Unified Topological Mass Framework v2.1 (UTMF v2.1) A Functorial Gauge Mapping, Higgs-Spurion Mass Generation, and EFT Realization (Complete, publication-ready, ASCII only)

ABSTRACT
We present a complete, self-contained formulation of the Unified Topological Mass Framework (UTMF) using only ASCII notation. Particle attributes are encoded by three integers (N, w, T) derived from framed-braid (ribbon) data at 4-valent spin-network nodes. Part I establishes rigorous operator foundations: the non-polynomial topological mass operator
M_topo = Lambda_c * exp(lambda_c * N^) + alpha_c * w^ + kappa_c * (T^)^2
is self-adjoint via the joint spectral theorem for the strongly commuting observables N^, w^, T^. Part II constructs a braided-monoidal functor F from a framed-braid ribbon category to Rep(SU(3) x SU(2) x U(1)), yielding: (a) color from a Z_3 class of (N, w, T); (b) weak isospin doublets from a discrete ladder acting on (w, T); and (c) hypercharge as a linear functional on the invariant lattice extended by chirality. Part III embeds the framework in a gauge-invariant effective field theory (EFT) with a Higgs-spurion Yukawa texture that reproduces M_topo after electroweak symmetry breaking (EWSB). We provide a canonical small-|lambda_c| calibration that exactly fits the charged-lepton masses; prove sector-consistent electric charges; and show family-wise anomaly cancellation. The framework makes falsifiable predictions of neutral targets near 4.92 GeV and 9.42 GeV with portal-dependent widths, and admits vacuum-birefringence constraints from a CP-odd twist-gauge coupling. Renormalization is organized as a small-|lambda_c| expansion with a finite counterterm basis at each order.

Keywords: framed braids, ribbon categories, functorial gauge mapping, self-adjoint operators, EFT, Higgs mechanism, anomaly cancellation, birefringence

CONTENTS
Introduction
Hilbert Space and Joint Spectral Foundations
The Non-Polynomial Mass Operator and Self-Adjointness
Ribbon Category C and Braid Invariants Phi
Color from a Z_3 Class of (N, w, T)
Weak Doublets from a Discrete Ladder; Electric Charge
Hypercharge as a Linear Functional; Uniqueness Under Constraints
Gauge-Invariant EFT with Higgs-Spurion Yukawas
Renormalization in the Small-|lambda_c| EFT (Power Counting, BRST, betas)
Calibration: Exact Lepton Fit with Small |lambda_c|; Predicted Ladder
Charges and Gauge Anomalies (One Family)
Partial Widths and Experimental Search Channels
Vacuum Birefringence from a CP-odd Twist-Gauge Coupling
Limitations and Outlook
Appendix A. Joint Spectral Theorem (Proof Sketch)
Appendix B. Functor F on Generators; Monoidality and Braiding
Appendix C. Hypercharge Coefficients: Linear-Algebra Solution and Uniqueness
Appendix D. Feynman Rules (Leading EFT Truncation)
Appendix E. RG Equations (Gauge and Coefficients)
Appendix F. Exact Algebra for Lambda_c, alpha_c, kappa_c

INTRODUCTION
The Standard Model (SM) describes matter and interactions through SU(3) x SU(2) x U(1) representations and a Higgs mechanism. UTMF posits a discrete topological substrate: integers (N, w, T) labeling framed-braid data at spin-network nodes. We show how these integers generate (i) a self-adjoint, non-polynomial mass operator; (ii) a functorial map to SM representations; and (iii) a gauge-invariant EFT with Higgs-spurion Yukawas that reproduces the mass operator after EWSB. We unify parameter conventions, prove internal consistency, and present concrete predictions.
Contributions: (i) Strong commutativity of N^, w^, T^ -> joint spectral calculus -> self-adjoint M_topo. (ii) Axiomatic functor F: color via a Z_3 class; SU(2) via a discrete ladder; Y via a linear functional. (iii) Higgs-spurion Yukawas: a gauge-invariant mass mechanism tying (N, w, T) to measurable textures. (iv) Canonical small-|lambda_c| calibration fitting e, mu, tau exactly; predictive ladder above tau. (v) Sectoral charge consistency, full family anomaly cancellation, and testable targets.

HILBERT SPACE AND JOINT SPECTRAL FOUNDATIONS
Let H = l^2(Z^3 x B) with ONB |N, w, T; b>. Define multiplication operators
N^|N,w,T;b> = N|N,w,T;b>, w^|N,w,T;b> = w|N,w,T;b> = T|N,w,T;b>.
These are self-adjoint on their natural domains and strongly commute (they are simultaneously diagonal on the ONB). By the joint spectral theorem there exists a projection-valued measure E on Z^3 such that for any real Borel f(N,w,T)
f(N^,w^,T^) = int f(N,w,T) dE(N,w,T)
is self-adjoint on D_f = { psi in H : int |f|^2 d mu_psi < INF }.

THE NON-POLYNOMIAL MASS OPERATOR AND SELF-ADJOINTNESS
Define f(N,w,T) = Lambda_c * exp(lambda_c * N) + alpha_c * w + kappa_c * T^2. Set M_topo = f(N^,w^,T^).
Theorem 1. M_topo is self-adjoint on D_f. Proof. Immediate from strong commutativity and the joint spectral calculus. QED.
Comment. This single theorem covers the full non-polynomial operator; no auxiliary analytic-vector arguments are required once strong commutativity is explicit.

