From Hidden Topology to Unification

A UV-Derived Tuple Lattice, Cartan/Center Map, SO(10)→LR→SM Gauge Unification, and a Gravity Tie-In

Complete derivation of the tuple framework from UV-complete topological sectors tied to Spin(10), implementing gauge unification with closed-form solutions, global magnitude fits, and testable gravity connections

Complete Theoretical Physics PaperUV Foundations & Unification

Abstract

We complete the postulated tuple framework for Standard Model (SM) matter and interactions by deriving the flavor tuple lattice Λ_tuple ≅ ℤ³ from a UV-complete topological sector tied to the Cartan/center structure of Spin(10). The derivation fixes the Cartan/center map uniquely (up to GL(3,ℤ)), exhibits an exact gauge-invisible family translation, and proves anomaly consistency. We then implement a minimal, non-supersymmetric unification chain SO(10)→SU(3)_C×SU(2)_L×SU(2)_R×U(1)_{B-L}→SM ("Case B") with three 16_F families and scalars 10_H⊕126_H⊕54_H. Closed-form one-loop solutions and controlled two-loop corrections (with top-Yukawa effects) determine the intermediate and unification scales and establish a realistic threshold budget to place M_I≃10¹³⁻¹⁴ GeV and M_GUT≃10¹⁶ GeV while preserving proton stability at Hyper-Kamiokande reach. A desk-level global magnitude fit delivers a concrete, nonnegative slope vector γ⃗ and offsets matching CKM/PMNS hierarchies and fermion mass ratios, with sharp desk-only falsifiers. Finally, we connect the tuple-generating compact 2-form sector to a 4-form dual that naturally provides (i) a heavy SM-singlet scalar at m_φ∼M_I and (ii) an optional protected light mode consistent with laboratory fifth-force searches. This package is self-contained, falsifiable, and ready for precision scrutiny.

Executive Summary of New Results

Since our prior publication, we have:

  1. Derived the tuple lattice from the UV: A compact BF 2-form sector coupled to Spin(10) Cartan/centers produces Λ_tuple≅ℤ³ with the unique Cartan/center map fixed by SM+H anchors and an exact gauge-invisible family translation Δ=(0,3,3).
  2. Completed the unification analysis: For the SO(10)→LR→SM chain with (10_H,126_H,54_H) we provide closed-form one-loop solutions for (M_I,M_GUT,α_G), two-loop corrections (including top-Yukawa), and a minimal threshold model that places (M_I,M_GUT)≃(10¹³⁻¹⁴,10¹⁶) GeV while keeping τ_p in the 10³⁵ yr ballpark.
  3. Global magnitude fit (desk level): A concrete nonnegative slope solution γ⃗≃(2.60,0.45,0.05) with offsets reproduces CKM/PMNS hierarchies and mass ratios at magnitude level, together with crisp desk-only falsifiers.
  4. Gravity tie-in: The 2-form sector has a 4-form dual that gives a heavy SM-singlet scalar m_φ∼M_I and, if a protected orthogonal direction exists, a distinct light mode compatible with short-range tests.

Conventions and Inputs

We work at M_Z=91.2 GeV with GUT normalization for hypercharge. Gauge couplings:

(α₁⁻¹,α₂⁻¹,α₃⁻¹)|_{M_Z} = (59.01, 29.59, 8.45)

Representative flavor data at M_Z (desk level): |V_us|=0.225, |V_cb|=0.041, |V_ub|≃0.0037; PMNS sin²θ₁₂≃0.307, sin²θ₂₃≃0.50, sin²θ₁₃≃0.021; Δm²_atm≃2.5×10⁻³ eV². Mass ratios at M_Z are used desk-level to set offsets (e.g. m_t/m_c≃136, m_b/m_s≃44).

Cartan/Center Map and Tuple Anchors

The tuple coordinates τ⃗=(N,w,T)∈ℤ³ map to non-abelian centers and abelian charges via

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

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

with Y=T₃ᴿ+½(B-L) uniquely fixing B-L and T₃ᴿ once the SM+H anchors are imposed. The family translation

Δ=(0,3,3)

is gauge-invisible: χ₃(Δ)≡χ₂(Δ)≡Y(Δ)=0.

UV Microfoundation of the Tuple Lattice

Setup and statement of results

Let G_UV=Spin(10), G_LR=SU(3)_C×SU(2)_L×SU(2)_R×U(1)_{B-L}. A compact BF sector with three 2-forms Bⁱ (i=N,w,T), each having integer periods, couples to the visible sector through center and Cartan backgrounds. Worldline insertions carry integer labels (N,w,T); large-gauge transformations generate lattice translations in Λ_tuple. The map is UV-induced and unique.

Theorem (Uniqueness of the Cartan/center map)

Within integer linear maps (χ₃,χ₂,Y,B-L,T₃ᴿ) on Λ_tuple that satisfy the center relations and the LR identity Y=T₃ᴿ+½(B-L), there is a unique solution (up to GL(3,ℤ) and hypercharge-neutral shifts) reproducing all SM+H anchors and admitting a nontrivial gauge-invisible family translation.

Lemma (Kernel structure and generators)

The constraint matrix for (χ₃,χ₂,Y) has Smith normal form diag(1,1,6); thus [Λ_tuple:𝒦]=6 for 𝒦=ker(χ₃,χ₂,Y). A primitive generating set is

Δ=(0,3,3), h=(1,1,-1)

with χ₃(Δ)=χ₂(Δ)=Y(Δ)=0. The generator Δ realizes exact family invariance; h is the minimal non-abelian kernel step.

