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

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 = Σj<k a_j, and define the affine correction sum and centeredresidue by Δ_0=0, Δₖ₊₁=3Δ_k+2A_k, s_k = cent_{2A_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 2A_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.