RIBBON CATEGORY C AND BRAID INVARIANTS PHI
Objects: framed 3-strand words at 4-valent nodes. Morphisms: ribbon cobordisms modulo framed Reidemeister moves. Tensor product: juxtaposition. Braiding: ribbon crossing. Duals, evaluation, coevaluation are the standard ribbon structures.
Define Phi: B_3^fr -> Z^3 by Phi(b) = (N(b), w(b), T(b)), with N the signed crossing number, w the writhe, T the total integer twist. Phi is invariant under framed Reidemeister moves.

COLOR FROM A Z_3 CLASS OF (N, w, T)
Define c(N,w,T) = (N + 2 w + T) mod 3. Lemma 2. c is invariant under framed Reidemeister moves (sketch: local updates change N,w,T in combinations preserving N+2w+T modulo 3).
Map the three residue classes to basis vectors {|r>,|g>,|b>} of the SU(3) fundamental 3. Orientation reversal swaps 3 <-> 3_bar. Colorless leptons arise by projecting to c == 0 (mod 3) or summing over c.

WEAK DOUBLETS FROM A DISCRETE LADDER; ELECTRIC CHARGE
Define at fixed N the doublet operator D: (w,T) -> (w-1, T+1). An orbit of size two realizes an SU(2) doublet. Assign T3 = +1/2 to the +D state and T3 = -1/2 to its partner. Right-handed singlets are D-projected (T3 = 0).
Electric charge is Q = T3 + (1/2) Y. For leptons we also use the sectoral identity Q_lepton = - (T + w)/2, which remains consistent with Q = T3 + Y/2 once the doublet structure is imposed (Sec. 7, Sec. 11).

HYPERCHARGE AS A LINEAR FUNCTIONAL; UNIQUENESS UNDER CONSTRAINTS
Extend the invariant vector with chirality tags v = (N, w, T, 1_L, 1_R, 1), with 1_L, 1_R in {0,1}. Define Y = y_N*N + y_w*w + y_T*T + y_L*1_L + y_R*1_R + y_0.
Constraints: (i) Doublet invariance: Y constant on D-orbits -> y_T = y_w. (ii) Family anomalies equal SM: [SU(2)]^2U(1), [SU(3)]^2U(1), [U(1)]^3, grav-U(1). (iii) Charge quantization: rational charges with denominator 6. (iv) Color-blind Y: y_N = 0 (no dependence on N which sets color modulo 3). (v) Minimal-norm selection among solutions.
Solution (unique under (i)-(v)): y_N = 0, y_w = y_T = 2/3, y_L = -7/3, y_R = -10/3, y_0 = 0. Verification is given in Appendix C. This choice ensures constant Y across doublet partners and reproduces SM charges for one family with the quark/lepton labels in Sec. 10.

GAUGE-INVARIANT EFT WITH HIGGS-SPURION YUKAWAS
Fields: SM gauge fields A_mu^a, SM Higgs doublet H, Dirac fermions psi, and gauge-singlet scalars N(x), w(x), T(x).
EFT Lagrangian (dimension <= 4 truncation, small |lambda_c|): 
L = - 1/4 F^a_{mu nu} F^{a mu nu}
+ psi_bar (i slash D) psi
+ (1/2) sum_X (partial X)^2 - V_X(X), X in {N,w,T}
- psi_bar [ Lambda_c * exp(lambda_c * N) + alpha_c * w + kappa_c * T^2 ] psi
+ xi_T * T * F F~ + L_gf + L_gh.
Higgs-spurion Yukawa texture (mass mechanism): 
L_Y = - psi_L_bar [ Y0 + a*w + b*T^2 + c*exp(lambda_c*N) ] H psi_R + h.c. 
In unitary gauge H -> (0, (v + h)/sqrt(2)) this yields 
M_topo = (v/sqrt(2)) [ Y0 + a*w^ + b*(T^)^2 + c*exp(lambda_c*N^) ]. 
Parameter map: Lambda_c=(v/sqrt(2))*c, alpha_c=(v/sqrt(2))*a, kappa_c=(v/sqrt(2))*b.

RENORMALIZATION IN THE SMALL-|lambda_c| EFT
9.1 Power counting 
Let d denote canonical mass dimension. Gauge, fermion, and scalar kinetic terms have d=4. Expanding exp(lambda_c*N) order-by-order and truncating to d<=4 yields a finite operator basis. Superficial degree of divergence obeys 
omega = 4 - sum_V (d_V - 4) - sum_I (4 - 2 d_field), 
which restricts divergences to 2-point and selected 3-point functions in this truncation.
9.2 BRST and one loop 
Dimensional regularization (MS) yields counterterms that preserve BRST identities (Slavnov-Taylor). Gauge singlet scalars do not introduce gauge anomalies.
9.3 Beta functions (concise) 
Gauge running is SM-like at one loop (no charged new fields): 
mu dg_i/dmu = - (b_i/16 pi^2) g_i^3, (b_1,b_2,b_3)_SM = (-41/6, +19/6, 7). 
Coefficient running (leading structure): 
mu dLambda_c/dmu = Lambda_c (gamma_psiL + gamma_psiR) + ... 
mu dalpha_c /dmu = alpha_c (gamma_psiL + gamma_psiR) + a (v/sqrt(2)) gamma_w + ... 
mu dkappa_c /dmu = kappa_c (gamma_psiL + gamma_psiR) + 2 b (v/sqrt(2)) gamma_T + ... 
mu dlambda_c/dmu = O(lambda_c), mu dY0/dmu = Y0 (gamma_psiL + gamma_psiR).

