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

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.

Erratum (v2). §2/Prop. 2.1: the point at ∞ is an irregular singular point of Poincaré rank 1, not regular; the ODE has two regular singular points {0, R} and an irregular point at ∞ (a regular-to-irregular connection problem), so the “three-regular-singular-point Fuchsian” description is corrected. §3: the Ince Fuchsian accessory count N_acc=(2d−1)(d−1) is withdrawn (it assumed three regular points); the corrected count, via the index of rigidity of the rank-2d connection with irr_∞(End∇)=2(2d−1), is P=d−1 accessory parameters, conditional on irreducibility and the apparent/semisimple local data at {0, R} (R is semisimple for γ∉ℤ and carries a single resonance-log for γ∈ℤ; both give the same count), with the d=1 confluent (Kummer) reduction giving the rigid P=0. The location R, exponent −γ, the s=R resonance-log, the connection coefficient C, the bounded null solution, and the transcendence conjecture (still a conjecture) are unaffected; EBR-I and EBR-Ib are unaffected.