The Synchronous Triad III - Constants, Horizons, and Consensus: The Bare 137 Electromagnetic Vertex, the 1836 Baryon-to-Lepton Simplex Law, 2n^2 Atomic Shell Closures, Evaporating Black Hole Horizons, the No-Dark-Matter Braiding Halo, and Multi-Obser

Completing the Synchronous Triad trilogy (sequel to Paper 1, 10.5281/zenodo.23884635, and Paper 2, 10.5281/zenodo.23908121), we establish twelve exact, parameter-free theorems in strictly division-free Lipschitz and Hurwitz integer quaternion arithmetic (Z[I,A,B,C] and L_Triad = 2H_Z)—with zero fractions, zero irrationals, zero continuum approximations, no external imaginary unit i = sqrt(-1), and zero bolted-on physical postulates. Rejecting the existence of any physical continuum, division field (Q, R, C), or completed N -> infinity limit, and uniting First-Gear additive wavefront propagation with Second-Gear coarse-grained worldline products via the exact Laurent Huygens-Kolmogorov convolution identity G(x,y,z)^(N_1+N_2) = G(x,y,z)^N_2 G(x,y,z)^N_1, we derive deductively from N = 0: (1) Unique Critical-Strip Selection of Shell_2, The Catalan (3^2, 2^3) Identity, & The Bare 137 Principal-Ideal Quotient Vertex Law: At the minimal untwisted carrier closure N = 3 (Omega_0^3 = +8I, Phi(X, 3) = 8Q(X, 3)), out of the four proper-time shells of L_3 with coherent norms {1, 3, 17, 27}, Shell_2 (X = (3,1,1), W = 9, Q = -3I + 2B + 2C, N(Q) = 17) is the unique shell lying strictly inside the critical strip 0 < T(Q)^2 < 4N(Q). Partitioning its W = 9 paths into Class I (3 permutations of {I,I,C} summing to +A+B+C) and Class II (6 permutations of {I,-A,-B} summing to -3I-A+B+C) proves exact longitudinal cancellation (+A - A = 0) along the face normal X = +3 (c_2 = 0), locking its scalar Coulomb quadrance c_1^2 = 3^2 = 9 and transverse polarization quadrance N(V_perp) = 2^3 = 8 at Catalan's consecutive perfect powers (3^2 - 2^3 = 1, 3^2 + 2^3 = 17, Delta I_nc = 9^2 - 17 = 64 = (2^3)^2). Under associative quaternion multiplication (upi_2)pi_1 = u(pi_2pi_1) across the norm-17 Lipschitz shell S_17 = Q_8 S_17^can (|S_17| = 144), the 17 non-backtracking canonical primes (pi_2 != pi_1^) generate 17 * 8 = 136 distinct primitive spinor states Y_gamma not in 17Z[I,A,B,C], whereas the 1 backtracking prime (pi_2 = pi_1^) yields u(pi_1^ pi_1) = 17u in 17Z[I,A,B,C] for all u in Q_8, collapsing all 8 backtracking products onto the single zero ideal coset [0I] in Z[I,A,B,C] / 17Z[I,A,B,C] generated by Y_0 = +17I. Because all 137 distinct channel representatives {Y_0, Y_1, ..., Y_136} share the exact same quadratic norm N(Y_gamma) = 17^2 = 289, summing the norm across the pointed ideal quotient yields sum_{gamma=0}^136 N(Y_gamma) = 136(289) + 1(289) = 137 * 289 = 137 N(+17I) (N_EM^(0) = 137 = 2^7 + 3^2 = 81 + 56). (2) Degree-Graded SO(3) Dirac-Maxwell Field Equations, Gaussian Domain-Wall Confinement, & The Brandt-Hecke Recurrence Discriminant Lock Delta_rec(T_17) = -4 Delta I_nc((3,1,1), 3) = -256: In the Laurent generating polynomial G(x,y,z) = -v_x v_y v_z I + (v_x u_y u_z A + u_x v_y u_z B + u_x u_y v_z C) (u_alpha = x_alpha + x_alpha^-1, v_alpha = x_alpha - x_alpha^-1), the 