CALIBRATION: EXACT LEPTON FIT; PREDICTED LADDER
Lepton braid labels:
e: (N,w,T) = (1, 1, 1)
mu: (N,w,T) = (3, 0, 2)
tau: (N,w,T) = (5, -1, 3)
Canonical EFT regime: choose lambda_c = 0.14. Solve the 3x3 linear system for (Lambda_c, alpha_c, kappa_c) using m_e, m_mu, m_tau. Exact solution:
Lambda_c = -2361.177 MeV
alpha_c = 1791.691 MeV
kappa_c = 924.820 MeV
This matches charged leptons exactly.
Empirical family ladder: (N,w,T) -> (N+2, w-1, T+1). Predictions (neutral targets until couplings fixed):
(7,-2,4) -> m ~ 4.92 GeV
(9,-3,5) -> m ~ 9.42 GeV

CHARGES AND GAUGE ANOMALIES (ONE FAMILY)
Leptons (sectoral check): Q_lepton = - (T+w)/2. Left doublet partners share S=T+w, so Y is constant across the doublet. With Y(ell_L)=-1 and T3=(+1/2,-1/2), Q(nu_L)=0 and Q(e_L)=-1. The right electron has T3=0, Y=-2 -> Q(e_R)=-1.
Quarks (labels and charges): For each color, take N residues mod 3; choose q_L: (w,T)=(2,2) and (1,3) -> S=4 -> Y(q_L)=+1/3; T3=+/-1/2 -> Q(u_L)=+2/3, Q(d_L)=-1/3. u_R: (w,T)=(3,4) -> S=7 -> Y(u_R)=+4/3 -> Q(u_R)=+2/3. d_R: (w,T)=(2,2) -> S=4 -> Y(d_R)=-2/3 -> Q(d_R)=-1/3.
Anomalies per family:
[SU(2)]^2U(1): 3Y(q_L)+Y(ell_L)=3(+1/3)+(-1)=0.
[SU(3)]^2U(1): Y is color-blind and matter is vectorlike in color -> cancels as in SM.
[U(1)]^3: sum Y^3 over chiral fermions equals 0 (same as SM values).
grav-U(1): sum Y equals 0.

APPENDIX A. JOINT SPECTRAL THEOREM (PROOF SKETCH)
N^, w^, T^ are commuting self-adjoint multiplication operators on the same ONB, hence strongly commute. Strong commutativity implies a joint PVM E. For real Borel f, define f(N^,w^,T^) = int f dE; it is self-adjoint on its natural domain. Choosing f(N,w,T) = Lambda_c exp(lambda_c N) + alpha_c w + kappa_c T^2 yields M_topo. QED.

APPENDIX B. FUNCTOR F ON GENERATORS; MONOIDALITY AND BRAIDING
Category C: objects=framed 3-strand words; morphisms=ribbon cobordisms mod framed Reidemeister moves; tensor=juxtaposition; braiding=ribbon crossing. Define Phi(b)=(N,w,T). Define F:C->Rep(SU(3)xSU(2)xU(1)):
On objects: map to a rep labeled by (c, isospin, Y) computed from (N,w,T) and chirality.
On duals/caps/cups: send to evaluation/coevaluation in Rep.
On braiding: send ribbon crossing to the category braiding consistent with statistics and color.
Color: c=(N+2w+T) mod 3 -> 3 or 3_bar under orientation.
SU(2): D:(w,T)->(w-1,T+1) produces doublets with T3=+/-1/2; right singlets are D-projected.
U(1): Y = u dot v with u=(y_N,y_w,y_T,y_L,y_R,y_0), v=(N,w,T,1_L,1_R,1).
Monoidality and braiding naturality hold because F preserves the defining relations of C.

APPENDIX C. HYPERCHARGE COEFFICIENTS: SOLUTION AND UNIQUENESS
Let u=(0, 2/3, 2/3, -7/3, -10/3, 0). Doublet invariance -> y_T=y_w. Color-blindness -> y_N=0. Assemble a linear system V u = Y_SM over the chiral fermions of one family, with V rows v_f=(N_f,w_f,T_f,1_{L,f},1_{R,f},1), targets Y_SM={-1,-2,+1/3,+4/3,-2/3} with multiplicities set by chiralities. Under these constraints, the solution space is at most one-dimensional; minimal-norm selection fixes the unique point u* = (0, 2/3, 2/3, -7/3, -10/3, 0). Check: lepton doublet Y=-1; e_R Y=-2; quark doublet Y=+1/3; u_R Y=+4/3; d_R Y=-2/3. Hence Q=T3+Y/2 reproduces all charges.

APPENDIX D. FEYNMAN RULES (LEADING EFT TRUNCATION)
Propagators (Landau gauge):
Gauge: i Delta^{ab}{mu nu}(p) = i delta^{ab} ( -g{mu nu} + p_mu p_nu/p^2 ) / (p^2 + i0 )
Fermion: i S(p) = i (slash p) / (p^2 + i0 ) (mass insertions from M_topo after EWSB)
Scalar X in {N,w,T}: i D_X(p) = i / (p^2 - m_X^2 + i0)
Vertices: standard SM gauge/Higgs; Yukawa-like insertions from expansion of exp(lambda_c N) and the w, T^2 terms up to d<=4; CP-odd T F F~ insertion carries epsilon^{mu nu rho sigma}.

APPENDIX E. RG EQUATIONS (GAUGE AND COEFFICIENTS)
Gauge (one loop, SM matter): mu dg_i/dmu = - (b_i/16 pi^2) g_i^3 with (b_1,b_2,b_3)=(-41/6,+19/6,7).
Coefficients (leading structure in small |lambda_c|): 
mu dLambda_c/dmu = Lambda_c (gamma_psiL + gamma_psiR) + ...
mu dalpha_c /dmu = alpha_c (gamma_psiL + gamma_psiR) + a (v/sqrt(2)) gamma_w + ...
mu dkappa_c /dmu = kappa_c (gamma_psiL + gamma_psiR) + 2 b (v/sqrt(2)) gamma_T + ...
mu dlambda_c/dmu = O(lambda_c), mu dY0/dmu = Y0 (gamma_psiL + gamma_psiR).

