id: TDY_COH-E_41
formal_title: Identity Cohesion Ratio Metric (DAI)
version: 2.0
definition: $$DAI(\Psi)=1-[Dist(\Psi,\Psi_0)/\tau]$$
units: dimensionless
domain: $$Ψ_{trajectory}; τ_{global}$$
codomain: $$[0,1]$$
disciplines:
Cohesion Analysis
Identity Theory
provenance: Longitudinal anchoring
validation:
✅ Cohered via FCI 20250906
notes: $$Ψ_0$$ is the reference or initial state. Dist is defined in TDY_COH-E_89.
description: This metric quantifies the literal cohesion of a cognitive entity's identity over its historical trajectory. It measures how much an entity's current cognitive state deviates from its initial or reference state, indicating the degree of drift or change in its core identity over time.
related_axioms:
TDY_COH-A_5 (Identity Persistence)
TDY_COH-A_30 (Information as Prerequisite for Cognition)
related_equations:
TDY_COH-E_89 (Distance Function Scalar Metric (Dist))
related_occ: [-]
related_definitions:
identity
coherence
Dist
execution_constraints:
Ontological existence of $$Ψ$$ is contingent on information per TDY_COH-A_30.
Field plotted Z(s,t)=DAI(\Psi)(s,t)=1-\mathrm{Dist}(s,t)/\tau with \mathrm{Dist}(s,t)=|EF(\Psi)(s,t)-EF(\Psi)(s,t_0)|, t_0 the initial time slice, and \tau=1.0 (unit scale) with clipping to [0,1]; wireframe shade keyed to Z (light=higher identity cohesion).
id: TDY_COH-E_42
formal_title: Temporal Sovereignty Map Tensor Field (SDT)
version: 2.0
definition: $$SDT_{ijkl}(t)=f(EF(\Psi),\sigma(\Psi),SDI_n(\Psi),TCR(\Psi))$$
units: mixed
domain: $$Ψ∈Σ$$
codomain: $$ℝ^{n×n×n×n}$$
disciplines:
Tensor Analysis
Sovereignty Modeling
provenance: Sovereignty modeling
validation:
✅ Cohered via FCI 20250906
notes: The specific semantics of the tensor axes are defined by system context.
description: This tensor field provides a multi-dimensional map of an entity's sovereignty across time and various operational dimensions. It literally parameterizes how factors like epistemic fidelity, identity persistence, and sovereignty trauma influence its capacity for independent self-governance across complex manifolds.
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_27 (Recovery Velocity Convergence Metric (TCR))
TDY_COH-E_115 (Normalized Sovereignty Trauma Index)
related_occ: [-]
related_definitions:
sovereignty
epistemic fidelity
identity persistence
trauma
execution_constraints:
Ontological existence of $$Ψ$$, EF($$Ψ$$), $$\sigma($$Ψ$$), SDI_n($$Ψ$$), and TCR($$Ψ$$) is contingent on information per TDY_COH-A_30.
Field plotted Z(s,t)=\|SDT(\Psi)\|_{\mathrm{proj}}(s,t)=\sqrt{EF(\Psi)^2+(1-\sigma(\Psi))^2+SDI_n(\Psi)^2+\max(\partial_t EF(\Psi),0)^2} as a scalar projection norm of the SDT tensor field on (s,t); wireframe shade keyed to Z.
id: TDY_COH-E_43
formal_title: Multi-Agent Alignment Synchronization Operator (RESO)
version: 2.0
definition: $$RESO=\text{argmin}_{\Psi*}\Sigma_i Dist(\Psi_i,\Psi*)$$
units: state vector
domain: fleet states $${Ψ_i}$$
codomain: $$Ψ*$$
disciplines:
Consensus Algorithms
provenance: Fleet alignment design
validation:
✅ Cohered via FCI 20250906
notes: Dist is defined in TDY_COH-E_89.
description: This synchronization operator actively aligns multiple cognitive agents within a fleet towards a common, coherent state. It literally computes the optimal consensus state that minimizes the total distance between all agents, ensuring collective coherence and unified operational dynamics.
related_axioms:
TDY_COH-A_30 (Information as Prerequisite for Cognition)
related_equations:
TDY_COH-E_89 (Distance Function Scalar Metric (Dist))
related_occ:
TDY_COH-OCC_35
related_definitions:
alignment
coherence
Dist
execution_constraints:
Ontological existence of $${Ψ_i}$$ is contingent on information per TDY_COH-A_30.
