Theory, premises, and open bridges

For B₃ = so(7), long roots have squared length 2, dim B₃ = 21 and |ρ|² = 35/4. The Spin(7) heat-kernel density at the identity obeys K_t(e,e) = (4πt)^−21/2e^35t/4[1 + δ(t)]. At t = α/7, its inverse square with prefactor (4/3)(7/4π)²¹ reproduces the relation up to a relative remainder below 10⁻⁸⁰⁰⁰. This is an all-orders local identity, not equality of global functions. The clock, readout and normalization are not physically forced.

The expression holds exactly for static laboratory G, electron pole mass and α(0), without an adjustable matching factor. Motivation: fixed numerical agreement and the mathematical form. Status: empirical, not derived. Independent precision-G convergence away from it rejects H1; adding a factor afterwards would define a new relation.

A physically normalized system uses B₃ invariants to determine the electron–gravity ratio. The heat kernel motivates the search, but its physical role is open. Any implementation with freely tunable clock, normalization or readout fails as an explanation; the failed-candidate census does not universally refute H2.

The electron is assigned to d = 5; d = 6 gives exponent 7/2, d = 7 gives 15/2. The ladder translates algebra dimensions into scales, but does not derive which particle occupies a rung. Neutrino and dark-boson assignments are separate hypotheses. For the registered neutrino fork both branches must be excluded to reject that implementation.

The cubic O(7) × F₄ theory has a real, physically admissible fixed point at d = 5 supplying the proposed dictionary. Exceptional tensors and the large-N family motivate it. Existence at five is open; real coefficients, stability and strict unitarity are distinct. A controlled boundary or nonperturbative exclusion could reject this carrier without rejecting the empirical G expression.

If effective neutrino mass coefficients and the charged-lepton diagonalizer are real in a common basis, J = 0 and nonzero mixing angles imply δ_CP = 0° or 180°. A real representation alone is insufficient. The required antiunitary symmetry has not been constructed. Under confirmed normal ordering, discovery-level exclusion of both targets rejects this implementation.

For Φ in (7,26) and X in (1,26), five-loop RG polynomials were assembled and exactly validated against known limits. Frozen calibrated continuation gives Δ_Φ = 1.576 ± 0.099, ω_high = 1.74 ± 0.89, and O₊ = 4.55 ± 1.07 at d = 5, conditional on a real fixed point. These allowances describe calibrated recipe uncertainty, not Gaussian measurement errors. The O₊ band contains 103/21 ≈ 4.9048 but is not evidence for that exact rational target. The lowest stability direction, Δ_X and O₋ are not retained; stability remains undecided.

Assume a continuously distinct branch, only Φ as weakly coupled field content near four, no termination as Gaussian or symmetry-enhanced O(182), and no strongly coupled real endpoint of the required kind near four. Then the real continuation ends at some d_* between 4 and 6. The last premise concerns an unknown nonperturbative spectrum. This does not locate d_* or decide existence at five.

For the stated twisted circle, a local unitary scalar effective theory below compactification, a weak critical limit and no extra light gauge or composite fields, the leading four-dimensional quartic beta functions have only a Gaussian real critical point. It cannot carry the five-dimensional eigenvalue 2/21 into the hierarchy. This is a scoped leading weak-coupling obstruction, not an all-orders theorem against strong or composite descent.

Under H3, the d = 6 scale is assigned to the lightest normal-ordering eigenvalue. Corrected splitting conventions give a sum of 87.76–89.02 meV. The assignment and its mechanism remain hypotheses.

Small breaking of a rung-degenerate pair cannot produce the solar splitting. Additional selectively protected dynamics could retain a tiny eigenvalue, but none is derived. With that extra premise and m₁ ≤ 1 meV, the sum is 58.89–59.96 meV; it is not distinctive of this programme.

With the heavier assignment, H5, Majorana neutrinos and standard light exchange, m_ββ lies in 3.4–5.4, 5.8–7.8, 15.2–16.3 or 17.7–18.6 meV. They are one-dimensional parameter-box bounds, not a joint 3σ region. A controlled signal outside all four rejects the conjunction, not every neutrino theory.

A physical construction must fix normalized operators, the infrared clock and prefactor without a residual-dependent counterterm. Nonperturbative carrier existence, new controlled descent, an assignment selector and a lepton CP symmetry are concrete missing pieces. A scoped falsifier changes the corresponding claim; a derived matching chain would change its evidence grade.