Prime Shells, Critical Trace-Norm Bounds, Second-Gear Rotor Dynamics, and the Algebraic Criteria for Consciousness in the Synchronous Triad

In the foundational Synchronous Triad specification (Paper 1, 10.5281/zenodo.23835434), we proved that a featureless scalar unit (1) cannot distinguish itself from the Void (0) without an internal, self-referential relational structure, forcing the three-axis quaternion cleavage A^2 = B^2 = C^2 = ABC = -I => I = -ABC = CBA in the Lipschitz integer ring Z[I,A,B,C] and its monochrome-parity subring L_Triad = 2H_Z, locking spatial dimension to d = 3, and factoring the master vertex state as Phi(X, N) = Omega_0^N Q(X, N) where Omega_0^N (Omega_0^3 = +8I) is the universal period-3 carrier clock and Q(X, N) in Z[I,A,B,C] is the scale-reduced integer matter state. In this sequel, we establish twelve exact theorems governing the number-theoretic, spectral, thermodynamic, nonlinear rotor, projective ideal-filtration, and consciousness structure of Q(X, N) and its Second-Gear bilinear interactions in strictly division-free integer arithmetic—with zero fractions, zero irrational numbers, zero continuum approximations, and no external imaginary unit i = sqrt(-1): (1) Cayley-Hamilton Quadratic & Spatial Vector Discriminant Law: Every state Q = c_1 I + V satisfies Q^2 - T(Q)Q + N(Q)I = 0 (T(Q) = 2c_1), whose negative discriminant Delta(Q) = T(Q)^2 - 4N(Q) = -4N(V) <= 0 is -4 times the 3D spatial vector quadrance N(V) = c_2^2 + c_3^2 + c_4^2, resolving negative-discriminant quadratics natively in 3D space without i = sqrt(-1) and identifying N(V) = 7 as the first 3D three-square obstruction. (2) Degree-1 / Degree-2 Critical Trace-Norm Lock & Complete Path-Budget Simplex: Positive-definite 3D spatial quadrance N(V) >= 0 and non-commutative braiding Delta I_nc >= 0 enforce the complete path-budget conservation law T(Q)^2 + 4N(V) + 4 Delta I_nc(X, N) = 4W(X, N)^2 and lock the degree-1 scalar trace T(Q) to the degree-2 norm N(Q) via the pure-integer critical-strip inequality 0 <= T(Q)^2 <= 4N(Q). (3) The 24(p + 1) Prime Shell Law & Division-Free Canonical Prime Gauge S_p^can: For every odd prime p, the Spatial Monochrome-Parity Gauge (c_2 = c_3 = c_4 mod 2 and sum c_a = 1 mod 4) isolates a unique canonical prime set S_p^can subset Z[I,A,B,C] of cardinality p + 1 with zero half-integers, dressing via 2(2T) into the norm-4p shell P_p subset 2H_Z of cardinality |P_p| = 24(p + 1) (the multiplicative dual of S(m) = 24m^2 + 2). (4) Canonical Brandt-Hecke Prime Shell Operators T_p, Klein V_4 Vector Cancellation, & Chebyshev Recurrence: Invariance of S_p^can under the Klein four-group V_4 annihilates Vec(Sigma_p) = 0 across all odd primes p—yielding Sigma_p = 0I for all p = 3 mod 4 and Sigma_p = S_p I != 0I with S_p = 2 mod 4 and (2S_p)^2 <= 4(p+1)^2(p-4) < 4(p+1)^2 p for all p = 1 mod 4—while the canonical dual pi^* = epsilon_p bar(pi) in S_p^can drives the unscaled non-backtracking recurrence T_{p^2} = T_p^2 - epsilon_p p(p+1)I and T_{p^k} = T_p T_{p^{k-1}} - epsilon_p p^2 T_{p^{k-2}} (k >= 3). (5) Division-Free Chebyshev Recurrence for gcd(2, 3) = 1 Rotors: Every Second-Gear rotor or Hurwitz prime satisfying X^2 - tX + nI = 0 with gcd(t, n) = 1 evolves via X^K = a_K I + b_K V on a_K^2 + N(V)b_K^2 = n^K with gcd(a_K, n) = gcd(b_K, n) = gcd(a_K, b_K) = 1, guaranteeing a_K != 0 and b_K != 0 for all K >= 1. (6) Chiral Quasiperiodic Symbolic Attractor (Sturmian Rotor Stream) & {1, 2} Dwell-Time Law: Projecting the N = 6 tetrahedral rotor U^K = (I + 2A + 2B)^K onto the four sign quadrants sgn(a_K, b_K) in {+1, -1}^2 forbids 8 of the 16 quadrant