Structural Invariants and External Connections of Recamán's Sequence

Structural Invariants and External Connections of Recamán's Sequence We analyze the decision dynamics of Recamán's sequence a (n) (OEIS A005132) through the binary direction sequence σn ∈ −1, +1 and its run-length encoding. 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. This separates the algebraic skeleton T (n) from the dynamical content W (n). E2 (Mod-2 Rigidity). For all n ≥ 0: a (n) ≡ T (n) (mod 2). The direction sequence is invisible modulo 2; parity follows the deterministic period-4 pattern of the triangular numbers, independently of all dynamical choices. This constitutes a 2-adic shadow structurally parallel to the role of ν2 (3q+1) in the Collatz/Syracuse problem. I2 (FWD-2 Exclusion). No forward run of (σn) has length exactly 2. If a forward run reaches length 2, the backward target at the next step collapses algebraically to a (n−2), which is already in the occupation set, forcing a third consecutive forward step. Admissible forward run lengths are 1 ∪ m ≥ 3. Tripartite Decomposition. Every step falls into exactly one of: BWD (both positivity and freshness hold), FWD (P) (positivity failure), or FWD (M) (congestion — freshness failure with positive target). FWD (M) is the unique locus of self-reference, carrying infinite memory through the occupation set Sn. Null-Model Benchmark. Under the null hypothesis 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 for empirical spectral diagnostics. Parity Obstruction. For m = 2k, k ≥ 2, the m-step return preserves parity: a (n+m) ≡ a (n) (mod 2). This is an unconditional algebraic obstruction to full mixing at powers of two. Empirical diagnostics (verified to N = 107). Modular equidistribution: discrepancy Dm 0. 94 for odd m; 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 δN ≈ 0. 113 at N = 107. Conjectural roadmap. A hierarchy of conjectural dependencies connects the Sloane coverage conjecture to congestion boundedness, run-length pseudorandomness, first-order phase balance, and modular diagnostics. The irreducible gap is identified as proving periodic decorrelation of σn against nontrivial test functions — the Recamán analogue of the Chowla/Sarnak obstruction in Collatz dynamics. Reproducibility. All computational claims are verified by the accompanying parallelized Python script (verifyᵣecamanᵥ075. py, 8-core adaptive, ~15s wall at N = 107). Machine-readable results are provided 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.