Signed Endpoint Mass in Tritium Beta Decay: A Nuisance-Null Differential Test and The Limits of Tachyonic Completion

This paper sets up a falsifiable, endpoint-only search for a negative neutrino mass-squared in tritium beta decay, and asks what would actually count as evidence for it. The motivation is the recurring negative central value returned by direct kinematic experiments — most recently KATRIN's m²_ν = −0.14 eV². Rather than claiming this points to a tachyonic ("faster-than-light") neutrino, the paper turns the question into a well-posed statistical and theoretical problem and shows where the literal tachyonic reading breaks down.

Phenomenologically, it introduces a common-offset mass ansatz that shifts all three neutrino mass eigenvalues by a shared negative term. This leaves the oscillation mass splittings exactly unchanged — so it stays consistent with oscillation data — while allowing the direct beta-decay endpoint parameter to go negative (corresponding to a scale µ₀ ≈ 0.374 eV). The signed endpoint spectrum is modeled by an analytic-continuation template, W√(W²+µ²), explicitly framed as a testable spectral/self-energy template, not a complete tachyonic field theory.

The central obstacle is a known degeneracy: any unmodeled Gaussian broadening biases the fitted mass-squared negative by roughly −2σ², so the reference value is mimicked by only about 0.265 eV of hidden experimental width. The paper's main contribution is a closed-form "nuisance-null" efficient score that projects the mass-square signal orthogonally to normalization, endpoint, background, and every variance-like response column — including a boundary (Dirac-delta) term arising from the endpoint cutoff — making the estimator first-order immune to this broadening degeneracy. A toy Fisher analysis quantifies what a real measurement would require: a 5σ-scale precision of σ(m²_β) ≈ 0.028 eV² needs an effective resolution near 0.24 eV (FWHM) over a roughly 10 eV endpoint window at KATRIN final-program statistics — far beyond the current ~2.8 eV integral MAC-E resolution. A finite-template check shows even the 0.24 eV figure is only a marginal threshold rather than a clean discovery forecast.

On the theory side, the paper proves a no-go result: no exact, causal, analytic single-pole model can reproduce the endpoint template all the way down to zero energy while also giving real, nonnegative relic-neutrino energies. A literal all-momentum tachyonic interpretation is therefore excluded under the stated assumptions; only a finite-band, flavor-universal self-energy can approximate the template over a finite experimental window, necessarily departing from the tachyonic form in the infrared.

The overall conclusion is deliberately cautious. Existing endpoint data do not constitute evidence for tachyonic neutrinos — the combined KATRIN/Mainz/Troitsk signed average is only about 1σ, and the negative sign can be produced by ordinary broadening. But a signed-endpoint search is now mathematically well posed, with concrete requirements on the observable, the resolution, and independent confirmation (for example, with atomic tritium).

The generative AI tools used in the preparation of this paper were Gemini 3.5 Flash, Opus 4.8 Max, and GPT-5.5 Pro. Gemini 3.5 Flash was used for conceptual brainstorming. Opus 4.8 Max and GPT-5.5 Pro were used for mathematical discussion, derivation checks, language editing, and structural refinement.