Neutrino Mass Sum Σm_ν = 0.089 eV, Normal Ordering Derived

Overview

V.T165/T225 and V.T189 together establish the neutrino mass predictions: the CF-asymmetric grid (Δpq = 1.16, Δpr = 0.87) at +7.4 ppm gives the exponent parameters; the sum Σm_ν = 0.089 eV follows from the σ-polarity matrix (p = 3.7, q = 4.8, r = 2.8). The normal hierarchy (m₁ < m₂ < m₃) is proven from p < q as a theorem (IV.R395), Lean-verified. Σm_ν = 0.089 eV is consistent with DESI Year 1 and represents a 4.5σ detection target for DESI full survey.

Parameter accounting (above-the-fold disclosure)

The framework’s “single posited master constant” framing on the homepage refers to the K0–K6 kernel layer where ι_τ = 2/(π+e) is the only continuous input. This page operates at the bridge layer, where matching observed neutrino structure introduces additional fitted parameters that are not derived from the kernel.

Concretely, the prediction Σm_ν = 0.089 eV here uses:

• 3 fitted real-valued exponent parameters: the σ-polarity matrix (p, q, r) = (3.7, 4.8, 2.8). These are bridge-layer fits, not derivations from K0–K6. They are chosen so that the implied individual masses m_i ∝ ι_τ^p_i sum to a value compatible with the observational neutrino-mass-sum upper bound.
• 2 grid parameters from V.T189’s CF-asymmetric grid (Δpq, Δpr) = (1.16, 0.87) — these are CF-derived (they come from the partial-quotient structure of ι_τ) and are not free fits.

Net cost-of-claim for Σm_ν = 0.089 eV: three real-valued parameters fitted at the bridge layer, plus the kernel posit ι_τ. The page’s framework-internal status is therefore:

• τ-effective: the sum Σm_ν ≈ 0.089 eV (under the fitted exponents) and the ordering m₁ < m₂ < m₃ (proven Lean-verified from p < q in IV.R395).
• Conjectural: the individual mass splittings — see the “completion status” paragraph below for the magnitudes by which the underlying Δm² values currently diverge from measurement.

This page is the project’s clearest counterexample to a naive reading of the “zero free parameters” framing, and it is surfaced explicitly here. A future-sprint task is either to (a) derive (p, q, r) from a kernel-level structural argument (which would retract the bridge-layer parameter cost) or (b) preserve the bridge-layer accounting and improve the individual-splittings derivation. The framework’s posture is that (a) is the eventual obligation; this page documents the current state honestly.

Detail

The τ-framework establishes neutrino mass predictions through two related results:

V.T189 (Wave 5A/7B) derives the CF-asymmetric grid (Δpq, Δpr) = (1.16, 0.87) from the CF structure of ι_τ. The key observation is that CF(ι_τ) = [0; 2, 1, 13, 3, …] is asymmetric (13 ≠ 3 ≠ 2), and this asymmetry is inherited by the neutrino mass exponent ratios. The grid optimum is at (203/175, 609/700) = (1.16, 0.87) at +7.4 ppm — the most precise neutrino prediction in the framework.

V.T165/T225 establish Σm_ν = 0.089 eV from the σ-polarity matrix parameters. The three exponents (p, q, r) = (3.7, 4.8, 2.8) give individual masses m_i ∝ ι_τ^{p_i}; summing gives Σm_ν = 0.089 eV. The normal ordering (IV.R395) follows from p < q (since p = 3.7 < q = 4.8 implies m_1 < m_2 which is necessary for normal hierarchy).

The completion status is partial because the individual mass splittings (Δm²₂₁ and Δm²₃₂) are off: Δm²₂₁ is 6.2× from measured; Δm²₃₂ is +22.9% off. The sum Σm_ν and normal ordering are τ-effective; the individual splittings are conjectural.

Result Statement

V.T165/T225: Σm_ν = 0.089 eV from (p,q,r) = (3.7, 4.8, 2.8). V.T189: CF-asymmetric grid (Δpq, Δpr) = (1.16, 0.87) at +7.4 ppm. Normal ordering proven from p < q (IV.R395), Lean-verified. Individual splittings remain conjectural.