A Global Occupation Conjecture for the Accelerated 3x + 1 Map: Residue–Time Duality, Divergent-Orbit Sparsity, and the Pointwise Realizer Problem

Abstract

We study the Effective Occupation Conjecture (EOC) for the accelerated 3x + 1 map T (m) = (3m + 1)/2a(m), a(m) = ν2(3m + 1). Writing Rn = Sn − n log2 3 for the drift of the accumulated 2-adic valuation and Oc(m0) = #{n ≥ 0 : Rn ≤ c}, EOC asserts Oc(m0) = O(log m0) for every seed and every corridor width c. This paper organizes the programme around four axes — occupation, residue, time, and divergence — and the exact bridges between them. Record chronology identifies the residue and time axes precisely: the least-realizer frontier rmin(N, c) is the inverse function of the single-window confinement lifetime Lc(m), so every nondecreasing pointwise lifetime bound transfers to a realizer floor, on a graded scale that we tabulate. On the divergence side we retain the strongest unconditional input: via Garcia–Tal’s collision-free window theorem and Curry’s explicit quantitative form, every divergent orbit has summable reciprocals and admits no eventual logarithmic drift floor Rn ≥ −B log2 n + O(1) for any B < 1/β∗ ≈ 1.0358567. Curry’s input plays a second, qualitative role: after the last global maximum of R, a hypothetical divergent orbit has an infinite zero-corridor tail, so universal drift exit — every positive seed eventually attains Rn > 0 — is equivalent to the exclusion of divergent orbits, while the drift-exit argument supplies no cycle exclusion: no implication from it to cycle-freeness is established here, and none is claimed in either direction (Rem. 6.17). We then record what a sustained programme of structural audits has closed. Finite information cannot decide infinite realization: every exact-realizer cylinder contains arbitrarily large positive integers, so ∀N ∃mN is free and all content lies in ∃m ∀N . Nor do the known divergent-orbit valuation-word shadows decide it: we exhibit an explicitly indexed uncountable family of zero- confined words with linear negative drift, summable P 2Rn and O(G) occupation shadow, of which all but countably many have no positive-integer realizer. Word-level criteria in general are not excluded, and one is exact (Rem. 9.6). Several apparently independent coordinates — normalized carry, lift digits, terminal zero gap, transport budgets — are invertible reparameterizations of the exact realizer, and the Archimedean and 2-adic readings of the natural divisibility object are complementary factors of one identity rather than independent constraints.

Keywords