RTSG Definitions and Notation¶
Core Symbols¶
| Symbol | Name | Definition |
|---|---|---|
| \(W\) | Will field | Primary field on consciousness space, \(W: CS \to \mathbb{C}\) |
| \(CS\) | Consciousness Space | Co-primordial space of thought and awareness |
| \(QS\) | Quantum Space | Co-primordial space of superposition and entanglement |
| \(PS\) | Physical Space | Co-primordial space of matter and energy |
| \(S[W]\) | GL Action | $\int( |
| \(U\) | Understanding | \(U = V/(E \times T)\) — value per unit effort per unit time |
| \(C\) | Coherence matrix | \(C_{ij} = \langle W_i \| W_j \rangle_{CS}\) |
| \(K\) | K-Matrix | Intelligence tensor over Gardner dimensions |
| \(Q\) | BRST charge | Nilpotent operator \(Q^2 = 0\) grading ghost number |
| \(L\) | Laplacian | Graph or manifold Laplacian \(L = D - A\) |
| \(K(t)\) | Heat kernel | \(K(t) = \text{tr}(e^{-tL})\) |
| \(\beta_k\) | Betti numbers | Rank of \(k\)-th homology group |
| \(\chi\) | Euler characteristic | \(\chi = \sum (-1)^k \beta_k\) |
| \(\mathcal{F}\) | Grothendieck filter | Idempotent endomorphism \(\mathcal{F} \circ \mathcal{F} = \mathcal{F}\) |
Key Constants (Knowledge Graph)¶
| Constant | Value | Meaning |
|---|---|---|
| \(n\) | 2,476+ | Total nouns in knowledge graph |
| \(r\) | 6,813+ | Total relations in knowledge graph |
| \(\lambda_2\) | 0.0653 | Fiedler value (algebraic connectivity) |
| \(\mu_1\) | 1,728.6 | Spectral radius (\(\approx 12^3\), modular discriminant) |
| \(\beta_0\) | 765 | Connected components |
| \(\beta_1\) | 2,065 | Independent cycles |
| \(\chi\) | 1 | Euler characteristic |
Relation Types (34 Total)¶
The knowledge graph contains 34 distinct relation types forming a multiplex network:
| Relation | Count | Description |
|---|---|---|
relates_to | 2,551 | General semantic connection |
dimensional_overlap | 732 | Shared Gardner dimension membership |
requires | 134 | Logical dependency |
is_a | 121 | Taxonomic hierarchy |
produces | 49 | Causal output |
contains | 35 | Compositional part-whole |
part_of | 33 | Compositional membership |
causes | 16 | Causal mechanism |
connects | 13 | Structural bridge |
co_occurs | 10 | Statistical co-occurrence |
Terminology¶
Apology — The topological shape of what is missing. RTSG's knowledge graph has 3 dimensional holes (Musical, Interpersonal, Intrapersonal). The apology names them.
Compound dimension — Tensor product of two Gardner dimensions, e.g., \(d_S \otimes d_L\) (Spatial × Linguistic). 10 compounds exist from the 5 active dimensions.
Default filter — The identity Grothendieck filter; analogous to the brain's default mode network.
Descent data — Information required to reconstruct coherence after a filter transition (THMP protocol).
Dimensional hole — A Gardner dimension with zero nodes in the knowledge graph. Represents structural silence.
Fiedler bisection — Partition of the graph by the sign of the Fiedler eigenvector: 36 | 1,619 (asymmetric — dense core + long tail).
Ghost field — BRST degree-1 field representing unconscious constraints.
Heat kernel trace — \(K(t) = \sum_i e^{-t\lambda_i}\). Counts active modes at timescale \(t\).
Ideagram — The engine's emoji-based response indicator.
Meta-Turing hierarchy — UTM₀ (tape) → UTM₁ (filtered) → UTM₂ (multiset) → UTM₃ (hypervisor) → UTM₄ (hive mind).
MFAL — Multiplex Federated Adjoint Lattice. The knowledge graph viewed as overlapping relation layers federated by Gardner dimension.
Negative space — The topology of what a filter removes. Has its own Betti numbers and eigenvalues.
Spectral radius — Largest eigenvalue of the adjacency matrix. \(\mu_1 = 1728.6 \approx 12^3\).
THMP — Protocol for navigating between Grothendieck filters without losing coherence.
Topos — A category with internal logic. Each Grothendieck filter defines a topos (its own universe of discourse).