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.