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

For the positive-coefficient polynomial continued fraction with b (n) = 3n² + 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.