# 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.