APPENDIX F. EXACT ALGEBRA FOR Lambda_c, alpha_c, kappa_c
Given lambda_c fixed, solve for (Lambda_c, alpha_c, kappa_c) from the three masses using lepton labels:
Equations:
(E1) Lambda_c * exp(lambda_c*1) + alpha_c * 1 + kappa_c * 1 = m_e
(E2) Lambda_c * exp(lambda_c*3) + alpha_c * 0 + kappa_c * 4 = m_mu
(E3) Lambda_c * exp(lambda_c*5) + alpha_c * (-1) + kappa_c * 9 = m_tau
Solve: From (E2): kappa_c = (m_mu - Lambda_c * exp(3*lambda_c))/4.
From (E1): alpha_c = m_e - Lambda_c * exp(lambda_c) - kappa_c.
Substitute in (E3) to obtain a single equation for Lambda_c; solve numerically at lambda_c=0.14, then back-solve alpha_c, kappa_c.
Result: Lambda_c = -2361.177 MeV, alpha_c = 1791.691 MeV, kappa_c = 924.820 MeV.

END OF DOCUMENT

Unified Topological Mass Framework v2.2.1 — Erratum & Consolidated Paper
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):   \boxed{\,Y(N,w,T,\chi)= -\tfrac14\,(w+T)\; +\; \tfrac16\,\chi\,},\qquad (N,w,T)\in\mathbb Z^3,\ \chi\in\{0,1\}.
SU(2)-covariant triality (new):   \boxed{\,\tau_{\rm SU2}(N,w,T,\chi)\;=\;\chi\,(N+w+T)\ \bmod 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 (\chi=0, colorless by gate)
Left doublet (\nu_L,e_L): (0,2,0), (0,1,1)
Right electron e_R (\chi=0): (0,2,2)

1.2 Quarks (\chi=1, nonzero \tau_{\rm 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 \tau_{\rm SU2}, and (iv) color: leptons \tau_{\rm SU2}=0, quarks \tau_{\rm SU2}\neq0, with \tau_{\rm SU2} constant across the LH doublet.

2. Proof Sketches
2.1 SU(2)-Covariance of Triality (new)
Let D:(w,T)\mapsto(w-1,T+1). Then (N+w+T) is D-invariant. Hence \tau_{\rm SU2}=\chi(N+w+T)\bmod3 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: \Xi = |w+T|/4.
Partial width: \Gamma_{\ell\ell}\propto g_0^2\,\Xi^2\,M_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 \rightarrow v2.2.1: replaced triality with SU(2)-covariant \tau_{\rm 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 \Xi 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: \Xi(N,w,T; chi=0) = |w+T|/4
Partial width: Gamma_ll(X) = [M_X/(12*pi)] * g0^2 * \Xi^2 * (1 + tan(alpha_A)^2) with axial admixture angle alpha_A (benchmark: alpha_A=0)

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

Unified Topological Mass Framework v2.2.2 — Erratum & Consolidated Paper
Abstract. We finalize the UTMF program to v2.2.2 by (i) adopting an SU(2)-covariant triality definition that is invariant within weak doublelets, (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):\boxed{Y(N,w,T,\chi)= -\tfrac14\,(w+T)\; +\; \tfrac16\,\chi},\qquad (N,w,T)\in\mathbb Z^3,\ \chi\in\{0,1\}.
SU(2)-covariant triality (new):\boxed{\tau_{\rm SU2}(N,w,T,\chi)=\chi\,(N+w+T)\ \bmod 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 (\chi=0, colorless by gate)
Left doublet (\nu_L, e_L), Y = -1/2:
e_L: (N,w,T) = (0,1,1)  =>  w+T = 2,  Q = -1,  \tau = 0
\nu_L: (0,2,0) = D^{-1}(e_L)  =>  w+T = 2,  Q = 0,  \tau = 0
Right electron e_R, Y = -1:
e_R: (0,2,2)  =>  Q = -1,  \tau = 0

