The branch exponent of the leading Borel singularity reads two coefficients, not one: a closed form, and the falsification that produced it

A short note reporting one closed-form identity and the falsification that produced it. The closed form. For a quasi-linear family with leading coefficients beta_d, beta_(d-1) and normalisation constant C, the branch exponent of the Borel transform at its leading singularity is beta_exp = -beta_(d-1) sqrt(-C) beta_d^(-3/2), equivalently -beta_(d-1) xi_0 / (2 beta_d), derived by exact symbolic substitution at d = 2, 3, 4, 5 in an implementation sharing no code with the numeric pipeline. The falsification. The conjecture beta_exp = -beta_d^(-(d+1)/d) is false in all three forms in which it can be stated: as a claim that beta_exp depends on beta_d alone, as the same claim with a free constant c(d), and as stated. Two independent kills: the factor beta_(d-1), and the exponent -3/2, which is d-independent and coincides with -(d+1)/d only at d = 2. The note reports the falsification at equal length with the result, because the way the conjecture survived is the more transferable finding. It agreed exactly -- not to eight digits, not to nineteen, but identically -- at the only family where anyone had looked, because that family sets two of the formula's parameters to one and minus one. A suppressed parameter is invisible at any precision from a single data point. Normalisation. Three conventions are audited from the sources' own text before any value is transported: two Borel normalisations coexist in the record stating beta_exp and differ by one unit in the branch exponent, which is a sign flip; two distinct quantities are written xi_0, a half-gap in the uniformiser and a full Stokes gap, numerically equal at the reference family; and two distinct objects are written rho. Every verdict in the note is stated as a magnitude or a ratio, which is invariant under the first of these. What is not claimed. No deposited result is corrected. No structural ground is supplied for any surface-type demotion -- a level-1 result would have been such a ground, and this is level 2; the relevant demotion stands on its own deposited grounds, which was checked by search rather than assumed. beta_exp and rho are not the same invariant, though they share inputs. And an integer-relation identity search would not have helped, because it would have succeeded: it would have returned the wrong law from the reference family at machine precision and stopped.