ABSTRACT
The Unified Tuple–Matrix Framework (UTMF) integrates a topological BF-like gravity sector with an SO(10) gauge unification and a tuple-curvature matter sector. Versions up to v4.0 established a quantum-complete candidate, derived flavor from tuples, and proposed topological cures for singularities. Here, v5.0 closes the phenomenology gap by: (i) deriving and instantiating renormalization-group (RG) flows with explicit one-loop β-functions for g₁₀, y, λc with a corrected quartic normalization; (ii) locating a quasi-invariant trajectory at M_GUT ≃ 2×10¹⁶ GeV and integrating to the electroweak scale; (iii) replacing the unified-gauge approximation below M_GUT with Standard Model (SM) two-loop running plus one-loop threshold matching and a linearized unification fit; (iv) presenting explicit dark-matter sectors (tuple solitons and a tuple-curvature axion-like particle), including a full-DM benchmark B1'; (v) detailing cosmological implications (flattened-quartic inflation, tuple-flavor leptogenesis, late-time acceleration) with falsifiable predictions; and (vi) providing full reproducibility in the appendices. We show that the normalization fix leaves low-energy gauge unification and the two-loop threshold fit robust.
1. Introduction
UTMF posits a discrete/topological substrate whose tuples and braids capture curvature and flavor, interfaced with a BF-type gravitational sector and SO(10) for gauge unification. The v4.0 line achieved: (1) topological singularity avoidance; (2) a path to UV completeness; (3) flavor-from-tuples mechanisms; (4) an integrated SO(10) bridge. Outstanding items—explicit RG flows, DM candidates, full cosmology, and falsifiability—are the focus here. A companion paper tests flavor predictions with IceCube data (in preparation).
Goals of v5.0
- Derive and instantiate the UTMF matter β-functions
- Perform a realistic EW-scale check with SM two-loop running and thresholds
- Define DM candidates with parameterizations and tests
- State cosmological predictions
- Present a concrete unification-fit procedure suitable for referee scrutiny
- Provide a falsification pack with benchmark planes
(Cosmology uses the reduced Planck mass M̄_P ≡ M_P/√(8π) ≈ 2.435×10¹⁸ GeV.)
2. UTMF Effective Sector and One-Loop RGEs
We adopt a renormalizable truncation sufficient for high-scale evolution:
where τᵃ (a=1,...,Nₜ) is a real multiplet in the adjoint of an internal tuple algebra (effectively O(Nₜ)-invariant quartic), ψ are SO(10) fermions, Yᵃ the tuple Yukawas, 𝒲 a CP-odd topological invariant, and λc and αc are UTMF couplings.
One-Loop β-Functions (MS̄ scheme)
With our quartic normalization ℒ ⊃ -λc/4(τᵃτᵃ)², the pure-scalar contribution is (16π²)β_λc^(pure) = 2(Nₜ+8)λc². For Nₜ=3, and for SO(10) with three 𝟏𝟔 families and one 𝟏𝟎_H:
With C₂(G₁₀)=8, T(𝟏𝟔)=2, C₂(𝟏𝟔)=45/8. We adopt a Yukawa alignment Yᵃ=yUᵃ with orthonormal Uᵃ, so Tr(Y†Y)=3y², Tr[(Y†Y)²]=3y⁴.
3. Quasi-Invariant Trajectory & Boundary at M_GUT
Along the quasi-fixed Yukawa ratio β_y=0 ⟹ y/g₁₀=3/2. Writing λc=k g₁₀² and imposing dk/d ln μ=0 yields:
On-Trajectory Boundary at M_GUT = 2×10¹⁶ GeV
y = 0.825
λc = 0.356
Convention note: with the alternate quartic convention λc → λc/(Nₜ+8) one finds λc=0.399 at the same M_GUT; low-energy predictions are unchanged when used consistently.
4. SM Two-Loop Running, Thresholds, and a Linearized Unification Fit
Below M_GUT we run (g₁,g₂,g₃) at two loops with Yukawa feedback and y at one loop (GUT-normalized g₁). Two-loop coefficients are given in Appendix C (see PDF appendices for the full matrices bᵢⱼ and dᵢ). Across each heavy threshold M_X:
At M_GUT encode GUT splittings via gᵢ(M_GUT)=(1+εᵢ)g₁₀ and linearize the fit to PDG targets at M_Z:
One-Loop Diagnostic (No Thresholds)
| g₁ | g₂ | g₃ | |
|---|---|---|---|
| Predicted | 0.4462996 | 0.7104159 | 1.626023 |
| Targets | 0.4614 | 0.6516 | 1.2172 |
| Rel. diff. | -3.3% | +9.0% | +33.6% |
Local Jacobian: J = diag(0.293869, 1.185259, 14.212147).
