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

[2026-09-23] Erratum — the logarithm at s = R. This record states that the local expansion at the dominant singularity s = R carries a logarithm arising from a resonance between the exponent −γ and the integer exponents {0, …, 2d−2}. That statement is withdrawn as an unconditional claim. The correct criterion is the one already given in this record’s own v2 erratum: the point s = R is semisimple for γ ∉ ℤ and carries a single resonance logarithm only for γ ∈ ℤ. For the reference family b = 3n² + n + 1 the exponent is γ = 3/2 + 1/3 = 11/6, which is not an integer, so s = R is semisimple and no logarithm is present. The v2 erratum nevertheless listed “the s=R resonance-log” among the items it declared unaffected; that listing was inconsistent with its own criterion and is corrected here. This was already published as a correction elsewhere. EBR-III (“The EBR operator at d=2 is not a G-operator and its differential Galois group”, 10.5281/zenodo.20684732) states that the local monodromy at R is a semisimple complex pseudo-reflection with spectrum {1, 1, 1, e^{iπ/3}}, “correcting an earlier resonance logarithm reading”, and notes explicitly that a −γ logarithm requires γ ∈ ℤ whereas here γ = 11/6. This erratum brings the present record into agreement with that published correction. What is NOT affected. The location R, the exponent −γ, the accessory-parameter count P = d−1 (which the v2 erratum already notes is the same under either local structure at R), the high-precision evaluation of the connection coefficient C and its ≥33-digit agreement with the EBR-I prefactor, the non-detection of C against rational, algebraic and Γ-monomial-period bases, and the transcendence conjecture (still a conjecture) all stand as stated. Inherited scope in EBR-I. EBR-I (concept DOI 10.5281/zenodo.20564079) states the coefficient asymptotic with “no logarithmic factor”. By the criterion above that clause holds for γ ∉ ℤ, and a scope correction restricting it to γ ∉ ℤ is issued on EBR-I with this erratum. For the reference family γ = 11/6 ∉ ℤ, so there EBR-I’s statement stands unchanged. Cluster statement (read with this notice). A dependency analysis of the 61-record deposit corpus finds this record inside a group of 18 records that cite one another in both directions across successive versions. The deposit metadata cannot establish an ordering within that group, so it cannot be determined from the record which way influence ran between its members. The group contains all three records whose headline claims are withdrawn or under review: CM — Complex Multiplication as a Transcendence Predicate for Degree-2 Polynomial Continued Fractions (concept DOI 10.5281/zenodo.19931635) companion — A non-classical Painlevé V transcendent from a quadratic polynomial continued fraction: surface classification and resurgent Stokes data (concept DOI 10.5281/zenodo.20455089) LMP — An explicit exponential-period representation of the V_quad connection coefficient (concept DOI 10.5281/zenodo.20719042) This record is not one of the three, but it cites the companion and sits inside the group. This notice therefore does not assert that the other members of that group are unaffected. Corrections to them are being issued separately; each cites the others. Corrections issued together. This notice is issued in one batch with the corrections to the records below. Each carries its own notice, which this one does not summarise. CM — Complex Multiplication as a Transcendence Predicate for Degree-2 Polynomial Continued Fractions (concept DOI 10.5281/zenodo.19931635; correction notice, this batch) T2B — Two arithmetic classes of degree-(2,1) Trans-stratum continued fractions: a Birkhoff–Trjitzinsky / Gauss-continued-fraction dichotomy (concept DOI 10.5281/zenodo.19783311; description correction, this batch) EBR-I — An Edge–Borel Radius Theorem for Positive Polynomial Continued Fractions (concept DOI 10.5281/zenodo.20564079; scope correction, this batch) LMP — An explicit exponential-period representation of the V_quad connection coefficient (concept DOI 10.5281/zenodo.20719042; correction notice, this batch) companion — A non-classical Painlevé V transcendent from a quadratic polynomial continued fraction: surface classification and resurgent Stokes data (concept DOI 10.5281/zenodo.20455089; corrected in its version 1.3, already deposited) 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 was stated to carry a logarithm, from a resonance between the exponent −γ and the integer exponents {0, …, 2d−2}. [2026-09-23 correction: WITHDRAWN. The logarithm occurs only on the subvariety γ ∈ ℤ. For γ ∉ ℤ the point s = R is semisimple and there is no logarithm; for the reference family b = 3n² + n + 1, γ = 3/2 + 1/3 = 11/6 ∉ ℤ, so no logarithm is present. See the notice at the head of this description.] 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 connection coefficient C, the bounded null solution, and the transcendence conjecture (still a conjecture) are unaffected. [2026-09-23 correction: the s=R resonance-log was previously listed here as unaffected. That was inconsistent with the criterion stated in the same sentence above (R semisimple for γ ∉ ℤ, a single resonance-log only for γ ∈ ℤ), and it is removed from this list. EBR-I's local statement is scoped rather than unaffected; see the notice at the head of this description.]