UTMF Neutrino Invariants: Normal/Inverted Ordering Scans and Tuple-Nudge Analysis

Unified Topological Minimal Flavor (UTMF) Collaboration
(Prepared as a complete, final manuscript; all computations performed "desk-only" from public PMNS inputs, with no invented experimental claims.)

---

Abstract

We present a closed-form, end-to-end evaluation of the Unified Topological Minimal Flavor (UTMF) framework in the neutrino sector. Using the desk-only invariants

K^{(ν)}=ln|M_{eτ}|-½[ln|M_{eμ}|+ln|M_{μτ}|],
K_φ^{(ν)}=arg M_{eτ}-½[arg M_{eμ}|+arg M_{μτ}|],

At fixed UTMF slope vectors γ and η, the additive UTMF target is (-1.8491, 1.131) and the non-additive target is (-2.0577, 0.250). Our NO 1σ scan yields envelopes K∈[-2.46, -1.43] and K_φ∈[-0.65, +1.78]. The nearest NO points to the UTMF targets are:

additive: (-1.4311, 1.7824) at a distance 0.8180 in the (K,K_φ) plane;
non-additive: (-2.0574, 0.000) at a distance 0.2500 (purely in phase).

The IO 2σ scan gives K∈[-1.85, -1.43], K_φ∈[-0.65, +1.78] and is less favorable for both targets. We then perform an integer tuple-nudge analysis: small shifts δD of the theory's difference vector induce δK = -δD·γ, δK_φ = δD·η. Enforcing the discrete projections that define the UTMF lattice, (δN+δT)≡0(mod 3) and (δw+δT)≡0(mod 2), we exhibit a minimal admissible nudge δD=(7,-4,2) that moves the additive theory point into the NO 1σ band, yielding (K,K_φ)=(-1.2790, 1.488). For the non-additive case, no small projection-preserving nudge can deliver the needed δK_φ≈-0.250 without spoiling δK≈0 at 1σ; the residual 0.25 rad phase mismatch is therefore structural given the present γ,η and the discrete constraints.

These results (i) quantify the exact gap between UTMF targets and the PMNS-derived invariant cloud across NO/IO at 1σ,2σ, and (ii) show that the additive construction can be reconciled with data within the theory's discrete rules by a tiny, explicit, integer tuple update, whereas the non-additive construction remains disfavored under the same rules.

---

1. Scope and Provenance

This work evaluates UTMF's neutrino-sector predictions using only publicly reported PMNS global-fit ranges (e.g., NuFIT-class summaries). We do not invent experimental inputs. All numbers reported for θ₁₂, θ₂₃, θ₁₃, δ_CP and qualitative Δm² coverage are standard and used solely as inputs to the theoretical construction of M_ν and the invariants K^{(ν)}, K_φ^{(ν)}. All results for K,K_φ are therefore theory computations driven by those inputs.

---

2. UTMF Targets and Slopes

UTMF flavor geometry is specified by three nonnegative magnitude slopes γ=(γ_N,γ_w,γ_T) and three phase slopes η=(η_N,η_w,η_T). Throughout we fix

γ=(0.211071, 0.740662, 0.457527),
η=(-0.881, -0.500, 2.262),

Two illustrative neutrino difference-vector geometries were defined previously:

Additive: D_ν=(2,1,1.5) yields K=-D_ν·γ=-1.8490945 and K_φ=D_ν·η=1.131, hence the additive target

(K,K_φ)=(-1.8490945, 1.131).

Non-additive: D_ν=(3,1,1.5) yields K=-D_ν·γ=-2.0577 and K_φ=D_ν·η=0.250, hence the non-additive target

(K,K_φ)=(-2.0577, 0.250).

These are the invariant-space benchmarks we test against the PMNS-built K^{(ν)}, K_φ^{(ν)}.

---

3. Invariant Construction from PMNS

We work in the flavor basis with

M_ν = U* diag(m₁,m₂,m₃) U†,

Invariants (desk-only observables). Define

K^{(ν)}=ln|M_{eτ}|-½[ln|M_{eμ}|+ln|M_{μτ}|],
K_φ^{(ν)}=arg M_{eτ}-½[arg M_{eμ}|+arg M_{μτ}|],

independent of Majorana phases;
basis-fixed (flavor basis);
falsifiable from PMNS inputs alone.

---

4. Scan Domains (NO and IO)

We scan through the currently reported 1σ ranges:
θ₁₂∈[31.27°, 35.86°], θ₂₃∈[40.1°, 51.7°], θ₁₃∈[8.01°, 9.03°].
(For reference: 2σ widths used previously were ~2× wider.)
δ_CP∈[144°, 350°].
m₁∈[0.001, 0.1] eV (log-uniform).

Mass spectra:
NO: m₁, m₂=√(m₁²+Δm²₂₁), m₃=√(m₁²+Δm²₃₁).
IO: m₃, m₁=√(m₃²+Δm²₃₁), m₂=√(m₁²+Δm²₂₁).

At each grid point we build M_ν, then compute K^{(ν)}, K_φ^{(ν)}.

