The Edge-Borel growth constant is a non-classical exponential period: an SL(2) monodromy on the Painleve III(D8) surface and a conditional transcendence theorem

v1.1 correction notice. This version corrects the identification of κ as a Stokes multiplier, withdraws the conditional transcendence theorem (Theorem 4) that the title refers to, and corrects four smaller statements, after an independent re-audit that recomputed the Stokes jump by two routes. The paper file, the title and the notes field are unchanged from v1.0; where any of them differs from this notice, this notice takes precedence. VERIFIED marks a numerical finding that is not proved. What stands. The value κ = 1.5394948485766410… is evaluated directly and correctly. The large-order amplitude A0 = lim Qn/((n!)23nn1/3), recomputed from the recurrence alone, agrees with the archived κ/Γ(4/3) to 124 digits, and κ = Γ(4/3)A0 is the transfer theorem applied to the (1−3z)−4/3 singularity of Φ(z) = ∑ Qnzn/(n!)2. The object that carries κ is the order-4 operator L that annihilates Φ: L is holonomic, its finite singular points are 0 and 1/3, and z = ∞ is an irregular singular point of slope 1/4. Proposition 2 (κ is an exponential period, STRUCTURAL) rests on the integrand chain y → Φ = y ⊙ I0(2√z) → AΦ, not on the bridge of item 2, and it stands, with κ read as a connection coefficient of L rather than of the H2 connection. κ is not the Stokes multiplier. Equation (12) asserts κ = Γ(4/3)A0 = AΦ = s*(B; x = 0), “the factor between s* and κ being the identity”. The script that records this bridge computes no Stokes multiplier: it makes no integration, ODE or continuation call, and the three identities it records relate κ, AΦ and C only. Computed directly, the lateral jump of the Borel-2 sum of y is σ times the recessive solution t−11/12e−2/√(3t)(1 + O(√t)), with σ = ±2πi·3−11/12·C = ±7.0134…i, where C = κ√π/Γ(4/3) = 3.0557068078904813…, so σ/κ ≈ ±4.5557i. VERIFIED by two routes: the large-order asymptotics of Qn, and a numerical Borel-2 summation whose jump ratio extrapolates to 1 within 10−8. The factor cannot be the identity in any normalisation in which the recessive solution is real for t > 0: σ is then purely imaginary, while κ is real. The bridge also names the wrong Stokes datum. The scalar-gauge step is correct for the homogeneous operator H2, whose Stokes matrices relate its two formal solutions of exponential type e±2/√(3t). But y solves the inhomogeneous equation H2y = −1, and the jump of item 2 is an entry of the Stokes data of the order-3 operator D∘H2, linking y to the recessive solution; it is not the off-diagonal entry of H2. This point is structural; it was not computed. Theorem 4 is withdrawn, not downgraded. Its hypothesis H4 certifies κ as “the non-vanishing off-diagonal connection entry” of the rank-2 connection H2. By items 2 and 3, κ is not that entry, so the argument does not reach κ. Its last sentence, “Consequently … C = κ√π/Γ(4/3) is transcendental as well”, does not follow even under H1–H4: √π/Γ(4/3) is itself transcendental, since π and Γ(1/3) are algebraically independent (Chudnovsky), so the transcendence of κ would not imply that of C. For the same reason, the statement in §1 that “a transcendence statement for any one of them is a transcendence statement for all three” (of κ, A0 and C) is false. The record contains no argument for the transcendence of κ or of C, conditional or otherwise, that survives these corrections. Whether either is transcendental is open. Smaller corrections. (a) §4.1: H2 does not have trivial determinant character. Its Wronskian is t−10/3, so the determinant maps GGal(H2) onto the cube roots of unity, and the group is not contained in SL(2). The Kovacic result SL(2) holds for the trace-normal form (7), which differs from H2 by an algebraic scalar factor, so the dimension 3 used in hypothesis H2 is unaffected. (b) Reference [7] is A. Its, O. Lisovyy and A. Prokhorov, Monodromy dependence and connection formulae for isomonodromic tau functions, Duke Math. J. 167 (2018), 