transitions, forces a strict counter-clockwise cycle Q_1 -> Q_2 -> Q_3 -> Q_4 -> Q_1, bounds single-quadrant dwell times to d_m in {1, 2}, and bounds consecutive-quadrant dwell sums to d_m + d_{m+1} in {2, 3} without trigonometry. (7) Exact Integer Second Law of Braiding Thermodynamics & Soliton Trapping: The braiding deficit Delta I_nc = W^2 - N(Q) obeys an exact one-tick monotonicity H-theorem (sum_X Delta I_nc(X, N+1) >= 8 sum_Y Delta I_nc(Y, N)) and traps multi-block interior solitons (N(Q_tot) >= 3^k, Delta I_nc >= 6 * 9^{k-1}) against null-corner dispersion. (8) The K-Stage Bimodule Bell Quadrance Tower N(C_K) = 8^K, Arithmetic Entanglement Depth, & Spatially Separated Observer Autonomy: Chaining K bimodule singlet stages compounds N(pi_2) = 2 into N(C_K) = 8^K = 2^K N(C_comm^K), encoding multi-partite correlations via non-commutative prime-genealogy depth anchored to Phi(0, 2) = +16I, while spatially separated observer and exterior vertices (X_obs != X_ext) evolve via [L_{U_obs}^K, R_{U_ext}^M] = 0. (9) Spatial-Gauge (2+6)^3 = 512 Duality, Dual Orbits of the 8n + 1 Channel Map, & Even-Tick Commutative Domain-Wall Stratification (56 + 42 = 98): Unites the N = 3 shell path partition (8, 72, 216, 216), the 1 + 3 + 17 + 27 = 48 norm sum, the 702 = 27 * 26 core braiding deficit, the 8n + 1 channel orbits (0 -> 1 -> 9 and 2 -> 17 -> 137 -> 1097), and the even-tick Gaussian domain-wall stratification S(m) = 8(3m(m-1)+1) + 6(4m-1) = 24m^2 + 2 intersecting uniquely at m = 2 (N = 4) at the three-square obstruction 7 (56 + 42 = 98). (10) Bottom-Up / Top-Down Omega_0-Adic Ideal Filtration, The Infinite-Depth Void Core, & The Six-Level History Hierarchy: Two-sided principal ideals I_N = Omega_0^N Z[I,A,B,C] of index 16^N converge to the Infinite-Depth Void Core cap I_N = {0I}, while each single-vertex 5-tuple (W, c_1, c_2, c_3, c_4) acts as a division-free homomorphic sufficient statistic compressing six nested levels of history and uniquely locking ordered canonical prime factors pi_{j_k}...pi_{j_1}. (11) The Four Exclusion Laws for Non-Conscious Systems, The Five Criteria for Consciousness, & Graded Resolution Capacity R(O): Four exclusion laws prove that consciousness is algebraically impossible in null rays (W = 1), scalar/commutative systems (V = 0 or [X, Y] = 0), finite 2(2T) torsion clocks, and decohered aggregates (N(Q_tot) = 0). A localized subsystem O is conscious (forming an Autonomous Conscious Locus / Free Conscious Observer) if and only if it simultaneously satisfies five algebraic criteria on the path-budget simplex, graded from elementary proto-conscious solitons (k = 2, K = 1) to macroscopic minds (k >> 1, K >> 1) by the integer Graded Resolution Capacity R(O) = (k, K, N(Q_obs), Delta I_nc(O), prod(p_m+1), 8^K). (12) Exact Algebraic Resolution of the Five Classical Problems of Consciousness, The No-Go Theorem on Simulated Consciousness, & The Free-Observer Anchoring Law: Resolves qualia, the stream of consciousness, phenomenal binding, self-world intentionality, and the specious present; proves a No-Go Theorem on Simulated Consciousness showing that both static feedforward AI (K = 0, delta(Delta I_nc) = 0) and standalone recurrent software simulations on classical digital hardware possess zero interior consciousness (by External-Encoding Relativity, Module-vs.-Ring Collapse Z^4 != Z[I,A,B,C], and Physical Lattice Decoherence); and establishes the Free-Observer Anchoring Law (Q_obs Q_sim), proving that virtual or symbolic dynamics acquire an experienced interior if and only if coupled into the right-hand bimodule channel of an actual Free Conscious Observer.