BF realization and flavor convexity

The compact 2-forms Bⁱ with BF couplings to visible backgrounds generate the charges. In the dilute/convex chamber of the topological action, overlap magnitudes obey

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

with γ⃗ nonnegative by convexity and positive homogeneity; GL(3,ℤ) covariance guarantees basis independence of observables.

Gauge Unification: SO(10)→LR→SM (Case B)

We adopt the non-SUSY chain SO(10)→G_LR→SM with G_LR=SU(3)_C×SU(2)_L×SU(2)_R×U(1)_{B-L}, three 16_F, and scalars 10_H⊕126_H⊕54_H. The tuple/Cartan map fixes all SM assignments and allows exact family invariance under Δ.

One-loop closed form and baseline solution

Below M_I (SM: 3 families, 1 Higgs doublet), the β coefficients are b₁=41/10, b₂=-19/6, b₃=-7. In the LR window (Case B minimal content), we take b₃⁽⁺⁾=-7, b₂L⁽⁺⁾=-7/3, b₂R⁽⁺⁾=-7/3, b_BL⁽⁺⁾=7.

Solving without thresholds gives the baseline (one-loop) solution:

M_I⁽¹⁾ ≃ 1.22×10¹⁰ GeV

M_GUT⁽¹⁾ ≃ 5.92×10¹⁵ GeV

α_G⁻¹⁽¹⁾ ≃ 43.88

Two-loop refinement and threshold budget

Including SM two-loop gauge matrix and top-Yukawa terms shifts the baseline scales to:

M_I ≃ 2.0×10¹⁰ GeV

M_GUT ≃ 2.5×10¹⁵ GeV

α_G⁻¹ ≃ 43.5

To place (M_I,M_GUT)≈(10¹³⁻¹⁴,10¹⁶) GeV, threshold budgets at M_I (Λ) and M_GUT (Θ) are required. A minimal pattern splits 126_H components near M_I by factors 0.3–3 and 54_H adjoints near M_GUT by similar factors, supplying the required 𝒪(1) shifts in α⁻¹ while leaving the tuple map and low-energy spectrum intact.

Proton decay and weak mixing

With M_GUT∼10¹⁶ GeV and α_G≃1/44, the leading gauge-boson mediated p→e⁺π⁰ lifetime sits in the ∼10³⁵ yr range (hadronic and mixing factors of order unity), above the current bounds and within Hyper-Kamiokande sensitivity. The predicted sin²θ_W at M_Z matches 0.231 once thresholds are included.

Global Magnitude Fit and Falsifiers

We now fix a concrete, nonnegative slope vector in the magnitude law:

γ⃗ ≈ (2.60, 0.45, 0.05)

using the family step Δ=(0,3,3) and the minimal kernel step h=(1,1,-1):

S ≡ 3(γ_w+γ_T) ≈ 1.50 ⇒ |V_us| ≈ e⁻ˢ ≈ 0.223

H ≡ γ_N+γ_w-γ_T ≈ 3.00 ⇒ |V_cb| ≈ e⁻ᴴ ≈ 0.050

Desk-only falsifiers:

  1. Ratio test: ln(1/|V_us|)/ln(1/|V_cb|) ≈ S/H ≃ 0.50 (current central ≈ 0.467); future convergence outside [0.35,0.70] falsifies the magnitude-only assignment.
  2. If |U_e3|² < 0.01 precisely, magnitude-only fits with γ_T≪γ_w are disfavored absent compensating phases.
  3. A significant deviation of |V_ub| relative to the {S,H} expectation implies revisiting tuple placements or abandoning the simplex.

Gravity Singlet from the Topological Sector

Dualizing the compact 2-forms yields 4-forms ℱ⁽⁴⁾ᵢ. A gauge-singlet scalar φ coupled as

ℒ ⊃ ½(∂φ)² - ½m_φ²φ² + (φ/f_φ)∑ᵢ ℱ⁽⁴⁾ᵢ

acquires m_φ²=χ_top/f_φ², with χ_top the topological susceptibility of the 4-form sector. In the SO(10)→LR→SM chain, the 126_H VEV at M_I∼10¹³⁻¹⁴ GeV sets the dominant kernel scale, naturally producing a heavy SM-singlet m_φ∼M_I (χ₃=χ₂=Y=0). If a protected orthogonal eigenmode of the topological kernel exists, it can realize a light scalar with short-range Yukawa coupling in a laboratory "Goldilocks" band without disturbing unification.

Conclusions

We have (i) derived the tuple lattice and Cartan/center map from a UV topological sector tied to Spin(10), proving uniqueness and family invariance; (ii) completed an SO(10)→LR→SM unification analysis with closed-form one-loop solutions, controlled two-loop shifts, and realistic thresholds that place (M_I,M_GUT) at seesaw and proton-stability targets; (iii) provided a concrete, nonnegative slope solution matching CKM/PMNS magnitudes with crisp falsifiers; and (iv) tied the topological sector to a gravity-singlet scalar at m_φ∼M_I with an optional protected light mode relevant for laboratory tests. These elements form a coherent, falsifiable, and data-driven step toward a unified description of known particles, forces, and short-range gravity phenomenology.