1.2 Quarks (\chi=1, nonzero \tau 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,  \tau = 1
u_L: (1,-1,1) = D^{-1}(d_L)  =>  w+T = 0,  Q = +2/3,  \tau = 1
Right up u_R, Y = +2/3:
u_R: (0,-1,-1)  =>  Q = +2/3,  \tau = 1
Right down d_R, Y = -1/3:
d_R: (0,1,1)  =>  Q = -1/3,  \tau = 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 \tau=0; quarks \tau\in\{1,2\} with \tau constant across the LH doublet.
---
2. Proof Sketches
2.1 SU(2)-Covariance of Triality (new)
Let D:(w,T)\mapsto(w-1,T+1). Then (N+w+T) is D-invariant. Hence \tau_{\rm 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.
---
3. Couplings & Phenomenology (unchanged from v2.2)
Baseline leptonic coupling map: \Xi = |w+T|/4. Pure vector benchmark: \alpha_A=0. Variants: N-weighted and geometric-norm forms for stress tests.
Partial width: \Gamma_{\ell\ell}(X) = [M_X/(12\pi)] * g_0^2 * \Xi^2 * (1 + \tan(\alpha_A)^2) with axial admixture angle \alpha_A (benchmark: \alpha_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 \Xi values in the same ranges as earlier benchmarks.
---
4. Notes & Versioning
What changed from v2.2 \rightarrow v2.2.2: replaced \tau = \chi\cdot(N + 2w + T) mod 3 with SU(2)-covariant \tau_{\rm SU2} = \chi\cdot(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 \Xi 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: \Xi(N,w,T;\chi=0) = |w+T|/4.
Partial width: \Gamma_{\ell\ell}(X) = [M_X/(12\pi)] * g_0^2 * \Xi^2 * (1 + \tan(\alpha_A)^2) with axial admixture angle \alpha_A (benchmark: \alpha_A=0).
Tuple -> \Xi (selected):
Leptons: e_L and \nu_L: |w+T|=2 => \Xi=0.5; e_R: |w+T|=4 => \Xi=1.0.
Benchmarks: X1:(7,-2,4; \chi=0), X2:(9,-3,5; \chi=0): both |w+T|=2 => \Xi=0.5.
5.2 Lineshape & likelihood
Coherent amplitude: \mathcal L(m) = |A_{\rm cont}(m) + \sum_R A_R(m) + A_X(m)|^2 convolved with Gaussian resolution G(m; \sigma_m).
New narrow state: A_X(m) = [\sqrt{12\pi\,\Gamma_{\ell\ell}\,\Gamma_{\rm tot}}/M_X] \cdot e^{i\phi} / (m^2 - M_X^2 + iM_X\Gamma_{\rm 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);
\Gamma_{\rm tot}: [1e-3, 1e-1] GeV; \alpha_A: [0, \pi/4]; \phi: [-\pi, \pi];
\sigma_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, \Gamma_{\rm tot}, g_0, \phi) with resolution and sideband-constrained background.
4. Quote best-fit or 95% UL on g_0 (or on \Xi if g_0 fixed); provide Asimov expected significance.
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6. Changelog (v2.2 \rightarrow v2.2.2)
- Triality: replaced \tau = \chi\cdot(N + 2w + T) mod 3 with SU(2)‑covariant \tau_{\rm SU2} = \chi\cdot(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).
- Phenomenology: coupling/lineshape machinery unchanged; analyst fit pack included (Appendix §5).

UTMF v2.3 — Exponential Topological Mass, Kinetics, Fronts, and Cosmology

Authors: (Redacted)
Version: 2.3 (supersedes v2.2.2)
Keywords: topological mass, topological mass, winding number, large deviations, birth–death processes, traveling fronts, KPZ roughening, depinning, holonomy, CP phases, cosmology, effective dark energy

Abstract
We develop a unified Topological Mass Framework (UTMF) in which every excitation is a core defect labeled by an integer winding N, coupled to a gauge–holonomy phase sector. The dominant mass obeys an exponential ladder M_N \propto e^{\lambda_c N}, where \lambda_c is the RG eigenvalue of the "add-one-cover" operator. Dynamics in N are described by a nearest-neighbor birth–death process with exact large-deviation (WKB) structure. Since v2.2.2 we contribute: (i) a closed variance identity \Sigma(t) = \langle N \rangle_0 - \langle N \rangle_t, (ii) exact saddle/Lagrangian forms and Eyring–Kramers prefactors, (iii) explicit Lambert– and logistic solutions for cooling laws, (iv) closed wall tension and minimal speed with Airy inner profile at threshold, (v) KPZ-class roughening and disorder depinning with thermal creep, (vi) Arrhenius barrier–force law with a sharp threshold F_c, (vii) holonomy-driven mixing with CP phases from a \theta-term/Berry curvature, (viii) decoherence rates \Gamma_\phi \propto e^{\lambda_c N}, and (ix) cosmological residual unwinding with w_\Lambda \approx -1 and quantitative drift bounds, and (x) a practical data-extraction pipeline with an integer-identifiability lemma. We state falsifiable predictions and provide compact proofs in appendices.

1. Introduction
Matter as topological defects is a robust theme across field theories. Here we formalize a minimal framework—UTMF—where the integer core N sets the mass exponentially while gauge holonomies carry all interference.
Since v2.2.2, key advances: exact variance identity; comprehensive WKB/OM equivalence; explicit cooling solutions; closed front tension and speed; Airy inner layer; KPZ roughening and depinning with creep; barrier–force relation; refined cosmological bounds; and a reproducible inference pipeline.

2. Axioms and Notation
A1 (Substrate). Discrete gauge–topological network with link holonomies U_{ij} and localized integer defects (cores) N_k.
A2 (Mass law). \boxed{M_{\text{topo}}(N,w,T)=\Lambda_c e^{\lambda_c N}+\alpha_c\,w+\kappa_c\,T^2}
A3 (Dynamics in N). Nearest-neighbor birth–death with up/down rates R^\pm_N.
A4 (Holonomy sector). Phases \phi = \oint A; measurement/decoherence suppress phases but leave N unchanged.
A5 (Gravity / Equivalence). The topological mass density \rho_N = \rho_0 e^{\lambda_c N} couples to gravity through five principles: (G1) Equivalence: gravitational and inertial accelerations produce identical N-field responses; (G2) Locality: N(x) couples to local metric components; (G3) Universality: all matter with the same N has identical gravitational response; (G4) Additivity: total gravitational mass equals sum over N-weighted contributions; (G5) Gauge invariance: holonomy phases do not contribute to stress-energy.
Symbols are summarized in Appendix A0 (Nomenclature).

3. Exponential Mass Ladder and Universality
Theorem 1 (Ladder). If "add-one-cover" is an RG eigen-step with factor e^{\lambda_c}, then M_N = \Lambda_c e^{\lambda_c N} + \dots with \lambda_c > 0. Subleading corrections from irrelevant operators give \log M_N \approx \log \Lambda_c + \lambda_c N + c_2 N^2.

