Neutrino Prediction Mode in the Unified Topological Mass Framework
Canon-synced, fully deterministic formulation with no tunable parameters and falsifiable predictions
We present a canon-synced, fully deterministic formulation of the neutrino sector within the Unified Topological Mass Framework (UTMF). This work transitions the framework from an exploratory, fit-capable regime to a strict prediction mode in which all adjustable degrees of freedom are eliminated and all outputs are falsifiable.
In Part I, we demonstrate that the minimal closure of the seesaw construction, in which the heavy sector is taken to be degenerate and flavor-aligned, leads to a rank-collapse of the effective Majorana mass matrix. This collapse produces a near-vanishing solar mass splitting, establishing that minimal closure is structurally inconsistent with experimental data rather than merely numerically imprecise.
In Part II, we resolve this failure by introducing a canon-consistent, non-degenerate heavy sector selected via an independent lexicographic rule and a Non-Collinearity (ND) Admissibility Axiom. The heavy sector is constructed as a distinct geometric entity in the tuple lattice, rather than as a deformation of the light sector. We prove that this construction stabilizes the rank of the neutrino mass matrix and yields a unique heavy-sector deformation parameter , fixed solely by geometric axioms and a single experimental scale anchor. No fitting or continuous tuning is permitted.
In Appendix A, we provide a necessity proof for the weighted flavor–heavy overlap kernel employed in the construction. We show that this kernel is the unique minimal extension that simultaneously preserves the light-sector magnitude invariant, prevents rank collapse, and introduces no additional free parameters. Any weaker or alternative extension either violates canon invariance or reintroduces degeneracy.
Together, these results establish a complete, closed-loop neutrino prediction package within UTMF that is mathematically consistent, physically constrained, and experimentally testable.
The Unified Topological Mass Framework (UTMF) provides a lattice-based, operator-defined origin for fermion masses and mixings. While earlier work established exploratory consistency with observed neutrino data, a critical open question remained: whether the framework could produce rigid, falsifiable predictions without parameter fitting.
This paper answers that question affirmatively. We construct a neutrino prediction mode in which all continuous freedom is removed, all inputs are canon-locked, and all outputs are fixed up to experimental scale anchors.
2.1 Minimal Seesaw Closure
The minimal closure assumes a degenerate heavy sector:
where is a single mass scale and is the identity matrix.
2.2 Rank Collapse
Under this assumption, the effective Majorana mass matrix becomes:
When is sourced from canonically aligned light-sector tuples, the matrix becomes nearly rank-1, leading to:
This implies after scale locking, which is catastrophically inconsistent with the observed solar mass splitting .
Conclusion (Part I):
The failure of minimal closure is structural, not numerical. Any viable prediction mode must prevent rank collapse by introducing a non-degenerate heavy sector.
3.1 Axiom ND (Non-Collinearity Admissibility)
Let denote the Dirac source matrix with column vectors . Define the normalized column coherence:
A heavy triple is ND-admissible if and only if there exists a fixed canon threshold such that:
3.2 Weighted Flavor–Heavy Kernel
Define the weighted norm on tuple space:
The flavor–heavy overlap kernel is then:
where are the light-sector tuples, are the heavy-sector tuples, and is a fixed phase vector.
3.3 Canonical Heavy Triple
The canonical heavy triple is defined as the lexicographically minimal triple in the heavy domain satisfying:
- Tuple distinction (all three tuples are different)
- Axiom ND (non-collinearity admissibility)
3.4 Uniqueness of the Heavy Deformation
If is ND-admissible, then and is generically .
For fixed ND-admissible heavy triple and fixed solar calibration target, there exists a unique deformation parameter such that:
Once the kernel, ND threshold, lexicographic rule, and scale anchors are fixed, the neutrino sector is fully determined with no tunable parameters.
The canonical construction yields the following rigid, falsifiable predictions:
Mass Ordering
Normal hierarchy with
Lightest Mass
Strong hierarchy with
Total Mass
Experimental Bounds
and well below current bounds
Falsifiability Statement:
These predictions are rigid and falsifiable. Discovery of inverted ordering, a non-zero , or total mass significantly different from would decisively falsify the canonical construction.
A.1 Purpose
We justify the weighted kernel as the unique minimal extension that preserves the light-sector invariant while stabilizing rank.
A.2 Invariant Preservation
The light-sector magnitude invariant is:
where is the neutrino tuple displacement and are the canonical weights. The weighted kernel preserves this invariant under the heavy-sector extension.
A.3 Failure of Alternatives
Dot-product kernels: Enforce collinearity and rank collapse
norms: Break additivity and invariant preservation
Unequal weights: Violate canon locking
Nonlinear kernels: Introduce free parameters
Conclusion (Appendix A):
The weighted kernel is the unique minimal choice satisfying:
- Rank stabilization
- Invariant preservation
- Parameter-free prediction mode
This paper establishes a complete, closed-loop neutrino prediction package within UTMF:
- Part I proves that minimal closure fails structurally due to rank collapse
- Part II resolves this via a non-degenerate heavy sector with unique
- Appendix A proves the weighted kernel is the unique minimal extension
The result is a neutrino sector that is mathematically consistent, physically constrained, and experimentally testable with no remaining free parameters.