Five Words, One Polygon Digit: memory, folded correlations, and arithmetic reconstruction

Five periodic binary words share a prime denominator, their Fourier intensities and their polygonal winding profiles. Seven consecutive digits distinguish the five classes. Exact Euler-tour formulas describe finite-memory fibres, including periodicity and reflection. The paper uses this example to study how memory and carefully placed observations recover arithmetic information that ordinary geometry loses. For every period with at most four distinct prime divisors, autocorrelation and one triple row folded about an occupied difference with an odd integer representative determine a Sidon support up to translation and reflection. Every generating edge therefore suffices. The proof excludes the final radical-210 case by comparing a nineteen-term upper bound for a shared signed relation with a twenty-term lower bound for a circuit involving seven; a positive-definiteness argument sharpens the upper bound to eighteen. At odd periods, rigidity extends through sixteen active prime axes. Any odd-period ambiguity requires at least 315 sites and an active reciprocal-prime sum greater than 433/465, while an unevaluated absolute bound limits its size. An exact binary lifting criterion and a collision-energy identity describe singly even periods and give an absolute support bound when few projected differences repeat. Removing the anchor gives an equivalent signed-convolution existence problem, and free reflection orbits yield cyclic torsion certificates. At odd torsion, a four-set realization requires at least 627 sites, including its two anchors. A ternary periodic remainder governs two-adic descent: any odd-anchor witness at an even radical with r ≥ 2 prime divisors has one of the same size and radical whose two-adic exponent is at most 2r + 2. A separate determinant argument transfers an s-site generating-edge witness to a period below 150 × 7^(s−2) in the same parity class, with a possible change of radical. Additional measurements finish reconstruction. Two adaptively incident folded rows give a stable linear-time decoder, while one row and pair counts leave a binary code with an exact optimal number of fourth-order probes. Controlled Fourier readings give sharp detected-photon bounds, and moment stabilizers treat affine templates with vanishing low moments. The general higher-prime classification remains open, including the even problem with five or more distinct prime divisors and odd periods outside the proved rigidity criteria. The project contains analytical proofs, sixteen vector figures, editable sources and eleven exact verification files.