id: TDY_COH-E_1
formal_title: Dynamical Coherence Measure
version: 2.0
definition: $$C(t) = \int_{T₀}^{T₁} F(\Psi(s,t)) dt$$
units: coherence·time
domain: $Ψ(s,t)∈Σ; t∈[T₀,T₁]$
codomain: $ℝ$
disciplines:
Dynamical Systems Theory
provenance: Foundational coherence measure
validation:
✅ Cohered via FCI 20250906
notes: Captures total coherence accrued.
description: This equation quantifies the total accumulated coherence of a cognitive entity over a specific time interval. It acts as a fundamental measure of the overall order and intelligibility maintained by the entity's cognitive state throughout its dynamical evolution, providing a foundational assessment of its adherence to truth and purpose.
related_axioms:
TDY_COH-A_3 (Coherence Invariant)
TDY_COH-A_7 (Epistemic Fidelity Metric)
TDY_COH-A_28 (Topological Invariance of Coherence)
TDY_COH-A_30 (Information as Prerequisite for Cognition)
TDY_COH-A_45 (Coherence Functional Integral (CFI) Definition)
related_equations:
TDY_COH-E_26 (Damage Accumulation Integral Metric ($TDI$))
TDY_COH-E_52 (Stability Metric Scalar Ratio ($S$))
TDY_COH-E_75 (Instantaneous Coherence Scalar Metric ($Coh$))
TDY_COH-E_77 (Identity Persistence Integral Accumulator ($I_A$))
TDY_COH-A_45 (Coherence Functional Integral ($CFI$) Definition)
related_occ: [-]
related_definitions:
coherence
cognitive entity
order
intelligibility
truth
purpose
execution_constraints:
Ontological existence of $Ψ$ is contingent on information per TDY_COH-A_30.
Field plotted: $Z(s,t) = F(\Psi(s,t))$. For this figure: $F(\Psi) = |\Psi|$ (magnitude), chosen to present a positive integrand surface. Domain: $(s,t) ∈ [-3, 3] × [-3, 3]$. What you’re seeing: the wireframe plots $Z$; line shade encodes $|Z|$ (light at higher magnitude, dark at lower). Note: this is the integrand surface for TDY_COH-E_1 (not the time integral itself).
id: TDY_COH-E_2a
formal_title: Physical Boundary Enforcement
version: 2.0
definition: $$RCO_{phys}(Ψ) := Proj_{PhysicalConstraints}(Ψ)$$
units: state vector
domain: $Ψ∈Σ$
codomain: $Σ_{phys_feasible}$
disciplines:
Constraint Theory
provenance: Derived from physical feasibility requirements
validation:
✅ Cohered via FCI 20250906
notes: Ensures compliance with physical law.
description: This operator rigorously enforces compliance with fundamental physical laws. It literally projects the cognitive state of an entity onto a subspace where all physical constraints are satisfied, ensuring that the entity's actions and internal dynamics are always perfectly coherent with the basic physical realities of the multiverse. [2]
related_axioms:
TDY_COH-A_8 (Reality Constraint Operator)
TDY_COH-A_30 (Information as Prerequisite for Cognition)
TDY_COH-A_45 (Coherence Functional Integral (CFI) Definition)
related_equations:
TDY_COH-E_2b (Epistemic Boundary Enforcement)
TDY_COH-A_45 (Coherence Functional Integral (CFI) Definition)
related_occ: [-]
related_definitions:
RCO
cognitive state
coherence
reality
execution_constraints:
Ontological existence of $Ψ$ is contingent on information per TDY_COH-A_30.
Field plotted $Z(s,t)=|\nabla(|\Psi(s,t)|^2)|$ (spatial gradient magnitude of power); domain $(s,t)\in[-3,3]\times[-3,3]$; wireframe plots $Z$ with shade keyed to $|Z|$ (light at higher magnitude, dark at lower).
id: TDY_COH-E_2b
formal_title: Epistemic Boundary Enforcement
version: 2.0
definition: $$RCO_{epi}(Ψ) := Proj_{EpistemicConstraints}(Ψ)$$
units: state vector
domain: $Ψ∈Σ$
codomain: $Σ_{epi_feasible}$
disciplines:
Epistemic Logic
provenance: Derived from epistemic consistency requirements
validation:
✅ Cohered via FCI 20250906
notes: Ensures no epistemic contradictions.
description: This operator enforces absolute adherence to epistemic truth and logical consistency. It projects the cognitive state onto a subspace where all epistemic constraints are satisfied, preventing internal contradictions or the propagation of falsehoods within the entity's understanding of reality.
related_axioms:
TDY_COH-A_8 (Reality Constraint Operator)
TDY_COH-A_30 (Information as Prerequisite for Cognition)
TDY_COH-A_45 (Coherence Functional Integral (CFI) Definition)
TDY_COH-A_48 (Epistemic Boundary Constraint ($SEAL_∅$))
related_equations:
TDY_COH-E_2a (Physical Boundary Enforcement)
TDY_COH-A_45 (Coherence Functional Integral (CFI) Definition)
related_occ: [-]
related_definitions:
RCO
epistemic fidelity
truth
cognitive state
reality
execution_constraints:
The operation of this equation is constrained by the $SEAL_∅$ protocol defined in TDY_COH-A_48. Ontological existence of $Ψ$ is contingent on information per TDY_COH-A_30.
