Phase 5 — litmus tests: from “signs of new science” to claims
Three novelty candidates came out of P3/P4. This phase runs the tests that decide them.
0. The novelty audit (literature check, done first)
| Candidate | Audit result |
|---|---|
| No-spiral theorem (force-free ⇒ radial-only nulls) | Downgraded, honestly. The MHD-relaxation literature (Fuentes-Fernández & Parnell, A&A 2012–13, spiral-null relaxation series) established dynamically that spiral nulls with spine current settle into genuinely non-force-free equilibria. Our 3-line pointwise proof is the elementary algebraic form of a dynamically-known fact. What remains ours: the statement as a catalogue audit, correctly scoped — any spiral null in a smooth force-free (bounded-α) extrapolation catalogue is a numerical force-free violation; non-force-free and data-driven MHD extrapolations are OUTSIDE the theorem and may host spiral nulls. The once-cited “radial-only Sun vs spiral-rich magnetotail” dichotomy is withdrawn: the T96 magnetotail census was retracted (S4b boundary audit — every root sits outside the model’s own validity domain). Claim class: clarification + audit corollary, not discovery. |
| δ(ε) = −ε² − (9/4)ε⁴ − … (gyro-refocusing delay) | Survives. Second-order guiding-centre theory (Littlejohn 1981; Brizard 1995; Hahm) contains the machinery (second-order gradient corrections to gyromotion) but the conjugate-time / caustic statistic itself is not a named output of that literature. Novelty contingent on the analytic derivation and a deeper GC-literature dive. |
| The scale-crossover observable ($w_4(r)$ plateaus 2/3/4; knee = pair separation) | Survives. Null-detection practice is pointwise (Poincaré index, trilinear, FOTE — Olshevsky et al. 2020); the nearest neighbour, multifractal local-dimension analysis in reconnection turbulence, is a statistical dimension of dissipation fields, not a per-point flux-reach exponent. No prior use of homogeneous-dimension scaling as a null/pair diagnostic found. |
1. T1 — THE RACE (decisive): crossover vs quadratic-fit root-finder
R2’s lesson, aimed at our own uniqueness claim: a model fit can also find sub-resolution pairs — fit a divergence-free quadratic field over the ball, root-find it, read the separation. If that beats the crossover everywhere, the crossover remains a beautiful observable but not a tool, exactly as classification fell to the linear fit.
Protocol (pre-registered). Input: the SAME noisy gridded $\mathbf B$ to both methods (no analytic potential — the crossover must construct its own $A$ from the ray/Poincaré gauge $A(\mathbf r) = -\mathbf r \times \int_0^1 s\,\mathbf B(s\mathbf r)\,ds$). Two legs:
- Pure fold — the field IS a divergence-free quadratic, i.e. the fit’s exact model class. Pre-registered prediction: the fit wins here, possibly totally. (If it does not, that itself is informative about noise conditioning.)
- Contaminated — fold + a smooth non-polynomial background (far point-charge field, 25% RMS over the ball): both methods misspecified; the crossover is model-free. Pre-registered prediction: open; this leg decides the claim.
Configurations: pair separations $2\sqrt\mu \in {0.4, 0.2, 0.1}$ on an $L=1$, $41^3$ grid (separations of 8, 4, 2 grid cells — resolved to marginal); noise $\sigma \in {0, 0.1, 0.25}$ (relative RMS over the info ball), $N=8$ draws. Metrics: relative separation error (median ± spread) and pair-vs-merged detection rate. The crossover’s knee-to-separation constant is calibrated ONCE on a single noiseless pure configuration, then frozen (disclosed).
Verdict rule: the crossover claim survives as a tool only if it beats the quadratic fit somewhere honest (the contaminated leg at realistic noise); otherwise it is demoted to conceptual observable — stated exactly so.
2. T2 — the series: pin $c_6$, derive $9/4$
- $c_6$ precision (cheap, decisive for the pattern): intercept fits of $z_2 = (z - 9/4)/\varepsilon^2$ on the high-precision δ data; verdict on $25/4$ and the odd-square conjecture $c_{2m} = (2m-1)^2/4$.
- Analytic derivation (time-boxed): perturbative solution of $\dot x = \cos\theta,\ \dot y = \sin\theta,\ \dot\theta = e^{\varepsilon x}$ around the circular orbit; conjugate time from the Jacobian zero of the exponential map; angle-average. Milestones: $c_2 = 1$ exactly at $O(\varepsilon^2)$; $c_4$ at $O(\varepsilon^4)$ by computer algebra. Partial progress documented as such.
3. T3 — catch a fold forming (real data)
Fetch an HMI LOS sequence across the AR11158 flux-emergence day (2011-02-13, the textbook emergence event); track the null census per frame in the emerging region; if a pair is born at our resolution, measure the knee migration. Honest expectation: birth at 1024²-scale resolution is not guaranteed; a null-count time series is the minimum deliverable.
(T4, MMS in-situ cross-match, is deferred: new data infrastructure, its own phase.)