The stack algebra

• 2026-09-09 [Proved] Stacking is Dirichlet convolution: the u-stack of the v-stack draws the inner scale n at scale kn with weight u(k) v(n), so the composite weight is u * v, exact at every scale m <= N under the hyperbolic cut kn <= N and only for m <= min(K, M) under a rectangular cut (10 of 16 scales differ above 24 in the 24 x 40 case at u = v = mu). The plain stack is zeta, the stack of stacks is zeta^2 with scale n drawn d(n) times, and 1 * mu = e collapses the Mobius stack of the plain stack to one layer. Checked by literal double stacking in exact rationals at N = 60 by three routes sharing no inner loop on u = v = 1, u = 1, v = mu, u = v = mu and u = 1, v = n^-1, 1102, 1102, 974 and 1102 nodes, zero mismatches. The group carries no RH content: RH sits at the inverse of 1 alone, whose b = 1 node is M(N). Witness: lab/py/stack-algebra convolution_check.
• 2026-09-09 [Proved] Every selected line stack has a closed-form node at denominator b: evens floor(N/lcm(2,b)), odds 0 at even b and ceil(floor(N/b)/2) at odd b, primes pi(N) at b = 1, 1 at prime b <= N and 0 elsewhere, squarefree the double divisor sum sum_{d^2 <= N/b, gcd(d,b) = 1} mu(d) sum_{e | b} mu(e) floor(N/(b d^2 e)), prime powers floor(log_p N) - i + 1 at b = p^i; zero mismatches against literal stacking at every b <= N for N = 30, 61, 200, 501; the primes-only stack at N = 501 lights b = 1 and the 95 primes, every prime node at brightness exactly 1. Witness: lab/py/stack-algebra selection_closed_forms.
• 2026-09-09 [Proved] The primes-only carpet stack fades at exactly the independent rate: distinct primes are coprime, so every layer pair has covariance exactly 0 by the gcd law, L Var of the L-layer mean is the mean of the per-layer variances identically and the ratio to independent layers is exactly 1 at every L; with 16 p^4 Var_p = 3p^4 - 4p^3 - 2p^2 + 4p - 1 = (p-1)^2 (3p-1)(p+1) the constant is c^2 = 3/16, c = sqrt(3)/4 = 0.4330127, approached from below at rate O(log log p_L / L), L Var reading 0.1429334753, 0.1595579958, 0.1833424270, 0.1869968711 at L = 5, 10, 100, 1000; the odd stack under the same estimator reads 0.2708541 and factor 1.202738 at L = 4000, converging to the lane's 1.2054. The criterion is pairwise coprimality, not primality, the odd primes being the densest uncorrelated selection by least-prime-factor injectivity; the squarefree-odd rival fails with Cov(C_15, C_21) = 284/99225 = 0.0028621819 and ratio 1.308596 over 1000 layers. Witness: lab/py/stack-algebra prime_carpet_variance, squarefree_carpet_variance.
• 2026-09-09 [Proved] The s-harmonic stack: weights n^-s are completely multiplicative, so the node a/b reads b^-s H_s(floor(N/b)) and tends to zeta(s)/b^s, and the total node mass sum_b phi(b) zeta(s) b^-s equals zeta(s-1) for s > 2, read at N = 16000 as 1.644872, 1.202057, 1.082323 against zeta(2), zeta(3), zeta(4). At s = 1 the renormalisation is Davenport's expansion at a = mu, sum mu(n)/n ((nx)) = -sin(2 pi x)/pi, so the renormalised Mobius stack of the sawtooth is one sine; truncated at n <= 10^5 over five rational x the max error is 5.49e-03, 1.37e-03, 2.08e-04 at cuts 10^3, 10^4, 10^5; the {nx} form differs by (1/2) sum mu(n)/n, whose vanishing is the prime number theorem, read as -0.00048723 at n <= 10^5; the identity is Verified through arXiv:2005.08279 equation 1.1, which quotes it, not at the 1937 source. Witness: lab/py/stack-algebra harmonic_stack, davenport_check.
• 2026-09-09 [Refuted] Everything inside the Dirichlet group of stacks is closed form: the group contains 1 and mu alike, so membership buys nothing and closed-form-ness is a property of the weight, not of the algebra; the group statement carries no RH content and RH sits at exactly one element, the inverse of 1. Witness: lab/py/stack-algebra.