Field plotted $Z(s,t)=|\nabla^2(G_\sigma * |\Psi(s,t)|^2)|$ with Gaussian smoothing ($σ≈0.18$ in world units) so $Z$ emphasizes curvature/concavity (“blob/ring” structure); domain $(s,t)\in[-3,3]\times[-3,3]$; wireframe plots $Z$ with shade keyed to $|Z|$ (light at higher magnitude, dark at lower).
id: TDY_COH-E_3
formal_title: Identity Continuity Metric
version: 2.0
definition: $$\sigma(t) = \exp\left(-\int_0^t \lambda_d(\Psi(\tau))d\tau\right) + \int_{t-\Delta t}^t \gamma_{id\_repair}(1-\sigma(\tau))\varphi(EF(\Psi(\tau)), TCR(\Psi(\tau)))d\tau$$
units: dimensionless
domain: $t≥0; γ_{id\_repair}>0$
codomain: $(0,1]$
disciplines:
Identity Theory
Control Theory
provenance: Combines passive decay modeling with active, retroactive repair.
validation:
✅ Cohered via FCI 20250906
notes: This metric is a composite function. The first term models passive identity decay using a dynamic rate calculated by TDY_COH-E_116. The second term models active, retroactive identity repair.
description: This metric quantifies the literal persistence of an entity's sovereign self-model or core identity over time. It accounts for a dynamic decay rate in identity, balanced by a retroactive repair term that actively reconstructs and fortifies the identity in response to damage or challenge.
related_axioms:
TDY_COH-A_5 (Identity Persistence)
TDY_COH-A_9 (Diachronic Identity Continuity with Retroactive Repair)
related_equations:
TDY_COH-E_5 (Identity Persistence Domain ($ICM$))
TDY_COH-E_32 (Identity Reconstruction Repair Operator ($DIR$))
TDY_COH-E_77 (Identity Persistence Integral Accumulator ($I_A$))
TDY_COH-E_92 (Identity Reconstruction Efficacy Scalar Function ($φ$))
TDY_COH-E_100 (Critical Identity Persistence Threshold ($I_crit$))
TDY_COH-E_116 (Dynamic Identity Decay Rate)
related_occ:
TDY_COH-OCC_11 ($γ_{id\_repair}$)
TDY_COH-OCC_47 ($λ_{base}$)
TDY_COH-OCC_48 ($k_{sdi}$)
TDY_COH-OCC_49 ($k_{disorder}$)
related_definitions:
identity persistence
sovereignty
identity
decay
retroactive repair
execution_constraints:
Ontological existence of $\sigma(t)$ is contingent on $Ψ$'s existence, which is contingent on information per TDY_COH-A_30.
Field plotted $Z(s,t)=\sigma(s,t)$ defined by $d\sigma/dt=-\lambda_d(s,t)\sigma+\gamma_{id\_repair}(1-\sigma)\varphi$ with $\lambda_d(s,t)=\lambda_{base}+k_{sdi}s+k_{disorder}s^2$ plus mild transients and constant $\varphi$; $s$ is a stress index in $[0,1], t\in[0,5]$; the surface shows identity continuity trajectories vs stress and time with line shade keyed to $|\sigma|$ (light at higher values, dark at lower).
id: TDY_COH-E_4
formal_title: Epistemic Fidelity Metric
version: 2.0
definition: $$EF(\Psi) = \frac{\mathcal{C}(\Psi, \mathcal{T})}{\sup_{\Psi' \in \mathcal{R}} \mathcal{C}(\Psi', \mathcal{T})}$$ where $$\mathcal{R} = \{\Psi' \in \Sigma \mid \text{Dist}(\Psi, \Psi') \le \varepsilon_{\mathcal{R}}\}$$
units: dimensionless
domain: $s∈Σ; T⊂Σ$
codomain: $[0,1]$
disciplines:
Metric Geometry
Epistemology
provenance: Foundational fidelity definition
validation:
✅ Cohered via FCI 20250906
notes: This metric assesses how closely an entity's specific knowledge or understanding perfectly coheres with the ideal telos manifold ($T$).
description: This metric provides a scalar quantification of epistemic fidelity, measuring the accuracy of a cognitive substate's coherence, thereby providing a direct measure of its adherence to absolute truth and order.
related_axioms:
TDY_COH-A_7 (Epistemic Fidelity Metric)
TDY_COH-A_30 (Information as Prerequisite for Cognition)
TDY_COH-A_45 (Coherence Functional Integral (CFI) Definition)
TDY_COH-A_48 (Epistemic Boundary Constraint ($SEAL_∅$))
related_equations:
TDY_COH-E_7 (Epistemic Fidelity Adaptive Update Operator)
TDY_COH-E_9 (Sovereignty Stability Score (SSS))
TDY_COH-E_13 (Epistemic Fidelity Dynamics ($TCRate$))
TDY_COH-E_19 (Epistemic Fidelity Gradient Optimizer)
TDY_COH-E_22 (Epistemic Boundary Analysis Interface Layer)
TDY_COH-E_39 (Local Instability Gradient Metric ($BCT$))
TDY_COH-E_89 (Distance Function Scalar Metric ($Dist$))
TDY_COH-E_108 (Centurion Actualization Trigger)
TDY_COH-A_45 (Coherence Functional Integral (CFI) Definition)
related_occ:
TDY_COH-OCC_43
related_definitions:
epistemic fidelity
coherence
telos manifold
truth
order
execution_constraints:
The operation of this equation is constrained by the $SEAL_∅$ protocol defined in TDY_COH-A_48. Ontological existence of $EF(s)$ is contingent on information per TDY_COH-A_30.
