Exponentially‑decreasing closure margin forthe Wasserstein–Fisher gap along φ^{2n} Gibbs directions

The Fisher information metric and the Wasserstein–Otto metric agree on Gaussian ϐluctuations but separate on non‑Gaussian directions. We study this separation perturbatively along the family of pure‑Hermite Gibbs deformations d???? ∝ exp(−?? ??2??(??)) d??0 of a Gaussian background ??0 , where ??2?? is the order‑2?? Hermite polynomial. The second‑order Rayleigh–Schrödinger expansion of the Wasserstein–Fisher gap takes the form Δ????(??) = ??0 (??) + ??1 (??) ?? + ??2 (??) ?? 2 + ??(?? 3 ), and the question whether Δ????(??) = 0 has a real root reduces, at this order, to whether the closure‑resistance ratio ??(??) ∶= ??1 (??) 2/[??0 (??) ??2 (??)] reaches the threshold 4. We give a Hermite–Wick fast algorithm that produces ??0 , ??1 , ??2 as exact ratio‑ nals for ?? = 2, … , 100 — a range out of reach of straight polynomial expansion. The ratio is strictly monotone decreasing on all 98 consecutive pairs, from ??(2) ≈ 1.022 down to ??(100) ≈ 3.5 × 10 −15 . A linear ϐit of log ??(??) on ?? ∈ [40, 100] yields ??(??) ≈ ?? ?? ?? with ?? = 1.998, ?? = 0.71192, and decay rate − log ?? = 0.339792. The log‑residual standard deviation is 2.4 × 10 −5 . A running‑window slope of width 8 agrees with the global ϐit to seven decimals on the last windows. The data resolve a hyperasymptotic decomposition ???? (??) ∼ ???? ⋅(??!) 2−?? ?? ?? ?? ⋅?? −???? with the integer factorial exponents ???? = 2 − ?? supported by Borel‑transform Aitken extrapolation. Saddle‑point analysis of the Hermite‑basis sum representa‑ tions identiϐies the three exponential rates in closed form: ??0 = 16 at the saddle ?? ∗ = 1/2 (Appendix A), ??1 = 54 at the previously unexpected saddle ?? ∗ = 2/3 (Appendix B), and ??2 = 256 at ?? ∗ = 1/2 (Appendix C). The factor 3 3 in ??1 arises 1from the three sum indices each contributing one log 3 at the 2/3 saddle of the ϐirst‑order Rayleigh–Schrödinger triple sum, an emergent pattern absent from ??0 and ??2 where the saddle is at 1/2 and only log 2 contributions remain. The Stirling sub‑leading exponents turn out to be a universal ??0 = ??1 = ??2 = 3 — the multi‑ dimensional Laplace prefactors at the three different saddles all collapse to the same power of ??, an emergent feature SAGE exact‑rational extraction conϐirms to within 10 −3 on the ?? = [50, 99] tail. The saddle prefactors are ??0 = −1/(4 √ 2 ??), ??1 = 1/(4 ?? 3/2 2 1/4 ), ??2 = 1/(8 ?? 2 ). Combining these gives the full closed‑form asymptotic ??(??) ∼ ?? ⋅ ( 729 1024 ) ?? , ?? = ?? 2 1 |??0 | ??2 = 2, with the power ?? −(2??1−??0−??2 ) = ?? 0 cancelling exactly under the universal ???? = 3. The empirical decay base ?? = 729/1024 = (3 3/2 5 ) 2 is identiϐied by saddle‑point analysis combined with a 98‑point Hermite–Wick extraction; the strict inequality ?? < 1 reduces to the integer comparison 3 3 = 27 < 32 = 2 5 . We treat ?? = 729/1024 as an extremely well‑supported conjectural closed form: the saddle Laplace prefactors ???? derived in Appendices A, B, C give the integer triple (??0 , ??1 , ??2 ) = (16, 54, 256) with ?? = ?? 2 1 /(??0 ??2 ), where ??0 = 16 enjoys a com‑ plete saddle support (Appendix A), while ??1 = 54 and ??2 = 256 rest on saddle analysis combined with a piece‑dominance argument (Appendices B, C) plus the 98‑point exact‑rational veriϐication at 10 −7 relative precision. The integer prefac‑ tor ?? = 2 matches the pure‑exponential ϐit on ?? = [40, 100] (giving ?? = 1.998) and the SAGE exact‑rational extraction at ?? = 100. Higher‑precision SAGE Aitken Δ² extraction at 500‑bit precision on the ?? = [50, 100] tail gives the invariant sub‑leading combination 2??1 − ??0 − ??2 ≈ −0.0243 (Aitken Δ² on raw difference sequence converges to −0.024306 over 15 iterations, while linear least‑squares ϐit on ??(??) over the same tail gives slope−0.024327 with residual standard devia‑ tion 6.15 × 10 −5 ; the two methods agree to 4 digits). The individual ???? extraction is representation‑dependent and the combination is the physically invariant quantity governing the 1/?? correction to ??(??). The reϐined data‑supported asymptotic is therefore ??(??) = 2 ⋅ (729/1024) ?? ⋅ (1 − 0.024326/?? + ??(1/?? 2 )), with the 1/?? coefϐicient non‑vanishing at the empirical value −0.024326 (4‑digit cross‑method agreement, 6‑digit linear‑LS agreement with the round‑7 ϐinding). The closed‑form identiϐication of −0.024326 as an elementary combination of saddle‑derivative invariants is recorded as Open Problem O8. Exact‑rational computations up to ?? = 100, together with saddle‑point analysis and indepen‑ dent SAGE 500‑bit veriϐication, provide strong evidence — but do not constitute a rigorous theorem — for the closed‑form expression above. With the ϐirst 49 data points a SAGE ore_algebra.guess search up to recurrence order 8 and polynomial‑coefϐicient degree 12 returns no holonomic recurrence, so the asymp‑ totic constants ?? and ?? are not accessible from a Zeilberger creative‑telescoping certiϐicate of bounded order and degree on those data. We close with geometry, RG, and ϐive open problems.