T2 — the closed form: δ(ε) = 1 − (2/π) K(2ε)

Reproduce: ../cosmic-web/.venv/bin/python scripts/run_p5_series.py (~15 s). Results → artifacts/p5_series.json. Supersedes the numerical value hunt of run_c4_precision.py (whose c₄ = 9/4 is now a corollary).

The result

For the magnetic contact structure with constant fractional gradient ($B = B_0 e^{\varepsilon x/r_L}$, $\varepsilon = \lvert\nabla\ln B\rvert\,r_L$), the angle-averaged θ-period $T$ obeys exactly

\[\frac{\langle T\rangle}{T^{\text{flat}}} \;=\; \frac{2}{\pi}\,K(2\varepsilon), \qquad\text{i.e.}\qquad \boxed{\;\delta(\varepsilon) \;=\; 1 - \frac{2}{\pi}K(2\varepsilon)\;}\]

(Notation: $T$ is the exact θ-period throughout; $t_c$ is reserved for the measured conjugate time, whose identification with $T$ is the conditional step.)

with $K$ the complete elliptic integral of the first kind (modulus convention $K(k) = \int_0^{\pi/2} d\phi/\sqrt{1-k^2\sin^2\phi}$). This is the period-average law. Reading the same expression as the angle-averaged conjugate (refocusing) time is conditional on the open conjugate-time = period identification (Conjecture A): verified to 1e-8 on this exponential profile, REFUTED off it (V1) — so the caustic reading is exponential-conditional, not established.

The chain (each link tested against the geodesic code)

Integral of motion (exact, one line). Along a geodesic, $\dot\theta = e^{\varepsilon x}$ and $\dot x = \cos\theta$, so $d\dot\theta/d\theta = \varepsilon\cos\theta$: $\dot\theta(\theta) = 1 + \varepsilon(\sin\theta - \sin\theta_0).$ The angular dynamics decouples — the system is integrable.
Per-angle θ-period (closed form); its identification with the conjugate time is the open step. $T(\theta_0) = \oint \frac{d\theta}{1+\varepsilon(\sin\theta-\sin\theta_0)} = \frac{2\pi}{\sqrt{(1-\varepsilon\sin\theta_0)^2 - \varepsilon^2}}.$ Verified: matches the Jacobian-zero conjugate times of the integrator to $\sim 10^{-8}$ relative, at ε = 0.1, 0.25, 0.4, all launch angles. (Analytic status: the period formula is exact; the identification of the conjugate time with the period is verified numerically to integrator precision — the remaining formal step is the Jacobian argument for this integrable family.)
The angle average is elliptic. $\frac{\langle T\rangle}{2\pi} = \frac{1}{2\pi}\oint \frac{d\phi}{\sqrt{(1-\varepsilon\sin\phi)^2-\varepsilon^2}} = \frac{2}{\pi}K(2\varepsilon).$ Verified two ways: symbolically, term by term through $O(\varepsilon^{12})$ (sympy — every coefficient $\binom{2m}{m}^2 4^{-m}$ matches, difference exactly 0); and numerically, $\delta$ from the full pipeline matches $1-(2/\pi)K(2\varepsilon)$ to $10^{-10}$–$10^{-9}$ absolute across ε = 0.05 … 0.45. The formal reduction of the integral is now WRITTEN OUT AND PROVEN (2026-07-12): phase-shift to cosine, t = tan(theta/2) factorises the radicand exactly into [t^2 + (1-2eps)][1 + (1+2eps)t^2], and Gauss’s INT_0^inf dt/sqrt((t^2+a^2)(t^2+b^2)) = K(k)/a with a = (1+2eps)^{-1/2}, b = (1-2eps)^{1/2} gives k^2 = 1 - (1-2eps)(1+2eps) = 4 eps^2 and prefactor 4 — i.e. exactly (2/pi)K(2eps), modulus convention, valid for eps < 1/2 (the factor 1-2eps > 0 is the domain; its vanishing is the critical gradient). Chain verified to machine precision + sympy: scripts/run_t2_reduction_check.py. The elliptic-reduction step (item 3) is CLOSED; the conjugate-time = period identification (item 2) remains the open formal step, unchanged.

