Appendix B — The 12 TE Equations in PA Notation

This appendix lists the twelve foundational equations of the Technology of Expressions (TE) and their translation into the formal notation of the Proportional Algebra.

#	TE Equation	TE Notation	PA Translation	PA Chapter
1.1	Collapse Function	E = Φ(C, I, K)	Φ: ℭ_h × 𝕀 × K → 𝒟. Collapse iff ρ(C, I, K) ≥ θ(C).	8.2
1.2	Decoherence Equation	D = Ψ(C, N, t)	D ∈ 𝒟. N = noise vector in fibre. t = τ (pulsation count). ⟨𝓚⁵⟩(D) < ⟨𝓚⁵⟩(E_original).	6.2, 12.2
1.3	Extended Semantic Derivative	dΦ/dt(I)	dΦ/dτ = lim_{n→∞} (Φ_{n+1} − Φ_n)/τ_n. Three regimes: >0 (evolution), =0 (inertia), <0 (degeneration).	12.5
1.4	Semantic Tension	T_sem = R − Φ(C, I, K)	T_sem = ρ(C, I) − ⟨𝓚⁵⟩(E). Tension = resonance minus achieved coherence. Residual = un-collapsed potential.	5.1, 6.2
1.5	Vertical Coherence	⟨𝓚⁵⟩_v(I) > θ_v	The coherence of the identity's trajectory satisfies the vertical threshold: Σ_i ⟨𝓚⁵⟩(E_i) / n ≥ θ_v. Recursive: R_{n+1} ⊇ R_n.	4.4.5, 3.8
1.6	Semantic Inertia	dΦ/dt → 0	dΦ/dτ = 0. Pulsation continues (τ > 0) but produces no change. The Remir repeats without restructuring. B_I is frozen.	12.5
1.7	Pulsation	T = τ(C ↔ E)	τ: ℭ_h × 𝒟 × 𝕀 → ℝ⁺. τ = d(C)/(ρ · plasticity) · τ₀. Time is generated, not parametric.	12.2
1.8	Expression–Content Asymmetry	E ≠ C;	E	<
1.9	Remir	ℛ(I) = (V_I, B_I)	V_I = finite set of semantic vectors. B_I: V_I × V_I → [-1,1]. Dominant vector λ(I) = eigenvector of B_I with max eigenvalue. 𝕀 is a non-commutative, non-associative algebra.	7.2, 7.6
1.10	Dominant Vector	λ(I) = argmax β(v, I)	λ(I) = eigenvector of B_I with largest eigenvalue λ_max. Determines collapse direction and trajectory bias.	7.3
1.13	Terminal Compatibility	Ξ(I, 𝒯, t) → [0,1]	Ξ measures ρ_K at the body-identity interface. Declining Ξ = declining capacity to collapse through the physical channel. High ρ_v + low ρ_K = identity "knows more, can say less."	16.5
1.14	Collective Field	ℭ_h = f({I₁...Iₙ})	⊗ⁿ_{k=1} I_k = ∪_{all resonant pairs} ℭ_shared. n-fold resonance. Emergent directions, resonance cascades, critical mass.	10.6

Key Transformations

Time (t) → Pulsation count (τ): All TE equations parametrised by t are re-parametrised by pulsation cycles. This eliminates the need for external time.
Coherence (abstract) → ⟨𝓚⁵⟩ (5-component function): All TE references to "coherence" become the formal ⟨𝓚⁵⟩ function with its five components.
Resonance (intuitive) → ρ (5-component metric): All TE references to "compatibility" or "resonance" become the formal ρ function with its five components and threshold θ.
Identity (functional description) → Remir algebra: All TE references to identity become the formal Remir ℛ(I) = (V_I, B_I) with its algebraic properties.