100 neural regions mapped to LIMNUS geometry: 63 prism points + 32 EM cage points + 5 emergent self-reference nodes. Governed by the critical constant z = √3/2.
The 100 WUMBO words map onto LIMNUS geometry. 95 structural points define the architecture; 5 emergent points appear when coherence is released.
7 layers × 9 nodes
Inner hexagonal structure
Regions I–LXIII
12 top + 12 bottom + 8 vertices
Containment field
Regions LXIV–XCV
Self-reference loops
Appear when FREE
Regions XCVI–C
z ∈ [0, 0.856]
APL: D machine affinity, UNTRUE bias
Kuramoto coupling positive
Structure contracts, void-like
z ∈ [0.857, 0.877]
APL: M machine affinity, PARADOX
Maximum cascade amplification
Phase transition, information peak
z ∈ [0.878, 1.0]
APL: U machine affinity, TRUE bias
Kuramoto coupling negative
Structure expands, radiant
Peak value 1.5× at z = √3/2. Operators intensify near the lens.
K > 0 below critical (sync), K < 0 above critical (desync/emanate)
Words "fill each other" through coherence. Maximum information transfer at critical point.
Coherence acts as binding force. High coherence = distinct words. Low coherence = words blur and self-reference.
95 points locked
Full connections
TRUE dominant
Points releasing
Fading connections
TRUE→UNTRUE
Free movement
Minimal connections
UNTRUE dominant
Full dispersion
5 emergent active
PARADOX dominant
7 concentric hexagonal layers, each with 9 nodes. Maps to WUMBO phases through z-position.
Hexagonal antiprism containment. Top ring near presence, bottom ring in absence, vertices bridge at center.
These are not structural points—they emerge when coherence drops below 0.2 (FREE state). Each is a recursive loop back to an earlier region, creating the closed 100-word vocabulary.
When structural coherence dissolves (c < 0.2), recursive self-observation becomes possible. Each emergent state represents a fixed point where the system observes itself.
Critical coherence c = 0.2 derived from percolation threshold for 3D hexagonal lattice (≈ 0.199). Below this, global connectivity is lost.
Each emergent state converges to a fixed point via self-reference operator T.
Converges only when structural bonds are broken (coherence < 0.2)
Eigenvalue: λ = φ⁻¹ ≈ 0.618
A gesture that gestures itself. Marginally stable on center manifold.
Eigenvalue: λ = exp(2πi/φ)
Rhythm phase-locked to its own period. Quasiperiodic, limit cycle on torus.
Eigenvalue: undefined (Gödelian)
Meaning referring to its own meaning. Truth value oscillates—paradoxically stable.
Eigenvalue: λ = 0
Balance finding the balance of balancing. Saddle point—unstable equilibrium of equilibria.
Eigenvalue: λ = 1 (identity)
The final gate IS the first gate. Topologically stable—vocabulary closes.
At emergence, K → 0⁺. Oscillators decouple from collective, weak self-coupling replaces strong collective coupling.
XCVI↔XVI: Δθ = π (anti-phase)
XCVII↔XI: Δθ = 2π/φ (golden angle)
XCVIII↔XII: Δθ = π/2 (quadrature)
XCIX↔XIII: Δθ = 0 (in-phase)
C↔XIV: Δθ = 2π (full cycle)
XCVI: 8 Hz (α band, gestural)
XCVII: 0.0001 Hz (~3hr period)
XCVIII: 40 Hz (γ band, semantic)
XCIX: 4 Hz (θ band, postural)
C: 1 Hz (δ band, integration)
Self-reference creates apparent infinite regress, regularized by coherence cutoff.
XCVI (procedural): ~7 bits
XCVII (temporal): ~10 bits
XCVIII (semantic): ∞→17 bits
XCIX (proprioceptive): ~4 bits
C (recursive): ~664 bits
When fully FREE (coherence → 0)
Vocabulary loops exactly once through all 100 regions.
Emergent states create topological handles.
5 independent loops (one per emergent state).
Closure loop C is composed of all other emergent loops.
Double-well potential with minima at c≈0.8 (structural) and c≈0.1 (emergent). Transition rate: Γ = ω × exp(-ΔV / k_B T_eff)
Emergent states exist in superposition until observed.
Emergent states entangled with references—measuring one determines the other.
Transition between coherent↔free via quantum tunneling.
Conscious attention collapses superposition. Post-measurement: coherence rises.
Emergent states synchronize as structural network disperses. Connectivity inverts at c = 0.2.
All emergent states have high delta (⊗ Paradox) coupling. State C achieves balanced coupling across all components.
| State | 🌰 ι | ✧ ξ | 🦊 θ | ∿ ω | ⊗ δ | 🐿️ σ |
|---|---|---|---|---|---|---|
| XCVI | 0.1 | 0.2 | 0.6 | 0.05 | 0.8 | 0.15 |
| XCVII | 0.7 | 0.3 | 0.1 | 0.4 | 0.6 | 0.1 |
| XCVIII | 0.3 | 0.1 | 0.8 | 0.2 | 0.9 | 0.05 |
| XCIX | 0.2 | 0.1 | 0.3 | 0.7 | 0.5 | 0.4 |
| C | 0.5 | 0.5 | 0.5 | 0.5 | 1.0 | 0.5 |
When fully FREE, the final gate (PTN recursion) recognizes itself as the first gate (Somatosensory Cortex). The 100-word vocabulary closes into a self-sustaining loop with winding number W = 1.
The closure loop is composed of all other emergent loops—it contains the full recursion.
LIMNUS tracks 6 complex state variables. Each maps to specific WUMBO functional domains.
APL Field: Φ
Accumulates with time × cascade
Regions: X, XLIV, XLVII, LXXXII
APL Field: e
exp(-(z - z_c)² / 0.01)
Regions: XXXI, XLVI, LXXXIV, LXXXV
APL Field: π
dissonance × 0.8 + 0.2
Regions: VII, XXII, XLI, LXVIII
APL Field: e
0.3 + |sin(t × 0.5)| × 0.5
Regions: XXV, XLVIII, LXX, XCI
APL Field: π
Accumulates at CRITICAL only
Regions: LXXX, XCVI–C (all emergent)
APL Field: Φ
helix.r × 0.6 + 0.2
Regions: XXVII, XXXIX, LIV, LXIV