# The leaning stack

- 2026-09-09 [Proved] The leaning stack shifts layer `n` of the line stack by a drift `t_n`, its lines sitting where `n x - theta_n` is an integer with `theta_n = n t_n mod 1`. The linear lean `theta_n = n delta` is a translation, lit set `F_Q + delta` and brightness `floor(N/b)`, 278 nodes and 0 mismatches at `N = 30`. The quadratic lean `theta_n = n^2 c/d` lights `a/b` iff `n (a d - b c n) = 0 mod b d`, the lit layers a union of `2^w` residue classes modulo the period `lcm(b, d*)` with `d*` the least `k` with `d | k^2` and `w` the number of primes with `v_p(b) <= v_p(d) < 2 v_p(b)`; 0 mismatches against literal stacking on 10240 point-drift pairs at `N = 60`, `b <= 20`, `d <= 16`, and 0 solution-set mismatches; `lcm(b, d*)` is a period and not always the least, the least being `lcm(b, d*)/2` exactly when `v_2(b) >= 1` and `v_2(d) = 2 v_2(b) - 1`, 417 of 16384 tuples with `b, d <= 20` halving and 0 breaches of the rule. Witness: lab/py/leaning-stack `linear_lean`, `quadratic_lean`, `brightness_form`, `minimal_period_law`, `adversarial`.
- 2026-09-09 [Proved] The leaning stack's lit set reads the numerator and its brightest node leaves the origin: at `delta = 1/4` the point `1/4` is lit by `n = 0, 1 mod 4` and `3/4` by `n = 0, 3 mod 4`, `B_61 = 31` against 30, 48 such pairs over `b <= 12` and `d` in 2, 4, 8, 9; the origin's brightness at drift `c/d` is `floor(N/d*)`, so at `delta = 1/2` the point `1/2` reads 60 at `N = 60` and the origin 30, the origin beaten at five of seven printed drifts; the origin's density `1/d*` is A019554, multiplicative with `a(p^e) = p^ceil(e/2)`, its Dirichlet series `zeta(2s+1) zeta(s+1)/zeta(2s+2)` recovered as `1.826902` against `1.826907` at `s = 1`; the phase census at prime drift denominator is a Legendre symbol, `n^2 c/p` taking `(p+1)/2` values with multiplicity `1 + (j c^-1/p)`, the centred twist `S(c, p) = (c/p) S(1, p)` holding coefficient by coefficient and the solution count being the Fourier sum of quadratic Gauss sums, 0 breaches at `p = 5, 7, 11, 13`. Witness: lab/py/leaning-stack `lit_set`, `origin_law`, `gauss_sums`.
- 2026-09-09 [Proved] Layers `m != n` of the quadratic lean share a lit point iff `lcm(m, n)(m - n) delta` is an integer, 0 criterion failures against literal intersection on 5280 pairs over `m < n <= 12` and every reduced drift with `d <= 16`, so at irrational drift no two layers ever coincide and brightness is at most 1 everywhere at every `N`, the translation twin of the dead-spin theorem; the 325 lit points of layers `1..25` are distinct at `sqrt 2 - 1` and `phi - 1`, the closest approach to `N = 120` being `9.202e-09`, and the near-coincidences follow Weyl's equidistribution of `n^2 delta`, star discrepancy `0.019450, 0.006421` and `0.022253, 0.009391` at `N = 1000, 10000` against `1/sqrt N`, four points and no exponent. Witness: lab/py/leaning-stack `sharing_law`, `adversarial`, `irrational_lean`.
- 2026-09-09 [Refuted] The lean is a weight on the scales: no weight `w` reproduces it, its brightness density `2^w/lcm(b, d*)` per unit `N` depends on the numerator while every weighted stack reads `sum_{k <= N/b} w(kb)`, denominator-only, and no weight makes a node brighter than the integer, which `x = 1/2` at `delta = 1/2` is; the lean is the first operation on this tree outside the Dirichlet group with a closed form. Witness: lab/py/leaning-stack `quadratic_lean`, `lit_set`.