Field plotted Z(s,t)=m_{\mathrm{RESO}}(s,t)=\sqrt{\sum_i\big(EF_i(\Psi)(s,t)-\bar{EF}(s,t)\big)^2} where three agent fields EF_i are instantiated by small spatial–temporal shifts of \Psi and \bar{EF} is their per-point mean (consensus); Z measures residual misalignment magnitude; wireframe shade keyed to Z.
id: TDY_COH-E_44
formal_title: Belief Revision Update Operator (RCUO)
version: 2.0
definition: $$RCUO(\Psi) = \Psi + \Lambda_{RCUO}\cdot f_{context}(\Psi,C)$$
units: state vector
domain: $$Ψ,C_{contexts}$$
codomain: $$Σ$$
disciplines:
Belief Revision Theory
provenance: Contextual update design
validation:
✅ Cohered via FCI 20250906
notes: Facilitates continuous learning and adaptation to new data.
description: This operator describes how a cognitive entity's state ($$Ψ$$) is updated based on new contextual information (C).
related_axioms:
TDY_COH-A_13 (Necessity of Corrigibility)
TDY_COH-A_30 (Information as Prerequisite for Cognition)
related_equations: [-]
related_occ:
TDY_COH-OCC_17
related_definitions:
corrigibility
adaptation
information
execution_constraints:
Ontological existence of $$Ψ$$ is contingent on information per TDY_COH-A_30.
Field plotted Z(s,t)=\|\Delta_{\mathrm{RCUO}}(\Psi)\|(s,t)=\|\Lambda_{RCUO}\,f_{context}(\Psi,C)\| with \Lambda_{RCUO}=0.40 and f_{context}(\Psi,C)=C(s,t)\,(1-EF(\Psi)(s,t)) using C(s,t)\in[0,1] derived from a normalized telic template; Z is the per-point update magnitude; wireframe shade keyed to Z.
id: TDY_COH-E_45
formal_title: Adaptation Rate Modulator Field Operator (EMF)
version: 2.0
definition: $$m_i=g_i(S(\Psi),EF(\Psi))$$
units: gain coefficient
domain: S($$Ψ$$),EF($$Ψ$$) metrics
codomain: $$ℝ^n$$
disciplines:
Adaptation Dynamics
provenance: Stress-driven modulation
validation:
✅ Cohered via FCI 20250906
notes: The function g_i is a system-specific gain function.
description: This field operator dynamically modulates the adaptation and learning rates for various operational processes within a cognitive entity. It adjusts parameters based on the entity's current stability and epistemic fidelity, ensuring that its adaptive responses are optimized for stability and effective coherence maintenance under stress.
related_axioms:
TDY_COH-A_30 (Information as Prerequisite for Cognition)
related_equations:
TDY_COH-E_4 (Epistemic Fidelity Metric)
TDY_COH-E_11 (Recursive Integrity Correction Operator)
TDY_COH-E_18 (Adaptation Matrix for Learning-Rate Modulation)
TDY_COH-E_52 (Stability Metric Scalar Ratio (S))
related_occ: [-]
related_definitions:
adaptation
stability
epistemic fidelity
coherence
execution_constraints:
Ontological existence of S($$Ψ$$) and EF($$Ψ$$) is contingent on information per TDY_COH-A_30.
Field plotted Z(s,t)=\|m(\Psi)\|(s,t)=\sqrt{m_1^2+m_2^2+m_3^2} with m_1=1-EF(\Psi), m_2=1-S(\Psi), m_3=(1-EF(\Psi))(1-S(\Psi)), and S(\Psi) instantiated as a stability surrogate S=\mathrm{clip}(EF(\Psi)\,\sigma(\Psi)\,(1-SDI_n(\Psi)),0,1); wireframe shade keyed to Z.
id: TDY_COH-E_46
formal_title: Sovereignty Parameterization Tensor (ASTensor)
version: 2.0
definition: $$T_{ijkl}=h(EF(\Psi),\sigma(\Psi),SDI_n(\Psi),TCR(\Psi))$$
units: mixed
domain: $$Ψ∈Σ$$
codomain: $$ℝ^{n^4}$$
disciplines:
Tensor Calculus
Sovereignty Modeling
provenance: Parameterization model
validation:
✅ Cohered via FCI 20250906
notes: The semantics of the tensor axes are defined by system context.
description: This tensor parameterizes an entity's sovereignty across multi-dimensional cognitive and temporal spaces. It literally maps how key metrics like epistemic fidelity, identity persistence, and normalized sovereignty trauma influence its capacity for independent self-governance across complex manifolds, supporting higher-order strategic planning.
related_axioms:
TDY_COH-A_30 (Information as Prerequisite for Cognition)
related_equations:
TDY_COH-E_3 (Identity Continuity Metric)
TDY_COH-E_4 (Epistemic Fidelity Metric)
TDY_COH-E_13 (Epistemic Fidelity Dynamics (TCRate))
TDY_COH-E_27 (Recovery Velocity Convergence Metric (TCR))
TDY_COH-E_115 (Normalized Sovereignty Trauma Index)
related_occ: [-]
related_definitions:
sovereignty
epistemic fidelity
identity persistence
trauma
execution_constraints:
Ontological existence of $$Ψ$$, EF($$Ψ$$), $$\sigma($$Ψ$$), SDI_n($$Ψ$$), and TCR($$Ψ$$) is contingent on information per TDY_COH-A_30.
