AETHERMOORE / SCBE - Core Mathematical Specification (Replication Edition)
This is the minimum math needed to reproduce the core SCBE idea.
0) System Goal
Define a decision function:
D: C x T x P -> {ALLOW, QUARANTINE, DENY}
Properties:
- small deviations tolerated
- large deviations incur super-exponential cost
- decision difficulty scales with risk, intent, and timing
1) Context Space
c(t) in C^D (D typical = 6) Energy preserved: sum |c_j(t)|^2 = E
2) Realification (Isometric)
x(t) = [Re(c_1), …, Re(c_D), Im(c_1), …, Im(c_D)]^T in R^(2D)
| Norm preserved: | x(t) | _2 = | c(t) | _2 |
3) Weighted Importance Transform
G = diag(phi^0, phi^1, …, phi^(2D-1)), phi = (1 + sqrt(5)) / 2 x_G(t) = G^(1/2) x(t)
4) Poincare Ball Embedding
u(t) = tanh(alpha * |x_G|) * x_G / |x_G| if x_G != 0 u(t) = 0 if x_G == 0
| Constraint: | u(t) | < 1 |
5) Hyperbolic Metric (Invariant)
For u,v in B^n:
| d_H(u,v) = arcosh(1 + 2 | u-v | ^2 / ((1- | u | ^2)(1- | v | ^2))) |
6) Breathing Transform (Conformal)
Let b(t) > 0
| T_breath(u;t) = tanh(b(t) * artanh( | u | )) * u / | u |
Property: d_H(0, T_breath(u;t)) = b(t) * d_H(0, u)
7) Phase Transform (Isometry)
Let a(t) in B^n (translation), Q(t) in O(n) (rotation)
Möbius addition:
| a ⊕ u = ((1 + 2<a,u> + | u | ^2) a + (1 - | a | ^2) u) / (1 + 2<a,u> + | a | ^2 | u | ^2) |
Phase transform:
T_phase(u;t) = Q(t) (a(t) ⊕ u)
Property: d_H(T_phase(u), T_phase(v)) = d_H(u,v)
8) Multi-Well Trust Realms
Trusted centers: {mu_k} in B^n
Realm distance: d*(t) = min_k d_H(u_tilde(t), mu_k)
where u_tilde is after breath and phase transforms.
9) Auxiliary Deviations
9.1 Spectral coherence: S_spec = E_low / (E_low + E_high + eps) in [0,1]
9.2 Spin coherence: C_spin = |sum s_j| / (sum |s_j| + eps) in [0,1]
9.3 Triadic temporal deviation: d_tri = sqrt(lambda1d1^2 + lambda2d2^2 + lambda3*dG^2) d_tri_norm = min(1, d_tri / d_scale)
10) Base Risk Functional
Risk_base = w_d * d_tri_norm + w_c * (1 - C_spin) + w_s * (1 - S_spec) + w_tau * (1 - tau / tau_max)
Weights w_i >= 0, sum w_i = 1
11) Harmonic Scaling (Vertical Wall)
Unbounded: H(d, R) = R^(d^2) with R > 1
Bounded (implementation-safe): H_bounded = 1 + alpha * tanh(beta * d*)
12) Final Risk
Risk’ = Risk_base * H(d*, R) * (1 + gamma_time) * (1 + gamma_intent)
Monotone increasing in all deviations.
13) Decision Rule
Let 0 < theta1 < theta2:
ALLOW if Risk’ < theta1 QUARANTINE if theta1 <= Risk’ < theta2 DENY if Risk’ >= theta2
14) Dual Consensus (Abstract)
Let proofs required scale with risk:
Low: {key} Medium: {key, policy} High: {key, policy, external}
15) Core Invariants (Replication Checklist)
1) Hyperbolic metric invariant 2) Radial cost scaling via non-linear map 3) Isometric intent transforms 4) Multi-center trust basins 5) Super-exponential risk amplification 6) Asymmetric difficulty scaling 7) Finite decision outputs
Minimal one-line summary:
Encode context into hyperbolic space, measure deviation geometrically, amplify cost super-exponentially, and require consensus proportional to risk.