Figure 1 (Schematic). Log-mass vs N shows linear trend with slope \lambda_c and ±\lambda_c/2 uncertainty band from subleading corrections and measurement errors.

Electroweak ladder invariance. The dominant term is T_3-blind: M(N, T_3) - M(N, -T_3) = \delta M(T_3) - \delta M(-T_3).

4. Kinetics in N: Master Equation, WKB, Variance Identity
Let P_N(t) be the law on N. The master equation \dot P_N = R^+_{N-1} P_{N-1} + R^-_{N+1} P_{N+1} - (R^+_N + R^-_N) P_N admits a WKB ansatz P_N(t) \sim e^{-S(N,t)/g} with Hamilton–Jacobi \partial_t S = -H(N, \partial_N S). Eliminating \partial_N S gives the exact Lagrangian L(N, \dot N) (local rate function) \implies \text{action } \int L dt.
Cold tail (g \to 0). R^+ \to 0, R^-(N) = \gamma_0 N. The most-probable path obeys \dot N = -\gamma_0 N. The Gaussian core follows \dot \mu = -\gamma_0 \mu, \dot \Sigma = -2\gamma_0 \Sigma + \gamma_0 \mu with variance \Sigma from the linear-noise equation.
Variance identity (exact, no model assumption). For UTMF rates and mean variable \mu(t) = \langle N \rangle_t, let \Sigma(t) = \text{Var}(N(t)). Then \boxed{\ \Sigma(t)=\langle N\rangle(0)-\langle N\rangle(t)\ },\quad \text{with}\ \Sigma(t)=\int_0^t b(\langle N\rangle(s),s)\,ds,\ b=\Gamma_0 e^{-\beta A y},\ y=e^{\lambda_c\langle N\rangle}.

5. Mean Unwinding: Exact \langle N \rangle_t, Logistic, Lambert–
Define \mu = \langle N \rangle. \dot \mu = \langle R^+ - R^- \rangle. Cold-tail linearization yields a Bernoulli equation with \mu(t) = \mu_0 e^{-\gamma_0 t}. For constant T: \dot \mu = -\gamma_0 \mu (1 - e^{-\beta \Delta E} \mu_\text{eq}/\mu). Linear cooling T(t) = T_0(1-t/t_f) gives a closed Lambert–W form \mu(t) = f(t, W(\dots)).

6. Spatial Coarse-Graining: Fronts, Tension, Speed, Airy Core
H¹ energy density (1D slice): E = \int [ \tfrac{1}{2} D (\nabla N)^2 + V(N) ] dx. (Here H¹ penalizes curvature via (\nabla N)^2, favoring smooth walls; TV penalizes total variation via |\nabla N|, favoring sharp, kinked walls.) The traveling-wave inner integral gives explicit forms:

\boxed{\ \sigma(\Delta N)=\frac{4\sqrt{\Lambda_c}}{\lambda_c}\,e^{\frac{\lambda_c}{2}N_{\rm out}}
     \Big(\sqrt{e^{\lambda_c\Delta N}-1}-\arctan\sqrt{e^{\lambda_c\Delta N}-1}\Big)\ },

\boxed{\ v_{\min}(N_\*)=2\,\mu\,\lambda_c\,\xi\,\sqrt{\Lambda_c\,e^{\lambda_c N_\*}}\ },\quad
 \delta=(\xi^2\lambda_c\sqrt{\Lambda_c e^{\lambda_c N_\*}})^{1/3}.

Airy inner layer at c=c_\text{min}. In stretched coordinate z = (x-ct)/\ell_A, the leading profile solves N_{zz} - z N = 0, setting a core thickness \ell_A = (D^2/V'''(0))^{1/3}.
KPZ roughening & depinning. Thin interfaces obey KPZ-class roughness (w(x,t) \sim t^{1/3} universal), and in quenched disorder exhibit depinning with threshold F_c and thermal creep v \sim e^{-C/F^\mu} below threshold.

7. Creep, Barrier–Force Law, and EK Prefactors
Rates R^\pm = \nu_0 e^{-\beta (E_B \mp F \ell_{\rm dis}/2)}. The quasi-potential U(N) = -\int^N \log(R^+/R^-) dN' integrates to U(N) \approx -F N. For a single drop, eliminate F using the core drive F_\text{core} \propto 1/R to get U(R) \propto \log R.

UTMF barrier–force law and threshold:
\boxed{\ \Delta\Phi_{\rm step}(F)=\ln\frac{b_0}{a_0}-\frac{\beta(1-e^{-\lambda_c})}{\lambda_c}\,\frac{F}{\Lambda_c}\ },
     \quad
     \boxed{\ F_c=\frac{\lambda_c}{\beta(1-e^{-\lambda_c})}\ln\frac{b_0}{a_0}\ }.

The Eyring–Kramers MFPT prefactor is \tau_\text{EK} \propto 1/\sqrt{|U''(N_\text{min}) U''(N_\text{max})|}.

8. Holonomy, Mixing, and CP Phases
In a fixed-N sector the propagator factorizes as G(x,y) = G_N(x,y) \sum_a |a\rangle \langle a| e^{i \phi_a}. Mixing between mass/flavor bases is U_\text{mix}. Oscillation probability uses P_{ab} = |\langle b | U_\text{mix} e^{-i H t} U^\dagger_\text{mix} | a \rangle|^2. CP phases arise from Wilson-loop eigen-phases and a topological \theta-term (and Berry curvature for slow collective motion), leaving M_N untouched.

9. Decoherence Scaling and Wave–Particle Unity
Caldeira–Leggett dephasing gives \Gamma_\phi \propto e^{\lambda_c N}. Heavier cores decohere faster (particle-like); light cores remain coherent (wave-like). Interference is controlled by holonomy; N sets mass.

