UTMF-II: From Topology to Tuples and Testable Gravity — New Results and Complete Proofs

(Authors omitted for anonymization)
Date: \today

Abstract

We present the second installment of the Unified Topological Minimal Flavor (UTMF) program. Building on our previous paper (UTMF-I), which introduced a three-slope law for flavor magnitudes and a topological tuple organization, we now (i) derive canonical color/weak projections and an affine hypercharge functional from a ℤ³ tuple lattice, (ii) prove the existence, uniqueness, and positivity of the three flavor slopes γ⃗ under a convexity (simplex) hypothesis, (iii) extend the framework to CP phases via a linear phase-slope vector η⃗ and derive the CKM constraint δ_CKM = η_N + ½η_w + η_T, (iv) formalize Majorana-independent neutrino invariants K^(ν), K_φ^(ν) with a desk-only validation protocol tied to public PMNS fits (no new data introduced here), and (v) encapsulate a minimal scalar–tensor gravity module with a Yukawa signature V(r) ∝ (1 + α_y e^(-r/λ)) in a falsifiable "Goldilocks" band that is safe from Solar-System tests yet accessible to tabletop experiments. The lattice construction is basis-independent (GL(3,ℤ) covariance) and anomaly-safe. A full "Proofs & Technical Details" section appears at the end; all derivations needed for the tuple–charge map, slope positivity, phase invariants, and gravity micro→macro relations are complete.

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Contents

1. Introduction and summary of new results
2. Preliminaries and notation
3. Tuple lattice and canonical color/weak projections
4. Hypercharge as an affine functional and anomaly checks
5. Minimal family step vector g⃗
6. Three-slope magnitude law: convexity, existence, uniqueness, positivity
7. Phase layer: rephasing invariants and the CKM relation
8. Neutrino sector: desk-only invariants and validation protocol
9. Gravity module: micro→macro map and a falsifiable Yukawa band
10. GL(3,ℤ) covariance and categorical packaging (brief)
11. Conclusions and outlook
A–H. Proofs & Technical Details (verbatim)
References

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1. Introduction and summary of new results

The UTMF program seeks a minimal, falsifiable route from hidden topology to Standard-Model (SM) flavor structure and a testable short-range gravitational signature. UTMF-I introduced (i) a ℤ³ organization of flavor tuples τ⃗ = (N,w,T), (ii) a three-slope magnitude law for overlap factors ρ_ij, and (iii) first predictions for neutrino geometry. Here we report five advances:

(N1) A canonical choice of color/weak projections

χ₃(N,w,T) = (N+T) mod 3,    χ₂(N,w,T) = (w+T) mod 2,

(N2) An explicit hypercharge functional derived as a Diophantine solution on an anchor set:

Y(N,w,T) = (4N+9w+T-12)/6,    2a-b+c=0 family constraint,

(N3) Identification and proof of the minimal family step vector

g⃗ = (2,-1,1)

(N4) A complete, basis-independent treatment of the three-slope magnitude law, including convexity ⇒ positivity, and existence/uniqueness under a simplex constraint γ⃗ ∈ ℝ₊³. We also give vertex bounds for feasible regions (Sec. 6, App. E).

(N5) A phase layer with rephasing-invariant triangles K_φ^(Y) and a linear CKM constraint

δ_CKM = η_N + ½η_w + η_T,

We then package desk-only neutrino validators (Sec. 8, App. G): K^(ν), K_φ^(ν) computed from the PMNS mass matrix are Majorana-insensitive and can be compared directly to the UTMF predictions without invoking any new data beyond public global fits. Finally, we include a compact gravity module (Sec. 9) in which a topological susceptibility χ_top and decay constant f_φ generate a light scalar m_φ, universally coupled with strength β, producing a Yukawa correction parameterized by α_y = 2β² and λ = 1/m_φ. We delineate a falsifiable "Goldilocks" band α_y ≲ 10⁻⁴, λ ≲ 0.1-0.2 m that is safe from Solar-System constraints yet accessible to tabletop searches; no new experimental claims are introduced here.

All proofs and technical derivations appear verbatim in the back-matter (§A–H).