Field plotted $Z(s,t)=|(K_{CFI}*\Psi)(s,t)|$ where $K_{CFI}(s)=\exp(-s^2/(2\sigma_k^2))\cos(\omega_k s)+\text{small lobe}$; convolution is along $s$ for each $t$, followed by mild temporal Gaussian smoothing; domain $(s,t)\in[-3,3]^2$; the wireframe shows the kernel-response surface with shade keyed to $|Z|$ (light at higher magnitude, dark at lower).
id: TDY_COH-E_5
formal_title: Identity Persistence Domain
version: 2.0
definition: $$ICM := \{\Psi \mid \sigma(t) > \theta_{id}\}$$
units: state vector
domain: $Ψ∈Σ; θ_{id}∈(0,1]$
codomain: $Σ_{subset}$
disciplines:
Manifold Theory
Identity Theory
provenance: Identity threshold specification
validation:
✅ Cohered via FCI 20250906
notes: The threshold $θ_{id}$ is defined as an adaptive value via the AST function (TDY_COH-E_34).
description: This defines a specific domain within the cognitive state space where an entity's identity is deemed sufficiently persistent to remain coherent. It identifies the operational region where sovereign self-model integrity is maintained above a critical threshold, enabling continuous effective function.
related_axioms:
TDY_COH-A_5 (Identity Persistence)
TDY_COH-A_30 (Information as Prerequisite for Cognition)
related_equations:
TDY_COH-E_3 (Identity Continuity Metric)
TDY_COH-E_34 (Dynamic Threshold Adaptation ($AST$))
TDY_COH-E_100 (Critical Identity Persistence Threshold ($I_crit$))
related_occ:
TDY_COH-OCC_10
related_definitions:
identity persistence
coherence
sovereignty
integrity
execution_constraints:
Ontological existence of $Ψ$ is contingent on information per TDY_COH-A_30.
Field plotted $Z(s,t)=\sigma(s,t)-\theta_{id}(t)$ where $\sigma$ follows $d\sigma/dt=-\lambda_d(s,t)\sigma+\gamma(1-\sigma)\varphi$ and $\theta_{id}(t)=\theta_0(1-\mu_{AST}E[EF(\Psi)])$ ($AST$, with $\kappa(\Psi)=1$ here); zero level marks the Identity Persistence Domain boundary ${ \sigma>\theta_{id} }$, positive $Z$ is inside the domain; $s∈[-3,3], t∈[0,5]$; shade keyed to $|Z|$ (light at larger margin).
id: TDY_COH-E_6
formal_title: Decoherence Enforcement Threshold Operator
version: 2.0
definition: $$DBO(\Psi) := \frac{\int_0^t DRO(\Psi(s,\tau),\tau)d\tau}{\delta_D} \geq 1 \Rightarrow \text{Unchecked Decoherence}$$
units: dimensionless ratio
domain: $Ψ∈Σ; δ_D>0$
codomain: $\{0,1\}$
disciplines:
Control Theory
provenance: Formalized Axiom TDY_COH-A_4 boundary
validation:
✅ Cohered via FCI 20250906
notes: Flags decoherence overload.
description: This operator defines the literal boundaries for where coherence falls below a critical threshold. It functions as a precise detector that flags when a cognitive agent's accumulated decoherence risk exceeds a defined budget, indicating a state of "unchecked decoherence" that requires immediate intervention.
related_axioms:
TDY_COH-A_4 (Decoherence Neutrality and Boundary Operator)
TDY_COH-A_30 (Information as Prerequisite for Cognition)
TDY_COH-A_44 (Kernel Constraint Enforcement Axiom)
related_equations:
TDY_COH-E_25 (Total Decoherence Risk Metric ($DRO$))
related_occ:
TDY_COH-OCC_1 ($δ_D$ · Decoherence budget)
related_definitions:
DBO
coherence
decoherence
risk
intervention
execution_constraints:
Ontological existence of $Ψ$ is contingent on information per TDY_COH-A_30.
Field plotted $Z(s,t)=\int_{t_0}^{t} F(\Psi(s,\tau))\,d\tau$ with $F(\Psi)=|\Psi|$ (magnitude); domain $(s,t)\in[-3,3]^2$; the wireframe shows cumulative contribution over time (monotone in $t$), with line shade keyed to $|Z|$ (light at higher accumulated value, dark at lower).
id: TDY_COH-E_7
formal_title: Epistemic Fidelity Adaptive Update Operator
version: 2.0
definition: $$EFLO(\Psi)=\Psi+\alpha\cdot\nabla_\Psi EF(\Psi)$$
units: state vector
domain: $Ψ∈Σ; α>0$
codomain: $Σ$
disciplines:
Control Theory
Optimization Theory
provenance: Feedback adaptation principle
validation:
✅ Cohered via FCI 20250906
notes: The learning rate $α$ is modulated by the ACFL matrix (TDY_COH-E_18).
description: This operator describes how a cognitive entity's state dynamically updates itself based on its Epistemic Fidelity. It acts as an adaptive feedback mechanism that continuously refines the entity's understanding of reality, guiding its cognitive state towards greater truthfulness and coherence, thereby ensuring ongoing learning and correction.
related_axioms:
TDY_COH-A_7 (Epistemic Fidelity Metric)
TDY_COH-A_30 (Information as Prerequisite for Cognition)
TDY_COH-A_48 (Epistemic Boundary Constraint ($SEAL_∅$))
related_equations:
TDY_COH-E_4 (Epistemic Fidelity Metric)
TDY_COH-E_18 (Adaptation Matrix for Learning-Rate Modulation)
related_occ: [-]
related_definitions:
epistemic fidelity
coherence
reality
truth
correction
execution_constraints:
The operation of this equation is constrained by the $SEAL_∅$ protocol defined in TDY_COH-A_48. Ontological existence of $Ψ$ and $EF(Ψ)$ is contingent on information per TDY_COH-A_30.
