Unified Topological Mass Framework v2.2.2 — Erratum & Consolidated Paper
Author: Dustin Beachy
Abstract. We finalize the UTMF program to v2.2.2 by (i) adopting an SU(2)-covariant triality definition that is invariant within weak doublets, (ii) retaining the derived hypercharge rule and charge normalization, and (iii) publishing a minimal, fully consistent tuple catalog for one Standard Model family. This closes the internal inconsistencies in v2.2 while preserving its phenomenology and fit templates.
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0. Conventions & Core Rules
Electric charge: Q = T3 + Y.
Hypercharge (derived): Y(N,w,T,χ)= -¼(w+T) + ⅙χ, (N,w,T)∈ℤ³, χ∈{0,1}.
SU(2)-covariant triality (new): τ_SU2(N,w,T,χ)=χ(N+w+T) mod 3
Minimality: choose small integers minimizing C = |N| + |w| + |T| subject to the above rules.
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1. Minimal Tuple Catalog (One SM Family)
1.1 Leptons (χ=0, colorless by gate)
Left doublet (ν_L, e_L), Y = -1/2:
e_L: (N,w,T) = (0,1,1)  =>  w+T = 2,  Q = -1,  τ = 0
ν_L: (0,2,0) = D^{-1}(e_L)  =>  w+T = 2,  Q = 0,  τ = 0
Right electron e_R, Y = -1:
e_R: (0,2,2)  =>  Q = -1,  τ = 0

1.2 Quarks (χ=1, nonzero τ with SU(2)-covariant triality)
Left doublet (u_L, d_L), Y = +1/6:
d_L: (1,-2,2)  =>  w+T = 0,  Q = -1/3,  τ = 1
u_L: (1,-1,1) = D^{-1}(d_L)  =>  w+T = 0,  Q = +2/3,  τ = 1
Right up u_R, Y = +2/3:
u_R: (0,-1,-1)  =>  Q = +2/3,  τ = 1
Right down d_R, Y = -1/3:
d_R: (0,1,1)  =>  Q = -1/3,  τ = 2

All tuples satisfy: (i) the hypercharge rule, (ii) Q = T3 + Y, (iii) SU(2) doublet structure with invariant w+T, and (iv) color: leptons τ=0; quarks τ∈{1,2} with τ constant across the LH doublet.
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2. Proof Sketches
2.1 SU(2)-Covariance of Triality (new)
Let D:(w,T)↦(w-1,T+1). Then (N+w+T) is D-invariant. Hence τ_SU2 is the same on both members of any doublet.
2.2 Hypercharge & SU(2) invariance
As S=w+T is D-invariant, Y is common to partners. Normalization reproduces all SM electric charges with the tuples above.
2.3 Anomalies (unchanged)
Using the SM field content with these assignments, the standard anomaly sums [SU(3)]^2U(1), [SU(2)]^2U(1), U(1)^3, and gravitational–U(1) cancel per family.
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3. Couplings & Phenomenology (unchanged from v2.2)
Baseline leptonic coupling map: Ξ = |w+T|/4. Pure vector benchmark: α_A=0. Variants: N-weighted and geometric-norm forms for stress tests.
Partial width: Γ_ℓℓ(X) = [M_X/(12π)] * g_0^2 * Ξ^2 * (1 + tan(α_A)^2) with axial admixture angle α_A (benchmark: α_A=0).
Lineshape: coherent sum with known resonances and continuum, Gaussian mass-resolution smearing, Poisson NLL fits; sideband-constrained backgrounds recommended.
These formulas apply unchanged; the approved tuples keep the relevant Ξ values in the same ranges as earlier benchmarks.
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4. Notes & Versioning
What changed from v2.2 → v2.2.2: replaced τ = χ·(N + 2w + T) mod 3 with SU(2)-covariant τ_SU2 = χ·(N + w + T) mod 3; updated quark tuples accordingly; retained hypercharge rule and all phenomenology.
Why it matters: ensures color representation is consistent across each weak doublet, eliminating the prior drift.
Next steps: extend catalog to higher generations; tabulate Ξ for benchmark states; provide detector-specific priors and full fit templates for Belle II / LHCb.
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5. Appendix: Analyst Fit Pack v1.0 (Belle II / LHCb)
5.1 Coupling quick-reference
Baseline leptonic coupling factor: Ξ(N,w,T;χ=0) = |w+T|/4.
Partial width: Γ_ℓℓ(X) = [M_X/(12π)] * g_0^2 * Ξ^2 * (1 + tan(α_A)^2) with axial admixture angle α_A (benchmark: α_A=0).
Tuple -> Ξ (selected):
Leptons: e_L and ν_L: |w+T|=2 => Ξ=0.5; e_R: |w+T|=4 => Ξ=1.0.
Benchmarks: X1:(7,-2,4; χ=0), X2:(9,-3,5; χ=0): both |w+T|=2 => Ξ=0.5.
5.2 Lineshape & likelihood
Coherent amplitude: ℒ(m) = |A_cont(m) + ∑_R A_R(m) + A_X(m)|^2 convolved with Gaussian resolution G(m; σ_m).
New narrow state: A_X(m) = [√(12π Γ_ℓℓ Γ_tot)/M_X] · e^{iφ} / (m^2 - M_X^2 + iM_X Γ_tot).
Fit with Poisson binned NLL; constrain background from sidebands via a Gaussian penalty on background parameters.
5.3 Default scan ranges
M_X: [4.80, 5.10] GeV and [9.30, 9.50] GeV (1–2 MeV steps or interpolation);
Γ_tot: [1e-3, 1e-1] GeV; α_A: [0, π/4]; φ: [-π, π]; σ_m: experiment prior (few MeV).
5.4 Minimal workflow
1. Choose tuple for X (e.g., X1 or X2).
2. Pick coupling model (baseline; optionally stress-test N-weighted / geometric variants).
3. Fit (M_X, Γ_tot, g_0, φ) with resolution and sideband-constrained background.
4. Quote best-fit or 95% UL on g_0 (or on Ξ if g_0 fixed); provide Asimov expected significance.
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6. Changelog (v2.2 → v2.2.2)
- Triality: replaced τ = χ·(N + 2w + T) mod 3 with SU(2)‑covariant τ_SU2 = χ·(N + w + T) mod 3 (invariant within weak doublets).
- Tuples: finalized one-family catalog — leptons: (0,2,0), (0,1,1), (0,2,2); quarks: LH (1,-2,2), (1,-1,1), RH (0,-1,-1), (0,1,1).
- Hypercharge & charge: unchanged (Y = -(w+T)/4 + χ/6, Q = T3 + Y).
- Phenomenology: coupling/lineshape machinery unchanged; analyst fit pack included (Appendix §5).