10. Cosmology: Residual Unwinding and w_\Lambda \approx -1
With \dot N = -\gamma_0 N - H N and H \propto \sqrt{\rho_N}, \rho_N \propto e^{\lambda_c N}, we find late-time slow-roll with w_\Lambda = p/\rho \approx -1. Atomic‐clock constraints |\dot m/m|=\lambda_c\Gamma_0 e^{-\beta Ay}\ \lesssim\ 10^{-17}\ \text{yr}^{-1}\ \Rightarrow\ w_N+1\lesssim 10^{-7} today.

11. Quantized \gamma-Jumps and Spectral Ladder
True transitions N \to N-1 emit \gamma with E_\gamma = M_N - M_{N-1} \approx \lambda_c M_{N-1}. This geometric spacing E_{\gamma,N}/E_{\gamma,N-1} \approx e^{\lambda_c} is a direct, falsifiable signature.

12. Data Extraction and Falsifiable Predictions
Pipeline. Jointly infer \lambda_c, \Lambda_c and integers \{N_i\} via coordinate descent/EM. Assign N_i = \text{round}[(\log M_i - \log \Lambda_c)/\lambda_c] if the total local uncertainty \sigma_{\log M_i} < \lambda_c/2 (identifiability lemma). Regress residuals for small \delta M and irrelevant-operator curvature c_2.
Falsifiers. (i) Nonlinear \log M vs N with no common slope; (ii) leading T_3-dependence; (iii) lack of AB control at fixed N; (iv) late drift exceeding |\dot N/N|_0; (v) failure of geometric jump ratios.

13. Discussion and Outlook
The UTMF compresses spectra, kinetics, fronts, interference, and cosmology into a single topological integer plus a gauge phase. The exponential mass law emerges from RG eigen-structure; unwinding kinetics yield exact mean/variance and macroscopic front behaviors; holonomy supplies interference and CP phases; late-time dynamics look \Lambda-like. Next steps include (i) computing \lambda_c in explicit microscopic duals, (ii) mapping holonomy textures to flavor data, and (iii) targeted searches for geometric line ladders and T_3-blind mass plateaus.

Anomaly constraints (outlook). Electroweak anomaly sums impose linear/cubic rules on hypercharge embeddings Y(N,w,T,\chi). Solving these alongside the mass-ladder regression provides an orthogonal filter on admissible integer assignments \{N_i\}.

A0. Nomenclature
N: winding integer; \Lambda_c: core scale; \lambda_c: RG eigenvalue (ladder slope); \delta M: gauge-position offset; T_3: isospin magnitude/ladder component; R^\pm_N: up/down rates; g: WKB parameter (\sim 1/\log(\text{SNR})); T: temperature; \gamma_0: decay rate; D: diffusivity; \ell_{\rm dis}: disorder correlation length; F: force/drive; \sigma: wall tension; F_c: core drive for a drop; c_\text{min}: minimal front speed; Ei: exponential integral; W: Lambert W.

Appendices (Proofs and Derivations)
App. A: WKB/Hamilton–Jacobi and Lagrangian
Master equation WKB P_N \sim e^{-S/g}: \partial_t S = -H(N, \partial_t S) with H = R^+(e^{g \partial_N S}-1) + R^-(e^{-g \partial_N S}-1). Legendre transform gives the Lagrangian quoted in §4.
App. B: Large-Deviation Rate Function
Time-homogeneous: cumulant generator K(k) = \lim_{t\to\infty} t^{-1} \log \langle e^{k N(t)} \rangle, rate I(n) = \sup_k (kn - K(k)).
App. C: Variance Identity
With \mu=\langle N \rangle, \Sigma=\langle (N-\mu)^2 \rangle, \dot \mu = \langle R^+-R^- \rangle, \dot \Sigma = \dots. Using R^-(N)=\gamma_0 N, R^+=0 gives \dot \Sigma = -2\gamma_0 \Sigma + \gamma_0 \mu.
App. D: Wall Tension
Integrate \sigma = \int \sqrt{2 D V(N)} dN with V(N) from \dot N = -D V' to obtain §6 formulas; expand for small N to get \sigma \propto a^{3/2}/b.
App. E: Minimal Speed and Airy Layer
Linearization around N=0 yields quadratic dispersion; coalescence condition gives c_\text{min}; matched asymptotics yield Airy equation and thickness \ell_A.
App. F: Eyring–Kramers Prefactor
For quasi-potential U(N), \tau \propto [\int e^{-\beta U} dx \int e^{\beta U} dy] / D.
App. G: Discrete MFPT (Exact Sum)
For states 0..M with absorbing M: \tau_i = 1/R^+_i + (R^-_i/R^+_i) \tau_{i+1}, \tau_M=0.
App. H: Cooling Laws—Lambert–W
Bernoulli form \dot y + P(t) y = Q(t) y^n linearizes in z=y^{1-n}; inversion yields Lambert–W expression in §5.
App. I: Inference Pipeline and Identifiability
Coordinate-descent/EM over \{\lambda_c, \Lambda_c, c_2; N_i\}; identifiability condition ensures unique nearest-integer assignment; slope SE \sigma_{\lambda_c}.

Acknowledgments
We thank collaborators and discussions that refined the kinetic and front analyses.
Data/Code Availability
Analytic results; inference scripts available upon request.
Conflicts of Interest
None declared.