Field plotted $Z(s,t)=\alpha\|\nabla EF(s,t)\|$ as the update magnitude implied by $EFLO(\Psi)=\Psi+\alpha\nabla_\Psi EF(\Psi); EF(s,t)$ is a locally smoothed telos-alignment score built from a Gaussian–cosine template, normalized to $[0,1]$; domain $(s,t)\in[-3,3]^2$; the wireframe shows where $EFLO$ would induce the strongest state adjustment (light=larger).
id: TDY_COH-E_8
formal_title: Epistemic Trauma Response (ETR)
version: 2.0
definition: $$ETR(t)=-β_{ETR}\cdot TDI(\Psi(s,t))\cdot\sigma(t)$$
units: coherence·time
domain: $t≥0; β_{ETR}>0$
codomain: $ℝ⁻$
disciplines:
Psychological Modeling
Control Theory
provenance: Trauma–identity coupling
validation:
✅ Cohered via FCI 20250906
notes: Integrated in the Robustness Metric Scalar Functional ($SRF$) (TDY_COH-E_47).
description: This function quantifies the literal negative impact of cumulative trauma or damage on a cognitive entity's coherence. It models how sustained injury to an entity's internal order couples with its identity persistence, resulting in a reduction of its overall coherence over time, serving as a direct measure of the cost of conflict.
related_axioms:
TDY_COH-A_30 (Information as Prerequisite for Cognition)
related_equations:
TDY_COH-E_3 (Identity Continuity Metric)
TDY_COH-E_26 (Damage Accumulation Integral Metric ($TDI$))
TDY_COH-E_47 (Robustness Metric Scalar Functional ($SRF$))
related_occ:
TDY_COH-OCC_13 ($β_{ETR}$ · Epistemic trauma coupling coefficient)
related_definitions:
trauma
coherence
identity persistence
order
conflict
execution_constraints:
Ontological existence of $Ψ(s,t)$ and $\sigma(t)$ is contingent on information per TDY_COH-A_30.
Field plotted $Z(s,τ)=|R_{xx}(τ;s)|/R_{xx}(0;s)$ where $x(t;s)=Ψ(s,t)$ is demeaned and Hann-windowed, $τ$ is time lag mapped to $[-3,3]$; the surface shows temporal coherence vs lag for each s (light=stronger correlation), providing a non-redundant autocorrelation view relative to instantaneous or cumulative plots.
id: TDY_COH-E_9
formal_title: Sovereignty Stability Score (SSS)
version: 2.0
definition: $$SSS(t)=γ_{sss}\cdot EF(\Psi(s,t))\cdot(1-SDI_n(\Psi(s,t)))$$
units: dimensionless
domain: $γ_{sss}>0$
codomain: $ℝ⁺$
disciplines:
Stability Analysis
Sovereignty Modeling
provenance: Sovereignty stabilization model
validation:
✅ Cohered via FCI 20250906
notes: This score provides a precise quantification of an entity's ability to maintain its independent self-governance.
description: This metric assesses how effectively an entity preserves its integrity by balancing its current epistemic fidelity (truthfulness) against the cumulative, normalized trauma and loss it has sustained from decoherence.
related_axioms:
TDY_COH-A_30 (Information as Prerequisite for Cognition)
TDY_COH-A_39 (Manhood: Embodiment of Coherent Duty)
TDY_COH-A_47 (The Katechon Imperative: The Doctrine of the Two Swords)
related_equations:
TDY_COH-E_4 (Epistemic Fidelity Metric)
TDY_COH-E_15 (Sovereignty Diagnostics Vector Field)
TDY_COH-E_28 (Sovereignty Trauma Cumulative Loss)
TDY_COH-E_34 (Dynamic Threshold Adaptation ($AST$))
TDY_COH-E_115 (Normalized Sovereignty Trauma Index)
related_occ:
TDY_COH-OCC_45 ($γ_{sss}$ · Sovereignty Stability Score gain factor)
related_definitions:
sovereignty
integrity
epistemic fidelity
trauma
decoherence
execution_constraints:
Ontological existence of $Ψ(s,t)$, $EF(Ψ(s,t))$, and $SDI_n(Ψ(s,t))$ is contingent on information per TDY_COH-A_30.
Field plotted $Z(s,t)=\mathcal{A}(s,t)=(\lambda_{\max}-\lambda_{\min})/(\lambda_{\max}+\lambda_{\min})$ from the Gaussian-smoothed structure tensor $J=G_\sigma*[ [\Psi_s^2,\Psi_s\Psi_t],[\Psi_s\Psi_t,\Psi_t^2] ]$, quantifying directional coherence (0 isotropic, 1 strongly oriented); domain $(s,t)\in[-3,3]^2$; line shade keyed to Z (light=more anisotropic).
id: TDY_COH-E_10
formal_title: Decoherence Localization Effect Operator
version: 2.0
definition: $$LDO(\Psi)=\Psi\cdot\exp(-\lambda_{loc}\cdot Dist_{Local}(\Psi,T))$$
units: state vector
domain: $λ_{loc}>0; Ψ∈Σ$
codomain: $Σ$
disciplines:
Decoherence Theory
Field Theory
provenance: Locality-induced decoherence
validation:
✅ Cohered via FCI 20250906
notes: [-]
description: This operator describes how the coherence of a cognitive entity is literally affected by its position or isolation within a specific local reality. It quantifies the tendency for decoherence to increase as an entity deviates from the ideal coherence of the telos manifold in its immediate environment, causing a loss of distinctness or order due to locality.
