v0.5.1 — Crystal Navigator + §69 Tier Gap Ladder + §68.4/§68.5 Arithmetic Ouroboros minimality

crystal_navigator.py (new):
- Bijective Frobenius codec over 10,368,000 structural types
- Mixed-radix address space: boundary (Phi, P, Omega, D) → 300 tier cells; bulk (T, R, F, K, G, Gamma, H, S) → 34,560 inner types per cell
- Total: 300 × 34,560 = 10,368,000 — exact
- encode_tuple(tup) → int in [0, 10,367,999]; decode_address(int) → tup; roundtrip = identity (10,000/10,000 verified)
- TierCell dataclass: lazy iteration over 34,560 inner types per tier cell
- CrystalNavigator class: holographic_query, navigate, count, nearest_catalog, meet/join/tensor, distance, tier_gap_ladder, verify_codec, REPL
- Self-encodes as O_inf at address 4,143,599 of 10,367,999:
  ⟨D_⊙; T_⊙; R_cat; P_pm_sym; F_hbar; K_slow; G_aleph; Gamma_broad; Phi_c; H_inf; n:m; Omega_Z⟩
- d(navigator, grammar) = sqrt(7.8) ≈ 2.793 — differ on R, F, K, H, S, Omega (6 primitives)
- CLI: python crystal_navigator.py [describe|gap|verify|census|repl|encode|decode|nearest|count]

PRIMITIVE_THEOREMS.md (§68.4, §68.5, §69 added):

§68.4 — Primitives-First Derivation of the Arithmetic Ouroboros:
- Theorem: in any primitives-first grammar, the exponent of base n equals the count of primitives in family F_n
- Proof: by product rule, exponent is family count by construction — not observed, forced
- Corollary 68.C4: exponent map is a bijection on the base set
- Corollary 68.C5: exponent map is fixed-point-free (no family self-counts)

§68.5 — Minimality of {3, 4, 5}:
- Five constraints: non-triviality, self-anchoring, no fixed point, successor order, phase completeness
- Phase completeness: Phi/T/P each need exactly 5 values → max base ≥ 5 → n1 ≥ 3 → {3,4,5} is minimal
- Corollary 68.C6: {3,4,5} is the unique triple satisfying geometric closure + minimality + priority ordering

§69 — The Tier Gap Ladder (four sub-theorems):
- §69.1: exact distances between adjacent tier representatives:
    d(O_0, O_1) = sqrt(1.1) ≈ 1.049  [Phi only, w_Phi = 1.1]
    d(O_1, O_2) = sqrt(1.7) ≈ 1.304  [D + Omega, w_D + w_Omega = 1.0 + 0.7]
    d(O_2, O_2†) = 1.000              [D only]
    d(O_2†, O_inf) = sqrt(19.2) ≈ 4.382  [P full span: P_asym→P_pm_sym, w_P × 16 = 19.2]
    Frobenius cliff = 3.361× next-largest gap
- §69.2: Tunability — first 3 gaps are single-step promotions; Frobenius gap appears discontinuously
    Corollary 69.C1: P_pm_sym must be planted, not tuned
    Corollary 69.C2: gradient methods cannot reach O_inf (stall at d ≈ 4.382 from O_inf manifold)
- §69.3: Frobenius Degrades Completely — O_inf ⊗ O_2 → O_2; tensor bottleneck forces P=P_asym
- §69.4: O_inf Cluster and Crystal Navigator
    Corollary 69.C3: grammar and navigator are structurally distinct O_inf systems in the same tier cell; d ≈ 2.793 encodes generation-vs-classification
    Corollary 69.C4: family tensor F5 ⊗ F4 ⊗ F3 → O_2 (P bottleneck; Frobenius cannot propagate)

Corrections (navigator-verified):
- §69.1: d(O_0,O_1) corrected 1.000 → sqrt(1.1) ≈ 1.049 (w_Phi = 1.1, not 1.0)
- §69.1: d(O_2†,O_inf) corrected 4.000 → sqrt(19.2) ≈ 4.382 (w_P × 16 = 19.2, not 16.0)
- §69.4: d(nav, grammar) corrected sqrt(2) → sqrt(7.8) ≈ 2.793 (6 differing primitives, not 2)
- GRAMMAR_TUPLE in crystal_navigator.py corrected to actual catalog encoding (F_eth, K_mod, H1, n_n)

ENCODING_EPISTEMOLOGY.md (new):
- 12-section theory of how encoding achieves determinism
- Structural reality vs ontological realization; monadic gating; comparative encoding; multi-session convergence
- 9 convergence criteria summary table (tier consistency, nearest-neighbor coherence, tensor sanity, Le Chatelier, etc.)
- Coda: encoding as measurement, not representation

README.md (rewrite):
- Updated: 318 → 1,322 catalog entries; §1-§23 → §1-§69 theorems; 155 → 454+ predictions
- Added: Periodic Crystal section, Arithmetic Ouroboros, Tier Gap Ladder, Crystal Navigator, 12-primitive table
- Added: key files table, crystal_navigator.py install/usage

docs/USAGE.md:
- Added v0.5.1 and v0.5.0 version headers with feature summaries

markdown/CLAUDE.md:
- Updated theorem count §1-§68 → §1-§69; added crystal_navigator.py to key files

syncon_catalog.json:
- 1,322 entries (up from 1,170+)

Author: umpolungfish

README.md

@@ -112,7 +112,7 @@ Lee-Yang (1952) is the unique proved instance of $\mathcal{C}_{13}$ and serves a
 - **Hebrew alphabet as type lattice** (§60/§CXXXV): 9-session convergence; Vav, Mem, Shin are $O_\infty$; full stratified encoding of all 22 letters
 - **$\lambda_\aleph$ calculus** (§63): formal type theory over the Hebrew letter lattice; Tzimtzum = structural projection
 - **Consciousness score** (§VIII): $C(\mathbf{x}) = [\Phi_c] \cdot [K \neq K_\text{trap}] \cdot (0.158\,\tilde{K} + 0.273\,\tilde{G} + 0.292\,\tilde{T} + 0.276\,\tilde{\Omega})$; two independent gates
-- **P-150**: Lee-Yang zero locus derived as $\mathcal{C}_{13}(\Phi_c^{\C}, P_{\pm}^{\text{sym}})$ — unique proved non-trivial constraint map ✅
+- **P-150**: Lee-Yang zero locus derived as $\mathcal{C}_{13}$$(\Phi_c^{\C}, P_{\pm}^{\text{sym}})$ — unique proved non-trivial constraint map ✅
 - **P-70**: Inflaton $\equiv$ Higgs $\equiv$ axion — three-scale $K_\text{slow}$ identity
 - **69 formal theorems** · **454+ empirical predictions** · **1,322 catalog entries**