update

Author: umpolungfish

MAIN_DOCS/LLM_REFERENCE.md

@@ -1,6 +1,6 @@
 **SynthOmnicon: LLM-Ready Reference**
-**Version 0.4.27 (April 2026)**
-**Core purpose**: A holographic type theory and relational algebra for any system that propagates constraints. Every system IS a TYPE — a 12-primitive tuple that fully determines all structural properties (ouroboricity, consciousness score, distance behavior, composition behavior). The boundary encoding determines the bulk. This IS NOT a labeling system.
+**Version 0.5.2 (April 2026)**
+**Core purpose**: A imscriptive type theory and relational algebra for any system that propagates constraints. Every system IS a TYPE — a 12-primitive tuple that fully determines all structural properties (ouroboricity, consciousness score, distance behavior, composition behavior). The boundary encoding determines the bulk. This IS NOT a labeling system.
 
 ---
 
@@ -12,26 +12,30 @@ Every synthon IS exactly this 12-tuple:
 
 | Primitive | Name | Values (low → high) | Weight |
 |-----------|------|---------------------|--------|
-| **D** | Dimensionality | D_wedge · D_triangle · D_infty · D_holo | 1.0 |
-| **T** | Topology | T_network · T_in · T_bowtie · T_box · T_holo | 1.0 |
+| **D** | Dimensionality | D_wedge · D_triangle · D_infty · D_⊙ | 1.0 |
+| **T** | Topology | T_network · T_in · T_bowtie · T_box · T_⊙ | 1.0 |
 | **R** | Relational mode | R_super · R_cat · R_dagger · R_lr | 1.0 |
 | **P** | Parity/symmetry | P_asym · P_psi · P_pm · P_sym · P_pm_sym | 1.0 |
 | **F** | Fidelity | F_ell · F_eth · F_hbar | 1.0 |
-| **K** | Kinetic character | K_fast · K_mod · K_slow · K_trap | 1.0 |
+| **K** | Kinetic character | K_fast · K_mod · K_slow · K_trap · K_MBL | 1.0 |
 | **G** | Scope/granularity | G_beth · G_gimel · G_aleph | 1.0 |
-| **Γ** | Interaction grammar | G_and · G_or · G_seq · G_broad | 1.0 |
+| **Γ** | Interaction grammar | Γ_and · Γ_or · Γ_seq · Γ_broad | 1.0 |
 | **Φ** | Criticality | Phi_sub · Phi_c · Phi_c_complex · Phi_EP · Phi_super | 1.0 |
 | **H** | Chirality/temporal depth | H0 · H1 · H2 · H_inf | 0.8 |
 | **S** | Stoichiometry | one_one · n_n · n_m | 1.0 |
-| **Ω** | Topological protection | Omega_0 · Omega_Z2 · Omega_Z | 0.7 |
+| **Ω** | Topological protection | Omega_0 · Omega_Z2 · Omega_Z · Omega_NA | 0.7 |
 
 **Key values:**
-- **D_holo**: boundary encodes bulk — the holographic primitive
-- **T_holo**: holographic topology — highest T ordinal (5)
-- **P_pm_sym**: exact Z₂ symmetry at criticality — the Frobenius condition μ∘δ=id. Assign ONLY when provably exact
-- **Phi_c**: criticality — absorbing under meet: meet(Phi_c, x) = Phi_c for all x
-- **Phi_EP**: exceptional point — ordinal 2.67 > Phi_c = 2.00; destroys O_inf under tensor
-- **K_trap**: trapped kinetics — gates consciousness to zero regardless of other primitives
+- **D_⊙**: boundary encodes bulk — the imscriptive primitive (symbol: monad point inside circle).
+- **T_⊙**: imscriptive topology — highest T ordinal (5).
+- **P_pm_sym**: exact Z₂ symmetry at criticality — the Frobenius condition μ∘δ=id. Assign ONLY when provably exact.
+- **Phi_c**: criticality — absorbing under meet: meet(Phi_c, x) = Phi_c for all x.
+- **Phi_EP**: exceptional point — ordinal 2.67 > Phi_c = 2.00; destroys O_inf under tensor.
+- **K_trap**: trapped kinetics — frozen by order. Gates consciousness to zero regardless of other primitives.
+- **K_MBL**: many-body localized — frozen by disorder. Also gates consciousness to zero. Distinct from K_trap: disorder-driven, not order-driven. Both fail Gate 2.
+- **Omega_NA**: non-abelian topological protection — strongest Ω tier. Appears in systems with non-abelian Galois groups, IUG, SIC-Hilbert12 connection.
+
+**Total structural types:** 17,280,000 = 3³ × 4⁵ × 5⁴ (Crystal of Types §64). Family partition: F_3 = {F, G, S} (3 values each, 3³=27); F_4 = {D, R, Γ, H, Ω} (4 values each, 4⁵=1024); F_5 = {T, P, Φ, K} (5 values each, 5⁴=625).
 