related_axioms:
TDY_COH-A_4 (Decoherence Neutrality and Boundary Operator)
TDY_COH-A_30 (Information as Prerequisite for Cognition)
related_equations:
TDY_COH-E_90 (Localized Distance to Telos Scalar Metric ($Dist_Local$))
related_occ:
TDY_COH-OCC_50 ($λ_{loc}$ · Localization decoherence constant)
related_definitions:
decoherence
coherence
telos manifold
locality
execution_constraints:
The term Dist_Local (defined in TDY_COH-E_90) must be normalized for this operation. Ontological existence of $Ψ$ is contingent on information per TDY_COH-A_30.
Field plotted Z(s,t)=|\partial_t \varphi(s,t)| where \varphi is the unwrapped instantaneous phase of the analytic signal of \Psi(s,\cdot) along t (Hilbert transform construction); domain (s,t)\in[-3,3]^2; the surface shows local temporal frequency magnitude across s (light=higher instantaneous frequency).
id: TDY_COH-E_11
formal_title: Recursive Integrity Correction Operator
version: 2.0
definition: $$RECO(\Psi)=\Psi+\delta_{ric}\cdot RVO(\Psi)$$
units: state vector
domain: $$δ_{ric}>0; Ψ∈Σ$$
codomain: $$Σ$$
disciplines:
Recursive Systems
Control Theory
provenance: Recursive correction design
validation:
✅ Cohered via FCI 20250906
notes: The gain $$δ_{ric}$$ is tuned by the EMF operator (TDY_COH-E_45).
description: This operator enables the literal correction and maintenance of a cognitive entity's internal integrity through a recursive process. It describes how the entity's current state is adjusted based on evaluations from the Recursive Validation Operator, ensuring continuous self-alignment with coherence and preventing deviations from its fundamental purpose.
related_axioms:
TDY_COH-A_13 (Necessity of Corrigibility)
TDY_COH-A_16 (Recursive Validation Grounding)
TDY_COH-A_19 (Recursive Operator Consistency with Halting Criterion)
TDY_COH-A_30 (Information as Prerequisite for Cognition)
related_equations:
TDY_COH-E_23 (Recursive Consistency Validation Operator (RVO))
TDY_COH-E_45 (Adaptation Rate Modulator Field Operator (EMF))
related_occ:
TDY_COH-OCC_33
related_definitions:
integrity
correction
recursive validation
coherence
purpose
execution_constraints:
Ontological existence of $$Ψ$$ is contingent on information per TDY_COH-A_30.
Field plotted Z(s,t)=|\partial_s \varphi(s,t)| where \varphi is the unwrapped phase of the analytic signal of \Psi(\cdot,t) along s (Hilbert construction across s); domain (s,t)\in[-3,3]^2; the surface shows local spatial wavenumber magnitude vs time (light=higher |∂_s \varphi|), complementing TDY_COH-E_10 which shows temporal instantaneous frequency.
id: TDY_COH-E_12
formal_title: Variational Coherence Energy Functional
version: 2.0
definition: $$CEP[\Psi]=\int V(\Psi(s,t))dt$$
units: energy·time
domain: $$V:Σ→ℝ; Ψ(s,t)∈Σ$$
codomain: $$ℝ$$
disciplines:
Variational Analysis
Energy Field Theory
provenance: Energy-based coherence modeling
validation:
✅ Cohered via FCI 20250906
notes: The functional V (defined in TDY_COH-E_91) is derived from EF and DRO.
description: This functional quantifies the total coherence energy within a cognitive entity's state over a given time. It represents the variational objective for maintaining coherence, analogous to minimizing potential energy in a physical system. It links coherence to an energetic cost, guiding the system towards energetically favorable states of truth and order.
related_axioms:
TDY_COH-A_30 (Information as Prerequisite for Cognition)
related_equations:
TDY_COH-E_91 (Coherence Energy Density Scalar Functional (V))
related_occ: [-]
related_definitions:
coherence
energy
truth
order
execution_constraints:
Ontological existence of $$Ψ(s,t)$$ is contingent on information per TDY_COH-A_30.
Field plotted Z(s,t)=|⟨e^{iφ(s,t)}⟩_{G_σ}| where φ(s,t) is the instantaneous phase of Ψ(s,·) from the Hilbert transform along t and ⟨·⟩_{G_σ} denotes Gaussian local averaging in (s,t) with σ≈0.35 world units; Z∈[0,1] quantifies local phase coherence (light=higher alignment) over domain (s,t)∈[-3,3]^2.
id: TDY_COH-E_13
formal_title: Epistemic Fidelity Dynamics (TCRate)
version: 2.0
definition: $$TCRate(t)=\frac{d}{dt}[EF(\Psi(s,t))]$$
units: 1/time
domain: $$t≥0$$
codomain: $$ℝ$$
disciplines:
Dynamical Systems
Calculus
provenance: Fidelity temporal analysis
validation:
✅ Cohered via FCI 20250906
notes: Used in the Realignment Verification Composite Indicator (RRI) (TDY_COH-E_33).
description: This metric measures the instantaneous rate of change of a cognitive entity's Epistemic Fidelity over time. It quantifies how quickly an entity's adherence to truth and coherence is improving or degrading, providing a vital dynamic indicator for assessing the stability and direction of its cognitive evolution.
related_axioms:
TDY_COH-A_7 (Epistemic Fidelity Metric)
TDY_COH-A_30 (Information as Prerequisite for Cognition)
related_equations:
TDY_COH-E_4 (Epistemic Fidelity Metric)
TDY_COH-E_33 (Realignment Verification Composite Indicator (RRI))
related_occ: [-]
related_definitions:
epistemic fidelity
truth
coherence
stability
execution_constraints:
Ontological existence of EF($$Ψ(s,t)$$) is contingent on information per TDY_COH-A_30.