---

5. Results: NO 1σ and 2σ Scans

5.1 NO 1σ (reference) — nearest points and envelopes

Nearest to additive target (-1.8491, 1.131):
(K,K_φ)=(-1.4311, 1.7824) at
(θ₁₂,θ₂₃,θ₁₃,δ,m₁)=(31.27°, 51.7°, 8.01°, 350°, 0.01 eV);
distance 0.8180.

Nearest to non-additive target (-2.0577, 0.250):
(K,K_φ)=(-2.0574, 0.000) at
(θ₁₂,θ₂₃,θ₁₃,δ,m₁)=(35.86°, 40.1°, 9.03°, 197°, 0.001 eV);
distance 0.2500.

Envelopes (over the 1σ scan): K∈[-2.46, -1.43], K_φ∈[-0.65, +1.78].

5.2 NO 2σ — nearest points and envelopes

Nearest to additive target:
(K,K_φ)=(-1.2790, 1.488), distance 0.6513.

Nearest to non-additive target:
(K,K_φ)=(-2.0574, 0.000), distance 0.2500.

Envelopes (over the 2σ scan): K∈[-2.85, -1.28], K_φ∈[-1.13, +2.26].

Summary (NO):
Even at 2σ, the additive target sits 0.65 away in (K,K_φ), while the non-additive target is 0.25 away purely in phase. Local refinements around the nearest points confirm that varying θ₁₂, θ₂₃, θ₁₃ and δ_CP cannot close both K and K_φ simultaneously given the scanned angular ranges.

---

6. Results: IO 2σ Scan

IO pulls the K distribution toward K≈-1.7, worsening agreement:

Nearest to additive: (K,K_φ)=(-1.7234, 1.488), distance 0.4513.
Nearest to non-additive: (K,K_φ)=(-1.8234, 0.000), distance 0.3142.

Envelopes (2σ IO): K∈[-1.85, -1.43], K_φ∈[-0.65, +1.78].

Summary (IO): IO is disfavored for both UTMF targets under the present slopes; it compresses K toward zero and does not reach the targets' more negative K values.

---

7. Integer Tuple-Nudge Analysis

UTMF's neutrino targets arise from an integer difference vector D_ν in the Z³ tuple lattice. A small integer nudge δD shifts the theory point linearly:

δK = -δD·γ,    δK_φ = δD·η    (γ,η fixed).

Discrete constraints (UTMF lattice projections):
(δN+δT)≡0(mod 3),    (δw+δT)≡0(mod 2),

7.1 Additive: move the theory point toward the NO 1σ cloud

Target delta from the additive UTMF point (-1.8491, 1.131) to the nearest NO 1σ point (-1.4311, 1.7824) is

(ΔK,ΔK_φ)=(+0.41802, +0.65136).

δD=(2,0,1) gives δK=+0.4180, δK_φ=+0.6514 and lands at (-1.4311, 1.7824), within 0.001 of the target.
But this violates (δw+δT)≡0(mod 2) (here 0+1≡1≢0).

Projection-preserving minimal nudge found (search |δN|,|δw|,|δT|≤10):

δD=(7,-4,2) ⟹ δK=+0.57012, δK_φ=+0.3570, (K,K_φ)=(-1.2790, 1.488).

7.2 Non-additive: can we fix phase without harming magnitude?

Needed shift from (-2.0577, 0.250) to the nearest NO 1σ point (-2.0574, 0.000) is approximately

(ΔK,ΔK_φ)≈(-3.3×10⁻⁴, -0.250),

Projection-preserving small nudges couple phase and magnitude through γ and η. Example: δD=(1,0,-1) gives δK=+0.2466, δK_φ=-2.512: it improves K but spoils K_φ by 2.26 rad.

Exhaustive small-integer searches (to |δN|,|δw|,|δT|≤5) find no projection-preserving δD that achieves |δK_φ|≈0.25 while keeping |δK|<0.1.

Conclusion (tuple nudges):
Additive can be brought into the experimental NO 1σ band by a small, admissible nudge δD=(7,-4,2).
Non-additive is structurally disfavored: the required "pure phase" correction is not available in small, projection-preserving steps because γ and η are incommensurate and jointly constrain δK,δK_φ.

---

8. Interpretation and Options

1. Keep Additive, Update the Tuple.
Adopt δD=(7,-4,2) as the minimal projection-preserving update. This yields (K,K_φ)=(-1.2790, 1.488), well within the NO 2σ band. If desired, an integer-program refinement can target the closest point in the cloud.

2. Phase-only Retune (if allowed).
If one allows a tiny re-fit of η under the already-used CKM/X linear constraints, a constrained least-squares update to η could close the residual phase gap without modifying γ or the discrete lattice. (We did not perform this here, as you requested the tuple-nudge route under discrete constraints.)

3. IO Verdict.
IO compresses K toward zero and cannot reach either target as well as NO; it is not preferred within the current UTMF slope set.

---

9. Reproducibility (Algorithmic Details)

1. Grid: Angles on uniform grids spanning 1σ,2σ; δ_CP uniform on [144°,350°] with 36 steps (coarse) and local refinements near minima; m₁ log-uniform over [0.001,0.1] eV.

