Centered-Residue Rigidity in Collatz Valuation Sequences: Tower-Sparse Returns and 2-Adic Logarithmic-Form Escape

Abstract

This paper supersedes the earlier pre-arXiv reduction manuscript "A Δ-CoreReduction Framework for the Collatz Conjecture" (v0.9.5, this same record).It retains the deterministic recurrence-rigidity framework of thatmanuscript and adds a substantially stronger unconditional result. For the accelerated odd-to-odd Collatz map, let a_k = ν2(3n_k+1),A_k = Σ_{j<k} a_j, and define the affine correction sum and centeredresidue by Δ_0=0, Δ_{k+1}=3Δ_k+2^{A_k}, s_k = cent_{2^{A_k}}(Δ_k). The paper first reproves, in sharpened form, the deterministicrecurrence bound #{k≤T : |s_k|≤B} = O_B(log* T), obtained fromcongruence rigidity and the Lifting-the-Exponent lemma alone, with noprobabilistic or ergodic input. This is the result previously reportedin v0.9.5. The principal new result removes the realizability question left openin v0.9.5. Using the exact divisibility 2^{A_k} | n_0 3^k + s_k togetherwith a specialization of Yu's theorem on 2-adic logarithmic forms, thepaper proves an effectively computable quantitative escape law:|s_k| ≥ exp(c* k / log k) for all sufficiently large k, for every fixedpositive odd seed n_0, with an effectively computable absolute constantc* independent of the seed. Consequently every fixed centered-residuestrip is visited only finitely often, and no fixed-residue abstracttower lock — of the kind whose realizability was left open in v0.9.5 —admits a positive-integer completion. The paper separates the periodic branch (a positive cycle, if oneexists, never enters the subcritical centered-residue corridor reachedby this escape theorem) from the nonperiodic divergent branch, andstates the exact remaining interface for each: - Branch I (aperiodic divergent orbits): the escape theorem proved here is unconditional. A Conditional Aperiodic Exclusion theorem is given: IF every hypothetical aperiodic nonconvergent orbit admits a subsequence with log|s_k| = o(k/log k) (an explicitly stated open "Entry" problem), THEN no such orbit exists. The paper states precisely why this Entry problem is not addressed by the present arithmetic machinery: it is a probability-to-structure transfer (converting typical/exceptional-orbit information into a pointwise guarantee for one fixed orbit) with no known general solution, not an arithmetic identity. - Branch II (nontrivial cycles): unaffected by this paper's escape theorem; requires a separate bound on the odd-step count of a hypothetical cycle, addressed in the companion trace-compressed normal-form literature. The paper does not claim a proof of the Collatz conjecture, a proof ofthe Entry problem, or an exclusion of nontrivial cycles. Independentspecialist review of the Yu-theorem specialization (Lemma 9.1) remainspending; the specialization's field, unit, height, and non-vanishinghypotheses are stated explicitly so they can be checked directly. A reproducibility package (exact-integer-arithmetic verificationscript, no floating point in any claim-bearing check) is included,covering the elementary and combinatorial results (Lemma 3.1, Theorem3.2, Lemma 6.1, Theorem 6.2, Lemma 8.1, Proposition 10.2, and theworked example n_0=27). It does not and cannot verify the Yu-theoremspecialization or the S-unit finiteness theorem, which rest on deepexternal transcendence-theoretic and Diophantine results; thislimitation is stated in the accompanying README. AI-assisted tools, including large language models, were used forexploratory drafting, proof-audit assistance, literature-searchassistance, language editing, LaTeX preparation, and the constructionand review of the exact-arithmetic verification script. Their outputswere treated as unverified suggestions. The author reviewed, revised,and approved all definitions, statements, proofs, computations,citations, and scope claims, and accepts full responsibility for thecontent.