Unified Topological Mass Framework v2.2.1 — Erratum & Consolidated Paper
Author: Dustin Beachy
Abstract. We finalize the UTMF program to v2.2.1 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.

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.

1. Minimal Tuple Catalog (One SM Family)
1.1 Leptons (χ=0, colorless by gate)
Left doublet (ν_L,e_L): (0,2,0), (0,1,1)
Right electron e_R (χ=0): (0,2,2)

1.2 Quarks (χ=1, nonzero τ_SU2 with SU(2)-covariant triality)
Left doublet (u_L,d_L): (1,-2,2), (1,-1,1)
Right up u_R: (0,-1,-1)
Right down d_R: (0,1,1)

All tuples satisfy: (i) the hypercharge rule, (ii) Q = T3 + Y, (iii) SU(2) doublet structure with invariant τ_SU2, and (iv) color: leptons τ_SU2=0, quarks τ_SU2≠0, with τ_SU2 constant across the LH doublet.

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=χ(N+w+T)mod3 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.

3. Couplings & Phenomenology (unchanged from v2.2)
Baseline leptonic coupling map: Ξ = |w+T|/4.
Partial width: Γ_ℓℓ∝g₀²Ξ²M_X.
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 values in the same ranges as earlier benchmarks.

4. Notes & Versioning
What changed from v2.2 → v2.2.1: replaced triality with SU(2)-covariant τ_SU2; 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.

5. Appendix: Analyst Fit Pack v1.0 (Belle II / LHCb)
5.1 Coupling quick‑reference
Baseline leptonic coupling factor: Ξ(N,w,T; chi=0) = |w+T|/4
Partial width: Gamma_ll(X) = [M_X/(12*pi)] * g0^2 * Ξ^2 * (1 + tan(alpha_A)^2) with axial admixture angle alpha_A (benchmark: alpha_A=0)

Tuple -> Ξ (selected):
Leptons: e_L and nu_L: |w+T|=2 => Ξ=0.5; e_R: |w+T|=4 => Ξ=1.0
Benchmarks: X1:(7,-2,4; chi=0), X2:(9,-3,5; chi=0): both |w+T|=2 => Ξ=0.5

6. Changelog (v2.2 -> v2.2.1)
Triality: replaced tau = chi*(N + 2w + T) mod 3 with SU(2)‑covariant tau_SU2 = chi*(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 + chi/6, Q = T3 + Y)