)}, above the trivial x^{alpha} by (1 - alpha)/2 for every digit set with alpha < 1, the same miss as square-root cancellation on the fourth-moment route.
• mobius.md the meter and its yardstick, the Mobius signs cancel the correlation no better than unstructured signs on the same support: over sixteen boxes across four families at two depths each, |Sigma_mu| divided by the random-sign root mean square stays in [0.0913, 0.7178] while the three support-matched draws stay in [0.0359, 1.5048], and both signed sums fall below the box representation count R at all sixteen while the unsigned Sigma_1 stays above it at all sixteen.
• mobius.md the meter and its yardstick, the apparent advantage of the Mobius function is its support: against three plain random sign vectors it sits below the band at eight cells, inside at seven and above at one, but against three random sign vectors on the squarefree integers it sits below at three, inside at nine and above at four, chi-square 0.375 against the uniform-rank null 4, 8, 4; |Sigma_mu| < |Sigma_lambda| at fifteen of sixteen cells and the support-matched control removes the whole gap. Sixteen cells against three draws exclude a large effect and nothing smaller: a true below-rate of 0.41 still gives three or fewer at probability 0.0556.
• mobius.md the meter and its yardstick, the bound has no floor over all bounded coefficients: a +-1 vector built by greedy single-sign flips against a known column drives the full quadratic form to 0.0006 of its own diagonal at F = {0,1,2,3}, base = 5, level = 7, M = 8, N = 2048, and the Cauchy-Schwarz bound to 0.0265 of the diagonal floor (1 - delta)(M R)^{1/2}, reaching 0.4909 of that floor inside the arc regime at M = N = 128; the diagonal is sign-blind but the off-diagonal can cancel it, so no floor statement holds beyond the coefficient vectors actually swept.
• mobius.md the meter and its yardstick, the off-diagonal correlation is a pure structure statistic: its expectation over a random column of the same density is zero at every coefficient vector, and the measured Sigma_1 falls from 67382 to 349 at F = {0,1}, base = 3, level = 14, from 41590 to 122 at base = 4, from 72359 to -27 at base = 5 and from 823318 to -2746 at one excluded digit of base = 10.
• mobius.md the meter and its yardstick, the balanced route returns the trivial box bound: inside the arc regime M, N >= x^{2/5} the margin alpha - log bound/log x for {0,1} at base = 3 falls 0.035675 at level = 12 to 0.027009 at level = 14 while the ratio of the bound to the box's own trivial bound rises 1.0134 to 1.0730, and over all admissible boxes the margin is already negative at level = 12; the random column of the same density beats the digit column at every cell.

• mobius.md the meter and its yardstick, the coefficient a real decomposition hands the method is not the coefficient the method would need: at the eight swept boxes with both sides above x^(2/5), the boxes the identity produces and where the box sits within a factor 8.73 of the cut, the Type II coefficient of Vaughan's identity takes values in {-1, 0, 1} at seven of the eight and its quadratic form stays between 0.6929 and 1.2045 of its own diagonal, where a sign vector engineered against the column reads 0.2043 on such a box, and no one of the eighty coefficient cells of the census falls below 0.3138.
• mobius.md digit strings across divisors, the fixed-fill wall: for F = {0,1} at base = 100 the worst normalized error over d <= 500 still reads 7.2 at level = 96, decaying by factor 0.9929 per digit, pinned at d = 303 | base^2 - 1.