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> = 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.

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