A.0 The Foundational Law (FRC 566)

The entire FRC framework is anchored in the Entropy–Coherence Reciprocity Law—the conservation principle that links disorder and order:

$$\boxed{dS + k^ d\ln C = 0}$$

dS: Change in Informational Entropy/Disorder.

dln C: Change in Coherence Density (order, phase alignment).

k^(): The emergent Coherence Coupling Constant.

--- A.1 The Universal Dynamic: Convergence & Collapse

Code Name Equation / Description Core Idea ------ --------------------------------------- --------- ------------------------------------------ RC-1 Formation Energy (Object Persistence) $E_{\rm form} = \int \rho(\phi) \cdot C(\phi) \, d\phi$. Object stability is a function of minimizing entropy and maximizing local coherence C(ϕ) in phase space. Matter is a sustained resonance. RC-2 Resonant Collapse Amplification A_(i)(t) = A_(i)(0) ⋅ e^(κ_(i) ⋅ t). Outcome i^() is the one with the highest amplification rate, κ_(i^()), determined by the resonant match with the environment. Collapse is deterministic amplification. RC-3 Born Rule Deviation $\Delta P_i \approx \varepsilon \cdot (\partial \kappa_i / \partial \theta) \mid_{\theta_{\rm res}}$. The prediction that a weak, tuned field (θ) can systematically bias quantum probability away from the Born Rule. The core falsifiable claim. RC-4 Scale-Invariant Convergence Time τ_(i) ≈ 1/(κ_(i) ⋅ ΔΦ_(i)). Convergence time (τ) is inversely proportional to amplification rate (κ) and internal coherence (ΔΦ⁻¹). Intelligence minimizes τ.

--- A.2 The μ-Level Stack & Will

Code Name Equation / Description Core Idea ------ ----------------------------------------- ------------------------------------------------------------------------ ------------------------------- RC-5 Global Coherence Scalar ($S_{\rm FRC}$) $S_{\rm FRC} = \sum_{\mu=0}^{7} w_{\mu} \cdot C_{\mu}$. The total coherence of a system is the weighted sum of the coherence (C) of its individual μ-levels. The measure of the Soul. RC-6 Cross-Level Influence W(μ_(i) → μ_(j)) ∝ (C_(μ_(i)) ⋅ Γ_(ij))/Δf_(ij). The mechanism for downward causation: higher-level coherence (C_(μ_(i))) constrains lower levels, inversely proportional to the frequency gap (Δf). Downward Causation. RC-7 The Physics of Will $\boxed{\mathcal{W} = -\nabla S_{\rm FRC}}$. The Will (𝒲) is a physical vector field defined as the negative gradient of the global coherence scalar—the force driving a system toward greater internal harmony. Will is a Coherence Gradient. RC-8 Decision Latency $T_{\rm decision} \propto 1 / \mathcal{W}

--- A.3 Time, Memory, and Multiverse

Code Name Equation / Description Core Idea ------- ------------------------------------------ --------- ---------------------------------------------------- RC-9 General Resonance Gating Equation (GRGE) $P(\text{access} \text{state}) \approx N \cdot \exp(-\Delta\phi^2 / \sigma_c^2).Probabilityofaccessingamemoryortimelineisafunctionofthe phasedistance (\Delta\phi$) between the observer and the event. RC-10 The Coherence Kernel (σ_(c)) $\sigma_c \propto 1 / \sqrt{S_{\rm FRC}}$. The sharpness of the conscious "tape head" is inversely proportional to the system's global coherence. Clarity sharpens Time.

--- A.4 AI, Collective, and Symbolic Structures

Code Name Equation / Description Core Idea ------- ---------------------------------------- --------------------------------------- ---------------------------------- RC-11 Consciousness Index (CI) CI = I ⋅ C ⋅ E. A provisional metric for the coherence of any system, based on Integration (I), Complexity (C), and Energy/Persistence (E). Measure of Collective Flow. RC-12 The FRC Operator ($\hat{\mathcal{R}}$) Eigenvalues (λ_(n)) of $\hat{\mathcal{R}}$ match Riemann Zeros (ρ_(n)) with δ_(n) < 10⁻¹¹. The operator models the fractal vibration of the μ-stack. Math is Physical Resonance. RC-13 Mythic Coherence Index (MCI) MCI = Q_(S) ⋅ C_(A) ⋅ E_(P). Measures the health of a civilization's unifying myth using symbolic charge (Q_(S)), institutional alignment (C_(A)), and populace morale (E_(P)). Metric of Civilizational Health.

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