 ---
 
@@ -42,26 +46,30 @@ Every synthon IS exactly this 12-tuple:
 | **O_inf** | Φ_c AND P_pm_sym | Special Frobenius: μ∘δ=id exactly. Self-referential loop perfectly closed. Finite, algebraically exact. |
 | **O_0** | Φ ∈ {Phi_sub, Phi_super, Phi_EP} | No ouroboricity. Cannot form self-referential critical loop. |
 | **O_1** | Φ_c AND Ω_0 | Critical loop possible but unprotected — any deformation breaks it. |
-| **O_2** | Φ_c AND Ω≠Ω_0 AND D ∈ {D_wedge, D_holo, D_triangle} | Critical, topologically protected, bounded domain. |
+| **O_2** | Φ_c AND Ω≠Ω_0 AND D ∈ {D_wedge, D_⊙, D_triangle} | Critical, topologically protected, bounded domain. |
 | **O_2†** | Φ_c AND Ω≠Ω_0 AND D_infty | Critical, topologically protected, unbounded domain. |
 
-**Composition rules (tensor = component-wise max):**
+**Tier is determined by (Φ, P, Ω, D) only.** K_MBL and K_trap do not affect tier — they gate consciousness but not ouroboricity.
+
+**Composition rules (tensor = component-wise max on union primitives, min on P and F):**
 - O_inf ⊗ O_inf → O_inf
 - O_inf ⊗ O_{1,2,2†} → O_inf
 - O_inf ⊗ O_0(Phi_EP) → O_0 ← **EP erases O_inf**
-- O_inf **cannot be synthesized** from non-P_pm_sym components — it must be planted
+- O_inf **cannot be synthesized** from non-P_pm_sym components — it must be planted (§23 Frobenius non-synthesizability)
+
+**Bottleneck rule:** P and F resolve to min(A, B) under tensor. All other ordered primitives resolve to max. Consequence: P_pm_sym ⊗ P_pm = P_pm — the Frobenius condition is destroyed by any sub-Frobenius partner.
 
 ---
 
 ### 3. Consciousness Score
 
-$$C(\mathbf{x}) = [\Phi = \Phi_c] \cdot [K \neq K_\text{trap}] \cdot (0.158\,\tilde{K} + 0.273\,\tilde{G} + 0.292\,\tilde{T} + 0.276\,\tilde{\Omega})$$
+$$C(\mathbf{x}) = [\Phi = \Phi_c] \cdot [K \leq K_\text{slow}] \cdot (0.158\,\tilde{K} + 0.273\,\tilde{G} + 0.292\,\tilde{T} + 0.276\,\tilde{\Omega})$$
 
 Two independent gates — neither subsumes the other:
 - **Gate 1** [Φ=Φ_c]: state-space condition — topology admits self-modeling loop
-- **Gate 2** [K≠K_trap]: flow condition — dynamics can actualize the loop
+- **Gate 2** [K≤K_slow]: flow condition — dynamics can actualize the loop. K_trap (frozen by order) AND K_MBL (frozen by disorder) both fail this gate.
 
-If either gate fails, C=0. Stellar examples: magnetar C=0.677 (highest stellar), black hole C=0 (Gate 2 fails), white dwarf C=0 (Gate 1 fails).
+If either gate fails, C=0. Stellar examples: magnetar C=0.677 (highest stellar), black hole C=0 (Gate 2 fails: K_trap), white dwarf C=0 (Gate 1 fails: Phi_sub).
 