Calibrated Two-Loop + Thresholds Fit
The calibrated two-loop + thresholds fit we present (with realistic GUT spectrum splittings) is:
5. Benchmark Predictions & Falsification Pack
B1 (tALP, sub-DM)
mₐ=10⁻⁶ eV, fₐ=5×10¹¹ GeV, cᵧ=1. Misalignment with M̄_P gives T_osc=6.879 GeV and:
If breaking is post-inflation with N_DW=1, add a string/wall contribution (scaling-network estimate).
B1′ (tALP, full DM)
mₐ=10⁻⁶ eV, fₐ=3.7×10¹² GeV, θᵢ=1, yielding Ωₐh² ≃ 0.12.
B2 (tuple-soliton)
m_sol=10 GeV, σ/m=0.5 cm²/g, R=16.84 fm with
B3 (inflation)
Flattened quartic:
For r=0.008 and N=50–60: nₛ ≃ 0.969 (50) to 0.974 (60), and αₛ ∼ -(1–4)×10⁻³.
Falsification Pack
The site's figures mirror the four planes in the PDF (see PDF appendices for full matrices and plots):
- tALP: (mₐ,fₐ) plane with full-DM line and B1/B1′; right axis gₐᵧ
- Solitons: (m_sol,σ/m) with target band and cluster shading; B2
- Inflation: (nₛ,r) for N∈[50,60] with a CMB-S4 guide at r=10⁻³
- Leptogenesis: wedge (M₁,m̃₁) with dashed efficiency guides and DI-bound limit curve
6. Cosmology: Inflation, Leptogenesis, Late-Time Acceleration
Inflation
Exact slow-roll expressions are standard for this potential; picking N∈[50,60] and modest α=𝒪(0.3–0.5) gives the values above.
Leptogenesis
With M₁ ∼ 3×10¹⁰ GeV and tuple-flavor CP phases, the Davidson–Ibarra bound gives ε₁^max=2.96×10⁻⁶. Taking κf=0.0208 yields ηB ≃ 6.0×10⁻¹⁰; κf∈(0.02,0.05) spans typical washout.
Late-time acceleration
Residual tuple curvature yields Λ_eff=Λc+ΔV_topo, or a light quintessence-like mode with mild w(z)>−1 linked to RG running of λc. A CPL fit (w₀,wₐ) follows once μq,fq are fixed.
7. Conclusion
UTMF v5.0 delivers explicit RG structure (with a documented quartic-normalization audit and IR insensitivity), a quasi-invariant GUT boundary, a two-loop thresholded unification fit with uncertainties, and testable DM and inflation benchmarks.
Immediate steps: refine the GUT spectrum to minimize |εᵢ|; produce the falsification-pack plots; complete the IceCube companion study.
References
Appendices
Appendix A: One-Loop Derivations (MS̄) and Quartic-Normalization Audit
Field content and conventions
We take a real O(N_t) multiplet τᵃ (a=1,...,N_t), three SO(10) fermion families in 16, and one scalar 10_H. The renormalizable truncation:
We use the reduced Planck mass M̄_P = 2.435×10¹⁸ GeV in cosmology.
Tensor mapping and the factor of 2
Writing the quartic as -1/4! λ_abcd τᵃτᵇτᶜτᵈ, the identification with -λc/4(τ²)² is:
With this normalization, the well-known O(N) one-loop result gives:
i.e. a factor of 2 relative to the common choice -λ̂/4!(τ²)² which yields (N_t+8)λ̂².
Aligned Yukawas and invariants
Write Yᵃ = y Uᵃ with orthonormal Uᵃ. Then S₁ ≡ Tr(Y†Y) = 3y² and S₂ ≡ Tr[(Y†Y)²] = 3y⁴.
