| ID | Title | Status | Statement |
|---|
AXZ-CSF-001 | Dead odd-square Ulam lane | PROVED | The lane C0(r)=(2r+1)^2 is composite for every r>=1 and therefore contains no primes after the trivial center. |
AXZ-MSQ-003 | Primitive 24k+1 compression gate | PROVED / LITERATURE-SUPPORTED | Primitive 3×3 magic-square-of-squares candidates satisfy cell congruence L_i ≡ 1 (mod 24); Lucas steps a and b must be multiples of 24. |
AXZ-MSQ-006 | Modulo-240 survivor lattice | FINITE_PROVED | Modulo 240 has 116 code-like nontrivial QR survivors; strict primitive compression leaves 50 survivor structures; center classes 25 and 145 account for 38/50. |
AXZ-MSQ-006-CRT | CRT product counts | FINITE_PROVED | Strict primitive mod 240 survivors combined with raw mod 7 and mod 11 QR survivors give 50×16×26=20,800 raw CRT product states; tightened nontrivial convention gives 12,000. |
AXZ-MSQ-008-CUBE | Cubic residue reductions | FINITE_PROVED | For cubic-residue Lucas gates, mod 7 and mod 9 each have exactly 4 nontrivial survivors, giving reductions 98.8338% and 99.4513%. |
AXZ-MSQ-010 | Closed additive-diamond reduction | PROVED REDUCTION | For center e=z^2, a 3×3 magic square of squares requires {a,b,a+b,|a-b|}⊂D(e), where D(e)={d>0:e−d and e+d are squares}. |
AXZ-MSQ-013 | Finite graph sweep through z≤1,000,000 | FINITE_NEGATIVE_RESULT | A finite sweep found 245,344 centers with valid D(e), 2,611,262 pair tests, zero 3-of-4 near-misses, zero additive diamonds, and zero valid square hits under the implemented gates. |
AXZ-MSQ-014.1 | Forced 24-divisibility lemma for centered square-pair differences | PROVED | If d∈D(z^2), then 24∣d. More sharply, if 2^s exactly divides z, then 24·4^s divides d. |