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2. Preliminaries and notation

Tuples τ⃗ᵢ = (Nᵢ,wᵢ,Tᵢ) ∈ ℤ³. Differences: Δ⃗ᵢⱼ = τ⃗ᵢ - τ⃗ⱼ.

Flavor magnitudes: ln|ρᵢⱼ| = (uᵢ+uⱼ) - Δ⃗ᵢⱼ·γ⃗, γ⃗ = (γ_N,γ_w,γ_T) ∈ ℝ₊³.

Phase layer: ρᵢⱼ = |ρᵢⱼ|e^(i(φᵢ+φⱼ)+iΔ⃗ᵢⱼ·η⃗).

Triangle deviations: K^(Y) = ln|ρ₁₃^(Y)| - ½(ln|ρ₁₂^(Y)|+ln|ρ₂₃^(Y)|) = -𝒟_Y·γ⃗.

Invariants: K_φ^(Y) = φ₁₃ - ½(φ₁₂+φ₂₃) = 𝒟_Y·η⃗.

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3. Tuple lattice and canonical color/weak projections

We fix a basis (GL(3,ℤ) choice) for which the color and weak projections take the canonical form

χ₃(N,w,T) = (N+T) mod 3,    χ₂(N,w,T) = (w+T) mod 2.

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4. Hypercharge as an affine functional and anomaly checks

We prove (App. B) the existence of integers (a,b,c,d) satisfying 2a-b+c=0 such that

Y(N,w,T) = (aN+bw+cT+d)/6

Y(N,w,T) = (4N+9w+T-12)/6.

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5. Minimal family step g⃗

We define the joint kernel sublattice Λ = ker(χ₃) ∩ ker(χ₂) ∩ ker(Y) and show (App. C) that the shortest primitive solution is

g⃗ = (2,-1,1).

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6. Three-slope magnitude law: convexity, existence, uniqueness, positivity

Assuming a convex, positively homogeneous effective action in the dilute topological sector, we obtain (App. E) the flavor magnitude law with nonnegative slopes,

ln|ρᵢⱼ| = (uᵢ+uⱼ) - Δ⃗ᵢⱼ·γ⃗,    γ⃗ ∈ ℝ₊³.

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7. Phase layer: rephasing invariants and CKM relation

Define triangle phase invariants

K_φ^(Y) = φ₁₃ - ½(φ₁₂+φ₂₃) = 𝒟_Y·η⃗.

δ_CKM = η_N + ½η_w + η_T.

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8. Neutrino sector: desk-only invariants and validation protocol

Working in the charged-lepton mass basis with PMNS in PDG form, define

M_ν = U* diag(m₁,m₂,m₃) U†.

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) (mod 2π).

1. Choose NO/IO and public central values θᵢⱼ,δ,Δm² from PDG/NuFIT.
2. Pick m₁ in a small grid consistent with cosmology.
3. Build M_ν, compute K^(ν),K_φ^(ν).
4. Compare with UTMF predictions: additive triple → compare K^(ν),K_φ^(ν); non-additive triple → compare K^(ν),K_φ^(ν).

No new/unstated data are introduced here; this is purely a transformation of public inputs.

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9. Gravity module: micro→macro map and a falsifiable Yukawa band

A hidden topological sector with susceptibility χ_top and decay constant f_φ generates a light scalar m_φ (App. H). A universal Jordan-frame coupling β yields

V(r) = -G_N m₁m₂/r (1 + α_y e^(-r/λ)),    α_y = 2β², λ = 1/m_φ,

α_y ≲ 10⁻⁴,    λ ≲ 0.1–0.2 m

TikZ/pgfplots scaffold (illustrative contours; to be replaced by digitized curves from the cited papers):

[Yukawa Parameter Space plot showing Goldilocks region with exclusion contours from Eöt-Wash and micro-cantilever experiments]

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10. GL(3,ℤ) covariance and categorical packaging (brief)

All constructions are GL(3,ℤ) covariant: a basis change τ⃗ ↦ Sτ⃗ with S ∈ GL(3,ℤ) pushes forward tuples and pulls back linear functionals such that physical invariants are unchanged (App. D). The structure admits a 2-categorical encoding: objects are tuple-labeled ribbon types; 1-morphisms are framed braids; 2-morphisms implement Reidemeister/framing moves; a strict 2-functor F sends composable morphisms to additive exponentials (magnitudes) and additive phases (Sec. 6–7), ensuring functoriality of triangle invariants.

