Arshad's Sieve
Theorem Register

Power Sums on Z₂₅₆

Arshad's Sieve is a modular power-sum framework derived from the geometry of the MPRC 3D+1D lattice on Z₂₅₆. It characterizes when the equation Σxᵢⁿ ≡ n² (mod m) has solutions, when it does not, and why the failures are structurally forced by the ring geometry.

Anchor, Disk Tower, Spillover Registry

The framework operates on the ring Z₂₅₆ with anchor A = 256 = 2⁸. The disk tower extends to primes ranked by proximity to A. The sin/cos LUT has 256 entries across 4 quadrants of 64.

Type-A, Type-B, Gap Splitting

Theorem 1 — Type-A Wall. Zₙ has a Type-A wall at n if and only if m | n². For m₁ = 256: wall period W = 16 = √256. The Type-A sequence on Z₂₅₆ is n = 16, 32, 48, 64, … At each such n the target n² ≡ 0 (mod 256) and charge routes to the next disk.

Proof. 256 | n² ⟺ 2⁸ | n² ⟺ 2⁴ | n ⟺ 16 | n. Target vanishes; spillover mandatory. ■

Theorem 2 — Type-B Wall at n = 8. No tuple (x₁,…,x₈) satisfies Σxᵢ⁸ ≡ 64 (mod 256).

Proof. For all odd x: x⁸ ≡ 1 (mod 32). Sum of 8 terms ≡ 8 (mod 32). Target 8² = 64 ≡ 0 (mod 32). Gap = 8 ≠ 0. Even x⁸ ≡ 0 (mod 32), cannot bridge. ■

Theorem 3 — Gap Splitting. When prime p saturates at n = pk, the residue on disk j satisfies Rⱼ(pk) = (mⱼ − p)² · k² (mod mⱼ).

Proof. p ≡ −(mⱼ − p) (mod mⱼ), so (pk)² ≡ (mⱼ − p)²k² (mod mⱼ). ■
Corollary: At n = 251, R₂₅₆ = 25, R₂₅₇ = 36. Their squares differ by 11 = 6² − 5².

Theorem 4 — Spillover Conservation. Cₖ + Cₖ' = mₖ at every saturated disk k.

Proof. Follows directly from the definitions of Cₖ and Cₖ'. ■

Theorem 5 — CRT Collision. k disks saturate simultaneously at n = lcm(w₁,…,wₖ). This forces introduction of disk k+1.

Proof. Simultaneous saturation requires n divisible by all wall periods. At such n, all active disks carry zero residue; a new disk is the only resolution. ■

Harmonic Structure, Partition, Non-Closure

Theorem 6 — Type-B Complete Characterization. On Z₂₅₆, exactly three Type-B walls exist: n ∈ {8, 12, 24}.

Theorem 7 — Type-B Harmonic Structure. All Type-B walls are rational multiples of W = 16: n = W/2, 3W/4, 3W/2. Every Type-B n is a multiple of W/4 = 4 (the quarter-period). The obstruction moduli {8, 16} are exactly {W/2, W}.

Theorem 8 — 16/9 Identity. The disk growth exponent is geometric and exact.

Theorem 9 — Disk Count Formula. The cumulative prime sum across d stacked disks satisfies C(d) ~ 2179.54 · d^(16/9). Minimum disk count: d(T) ≈ ((T − 4422) / 2179.54)^(9/16).

Theorem 10 — Type-AB Wall. n is a Type-AB wall iff 64 | n and n ≢ 0 (mod 256). The group exponent of (ℤ/256ℤ)* equals 64 = 2⁶. Type-AB walls: n = 64, 128, 192, 320, …

Theorem 11 — Spiral Non-Closure. When spillover is broken at n = 128, the trajectory f(n) = (n² − 128) mod 256 never equals 0. Charge traces a non-closing spiral through Z₂₅₆.

Proof. Closure requires 128 ∈ QR₂₅₆. By 2-adic descent all cases exhaust: n odd → n² ≡ 1 (mod 8) ≠ 0; n even → m² ≡ 32 (mod 64); j odd → impossible; j even → i² ≡ 2 (mod 4) — impossible. ■
Corollary: normal closure = 720° (2π); broken spillover at n=128 = 1440° (4π) — the discrete spinor condition.

Theorem 12 — Z₂₅₆ Three-Region Partition. The ring partitions into exactly three disjoint regions.

Proof. |QR₂₅₆| = 44 by direct computation. Shift preserves size. QR₂₅₆ ∩ (QR₂₅₆+128) shares 42 elements but not {0,128}. |Δ| = 256 − 46 = 210. ■

Three Open Questions

O1 — Witness Existence for All n. Prove: for every n ≥ 3, ∃ integers (x₁,…,xₙ) such that Σxᵢⁿ ≡ n² (mod mⱼ*). Verified: n = 3..16, 32, 48, 64. Partial proof: Weil (1948) character sum bounds close the prime-disk case. Gap: the Z₂₅₆ direct case for non-wall, non-obstructed n.

O2 — Type-B Walls Beyond n = 64. The harmonic structure (T7) suggests no Type-B walls exist beyond n = 24. Candidates n = 40 and n = 48 were verified clear. A formal proof of termination, or a counterexample, remains open.

O3 — Type-B Characterization on Extended Disks. Each stacked disk mₖ (primes near 256) has its own wall structure. The Type-B obstructions on Z₂₅₁, Z₂₅₇, Z₂₆₃ are not yet characterized. Does T7 generalize to prime moduli, or is it specific to the 2-power anchor 256?