The Gap–m–Gap picture: a Planck-frame reading of the Dirac clock and a falsifiable neutrino-mass prediction for JUNO

We re-read known quantum physics as a film of Planck frames: a particle of mass m advances its phase by θ = m/mP at every Planck time tP, one full turn being the quantum h; the unit mass M = 2πmP is the mass whose Compton wavelength is one Planck length. The frame rule is the Dirac quantum walk (coin angle θ + chiral shift), whose continuum limit is the 1+1 Dirac equation, with exact lattice dispersion cosω = cosθ cos k, Zitterbewegung at 2mc2/ℏ, and a double slit with a graded which-path detector obeying Englert's V2 + D2 ≤ 1. The maximal-mass bound of the Dirac automaton of Bisio, D'Ariano and Tosini (sinθ ≤ 1) reads θ ≤ π/2 (m ≤ M/4) in the angle variable; we read it as a Nyquist condition and note that the author's original drawing sits on that boundary. Collapse is read as a clock tick plus rare Bell–Tumulka flashes (rate ∝ m, ε ~ 10-39 per Compton turn, within the window allowed by collapse-model bounds). The simulations use a global lattice in the rest frame of the source; with frames counted in proper time the clock phase is Lorentz invariant by construction, but a covariant version of the full update rule is not constructed and invariance remains an assumption. We state honestly what is a rewriting of de Broglie, Dirac and Schrödinger, and what is new. The one falsifiable number is in the neutrino sector. The neutrino families are placed on a ring enclosing one Planck radian of total phase (mPc2tP/ℏ = 1). By Bloch's theorem this gives the three-phase (Koide–Brannen) form √mk = A[1 + a cos(δ + 2πk/3)]; only the total phase matters (gauge theorem). The neutrino families are three (LEP: Nν = 2.9840 ± 0.0082; Janot–Jadach: 2.9963 ± 0.0074), and the Planck radian is divided among them: δν = 1 rad/3. With amplitude a = 1 (motivated: the largest amplitude keeping all roots positive, and the equipartition of the frame walk) no free parameter is left except the scale. Comparison with data: the ratio Δm221/Δm231 = 0.029972 is compatible with JUNO at 207 days (0.02945 ± 0.00044, +1.2σ); only N = 3 is compatible, and the data correspond to N = 3.03 ± 0.02. Predictions (quantities not yet measured): masses m1 = 0.789, m2 = 8.71, m3 = 50.10 meV, Σmν = 59.6 ± 0.3 meV, mβ = 8.81 meV, mββ = 0.96–4.21 meV, normal ordering, Δm232 = (2.434 ± 0.025)×10-3 eV2, and JUNO's final value at full precision, Δm221 = (7.52 ± 0.08)×10-5 eV2, against 7.445 predicted by Alikhanov's relation (R = 17 − 12√2 = 0.029437) and 7.75 by Brannen's phase (δν = 2/9 + π/12, R = 0.03084), the origin of the three-phase form, kept as a comparison and now 3.2σ from JUNO. If the JUNO central value holds, Brannen's value is excluded at 5σ in 2027; JUNO separates ours from Alikhanov's between 2027 (3σ) and about 2031 (final precision). We respect the DESI DR2 Bayesian bound (64.2 meV); the Feldman–Cousins bound (53 meV) excludes every normal-ordering spectrum. Limits stated: the rule holds only in the ℏ convention, the charged leptons (δ = 2/9) have no analogous reading that survives the quarks, lepton mixing is not explained, and the Majorana sector needs extra structure. Explicit falsification criteria and the full Python code that regenerates every computed number and figure are included. Riassunto. Le famiglie di neutrini sono tre (LEP) e il radiante di Planck si divide fra le tre famiglie: δν = 1 rad/3, con ampiezza 1 (motivata). Il rapporto R = 0,029972 è compatibile con JUNO a 207 giorni (+1,2σ). Previsioni: masse 0,789 / 8,71 / 50,10 meV, somma 59,6 ± 0,3 meV, mβ 8,81 meV, mββ 0,96–4,21 meV, ordinamento normale, e il valore finale di JUNO Δm221 = (7,52 ± 0,08)×10-5 eV2 (Alikhanov 7,445; Brannen 7,75), atteso fra il 2027 e il 2031. Tutti gli script Python per rigenerare ogni numero e figura sono allegati. A first, informal version of the intuition was deposited by the author in December 2024 (doi:10.5281/zenodo.14551679).