Structural Invariants, Memory, and Modular Diagnostics of Recamán's Sequence

Structural Invariants, Memory, and Modular Diagnostics of Recamán's Sequence We analyze the decision dynamics of Recamán's sequence \ (a (n) \) (OEIS A005132) through the binary direction sequence \ (ₙ \-1, +1\ \) and its run-length encoding. The exact recurrence is deterministic on historical states \ (Hₙ = (a (n), Sₙ) \), not on \ (a (n) \) alone; all modular and spectral diagnostics are treated as projections of this exact dynamics. Unconditional results. E1 (Triangular Identity). For all \ (n 0 \): \ (a (n) = T (n) - 2W (n) \), where \ (T (n) = n (n+1) /2 \) is the \ (n \) -th triangular number and \ (W (n) \) is the sum of backward-step indices. E2 (Mod-2 Rigidity). For all \ (n 0 \): \ (a (n) T (n) 2 \). The direction sequence is invisible modulo 2; parity follows the deterministic period-4 pattern of the triangular numbers. This constitutes a 2-adic shadow structurally reminiscent of the role of \ (₂ (3q+1) \) in the Collatz/Syracuse problem. I2 (FWD-2 Exclusion). No forward run of \ ( (ₙ) \) has length exactly 2. If a forward run reaches length 2, the backward target collapses algebraically to \ (a (n-2) S \), forcing a third consecutive forward step. Tripartite Decomposition. Every step falls into exactly one of: BWD, FWD (P) (positivity failure, locally decidable), or FWD (M) (congestion — the unique locus of historical self-reference, requiring the full occupation set \ (S₍-₁ \) ). Projection Incompleteness. No representation that collapses two historical states with different freshness outcomes can factor the exact recurrence. This is an obstruction criterion: it does not assert that two such histories occur in the single realized trajectory, but that any proposed projected closure must rule out freshness collisions. Null-Model Benchmark. Under \ (H₀ \) (i. i. d. Rademacher signs independent of phase), the phase-averaged transition matrix on \ (Z/mZ \) is exactly \ (J/m \) (uniform), with spectral gap 1. This provides a calibration reference, not a generative model. Parity Obstruction. For \ (m = 2ᵏ \), \ (k 2 \), the \ (m \) -step return preserves parity: \ (a (n+m) a (n) 2 \). Empirical diagnostics (verified to \ (N = 10⁷ \) ). Modular equidistribution: discrepancy \ (Dₘ 0. 94 \) for the tested odd moduli; halved to \ (0. 49 \) at \ (m = 2ᵏ \), consistent with E2. Regime fractions: BWD = 50. 0%, FWD (P) = 28. 8%, FWD (M) = 21. 2%. Run-length statistics: 99. 94% unit runs; FWD lengths in \ (\1, 3, 4, 5, 6\ \) ; length 6 first attained at \ (N = 10⁷ \). Coverage density \ (N 0. 113 \) at \ (N = 10⁷ \). Conjectural roadmap and the reconstructive gap. A hierarchy of conjectural dependencies connects the Sloane coverage conjecture to congestion boundedness, run-length pseudorandomness, first-order phase balance, and modular diagnostics. The key gap is reconstructive: the projected process modulo \ (m \) does not retain the exact historical set \ (Sₙ \), so even strong modular diagnostics cannot decide whether a future backward candidate is genuinely fresh. Reproducibility. All computational claims are verified by the accompanying parallelized Python script (verifyᵣecamanᵥ075. py, 8-core adaptive via Unix fork, ~15s wall at \ (N = 10⁷ \) ). Machine-readable results in VERDICTᵥ075. json. Epistemic discipline. All results are classified into four levels: unconditional theorems, null-model theorems, empirical diagnostics, and conjectural dependencies. No empirical observation is promoted beyond its verified status.