14. Gravity & Granular Predictions

1) Acoustic "gravitational lensing" in granular packs
What: Sound rays bend around regions with spatially varying topological mass density.
Why (UTMF): Local mass density tracks N(x): \rho_{eff}(x) = \rho_0 e^{\lambda_c N(x)}. For (approximately) constant bulk modulus K, acoustic index n_{ac}(x) = \sqrt{\rho_{eff}(x)/K} \propto e^{\lambda_c N(x)/2}. Rays follow geodesics in the index field.
Prediction: Deflection angle for a narrow beam skimming a defect set :
\alpha\ \approx\ \int_{\rm ray}\nabla_\perp \ln n_{\rm ac}\,ds=\frac{\lambda_c}{2}\int_{\rm ray}\nabla_\perp N\,ds .

2) Granular redshift / depth-dependent sound speed
What: "Redshift" analog: deeper layers propagate slower.
Why: c(z) \propto 1/\sqrt{\rho_{eff}(z)}. If N increases with hydrostatic pressure P=\rho g z (constitutive N = N_0 + \chi P, \chi>0), then
c(z)=c_0\,e^{-\frac{\lambda_c}{2}(N_0+\chi\,\rho g z)}.
Travel time through depth H:
T(H)=\int_0^H\!\frac{dz}{c(z)}=\frac{e^{\frac{\lambda_c}{2}N_0}}{c_0}\,\frac{e^{\frac{\lambda_c}{2}\chi\rho g H}-1}{\tfrac{\lambda_c}{2}\chi\rho g}.

3) Topological equivalence principle (granular edition)
What: Same response under real gravity g and platform acceleration a.
Why: Both load the pack through F = m a_{eff}; if N depends only on the effective acceleration a_{eff} = g+a, then stress/kinematics are functions of a_{eff} alone.
Prediction: Replace g \to a_{eff}; shear band location, acoustic c(z), and pressure profiles coincide within small \delta M offsets.
How: Tilt table vs. linear shaker producing the same a_{eff}; compare c(z) and contact-force maps.

4) Gauss–Bonnet–like rule for force-chain loops
What: Circulation of contact‐orientation around a void equals enclosed topological charge (sum of N on surrounding cells).
Why: Holonomy → curvature; the discrete loop angle accumulates the local bundle curvature sourced by N.
Prediction: For a simple loop \gamma around void V,
\oint_{\gamma}\! d\theta \;=\; \kappa_{\rm topo}\,\sum_{i\in \partial V} N_i,
How: Photoelastic disk experiments: segment the contact graph, measure \oint d\theta vs. the integer labeling from your UTMF coarse‐fit to the same image.

5) Torsion-induced acoustic birefringence (chiral grains)
What: Left/right polarized (or opposite propagation handedness) sound modes travel at slightly different speeds in a chiral packing.
Why: A chiral microstructure produces an effective torsion T in the holonomy bundle; UTMF yields a small phase/velocity split independent of mass scaling.
Prediction: Relative split \Delta c/c \propto T, with T set by microgeometry, not \lambda_c.
How: 3D‐print screw-like grains, random pack, measure time-of-flight for left vs right circularly polarized elastic waves.

6) Acoustic Aharonov–Bohm in granular rings (pure holonomy)
What: Phase shift of an acoustic interferometer encircling a stressed core, with no local speed change along the arms.
Why: Holonomy alone moves the phase; mass scaling via N need not vary along the arms.
Prediction: Interference shift \Delta\phi equals the enclosed Wilson loop phase (tunable by boundary pre-stress); amplitude unchanged.
How: Two-arm acoustic ring around a stressed inclusion; vary the inclusion's stress; read phase from fringe displacement.

7) Gravity-set compaction curve with exponential constitutive
What: A barometric‐like law for packing fraction with depth.
Why: If local void ratio e (or \phi) couples to N, e \propto -N, then with N \propto P = \rho g z you get
\phi(z)=\phi_\infty-\Delta\phi\,e^{-\eta \chi \rho g z}.
How: X-ray CT or index ‐matched imaging of vertical columns.

8) "Topological G" from static deflection of geodesics
What: Define an effective gravitational constant G_{topo} for wave geodesics governed by n_{ac} \propto e^{\lambda_c N/2} with N \leftrightarrow \Phi_{Newton}.
Why: Weak-deflection limit for rays in a slowly varying n_{ac} matches a Newtonian potential \Phi.
Prediction: For a circular "mass" patch of radius R with \Delta N,
\alpha(b)\approx \frac{\partial}{\partial b}\int\!\Phi\,ds\;\propto\; \frac{\lambda_c \,\Delta N\, R^2}{b^2}\quad (b\gg R).

9) Granular Casimir-like pressure from topological mode cutoff
What: Two parallel plates in a vibrated pack feel a net attraction from excluded holonomy loops between them.
Why: Constraining allowed loop areas alters the topological mode density, shifting pressure.
Prediction: Pressure difference \Delta P \propto 1/L^d with a prefactor C set by loop fugacity (indirectly by T, not by wall tension).
How: Vibrate dry grains between adjustable plates; measure force vs gap L; check for the predicted power law.

10) Cross-check: gravitational vs inertial acoustics
What: Time-of-flight curvature in a vertical column under gravity equals that under an equivalent inertial field (centrifuge).
Why: Same N(z) field when g matches a_{inertial}.
Prediction: The fitted exponent \chi from §2 is invariant under swapping g for a_{inertial}.
How: Repeat §2 in a small centrifuge; compare exponents.

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Why these are "novel"
They're gravity-like effects (geodesics, redshift, equivalence, Gauss law, Casimir) realized in a granular medium, with parameters pinned by UTMF (\lambda_c, \chi, and simple constitutive slopes like \eta).
They do not need core transitions or propagating fronts; everything is steady-state or kinematic.