3D Dirac vector gradient is of degree 1 in the finite differences v_alpha whereas the chiral octant scalar is of degree 3, so G(x,y,z)^2 automatically isolates the degree-2 SO(3)-isotropic 3D Laplacian symbol while confining even-tick domain walls XYZ = 0 to Z[I, e_alpha] and coordinate axes XY = YZ = ZX = 0 to ZI. On S_17^can, octahedral symmetry reduces the prime sum to Sigma_17 = S_17 I = -30I (S_17 = -30), and the Chebyshev recurrence S_{17^k} - S_17 S_{17^{k-1}} + 289 S_{17^{k-2}} = 0 has recurrence discriminant Delta_rec(T_17) = (-30)^2 - 4(289) = -256 = -4 * 64 = -4 Delta I_nc((3,1,1), 3), locking the global Hecke half-trace |-30/2| = 15 and local vertex braiding root sqrt(64) = 8 into 15^2 + 8^2 = 17^2 = 289. (3) The Even-Tick Origin Scalar Condensate (N = 2 and N = 4), The 13-Axis Causal Cube Partition, & The Coarse-Graining Sequence of Electroweak Mixing Partitions (144 : 216 -> 8 : 17 -> 3 : 7 -> 3 : 13): At even ticks N = 2m, the O_h-invariant origin state Phi(0, 2m) = 2^(2m) Q_even(0, 2m) in ZI forms a pure scalar condensate with (W, Q_even, N(Q_even), Delta I_nc) = (8, +4I, 16, 48) at N = 2 and (216, +36I, 1296, 45,360) at N = 4, while the electroweak channel partition progresses monotonically across coarse-graining scales from 144 : 216 = 2 : 3 (the 144 SU(2) weak mixing holonomies inside 216 = 24 + 144 + 48) to 8 : 17 (single-site Shell_2 vertex quadrance) to 42 : 98 = 3 : 7 (N = 4 causal shell S(2)) down to 3 : 13 on the 26-neighbor unit causal stencil S(1) = 26 = 2 * 13 (13 = 3_axial + 6_planar + 4_octant = 1_U(1) + 3_SU(2) + 9_U(3)). (4) The Homogeneous Degree-2 Simplex-Hadamard Identity 4N(Q) = T(Q)^2 + sum_{g in V_4} q_parallel(gQ)^2, The Three Independent Generations {1, 841, 1521} on Branch A, & The A_4 = V_4 x| Z_3 Derivation of the Bare 1836 = 576 + 1260 Baryon-to-Lepton Simplex Law: Across the Klein four-group V_4 = {g_0, g_x, g_y, g_z}, the longitudinal projections q_parallel(gQ) = -<D, gQ> obey q_parallel(pi_0) = -(q_parallel(pi_x) + q_parallel(pi_y) + q_parallel(pi_z)) (leaving strictly 3 independent generations) and unite with the Path-Budget Simplex into the homogeneous degree-2 quadratic form 4N(Q) = T(Q)^2 + sum_{g in V_4} q_parallel(gQ)^2. Multiplying pi_1 in S_3^can and pi_2 in S_17^can by sigma^2(pi_2) splits the two non-commutative orderings into Branch A (Q_212 = 16I - 15A + 5B + 19C), where the three independent sign-flipped generations (pi_y, pi_z, pi_x) have degree-2 quadrances {1^2, 29^2, 39^2} = {1, 841, 1521} (and dependent sum q_parallel(pi_0)^2 = 81), and Branch B (Q_122 = 8I - 7A - 15B + 23C, {225, 961, 2025}). Because the Shell_3 Baryon Core Q_3 = 3D is a Z_3-invariant color singlet under A_4 = V_4 x| Z_3, it traces symmetrically over all |V_4| = 4 orbit terms, yielding on Q_32 = Q_{Shell_3} Q_{Shell_2} = -12I - 9A - 15B - 3C (N(Q_32) = 27 * 17 = 459) the sum M_{p/e}^(0) = 4N(Q_32) = 576 + 1260 = 17 * 108 = 1836 relative to the single-character first-generation lepton quadrance q_parallel(pi_y)^2 = 1. (5) Yukawa Exponential Screening (W^(k)^2 = 3^k N(Q^(k))), The (L^infty, L^1) Theorem of (n, l, n+l), & The Exact Bijection Between