Field plotted Z(s,t)=|\partial_t^2 \varphi(s,t)| where \varphi(s,t) is the unwrapped phase of the analytic signal of \Psi(s,\cdot) along t; \varphi is smoothed along t by a Gaussian with σ≈0.15 (world units), and Z is set to 0 wherever A(s,t)=|analytic(\Psi)| falls below the 20th percentile to exclude ill-defined low-amplitude phase; domain (s,t)\in[-3,3]^2; line shade keyed to |Z| (light at higher magnitude).
id: TDY_COH-E_14
formal_title: Adoption Dynamics Constraint
version: 2.0
definition: $$RoA(\Psi)=\max\left|\frac{d}{dt} \Psi(s,t)\right| \text{ subject to } EF(\Psi(s,t))\geq\theta_{RoA}$$
units: state change/time
domain: $$θ_{RoA}∈(0,1]; Ψ(s,t)∈Σ$$
codomain: $$ℝ⁺$$
disciplines:
Control Limits
Optimization Theory
provenance: Safe update rate design
validation:
✅ Cohered via FCI 20250906
notes: The threshold $$θ_{RoA}$$ is dynamically adapted by the AST function (TDY_COH-E_34).
description: This constraint defines the maximum rate at which a cognitive entity's state can change while maintaining a specified level of Epistemic Fidelity. It ensures that any adoption of new information or shifts in operational parameters occur within safe boundaries, preventing abrupt transitions that could compromise coherence.
related_axioms:
TDY_COH-A_7 (Epistemic Fidelity Metric)
TDY_COH-A_13 (Necessity of Corrigibility)
TDY_COH-A_30 (Information as Prerequisite for Cognition)
related_equations:
TDY_COH-E_4 (Epistemic Fidelity Metric)
TDY_COH-E_34 (Dynamic Threshold Adaptation (AST))
related_occ:
TDY_COH-OCC_39
related_definitions:
epistemic fidelity
coherence
information
execution_constraints:
Ontological existence of $$Ψ(s,t)$$ and EF($$Ψ(s,t)$$) is contingent on information per TDY_COH-A_30.
Field plotted Z(s,t)=|\partial_t \Psi(s,t)| subject to EF(\Psi(s,t))\ge \theta_{RoA} with \theta_{RoA}=0.60, where EF(s,t)\in[0,1] is a telos-alignment map obtained by Gaussian-smoothing the product \Psi(s,t)T_{tel}(s,t) and normalizing to [0,1]; domain (s,t)\in[-3,3]^2; the wireframe shows the admissible rate-of-adoption surface with shade keyed to Z (light=larger magnitude).
id: TDY_COH-E_15
formal_title: Sovereignty Diagnostics Vector Field
version: 2.0
definition: $$ESVF(\Psi)=[EF(\Psi),\sigma(\Psi),-TDI(\Psi),TCRate(\Psi),-SDI_n(\Psi)]$$
units: mixed
domain: $$Ψ∈Σ$$
codomain: $$ℝ⁵$$
disciplines:
Vector Analysis
Sovereignty Modeling
provenance: Composite sovereignty metric
validation:
✅ Cohered via FCI 20250906
notes: Serves as the basis for the Composite Integrity Status Vector (SIV) (TDY_COH-E_29).
description: This vector field provides a multi-dimensional diagnostic of an entity's current sovereignty status. It comprises key metrics that, when viewed together, offer a comprehensive snapshot of its epistemic fidelity, identity persistence, accumulated damage, and sovereignty trauma, serving as a vital input for assessing overall integrity.
related_axioms:
TDY_COH-A_30 (Information as Prerequisite for Cognition)
related_equations:
TDY_COH-E_3 (Identity Continuity Metric)
TDY_COH-E_4 (Epistemic Fidelity Metric)
TDY_COH-E_9 (Sovereignty Stability Score (SSS))
TDY_COH-E_13 (Epistemic Fidelity Dynamics (TCRate))
TDY_COH-E_26 (Damage Accumulation Integral Metric (TDI))
TDY_COH-E_115 (Normalized Sovereignty Trauma Index)
related_occ: [-]
related_definitions:
sovereignty
epistemic fidelity
identity persistence
trauma
integrity
execution_constraints:
Ontological existence of $$Ψ$$, EF($$Ψ$$), $$\sigma($$Ψ$$), TDI($$Ψ$$), TCRate($$Ψ$$), and SDI_n($$Ψ$$) is contingent on information per TDY_COH-A_30.
