stack-algebra

• Stacking a stack is Dirichlet convolution on the layer weights, and this study checks that by literal stacking, then reads off the closed form of every selected stack.
• The w-weighted line stack over scales 1..N gives node a/b, denominator b, the brightness sum_{k <= N/b} w(kb); the u-stack of the v-stack draws, at scale kn, the weight u(k) v(n).
• Under the hyperbolic cut kn <= N the composite scale weight is exactly (u * v)(m) for every m <= N, so the two pictures agree node for node.
• Checked at N = 60 on u = v = 1, u = 1, v = mu, u = mu, v = mu and u = 1, v = n^-1: 1102, 1102, 974 and 1102 nodes, zero mismatches by three routes that share no inner loop.
• The three routes are explicit copy placement, the line at j/m inside copy i of k landing at ((i-1)m + j)/(km); the composite scale weight built from the hyperbolic cut; and the node formula sum_{k <= N/b} (u * v)(kb).
• The rectangular cut 24 x 40 is exact at every scale m <= 24 and wrong above it, 16 of the 16 scales from 25 to 40 differing for 1 * 1, 1 * mu and 1 * n^-1 and 10 of 16 for mu * mu.
• 1 * mu = e: the Mobius stack of the plain stack is one layer, weight 1 at scale 1 and nothing else.
• Closed-form node brightness of the five selections, checked against literal stacking at every b <= N for N = 30, 61, 200, 501, zero mismatches:

- evens floor(N / lcm(2, b)) - odds 0 at even b, else ceil(floor(N/b) / 2) - primes pi(N) at b = 1, 1 at prime b <= N, else 0 - squarefree 0 unless b is squarefree, else sum_{d^2 <= N/b, gcd(d,b) = 1} mu(d) sum_{e | b} mu(e) floor(N/(b d^2 e)) - prime powers sum_p floor(log_p N) at b = 1, floor(log_p N) - i + 1 at b = p^i <= N, else 0

• The primes-only line stack lights 96 denominators at N = 501: b = 1 and the 95 primes, every prime node at brightness exactly 1.
• The s-harmonic stack, weights n^-s: node a/b reads b^-s H_s(floor(N/b)), tending to zeta(s)/b^s; total node mass sum_b phi(b) zeta(s) b^-s = zeta(s-1) for s > 2, read at N = 16000 as 1.644872, 1.202057 and 1.082323 for s = 3, 4, 5 against zeta(2), zeta(3), zeta(4).
• The carpet layer covariance is rebuilt from scratch by exact rational integration on the lcm grid and matched to the gcd closed form at (3,5), (5,7), (3,9), (15,21), (5,15).
• The primes-only carpet stack has zero covariance at every layer pair, so L * Var of the L-layer mean, with L the layer count, is the mean of the per-layer variances exactly, ratio to the independent value exactly 1 at every L; the values run 0.1429334753, 0.1595579958, 0.1833424270, 0.1869968711 at L = 5, 10, 100, 1000, rising to 3/16.
• Per-layer exactness: 16 p^4 Var_p = 3p^4 - 4p^3 - 2p^2 + 4p - 1, no breach over the first 1000 odd primes, so the limit constant is exactly 3/16 and c = sqrt(3)/4 = 0.4330127.
• The tree's full odd stack over the same estimator reads L * Var = 0.2708541 at L = 4000, ratio 1.446579 and c factor 1.202738, still climbing.
• The squarefree-odd stack is not uncorrelated: Cov(15, 21) = 284/99225 = 0.0028621819, Pearson 0.0165610084, and 64087 of its first 1000 layer pairs share a factor, giving ratio 1.308596.
• Davenport's expansion at a = mu, sum_{n >= 1} mu(n)/n ({nx} - 1/2) = -sin(2 pi x)/pi, truncated at n <= 10^5 over five rational x: max error 5.49e-03, 1.37e-03, 2.08e-04 at cuts 10^3, 10^4, 10^5.

RUN

• uv run python research/lab/py/stack-algebra/stack_algebra.py
• From the repo root. One core, under two seconds.
• Domain is the full source domain: N = 60 for the convolution check, N = 30, 61, 200, 501 for the selections, N = 1000, 4000, 16000 for the harmonic stack, L up to 4000 layers for the variance, n <= 10^5 for Davenport.
• Nothing is written to disk.

WITNESSES

• The convolution theorem for weighted stacks and its hyperbolic cut, with the four checked pairs and their node counts.
• 1 * mu = e, the Mobius stack of the plain stack collapsing to a single layer.
• The five selection closed forms and the primes-only stack's 96 lit denominators at N = 501.
• The s-harmonic node form and the total node mass zeta(s-1).
• The primes-only carpet stack fading at exactly the independent rate, ratio 1 at every L, constant 3/16, c = sqrt(3)/4.
• The squarefree-odd stack failing that property, Cov(15, 21) = 284/99225.
• Davenport's mu expansion checked numerically at truncation 10^5.

NOTE

• Every layer pair here is a carpet pair, so the gcd covariance law is used as a landed theorem and re-derived only at the five spot-check pairs.
• The exact rational L * Var at L = 5 is 36324973301545304/254139019762753125; beyond that the numerators run to thousands of digits and only the float is printed.
• The {nx} form of Davenport's identity differs from the sawtooth form by (1/2) sum mu(n)/n, which the script reads as -0.00048723 at n <= 10^5.