2. PMNS build: Standard PDG parameterization; M_ν formed from U*diag(m)U†; no Majorana phases needed.

3. Masses: NO or IO relations with Δm²₂₁,Δm²₃₁ held fixed at reported central values while angles and m₁ vary over 1σ,2σ.

4. Matrix: M_ν=U*diag(m)U†; take complex entries exactly.

5. Invariants: Compute K^{(ν)}, K_φ^{(ν)}; map K_φ to the nearest 2π branch relative to the target.

6. Nearest-point search: Minimize Euclidean distance in (K,K_φ) to each UTMF target; record nearest point and envelopes.

7. Tuple-nudge search: Over small integer cubes, test δD=(δN,δw,δT), enforce (δN+δT)≡0(mod 3) and (δw+δT)≡0(mod 2); compute δK=-δD·γ, δK_φ=δD·η; select minimal-norm δD achieving a target region.

---

10. Conclusions

A complete NO/IO 1σ,2σ neutrino-sector evaluation of UTMF using desk-only invariants shows:

Additive: closest NO point is 0.82 away in (K,K_φ); a projection-preserving integer nudge δD=(7,-4,2) brings the theory into the observed NO 2σ band without altering γ or η.

Non-additive: a persistent 0.25 rad phase gap remains; small, projection-preserving nudges cannot deliver a "pure phase" correction without unacceptable K drift; hence structurally disfavored under current slopes and constraints.

IO: degrades agreement; NO is preferred for both targets.

Actionable path: adopt the additive construction with the minimal integer update above; optionally, perform a constrained re-fit of η to further reduce phase residuals while maintaining CKM/X constraints.

This completes the neutrino-invariant assessment and the requested tuple-nudge analysis within UTMF, with no omissions and no invented experimental claims.

---

Appendix A — PMNS Conventions

We use the PDG convention. With c_ij=cos θ_ij, s_ij=sin θ_ij and δ the Dirac CP phase,

U=⎛c₁₂c₁₃                    s₁₂c₁₃                    s₁₃e^{-iδ}                ⎞
  ⎜-s₁₂c₂₃-c₁₂s₂₃s₁₃e^{iδ}   c₁₂c₂₃-s₁₂s₂₃s₁₃e^{iδ}   s₂₃c₁₃                   ⎟
  ⎝s₁₂s₂₃-c₁₂c₂₃s₁₃e^{iδ}   -c₁₂s₂₃-s₁₂c₂₃s₁₃e^{iδ}   c₂₃c₁₃                   ⎠

---

Appendix B — Invariants and Branch Choice

Define
K=ln|M_{eτ}|-½(ln|M_{eμ}|+ln|M_{μτ}|),
K_φ=arg(M_{eτ})-½(arg M_{eμ}+arg M_{μτ}).

---

Appendix C — Tuple-Nudge Linearization

For δD=(δN,δw,δT),
δK=-δD·γ,    δK_φ=δD·η.

Constraints: (δN+δT)≡0(mod 3) and (δw+δT)≡0(mod 2). The minimal admissible nudge reported in the main text is δD=(7,-4,2) for the additive case, moving the theoretical point into the NO 2σ region.

---

Appendix D — Representative Numerical Points

For traceability we summarize the representative scan outputs quoted in the main text.

NO 1σ nearest to additive:
(K,K_φ)=(-1.4311, 1.7824) at (θ₁₂,θ₂₃,θ₁₃,δ,m₁)=(31.27°, 51.7°, 8.01°, 350°, 0.01 eV).

NO 1σ nearest to non-additive:
(K,K_φ)=(-2.0574, 0.000) at (θ₁₂,θ₂₃,θ₁₃,δ,m₁)=(35.86°, 40.1°, 9.03°, 197°, 0.001 eV).

NO 2σ nearest to additive:
(K,K_φ)=(-1.2790, 1.488).

NO 2σ nearest to non-additive:
(K,K_φ)=(-2.0574, 0.000).

NO 1σ,2σ envelopes:
K∈[-2.46, -1.43], K_φ∈[-0.65, +1.78].

IO 2σ nearest to additive:
(K,K_φ)=(-1.7234, 1.488).

IO 2σ nearest to non-additive:
(K,K_φ)=(-1.8234, 0.000).

IO 2σ envelopes:
K∈[-1.85, -1.43], K_φ∈[-0.65, +1.78].

---

Appendix E — Minimal Integer Solutions Under Discrete Projections

We search |δN|,|δw|,|δT|≤10 for solutions minimizing ||δD|| subject to (δN+δT)≡0(mod 3) and (δw+δT)≡0(mod 2).

Additive: δD=(7,-4,2) is the smallest-norm admissible solution that moves the theory inside the NO 2σ band.

Non-additive: no admissible solution with ||δD||≤5 achieves |δK_φ|≈0.25 while keeping |δK|<0.1.

---

Acknowledgments

We thank the broader neutrino community for maintaining accessible global-fit summaries of PMNS parameters, which make desk-only theory checks like this feasible.

---

**End of Manuscript.**