One-loop β-functions (UTMF matter sector)
For N_t=3, C₂(G)=8, T(16)=2, C₂(16)=45/8:
Quasi-invariant trajectory and GUT boundary
Imposing β_y = 0 ⟹ y/g₁₀ = 3/2. Let λc = k g₁₀²; stationarity (dk/d ln μ = 0) gives:
We take the positive root. With g₁₀ = 0.55, g₁₀² = 0.3025:
Convention note: In the alternate convention (pure term (N_t+8)λ²), k₊=1.321⟹λc=0.399 for the same g₁₀; IR predictions are insensitive provided a single convention is used consistently.
Appendix B: SM Channel Coefficients for (16π²)β_y (GUT-normalized g₁)
For a Dirac-like Yukawa y coupling to a given SM channel (with g₁ in SU(5) normalization):
For mixed participation, use weights (w_u,w_d,w_e,w_ν) (sum to 1) and define effective coefficients.
Appendix C: Two-Loop SM Gauge Running with Yukawa Feedback
Let t ≡ ln μ. The two-loop RGEs for (g₁,g₂,g₃) (one Higgs doublet, three families) are:
with b_i = (41/10, -19/6, -7), coefficient matrix b_ij, and (for up-type dominance) d_i = (17/10, 3/2, 2).
Appendix D: Linearized Unification Fit (Jacobian and One-Loop Diagnostic)
Setup
M_GUT = 2×10¹⁶ GeV, M_Z = 91.1876 GeV, g₁₀ = 0.55, y = 0.825, λc = 0.356. One-loop/no-threshold baseline:
Targets (GUT-normalized): (0.4614, 0.6516, 1.2172).
Jacobian
Define ε_i by g_i(M_GUT) = (1+ε_i)g₁₀ and linearize. Numerically (via |δε| = 10⁻⁴ nudges):
Our calibrated two-loop+thresholds fit used in the text is:
Appendix E: Exact Slow-Roll for the Flattened Quartic and Isocurvature
Potential and slow-roll parameters
Use χ ≡ φ/M̄_P and:
Exact points for r = 0.008 (ε★ = 5×10⁻⁴)
| N | α | χ★ | χ_end | n_s | r | λ_eff |
|---|---|---|---|---|---|---|
| 60 | 0.31540084 | 7.231203 | 1.581304 | 0.973761 | 0.008000 | 1.1133×10⁻¹⁰ |
| 55 | 0.40612500 | 6.657427 | 1.639 | 0.971500 | 0.008000 | 1.8283×10⁻¹⁰ |
| 50 | 0.53607269 | 6.078771 | 1.708 | 0.968787 | 0.008000 | 3.1549×10⁻¹⁰ |
Inflationary Hubble and isocurvature suppression
At N=60, χ★=7.2312⟹φ★=1.761×10¹⁹ GeV; an axion-like mode with m_a²(φ)=m_a²+κ_a φ² satisfies the "heavy-during-inflation" condition if:
Appendix F: Numerics, Misalignment Abundance, Soliton Geometry, Leptogenesis
Integrator details
t = ln μ, with ln(M_GUT/M_Z) = 33.0215897455. Step h = [ln M_Z - ln M_GUT]/6000 = -5.5035983×10⁻³.
Above M_GUT: one-loop (g₁₀,y,λc); below: two-loop (g₁,g₂,g₃) and one-loop (y,λc). Absolute/relative tolerances 10⁻⁹; explicit RK4.
ALP misalignment (B1) with reduced Planck mass
H(T) = 1.66√g★ T²/M̄_P; 3H(T_osc) = m_a gives:
Present abundance:
Tuple-soliton geometry (B2)
Given m_sol = 10 GeV and σ/m = 0.5 cm²/g:
Leptogenesis (quick check, Davidson–Ibarra bound)
For M₁ = 3×10¹⁰ GeV:
Taking κ_f = 0.0208, ε₁ = ε₁^max gives:
consistent with observation.
Appendix G: Implementation Notes (Reproducibility Aids)
ODE integration (sketch)
A minimal RK4 loop over μ with fixed log-step h:
for (k = 0; k < Nsteps; ++k) {
mu = mu0 * exp(k*h);
beta1 = beta(g, y, lambda); // pack all betas at scale mu
// standard RK4 staging on vector (g1,g2,g3; y; lambda)
}Finite differences for J: nudge one ε_j at M_GUT by ±δ, rerun to M_Z, take the central derivative with δ = 10⁻⁴.