A Newton-polygon formal-level baseline for the Birkhoff-Trjitzinsky exponent on the SIARC polynomial continued-fraction Wallis stratum

A standalone Zenodo note that consolidates the SIARC T1 Phase 2verdict UPGRADE_PARTIAL_FORMAL_LEVEL (bridge sessionT1-BIRKHOFF-PHASE2-LIFT-LOWER, commit 37c939f, 2026-05-04) into asingle citable artefact: a Newton-polygon formal-level baselinefor the Birkhoff-Trjitzinsky (B-T) asymptotic exponent A on theSIARC polynomial continued-fraction (PCF) Wallis stratum. Triple framing (the note's load-bearing partition): PROVEN (formal level): For the SIARC PCF Wallis recurrence atpolynomial degree d in [2, 8] and any of the three SIARCconventions (alpha-direction, symmetric, delta-direction), thenaive Wimp-Zeilberger Newton-polygon ansatz yields the formalexponent A_naive = 2d - deg_a in {d-1, d, d+1}. This isTheorem 1.1 of the note. The proof is by direct symbolicbalance analysis on the formal series Q_n ~ exp(A_naive * n1/2+ ...) ansatz, derived in Section 2 and closed by per-degreecase analysis in Section 3 across the three conventions. VERIFIED (literature-citation): At d = 2 the formal-levelprediction A_naive = 3 (alpha-direction) coincides with theLOWER branch of PCF-1 v1.3 (DOI 10.5281/zenodo.19937196),Section 6 Theorem 5, V_quad family QL01-QL26, verified at 80algebraic digits. The B-T 1933 Sections 7-9 existence andfactorization machinery (Theorem II, Lemma 10, FundamentalTheorem) is verified against the SIARC stratum gates C.2.1through C.2.4 (non-resonance match, non-degeneracy match, d-rangeapplicability, mu-units consistency) in the bridge Phase Cliterature verification. STRUCTURAL FRAMING (not closure): At d >= 3 the empirical PCF-2v1.3 (DOI 10.5281/zenodo.19963298) record R1.1 + R1.3 + Q1yields A_fit ~ 2d (50/50 cubic representatives, 60/60 quarticrepresentatives) which lies strictly outside the formal baseline{d-1, d, d+1}. Proposition 1.2 of the note frames this gap withtwo non-mutually-exclusive structural mechanisms - (i') aborderline-locus condition where the SIARC stratum sits on thedeg_a = 2*deg_b boundary of the Wimp-Zeilberger normal case, and(ii') a definitional translation under which the empiricalfit-protocol of [PCF-2 v1.3] measures a different exponent thanA_naive. Distinguishing (i') from (ii') is the open content ofSection 6 Q1. CONJECTURED (Conjecture B4): The note states Conjecture B4 of[PCF-2 v1.3] (A = 2d for representatives in the SIARC stratum ateach degree d >= 2) verbatim. The note does NOT claim to prove,partially prove, or refute Conjecture B4. The formal-levelbaseline is consistent with - but does not establish - A = 2d.Section 6 Q4 names the sectorial upgrade gap (Wasow Section X.3Theorem 11.1, Adams 1928, Turrittin 1955, Immink 1984) thatwould, jointly with the closure of (i') or (ii'), turn theformal-level baseline into a proof of A = 2d on the SIARCstratum. Cross-cited Zenodo deposits and DOIs: - PCF-1 v1.3 ........... 10.5281/zenodo.19937196- PCF-2 v1.3 ........... 10.5281/zenodo.19963298- D2-NOTE v2.1 ......... 10.5281/zenodo.20015923- CT v1.3 .............. 10.5281/zenodo.19899996- T2B v3 ............... 10.5281/zenodo.19915689- umbrella v2.0 ........ 10.5281/zenodo.19965041 Primary literature anchors: - Birkhoff & Trjitzinsky 1933, Acta Math. 60, DOI 10.1007/BF02398269 ("Analytic theory of singular difference equations"), Sections 7-9 plus Section 11 Theorem III.- Wimp & Zeilberger 1985, J. Math. Anal. Appl. 111, "Resurrecting the asymptotics of linear recurrences".- Wasow 1965, "Asymptotic expansions for ordinary differential equations", Section X.3 Theorem 11.1.- Adams 1928, Annals of Math. 30 (1), "On the asymptotic expansions of the solutions of linear difference equations".- Turrittin 1955; Immink 1984; Costin 2008; Birkhoff 1930; Loday-Richaud 2016 (related literature). AEAL (Author-Evidence-Aided Logging) schema: 10 entries inclaims.jsonl. Six entries are literature-citation carry-forwardfrom the bridge T1-BIRKHOFF-PHASE2-LIFT-LOWER session (Phase Adeg 2/3/4 baselines, Phase B sweep d in [5,8], Phase C foursub-gates, Phase D verdict). Two entries are literature-citationanchors to PCF-1 v1.3 Section 6 Theorem 5 lower branch and PCF-2v1.3 R1.1 + R1.3 + Q1 empirical record. One entry is thecomputation claim recording the three-pass pdflatex + bibtexbuild of bt_baseline_note.pdf (8 pages, 0 unresolved citations).One entry is the reading claim recording the rubber-duck self-review and forbidden-verb hygiene scan, with the four-wayepistemic partition (PROVEN / VERIFIED / STRUCTURAL FRAMING /CONJECTURED) confirmed intact in every section. Related identifiers metadata for Zenodo: - isCitedBy / isContinuationOf / isPartOf: PCF-1 v1.3 (10.5281/zenodo.19937196) PCF-2 v1.3 (10.5281/zenodo.19963298) D2-NOTE v2.1 (10.5281/zenodo.20015923) CT v1.3 (10.5281/zenodo.19899996) T2B v3 (10.5281/zenodo.19915689) umbrella v2.0 (10.5281/zenodo.19965041)- isSupplementTo: siarc-relay-bridge sessions/2026-05-03/T1-BIRKHOFF-PHASE2-LIFT-LOWER (commit 37c939f). Build provenance: - pdflatex (MiKTeX 25.12); bibtex (BibTeX 0.99e), bibliography style 'plain', 15 entries, 0 warnings.- bt_baseline_note.pdf SHA-256 23022f0de77ac8388ed584b2196c0ab995cd8cf18b2dd71efbc0488a0f6e5b7c- bt_baseline_note.tex SHA-256 6746692c517dc25238473e819527c5682465cdc9e1def69d1f6df31c1014d51b- annotated_bibliography.bib SHA-256 589f6e4a0b29de401229d60a6252de15436f355a2fa8b3ac04907779c2de923d Author-AI disclosure: This note was prepared by the author withAI assistance (GitHub Copilot, Microsoft; and Anthropic Claude ina separate session) for code generation, numerical computations,LaTeX drafting, and provenance bookkeeping under the SIARC(Self-Iterating Analytic Relay Chain) protocol. Allload-bearing claims are anchored to AEAL entries with SHA-256provenance to either bridge-session computational artefacts orverbatim literature quotations. The author bears fullresponsibility for the content and the partitioning of claimsacross the four-way epistemic partition. License: CC-BY 4.0.