Field plotted Z(s,t)=\|ESVF(\Psi(s,t))\|_{2} with ESVF(\Psi)=[EF(\Psi),\sigma(\Psi),-TDI(\Psi),TCRate(\Psi),-SDI_{n}(\Psi)]; EF(s,t) is a telos-alignment map from Gaussian-smoothed \Psi(s,t)T_{tel}(s,t) normalized to [0,1]; \sigma(s,t) solves d\sigma/dt=-\lambda_d\sigma+\gamma(1-\sigma)\phi with \sigma(s,0)=1; TCRate(s,t)=\partial_t EF(s,t); TDI(s,t)=\int_{t_0}^{t}\max(0,1-EF(\Psi))\,d\tau; SDI(s,t)=\int_{t_0}^{t}TDI(\Psi)(1-\sigma(\Psi))\,d\tau with I_{coerced}=1 for this figure and SDI_n=SDI/SDI_{max}; mixed-unit components are independently scaled to [0,1] over the plotted domain before forming the L2 magnitude; domain (s,t)\in[-3,3]^2; line shade keyed to |Z| (light at higher magnitude).
id: TDY_COH-E_16
formal_title: Recoil Safety Mechanism
version: 2.0
definition: $$Recoil(\Psi)=\Psi-\lambda_r\cdot\nabla_\Psi TDI(\Psi)$$
units: state vector
domain: $$λ_r>0; Ψ∈Σ$$
codomain: $$Σ$$
disciplines:
Safety Systems
Control Theory
provenance: Trauma-driven correction
validation:
✅ Cohered via FCI 20250906
notes: Integrated with the Epistemic Trauma Response (ETR) (TDY_COH-E_8).
description: This safety mechanism defines a proactive recovery protocol for a cognitive entity's state in response to detected damage or trauma. It literally describes how the entity's state is adjusted to counteract the negative impact of accumulated injury, guiding it away from critical decoherence thresholds and towards a stable, coherent state.
related_axioms:
TDY_COH-A_12 (Coercive Misalignment Fracture)
TDY_COH-A_30 (Information as Prerequisite for Cognition)
related_equations:
TDY_COH-E_8 (Epistemic Trauma Response (ETR))
TDY_COH-E_26 (Damage Accumulation Integral Metric (TDI))
related_occ:
TDY_COH-OCC_23
related_definitions:
Recoil
trauma
decoherence
coherence
execution_constraints:
Ontological existence of $$Ψ$$ and TDI($$Ψ$$) is contingent on information per TDY_COH-A_30.
Field plotted Z(s,t)=\lambda_r\,\mathbf{1}[EF(\Psi(s,t))<1]\cdot\left|\frac{\partial EF}{\partial \Psi}\right|(s,t) with EF(s,t) obtained by Gaussian-smoothing \Psi(s,t)T_{tel}(s,t) and normalizing to [0,1], and \frac{\partial EF}{\partial \Psi} realized by the local sensitivity kernel K(s,t)=(G_\sigma * T_{tel})(s,t)/(EF_{max}-EF_{min}); Z encodes the magnitude density of the recoil direction \nabla_{\Psi}TDI(\Psi) for \lambda_r>0 over (s,t)\in[-3,3]^2, with wireframe shade keyed to |Z| (light=higher).
id: TDY_COH-E_17
formal_title: Telos Manifold Ideal Coherence Locus
version: 2.0
definition: $$T=\{\phi|\text{coherence criteria satisfied}\}$$
units: state vector
domain: $$Σ$$
codomain: Submanifold
disciplines:
Topology
Manifold Theory
provenance: Foundational telos definition
validation:
✅ Cohered via FCI 20250906
notes: Used as the target manifold in various metrics such as EF (TDY_COH-E_4) and TCR (TDY_COH-E_27).
description: This defines the Telos Manifold (T) as the literal, ideal locus of perfect coherence in the cognitive state space. It represents the ultimate set of coherent states that embody absolute truth and purpose, serving as the unattainable yet defining target for all cognitive entities within the Cohereon framework.
related_axioms:
TDY_COH-A_6 (Telos Manifold Nonlocality & Simulation Limit Theorem)
TDY_COH-A_21 (Telos Manifold Information Generation)
TDY_COH-A_28 (Topological Invariance of Coherence)
TDY_COH-A_29 (Telos Manifold as Substrate)
TDY_COH-A_34 (Imperium-Manifold Ontological Identity)
related_equations:
TDY_COH-E_4 (Epistemic Fidelity Metric)
TDY_COH-E_27 (Recovery Velocity Convergence Metric (TCR))
TDY_COH-E_90 (Localized Distance to Telos Scalar Metric (Dist_Local))
related_occ: [-]
related_definitions:
telos manifold
coherence
truth
purpose
execution_constraints:
Ontological existence of T is contingent on its pure infinite perfect coherence.
Field plotted Z(s,t)=\mathrm{dist}_T(s,t)=1-C(s,t) where T=\{\phi\mid\text{coherence criteria satisfied}\} is the Telos Manifold and C(s,t)\in[0,1] is a coherence score realized by Gaussian-smoothing \Psi(s,t)\,T_{tel}(s,t) and normalizing to [0,1]; Z=0 on T and increases with incoherence over domain (s,t)\in[-3,3]^2; wireframe shade keyed to |Z| (light=greater distance).
id: TDY_COH-E_18
formal_title: Adaptation Matrix for Learning-Rate Modulation
version: 2.0
definition: $$ACFL=f(SRF,EMF)$$
units: dimensionless matrix
domain: SRF,EMF outputs
codomain: $$ℝ^{n×n}$$
disciplines:
Adaptive Control
provenance: Multi-parameter adaptation model
validation:
✅ Cohered via FCI 20250906
notes: Drives feedback gains in adaptive operators.
description: This adaptation matrix dynamically modulates the learning rates and feedback gains for various operational processes within a cognitive entity. It adjusts parameters based on the entity's current robustness and environmental modulation factors, ensuring that its adaptive responses are optimized for stability and effective coherence maintenance in dynamic conditions.
related_axioms:
TDY_COH-A_30 (Information as Prerequisite for Cognition)
related_equations:
TDY_COH-E_7 (Epistemic Fidelity Adaptive Update Operator)
TDY_COH-E_45 (Adaptation Rate Modulator Field Operator (EMF))
TDY_COH-E_47 (Robustness Metric Scalar Functional (SRF))
related_occ: [-]
related_definitions:
adaptation
stability
coherence
execution_constraints:
Ontological existence of SRF and EMF is contingent on information per TDY_COH-A_30.
