For CRT channel Z/p^2, the p-th power map compresses to exactly p distinct outputs. The chain primes are self-resonance fixed points: K->K, E->E, b->b. D generates K. Total (axiom prime, channel) resonance pairs = GATE = 13.
Sigma-graph degree at ring level k = 2k-1. At TRANS (k=7): degree = GATE = 13, majority threshold = b = 7. Next degree 15 = K*E would be composite -- the chain MUST stop.
PASS 11/11
SELF-RESONANCE + DEGREE CONSENSUS
Z/p^2 raised to p -> p outputs (chain fixed points). Resonance count = GATE = 13. Degree-consensus: TRANS threshold = b out of GATE.
225. CMB Triangle Curvature
Random triangles on the CRT torus have angle sum > pi. The excess grows with ring level and fattening. Small triangles (near the identity) are flat. The discrete torus has positive topological curvature from periodic boundary wrap-around.
PASS 7/7
CMB TRIANGLE CURVATURE
Positive Discrete Curvature Theorem: on T^k, random triangle angle excess is positive and grows monotonically with ring level. Z/8 (even) exactly flat. Small triangles flat, equidistant triangles maximally curved (3pi). Fattening creates the largest curvature jump. The excess is topological (periodicity), not Riemannian.
226. 7ch TRANS Curvature
All three curvature frameworks (Ollivier-Ricci, Gauss-Bonnet, Forman-Ricci) extend to TRANS with 7 channels. Per-channel CRT decomposition: 7 small curvature computations instead of one huge one. The curvature numerators form their own axiom chain.
PASS 14/14
7ch TRANS CURVATURE EXTENSION
Curvature Numerator Theorem: q_i-2 maps each CRT channel to an axiom-native value. E-channel = cross-lattice kappa (23). b-channel = universal boundary (47). ALIVE sum = ANSWER (42). Thin sum = E*b = 35 (490 split in curvature). Hamming degree = E^3. Forman: F_b=-HYDOR.
227. P-adic Ultrametric
Each CRT channel Z/p^e has a natural p-adic metric. All triangles are isosceles (exhaustive). The product ring has tau(N) distinct p-adic distance profiles -- one per divisor. Tree depth per channel = Pareto exponent; total = 12 = heartbeat.
PASS 11/11
P-ADIC ULTRAMETRIC GEOMETRY
Product-of-Trees Theorem: the product ring Z/N has tau(N) distinct p-adic distance profiles (one per divisor). Building equation: tau*D^2 = |Idem|*K^3. Each channel is a p-ary tree with depth = Pareto exponent. Total depth = 12 = heartbeat. All triangles isosceles (exhaustive on 7 channels).
PASS 13/13
CURVATURE SCALE THEOREM
Cross-lattice kappa undergoes a phase transition at fattening: 0 (squarefree) -> 23/27 (fat), then invariant under thin extension. D-K resonance: D^3-D = K^2-K = D*K = 6 (identical curvature gain, root: K=D+1). 490 split in gain: DEAD = D^2*ESC = 68, ALIVE = D*K = 6. b gain = 42 = ANSWER. K amplification = b (only integer). 71/71 verified.
229. CURVATURE-LEARNING THEOREM
Curvature governs quadratic degeneracy. Zero-class product = DATA = 210.
Ollivier-Ricci curvature governs squaring degeneracy per CRT channel. The zero-class of squaring at each fat channel equals its base prime: zero(Z/D^3)=D, zero(Z/K^2)=K, zero(Z/E^2)=E, zero(Z/b^2)=b. Their product = DATA = 210.
PASS 9/9
CURVATURE-LEARNING THEOREM
Ollivier-Ricci curvature numerator q_i-2 governs squaring degeneracy per CRT channel. Zero-class at each fat channel = its base prime. Product = D*K*E*b = DATA = 210. DEAD zeros = D*b = 14, ALIVE = D*K = 6. Fat channels have fewer QR classes (more collisions) -> easier learning. 43/43 verified.
231. CONFORMAL DIMENSION THEOREM
conf(d)=(d+1)(d+2)/2 is b-smooth for d=1..7. Physical spacetime projects to depth. HYDOR = conf(GATE).
The conformal Lie algebra dimension conf(d) = (d+1)(d+2)/2 factors entirely from DATA primes {2,3,5,7} for d=1..7. The sequence conf(1..7) = K, D*K, D*E, K*E, K*b, D^2*b, D^2*K^2 cycles through axiom prime pairs. At d=4 (physical spacetime): 15 mod 8 = b. At d=b=7: 36 mod 9 = 0 (K-void). At d=GATE: 105 = HYDOR. At d=ESCAPE: 171 = K^2*19 breaks axiom-smoothness.
PASS 8/8
CONFORMAL DIMENSION THEOREM
conf(d)=(d+1)(d+2)/2 is b-smooth for d=1..7: K, D*K, D*E, K*E, K*b, D^2*b, D^2*K^2. Physical spacetime conf(4)=K*E=15 projects to b in D-channel; depth conf(7)=D^2*K^2=36 is void in K-channel. conf(GATE)=HYDOR=105. Smoothness breaks at ESCAPE: conf(17)=K^2*19. Poincare(D*E)+observer(E)=conformal(K*E). test_conformal_group.ax 67/67.
232. TRAPPED SURFACE CONFORMAL THEOREM
Per-channel trapped count T(p,e) = p^{e-1} - 1. Sum = K*E = 15 = conf(4). Four paths to 15.
Penrose trapped surface analog: elements with 0 < v_p(n) < e_p are trapped -- squaring drives them to zero (singularity). T(D,3)=3, T(K,2)=2, T(E,2)=4, T(b,2)=6. Depth-1 channels: T=0 (unit or dead). Sum = K*E = conformal dimension of R^{3,1}. Four independent paths: trapped sum, f(E)-D^2, K*(b-D), conf(4).
PASS 6/6
TRAPPED SURFACE CONFORMAL THEOREM
T(p,e) = p^{e-1}-1 trapped elements per CRT channel. Depth-1: T=0 (meadow). Sum = K*E = 15 = conf(4). Four algebraic paths converge: trapped counts, depth quadratic f(E)-D^2, chain arithmetic K*(b-D), conformal dimension. Fattening THIN->DEEP creates all trapped surfaces. test_trapped_surface.ax 35/35.
233. SPINOR VALENCE CHAIN THEOREM
Symmetric spinor valence n -> n+1 components. Prime component counts = axiom chain. D=2 IS the spinor dimension.
Penrose two-component spinors: valence n gives n+1 independent components. Filtering by PRIME component count: 2=D (spinor), 3=K (EM field), 5=E (Weyl/gravity), 7=b, 11=L, 13=GATE, 17=ESCAPE. Valences = phi(p) = p-1, all axiom-smooth. Gaps = D-powers. Gravity valence D^2 = EM valence D squared.
PASS 7/7
SPINOR VALENCE CHAIN THEOREM
Symmetric spinor valence n -> n+1 components. Prime component counts = axiom chain {D,K,E,b,L,GATE,ESCAPE}. Valences = totients phi(p) = {1,D,D^2,D*K,D*E,D^2*K,D^4}, all axiom-smooth. Gaps = D-powers. EM=valence D, gravity=valence D^2. D=2 IS the spinor dimension. test_spinor_chain.ax 33/33.