Between the Landmarks: Figurate Resonance in the Arithmetic Atlas

Between the Landmarks develops figurate resonance, a mathematical framework arising from the Arithmetic Atlas. Polygonal sequences divide the number line into chambers between consecutive landmarks, and the normalized Memory of an integer records its fractional position within such a chamber. Figurate resonance occurs when an integer occupies the same strictly interior normalized position on two polygonal Roads. The paper derives a master Diophantine equation, a universal denominator sieve, and an exact parameterization by chamber widths. It distinguishes factor-forcing, prime-permitting, and resonance-forbidden Road pairs. Every square–triangular interior resonance is proved composite, with a proper factor recovered from the reduced ratio of chamber widths. Other Road pairs admit prime resonances, while triangular–hexagonal interior resonance is proved impossible. Exact computation finds 28 square–pentagonal resonances through square index 100,000,000, including two large primes organized within a generalized Pell corridor. Independent Pocklington certificates verify the primality claims. The work develops the Arithmetic Atlas from an explanatory representation into a comparative theory of positions between landmarks. It is presented as a mathematical continuation of Who Saw the Atlas? Arithmetic as Atlas and Human-AI Collaboration.