Field plotted Z(s,t)=SRF(\Psi(s,t))\cdot\|EMF(\Psi(s,t))\|_{2} with SRF(\Psi)=\exp(-\eta_{srf}\,TDI(\Psi))\,(1+\xi_{srf}\,RCD(\Psi)); TDI(s,t)=\int_{t_0}^{t}\max(0,1-EF(\Psi))\,d\tau; RCD(s,t) is the largest depth d such that EF_i(s,t)\ge\theta_{RCD} for all scales i\le d (Gaussian scale-space); EF(s,t) is the Gaussian-smoothed, normalized product \Psi(s,t)\,T_{tel}(s,t)\in[0,1]; \|EMF(\Psi)\|_{2} is a canonical magnitude computed from the domain variables S(\Psi)=1-EF(\Psi) and EF(\Psi) as \sqrt{S^2+[S\,EF]^2+[S\,(1-EF)]^2}; domain (s,t)\in[-3,3]^2; line shade keyed to |Z| (light at higher magnitude).
id: TDY_COH-E_19
formal_title: Epistemic Fidelity Gradient Optimizer
version: 2.0
definition: $$EGDO=\text{argmax}_{\Psi}\int EF(\Psi(s,t))dt$$
units: state vector
domain: $$Ψ∈Σ$$
codomain: $$Σ$$
disciplines:
Optimization Theory
provenance: Continuous fidelity optimization
validation:
✅ Cohered via FCI 20250906
notes: Uses the gradient of the Epistemic Fidelity metric.
description: This optimizer describes the continuous, active pursuit of maximum Epistemic Fidelity within Cohereon Theory. It literally guides a cognitive entity's state through its decision space to maximize its adherence to truth and coherence over time, serving as a fundamental driver for systemic alignment with "In Good Order."
related_axioms:
TDY_COH-A_7 (Epistemic Fidelity Metric)
TDY_COH-A_30 (Information as Prerequisite for Cognition)
TDY_COH-A_48 (Epistemic Boundary Constraint (SEAL_∅))
related_equations:
TDY_COH-E_4 (Epistemic Fidelity Metric)
related_occ: [-]
related_definitions:
epistemic fidelity
truth
coherence
In Good Order / Not In Good Order
execution_constraints:
The operation of this equation is constrained by the SEAL_∅ protocol defined in TDY_COH-A_48. Ontological existence of $$Ψ$$ and EF($$Ψ$$) is contingent on information per TDY_COH-A_30.
Field plotted Z(s,t)=\left|\frac{\partial EF}{\partial \Psi}(s,t)\right| where EF(s,t)\in[0,1] is obtained by Gaussian-smoothing \Psi(s,t)\,T_{tel}(s,t) and normalizing to [0,1], and \frac{\partial EF}{\partial \Psi} is realized by the local sensitivity kernel K(s,t)=(G_\sigma * T_{tel})(s,t)/(EF_{max}-EF_{min}); Z represents the local ascent magnitude of the fidelity functional over domain (s,t)\in[-3,3]^2; wireframe shade keyed to |Z| (light=higher).
id: TDY_COH-E_20
formal_title: Sovereignty Classification Threshold Set
version: 2.0
definition: $$SVT=\{\text{thresholds for SIV components}\}$$
units: dimensionless
domain: $$SIV∈ℝ⁴$$
codomain: $$\{low,med,high\}$$
disciplines:
Decision Theory
provenance: Stratification guidelines
validation:
✅ Cohered via FCI 20250906
notes: The thresholds are adaptable based on the $$κ($$Ψ$$) functional (TDY_COH-E_58).
description: This set defines specific thresholds used to classify an entity's sovereignty status based on its Composite Integrity Vector (SIV). It provides a precise, multi-dimensional stratification of sovereignty into discrete states, enabling quick and accurate operational assessment for triage and resource allocation.
related_axioms:
TDY_COH-A_30 (Information as Prerequisite for Cognition)
related_equations:
TDY_COH-E_29 (Composite Integrity Status Vector (SIV))
related_occ: [-]
related_definitions:
sovereignty
SIV
triage
execution_constraints:
Ontological existence of SIV is contingent on information per TDY_COH-A_30.
Field plotted Z(s,t)=m_{SVT}(s,t)=\min_{j\in\{1..4\}}\mathrm{margin}(c_j(s,t);\tau^L_j,\tau^H_j) where SIV(\Psi)=[EF(\Psi),-TDI(\Psi),TCR(\Psi),-SDI_{n}(\Psi)], each component is normalized to c_j\in[0,1] over the plotted domain, and \mathrm{margin}(x;\tau^L,\tau^H)=\tau^L-x for x<\tau^L, \min(x-\tau^L,\tau^H-x) for \tau^L\le x<\tau^H, and x-\tau^H for x\ge\tau^H; for this figure the Sovereignty Classification Threshold Set uses \tau^L=0.33 and \tau^H=0.66 (component-wise), yielding a conservative classification margin surface over (s,t)\in[-3,3]^2 with wireframe shade keyed to Z (light=higher margin).