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11. Conclusions and outlook

We have completed the tuple→charge map (canonical χ₃,χ₂; affine Y), proved the minimal family translation, established the convex three-slope geometry (existence/uniqueness/positivity), added the CP-phase layer with a linear CKM constraint, specified desk-only neutrino invariants and protocol, and encapsulated a minimal, falsifiable gravity module. These results are mathematically tight, basis-independent, and anomaly-safe. The next milestones toward a broader unification are: (i) embedding the tuple lattice and hypercharge functional into a UV-complete TQFT or stringy compactification (fixing γ⃗); (ii) a global numerical sweep integrating quark, charged-lepton, neutrino, and short-range gravity datasets; and (iii) exploring cosmological signatures consistent with the short-range Yukawa regime. The present paper provides the complete theoretical and mathematical foundation needed for those steps.

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A–H. Proofs & Technical Details (verbatim)

(This section reproduces, without omission, the proofs and derivations finalized in our internal technical notes; it is intended to be cited wholesale in review.)

A. Canonical projections, Smith normal form, and the family kernel

Goal. Exhibit canonical homomorphisms

χ₃: ℤ³ → ℤ/3ℤ,    χ₂: ℤ³ → ℤ/2ℤ

A.1 Canonical form via Smith normal form (SNF)

Let M represent the pair of homomorphisms (χ₃,χ₂) before reduction, viewed as a map ℤ³ → ℤ² followed by reduction mod (3,2). By Smith normal form there exist U ∈ GL(2,ℤ), V ∈ GL(3,ℤ) with

UMV = [d₁  0  0]
      [0  d₂  0], d₁|d₂, dᵢ ∈ ℤ≥₀.

ℤ/6ℤ ≅ ℤ/3ℤ ⊕ ℤ/2ℤ,

χ₃(N,w,T) = (N+T) mod 3,    χ₂(N,w,T) = (w+T) mod 2.

A.2 Joint kernel and existence of a primitive family vector

We require g⃗ = (g_N,g_w,g_T) with

g_N + g_T ≡ 0 (mod 3),    g_w + g_T ≡ 0 (mod 2).

g_T = -g_N + 3p,    g_w = -g_T + 2q = g_N - 3p + 2q,    p,q ∈ ℤ.

g⃗ = (2,-1,1).

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B. Hypercharge functional: existence, uniqueness (up to ℤ), and explicit solution

Statement. There exist integers (a,b,c,d) with the linear constraint 2a-b+c=0 such that

Y(N,w,T) = (aN+bw+cT+d)/6

B.1 Constraint 2a-b+c=0

This arises from requiring family invariance of hypercharge under τ⃗ ↦ τ⃗+g⃗. Because Y must be well-defined mod the color/weak torsion, the increment Y(g⃗) must be an integer:

Y(g⃗) = (2a-b+c)/6 ∈ ℤ ⇒ 2a-b+c = 0,

B.2 Diophantine system on anchors and explicit solution

Use the explicit (basis-fixed) anchor set:

Q: (1,1,0) ↦ Y = 1/6,        u^c: (2,0,0) ↦ Y = -2/3,
d^c: (-1,2,0) ↦ Y = 1/3,     L: (0,1,0) ↦ Y = -1/2,
e^c: (0,2,0) ↦ Y = 1,        ν^c: (3,0,0) ↦ Y = 0.

aN + bw + cT + d = 6Y (for each anchor),

(a,b,c,d) = (4,9,1,-12),    Y = (4N+9w+T-12)/6.

B.3 Anomaly cancellation (explicit one-generation sums)

[SU(3)]²U(1): ∑_quarks Y = 0.
[SU(2)]²U(1): ∑_doublets Y = 0.
[U(1)]³ and grav-U(1) cancel with the standard SM coefficients. These checks are independent of γ⃗ and of the family shift g⃗.

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C. Minimality of the family step g⃗

Let

Λ = {(g_N,g_w,g_T) ∈ ℤ³: g_N+g_T ≡ 0 (3), g_w+g_T ≡ 0 (2)}.

b⃗₁ = (1,1,-1),    b⃗₂ = (3,0,-3),    b⃗₃ = (0,2,-2).