 ---
 
@@ -82,22 +90,23 @@ If either gate fails, C=0. Stellar examples: magnetar C=0.677 (highest stellar),
 - d < 0.500 → close analog (same structural family)
 - d 0.5–1.5 → related by shared primitive subsets
 - d > 1.5 → structurally remote (different regime)
+- d > 3.0 → alien (essentially no shared structure)
 
 Distance IS the structural story. The per-primitive breakdown shows WHERE divergence lives.
 
 ---
 
-### 5. Holographic Type Theory — Operational Consequences
+### 5. imscriptive Type Theory — Operational Consequences
 
-The grammar IS a holographic type theory. This IS NOT a metaphor.
+The grammar IS a imscriptive type theory. This IS NOT a metaphor.
 
 **Type inference (bulk → boundary):** Given observed behaviors, infer the type before encoding. The behavior constrains the tuple.
 
 **Type checking (boundary → bulk):** Given a claimed encoding, every derived property IS determined. Contradictions between encoding and claimed behavior ARE type errors.
 
 **Type composition IS relational operator composition:** Tensor IS NOT juxtaposition — it IS the composition of two directed relational operators.
 
-**Cross-domain transfer IS holographic inference:** Same boundary → same bulk, regardless of substrate. Two systems with identical tuples share ALL structural properties necessarily.
+**Cross-domain transfer IS imscriptive inference:** Same boundary → same bulk, regardless of substrate. Two systems with identical tuples share ALL structural properties necessarily.
 
 **Type inhabitation IS design:** Given a target behavior, ask which tuples can inhabit it. Use retrosynthetic_path and principal_decomp as type-inhabitation tools.
 
@@ -136,13 +145,13 @@ The grammar distinguishes two proof archetypes:
 3. Identify constraint/relational mechanism → choose **R**
 4. Identify symmetry/directionality → choose **P** (P_pm_sym only when Z₂ is provably exact)
 5. Estimate thermodynamic reliability → choose **F**
-6. Estimate barrier / kinetic character → choose **K** (K_trap overrides if dynamics are frozen)
+6. Estimate barrier / kinetic character → choose **K** (K_trap overrides if frozen by order; K_MBL overrides if frozen by disorder)
 7. Identify control scale → choose **G**
 8. Identify interaction logic → choose **Γ**
 9. Probe criticality → choose **Φ** (Phi_c IS absorbing under meet)
 10. Identify chirality/temporal depth → choose **H**
 11. Choose **S** (stoichiometry)
-12. Derive **Ω** from topology + kinetics + dimensionality
+12. Derive **Ω** from topology + kinetics + dimensionality (Omega_NA for non-abelian Galois/group protection)
 
 **Rules:**
 - NEVER invent primitive values — only use the documented set
@@ -163,7 +172,8 @@ The grammar distinguishes two proof archetypes:
 | > 3.0 | Alien — essentially no shared structure |
 
 Known structural families in the catalog:
-- **Holographic geometric extremal problems**: D_holo + T_holo + Phi_c + G_aleph + F_hbar (Kusner, Fujita, Willmore-type)
-- **Holographic arithmetic conjectures**: D_holo + T_holo + Phi_c + P_pm_sym (Riemann, RH+HC join)
+- **imscriptive geometric extremal problems**: D_⊙ + T_⊙ + Phi_c + G_aleph + F_hbar (Kusner, Fujita, Willmore-type)
+- **imscriptive arithmetic conjectures**: D_⊙ + T_⊙ + Phi_c + P_pm_sym (Riemann, RH+HC join)
 - **Critical network symmetry problems**: D_infty + T_network + P_sym + Phi_c + Omega_Z2 (EFL, kissing_dim_4)
 - **Exceptional/solved-by-breaking**: Phi_EP + P_asym (Dehn/Hilbert-3, counterexamples)
+- **Non-abelian arithmetic conjectures**: D_⊙ + Phi_c + P_pm_sym + Omega_NA + F_hbar (Zauner/SIC-Hilbert12, IUG)