1347–1432. (c) Reference [3], EBR-III, has the concept DOI 10.5281/zenodo.20684732. (d) §1 calls CEBR “the EBR-III connection coefficient”. EBR-III calls it the coefficient-asymptotics prefactor and reserves “connection coefficient” for a different constant; the values agree. This notice concerns the κ-bridge, Theorem 4 and the items above. The Painlevé III(D8) selection (§3), the monodromy point (§6), the integer-relation nulls (§7) and the test against the algebraic locus (§8) were not re-audited. For the positive-coefficient polynomial continued fraction with b(n) = 3n^2 + n + 1, the Edge–Borel growth constant is governed by a single transcendental number κ = 1.5394948485766410…, the Stokes/connection constant of a rank-2 ordinary differential operator H2 that arises by a Borel–2 reduction of the generating function. We identify κ inside the isomonodromy and period frameworks and assemble, with explicitly graded hypotheses, a conditional transcendence theorem for it. We establish, at increasing evidential strength: (i) H2 has singular set {0, ∞} with both points irregular of ramified slope 1/2; its index of rigidity is 0 (non-rigid, moduli dimension 2), and the local ramification data (2, 2) select, under a surface-type selector (RULE S) we state precisely, the Sakai surface D8(1) — equivalently Painlevé III(D8); the selector is computed, not pattern-matched. (ii) The differential Galois group is GGal(H2) = SL(2), by an exact Kovacic computation closing the imprimitive case through reducibility over the unique quadratic cover, so H2 is irreducible with no Liouvillian solutions. (iii) On the convergent Borel–2 transform Φ, the constant κ is realised constructively as the connection coefficient AΦ of an order-4 operator with an irregular point of slope 1/4, computed by a monodromy spectral projector to 129 digits and agreeing with two further independent channels (a large-order asymptotic channel using only the integer sequence Qn, and a frozen-composition channel); this places κ in the ring of exponential (rapid-decay) periods of the connection, with κ = Γ(4/3)·A0. (iv) We determine the monodromy point (tr M0, κ) with tr M0 = −51.0655631399546… (hyperbolic, hence irreducible), survey the solved D8 connection problems and find that none computes κ (a tau-side/Lax-side distinction we keep explicit), record two positive-control-validated integer-relation nulls, and verify the point lies off the algebraic Painlevé III(D8) locus. Under the Fresan–Jossen period conjecture for exponential motives, a differential-to-motivic dimension comparison, and a verified non-degeneracy datum, we conclude κ ∉ ℚ-bar, hence the EBR connection coefficient C = κ√π/Γ(4/3) is transcendental. We are explicit, throughout and in both directions, that this is a conditional statement: neither the large Galois group, nor the non-rigidity, nor any integer-relation null proves transcendence, and a closed form — had one appeared — would have argued the opposite. The conditional theorem is the program's ceiling; unconditional transcendence of C/κ remains a conjecture whose only route is a separate period analysis. Each hypothesis is individually graded (STRUCTURAL exponential-period membership; a CONJECTURED-motivic / VERIFIED-differential dimension input; the external CONJECTURED period conjecture; a VERIFIED period-count datum), and a first-class gap list records every place the chain relies on something unproven. This is the fourth entry in the Edge–Borel Radius (EBR) series. PROVEN is reserved for the Lean 4 / Mathlib cores established in EBR-III; this paper introduces no new Lean core but flags its finitary identities (the order-4 L-operator and indicial data, the H2→B gauge-chain identity, the Kovacic emptiness certificate, the de Rham dimension counts) as Lean-core candidates. A complete reproducibility package (reproducer scripts, results JSONs with canonical SHA-256 self-hashes, a one-command verifier, and a claims-ledger snapshot) accompanies the deposit; the three independent κ channels are each separately reproducible.