Corollaries

The Taylor coefficients are squared normalised central binomials: $-\delta(\varepsilon) = \sum_{m\ge1} \left[\binom{2m}{m}2^{-m}\right]^2 \varepsilon^{2m} = \varepsilon^2 + \tfrac94\varepsilon^4 + \tfrac{25}4\varepsilon^6 + \tfrac{1225}{64}\varepsilon^8 + \cdots$ The measured $c_4 = 2.2497 \pm 0.0009$ is now a consequence; the fitted $c_6$ was always a model-dependent nuisance (6.18–6.57 across the fit models in c4_precision.json; the fitted $c_8 \approx 23.6$ likewise scatters against the exact $1225/64 \approx 19.14$) — high-order coefficients on a 10-point window are unstable, and the closed form is verified directly against $\delta$ to $10^{-10}$, superseding the fits. The earlier “odd-square” guess ($c_6 = 25/4$ coincides, $c_8$ does not) is corrected by the exact law.
A critical gradient exists: $\varepsilon_c = 1/2$. $K$ diverges at unit modulus, so the mean period diverges as $\varepsilon \to 1/2$; per angle, the slowest launch direction ($\sin\theta_0 = 1$) has $T = 2\pi/\sqrt{(1-\varepsilon)^2-\varepsilon^2} = 2\pi/\sqrt{1-2\varepsilon}$, diverging at exactly $\varepsilon = 1/2$. Verified on the measured conjugate times (their tracking of $T$ is the Conjecture-A step): $t_c/2\pi = 3.162$ at ε=0.45 and $7.071$ at ε=0.49 (both matching $1/\sqrt{1-2\varepsilon}$); above the critical gradient, for exactly the launch band sin θ0 ≥ (1−ε)/ε, no conjugate point is detected within the eight-period integration window (finite-horizon evidence, run_r6_supercritical.py), while every other angle keeps $t_c = T$. Physically, three separate claims: the mean period diverges — a theorem; for the slowest launch band, refocusing is not observed on any integrated horizon once the field changes by more than half across a Larmor radius — a finite-horizon measurement; and that the launch-averaged caustic opens with the period is Conjecture A — the rest of the beam keeps refocusing.
The inversion of Part 3 upgrades from leading-order to exact for the period-average δ on this exponential profile: $\varepsilon = \tfrac12 K^{-1}!\big(\tfrac\pi2(1-\delta)\big)$ (as a caustic statement, conditional on Conjecture A; general 1D profiles carry the $1-\tfrac34\beta$ calibration).

Novelty status (per the P5 audit)

Second-order guiding-centre theory contains the machinery but not this statistic; we found no statement of this closed form in the checked literature. It is also a natural fit to the sub-Riemannian lineage: the SE(2) sub-Riemannian geodesics (Sachkov–Moiseev) are governed by Jacobi elliptic functions, and here the magnetic contact structure’s caustic turns out to be governed by the complete elliptic integral. Claim class: exact result, candidate-new, pending one more targeted literature pass on gyro-period integrals in exponential field profiles.

Post-review addenda (pointers)

The chain above was subsequently sharpened by four results recorded elsewhere: step 2 (elliptic reduction) is PROVEN (run_t2_reduction_check.py; D5/article §3.5); the caustic-vs-period distinction became load-bearing — the identification t_c = T is EXPONENTIAL-SPECIFIC (V1 counterexample, V1-profile-law.md), the caustic profile law c2 = 1 − (3/4)β is DERIVED (run_o6_caustic_c2.py, article Prop 4.2), and above the critical gradient exactly the launch band sin θ0 ≥ (1−ε)/ε shows no conjugate point within the eight-period integration window (finite-horizon evidence) while all other angles keep t_c = T (run_r6_supercritical.py). The “one more targeted literature pass” promised above has run: machinery found (Northrop, Littlejohn), statistic/closed form/critical gradient not found — recorded as a bounded search, D5 + article §6.