The EBR Amplitude as a Connection Coefficient: Characterization and a Rigidity Dividing-Line Conjecture

ERRATUM NOTICE — this is version 1.0, and it has been superseded. A corrected version of this work is published as version 1.2 at 10.5281/zenodo.20571232; the concept DOI 10.5281/zenodo.20566465 always resolves to the latest version. The correction: the point at s = ∞ is an irregular singular point of Poincaré rank 1, not a regular one. The ODE therefore has two regular singular points {0, R} and an irregular point at ∞ — a regular-to-irregular connection problem — so the description of it as an order-2d Fuchsian ODE with “exactly three regular singular points” is withdrawn wherever it occurs in this version, including the abstract below, §5, and the §7 grade table, which graded that structure Established. The Ince accessory-parameter count N_acc = (2d−1)(d−1) of §3 is withdrawn with it: the corrected count, via the index of rigidity, is P = d−1, conditional on irreducibility and the local data at {0, R}, with the d = 1 Kummer reduction giving the rigid P = 0. Unaffected: the location R, the exponent −γ, the s = R resonance-log, the connection coefficient C, the bounded null solution, and the transcendence conjecture, which remains a conjecture; EBR-I and EBR-Ib are unaffected. The PDF attached to this version still carries the uncorrected wording — the files of a published version are immutable — so this notice is the only correction that reaches a reader who visits this record, and nothing reaches a reader who has already downloaded the v1.0 PDF. EBR-I established, for positivity-hypothesis polynomial continued fractions, that the physical Borel object G(s) = Σ Q_n sⁿ/(dn)! is holonomic with dominant singularity at s = R = dᵈ/β_d and local form G(s) ∼ A(1 − s/R)^{−γ}, γ = (d+1)/2 + b_{d−1}/β_d, leaving the amplitude A = C·Γ(γ) (with C the coefficient prefactor) uncomputed. Here we characterize C structurally. We prove that G satisfies an order-2d Fuchsian ODE with exactly three regular singular points {0, R, ∞} and an explicit Riemann scheme, and that C is the connection-matrix entry linking the exponent-0 Frobenius solution at s = 0 to the dominant (exponent −γ) solution at s = R. The connection problem is generically non-rigid: its Ince accessory-parameter count is N_acc = (2d−1)(d−1), vanishing only at d = 1 (the rigid, hypergeometric degree) and positive for all d ≥ 2. We compute C to high precision by independent analytic continuation, reproducing the EBR-I coefficient prefactor to ≥33 digits and confirming its dependence on the constant term and leading coefficient of b; the local expansion at s = R is found to carry a logarithm, from a resonance between the exponent −γ and the integer exponents {0, …, 2d−2}, refining the EBR-I leading-order local form. We show C is generically not a Γ-quotient (consistent with its non-detection against rational, algebraic, and Γ-monomial-period bases) and that the EBR connection problem is the same flavor as — but a distinct ODE from — the Painlevé-V connection datum σ_conn of the companion V_quad analysis. We conjecture a rigidity dividing line: C is elementary exactly on rigid or special members, and a genuine transcendental period in the generic non-rigid case. A proof of transcendence is not given and is identified as the principal open problem. Grade statement (read before the results). The connection-coefficient characterization — the order-2d Fuchsian structure, the three singular points, the Riemann scheme and Fuchs relation, and the identification of C as a specific connection-matrix entry — is established symbolically and verified at d = 2, 3, 5, 7. The numerical value of C is computed by independent analytic continuation to ≥33 digits at d = 2, 3, 5, reproducing the EBR-I prefactor and its c- and β_d-dependence. The transcendence of C, and the rigidity dividing-line, are stated as conjectures: the non-rigidity count N_acc > 0 establishes the generic connection problem is not rigid, but does NOT prove that this specific connection entry is non-elementary (a non-rigid family may have special elementary members, and a single entry may simplify). No proof of non-elementarity, no explicit period integral, and no irreducibility/monodromy argument is given. That distinction is maintained throughout.