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 \( \sigma_n \in \{-1, +1\} \) and its run-length encoding. The exact recurrence is deterministic on historical states \( H_n = (a(n), S_n) \), 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 \( n \geq 0 \): \( a(n) = T(n) - 2W(n) \), 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 \( n \geq 0 \): \( a(n) \equiv T(n) \pmod{2} \). 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 \( \nu_2(3q+1) \) in the Collatz/Syracuse problem. I2 (FWD-2 Exclusion). No forward run of \( (\sigma_n) \) has length exactly 2. If a forward run reaches length 2, the backward target collapses algebraically to \( a(n-2) \in S \), 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 \( S_{n-1} \)). 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 \( H_0 \) (i.i.d. Rademacher signs independent of phase), the phase-averaged transition matrix on \( \mathbb{Z}/m\mathbb{Z} \) is exactly \( J/m \) (uniform), with spectral gap 1. This provides a calibration reference, not a generative model. Parity Obstruction. For \( m = 2^k \), \( k \geq 2 \), the \( m \)-step return preserves parity: \( a(n+m) \equiv a(n) \pmod{2} \). Empirical diagnostics (verified to \( N = 10^7 \)). Modular equidistribution: discrepancy \( D_m < 10^{-5} \) for all tested moduli. Phase-averaged spectral gap \( > 0.94 \) for the tested odd moduli; halved to \( \sim 0.49 \) at \( m = 2^k \), 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 \( N = 10^7 \). Coverage density \( \delta_N \approx 0.113 \) at \( N = 10^7 \). 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 \( S_n \), 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 (verify_recaman_v075.py, 8-core adaptive via Unix fork, ~15s wall at \( N = 10^7 \)). Machine-readable results in VERDICT_v075.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.