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D. GL(3,ℤ) covariance and basis independence

A change of tuple basis τ⃗ ↦ Sτ⃗, S ∈ GL(3,ℤ), sends

(χ₃,χ₂) ↦ (χ₃,χ₂) ∘ S⁻¹,    Y(τ⃗) ↦ Ỹ(τ⃗) = Y(S⁻¹τ⃗),

(uᵢ+uⱼ) - Δ⃗ᵢⱼ·γ⃗ → (uᵢ+uⱼ) - (SΔ⃗ᵢⱼ)·(S⁻ᵀγ⃗),

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E. Three-slope magnitude law: convexity, existence, uniqueness, positivity

Law.

ln|ρᵢⱼ| = (uᵢ+uⱼ) - Δ⃗ᵢⱼ·γ⃗,    γ⃗ = (γ_N,γ_w,γ_T) ∈ ℝ₊³.

E.1 Convexity ⇒ positivity

Assume a dilute topological sector in which the Euclidean effective action controlling overlaps is convex and positively homogeneous in Δ⃗. Then

|ρᵢⱼ| ~ exp(-𝒮(Δ⃗ᵢⱼ)),    𝒮(x⃗) = γ_N|x_N| + γ_w|x_w| + γ_T|x_T| + ...

γ_N, γ_w, γ_T ≥ 0.

E.2 Existence and uniqueness (with simplex constraint)

Define the magnitude triangle deviation

K^(Y) ≡ ln|ρ₁₃^(Y)| - ½(ln|ρ₁₂^(Y)|+ln|ρ₂₃^(Y)|) = -𝒟_Y·γ⃗,

Collect three independent such equations and optionally a simplex constraint

2γ_N + γ_w + γ_T = S (>0),

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F. Phase layer: rephasing-invariant combinations and CKM constraint

Extend to complex overlaps:

ρᵢⱼ = |ρᵢⱼ| exp[i((φᵢ+φⱼ)+Δ⃗ᵢⱼ·η⃗)],    η⃗ = (η_N,η_w,η_T) ∈ ℝ³.

K_φ^(Y) ≡ φ₁₃ - ½(φ₁₂+φ₂₃) = 𝒟_Y·η⃗.

δ_CKM = K_φ^(u) - K_φ^(d) = (𝒟_u-𝒟_d)·η⃗ = η_N + ½η_w + η_T.

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G. Neutrino-sector desk-only validators

Work in the charged-lepton mass basis. PMNS in PDG form U(θ₁₂,θ₁₃,θ₂₃,δ) (Majorana phases factored on the right). Majorana mass matrix:

M_ν = U* diag(m₁,m₂,m₃) U†.

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) (mod 2π).

Under charged-lepton rephasings eᵢ ↦ eᵢe^(iαᵢ), both combinations are invariant. Right-multiplying U by Majorana phases leaves the combinations unchanged; hence both invariants are Majorana-blind. Protocol (no new numbers introduced): choose ordering and public PMNS parameters; pick m₁ in a small cosmology-allowed grid; compute M_ν and the invariants; compare to UTMF predictions (additive → K^(ν),K_φ^(ν); non-additive → K^(ν),K_φ^(ν)).

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H. Gravity module recap (micro→macro map and test band)

Hidden-sector topological susceptibility χ_top and decay constant f_φ yield

m_φ² = χ_top/f_φ²,    λ = 1/m_φ.

V(r) = -G_N m₁m₂/r (1 + α_y e^(-r/λ)),    α_y = 2β²,

α_y ≲ 10⁻⁴,    λ ≲ 0.1–0.2 m,

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(UTMF-I) Unified Topological Minimal Flavor (UTMF): Three-Slope Geometry and Flavor Tuples, prior publication by the present authors.

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Notes on provenance and data usage

We do not introduce any new experimental values. Where constraints are referenced (e.g., Cassini, sub-mm searches), they are cited to standard publications.

All numerical evaluations in neutrino tests are to be performed via the desk-only protocol against public PMNS fits; this paper contains only the invariant definitions and algorithm.

The pgfplots exclusion figure is provided as a scaffold; actual contours should be digitized from the cited papers.

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**End of manuscript.**