Manhattan-Norm Diagonals (n + l = N + 1) and Bipartite Radial Strata (2n_N^2 = 2, 8, 8, 18, 18, 32, 32): Canonical quark-antiquark pairs pi, pi^* in S_3^can form the neutral color-singlet meson pipi^ = -3I, whose k-step spatial chain along Shell_1 exponentially screens coherent norm relative to path-pair weight as W^(k)^2 = 9^k = 3^k N(Q^(k)). On the n-th cleaved causal shell S(n) - 2 = 24n^2 = 12 * (2n^2), n = (1/2)max|X_alpha| is the 3D L^infty norm, l = (1/2)(|Y|+|Z|) in {0, ..., n-1} is the 2D transverse L^1 norm with 2l + 1 diamond sites per tetrahedral sector, and n + l = (1/2)(|X|+|Y|+|Z|) is the 3D L^1 Manhattan norm, whose constant-norm diagonals n + l = N + 1 biject with the bipartite radial strata M_N (n_N = floor(N/2) + 1) and sum to sum_{n+l=N+1} 2(2l+1) = 2n_N^2 = 2, 8, 8, 18, 18, 32, 32 across N = 1..7. (6) The Antipodal Basis Existence Theorem for Closed Pauli-Braided Evaporating Horizon Shells dB_n, Surface Area Capacity S(n) = 24n^2 + 2, & Kinematic Proper-Time Tensor Decomposition: Factoring the First-Gear recurrence into the 4-basis-element antipodal difference Q(X, N) = sum_{s_x=+1} sigma^r(q(s))(Q(X-s, N-1) - Q(X+s, N-1)) proves that any two-sided converging shell configuration dynamically forms a closed horizon shell dB_n where Q(X_hor, N) = 0I, Delta I_nc = 64w^2, and 3-axis Pauli braiding sum N(V_1 x V_2) = 12w^2 > 0 hold simultaneously across all S(n) = 24n^2 + 2 surface vertices, halting outgoing spinor currents (J_out = 0I) at tick N before unitarily evaporating across subsequent ticks (2W_hor W_ext > 0). Decomposing tau_{alpha alpha} = N^2 - X_alpha^2 under Z_3 yields the isotropic trace T_tau (L_9pt T_tau = +96, with equal temporal and spatial second differences Delta_N^2(3N^2) = +6 = Delta_s^2(Q_X), doubling null-step curvature to 6 + 6 = 12) and the two traceless quadrupole modes h_+ = Y^2 - X^2 and h_x = X^2 + Y^2 - 2Z^2 (L_9pt h_{+,x} = 0). (7) The Period-3 No-Dark-Matter Braiding Halo Cycle (1 : 0 : 1 -> 1 : 2 : 3 -> 11 : 114 : 125), The 27^26-27^27 Gravitational Hierarchy, & Cosmological Braiding Expansion: Across the three carrier phases r = N mod 3 in {1, 2, 0} of the period-3 clock Omega_0^3 = +8I, the global partition sum N(Q) : sum Delta I_nc : sum W^2 is 1 : 0 : 1 at N = 1 (100% luminous), 1 : 2 : 3 at N = 2 (33.33% luminous vs. 66.67% braiding halo), and 11 : 114 : 125 at N = 3 (8.80% luminous vs. 91.20% braiding halo), bounding each elementary soliton period within [8.80%, 33.33%] (geometric mean 17.1%) and accounting for galactic and cosmological mass discrepancies via the non-luminous braiding halo Delta I_nc with zero dark matter particles, while k = S(1) = 26 to k = W_core = 27 Baryon Core stages compound W_tot^2 / N(Q_tot) to 27^k in [3^78, 3^81] (~ 10^37-10^38). (8)-(12) Stationary Proper-Time Action, Brandt-Epstein Spectral Lock Over Q_N, All-Tick Graded Chiral Coset Arrow of Time Phi(X, N) in I_N \ (I_{N-1} bar(I)_1), Multi-Observer Consensus & Adjoint Irreversibility Against Wigner's Super-Observer, and Biological (310 K Sparse Molecular Constellation, M = 1) vs. Synthetic (3-Axis Spin-Orbit-Locked Cooper Soliton, M >> 1) Conscious Architectures.