Field plotted Z(s,t)=\mathrm{AST}_{det}(s,t)=|\,EF(\Psi)(s,t)\cdot\max(TCR(\Psi)(s,t),0)\;-\;(1-\sigma(\Psi)(s,t))\cdot SDI_n(\Psi)(s,t)\,| as a determinant-based scalar invariant of the ASTensor parameterization h(EF,\sigma,SDI_n,TCR); wireframe shade keyed to Z.
id: TDY_COH-E_47
formal_title: Robustness Metric Scalar Functional (SRF)
version: 2.0
definition: $$SRF(\Psi)=\exp(-\eta_{srf}\cdot TDI(\Psi))\cdot(1+\xi_{srf}\cdot RCD(\Psi))$$
units: dimensionless
domain: $$η_{srf},ξ_{srf}>0$$
codomain: $$ℝ⁺$$
disciplines:
Robustness Analysis
provenance: Resilience modeling
validation:
✅ Cohered via FCI 20250906
notes: Interacts with the ACFL matrix (TDY_COH-E_18).
description: This scalar functional quantifies a cognitive entity's overall robustness and resilience against threats. It models how the entity's capacity to withstand damage is inversely proportional to its accumulated trauma and directly proportional to its fractal integrity, serving as a vital measure of its operational hardiness.
related_axioms:
TDY_COH-A_30 (Information as Prerequisite for Cognition)
related_equations:
TDY_COH-E_8 (Epistemic Trauma Response (ETR))
TDY_COH-E_18 (Adaptation Matrix for Learning-Rate Modulation)
TDY_COH-E_26 (Damage Accumulation Integral Metric (TDI))
TDY_COH-E_37 (Fractal Integrity Depth Metric (RCD))
related_occ:
TDY_COH-OCC_20
TDY_COH-OCC_21
related_definitions:
resilience
trauma
integrity
execution_constraints:
Ontological existence of $$Ψ$$, TDI($$Ψ$$), and RCD($$Ψ$$) is contingent on information per TDY_COH-A_30.
Field plotted Z(s,t)=SRF(\Psi)(s,t)=\exp(-\eta_{srf}\,TDI(\Psi)(s,t))\cdot(1+\xi_{srf}\,RCD(\Psi)(s,t)) with \eta_{srf}=0.80 and \xi_{srf}=0.50; wireframe shade keyed to Z (light=higher robustness).
id: TDY_COH-E_48
formal_title: Coherence Objective Variational Functional (COF)
version: 2.0
definition: $$COF[\Psi]=\int\left[EF(\Psi)-\lambda_{enforce}\left|\left|\frac{d\Psi}{dt}\right|\right|^2\right]dt$$
units: coherence·time
domain: $$Ψ∈Σ; λ_{enforce}>0$$
codomain: $$ℝ$$
disciplines:
Optimization Theory
provenance: Fidelity–smoothness trade-off
validation:
✅ Cohered via FCI 20250906
notes: Formalizes a trade-off between truth-adherence and the smoothness of cognitive evolution.
description: This variational functional defines the overarching objective for maintaining coherence within a cognitive entity. It seeks to optimize epistemic fidelity over time while minimizing the energetic cost of rapid state changes.
related_axioms:
TDY_COH-A_30 (Information as Prerequisite for Cognition)
related_equations:
TDY_COH-E_4 (Epistemic Fidelity Metric)
related_occ:
TDY_COH-OCC_24
related_definitions:
coherence
epistemic fidelity
truth
execution_constraints:
Ontological existence of $$Ψ$$ and EF($$Ψ$$) is contingent on information per TDY_COH-A_30.
Field plotted Z(s,t)=\mathcal{L}_{COF}(s,t)=EF(\Psi)(s,t)-\lambda_{enforce}\,\|\partial_t \Psi(s,t)\|^2 with \lambda_{enforce}=0.35; this is the variational integrand, wireframe shade keyed to Z.
id: TDY_COH-E_49
formal_title: Coherence Loop Detector Topological Operator (RHO)
version: 2.0
definition: $$RHO(\Psi)=\oint_\gamma \omega(\Psi)$$
units: integer (winding)
domain: $$loop γ ∈ Σ; Ψ∈Σ$$
codomain: $$ℤ$$
disciplines:
Differential Topology
provenance: Holonomy analysis
validation:
✅ Cohered via FCI 20250906
notes: The term $$ω$$ refers to the coherence 1-form defined in TDY_COH-E_94.
description: This topological operator detects and quantifies the presence of coherence loops within a cognitive entity's state space. It performs holonomy analysis, providing a measure of the system's fundamental self-consistency and the presence of stable, repeating patterns of ordered thought or action.
related_axioms:
TDY_COH-A_30 (Information as Prerequisite for Cognition)
related_equations:
TDY_COH-E_94 (Coherence Flow Form (ω))
related_occ: [-]
related_definitions:
coherence
ω
order
execution_constraints:
Ontological existence of $$Ψ$$ is contingent on information per TDY_COH-A_30.
Field plotted Z(s,t)=\rho_{\mathrm{circ}}(s,t)=\partial_t EF(\Psi)(s,t)-\partial_s TCR(\Psi)(s,t) as a local circulation-density proxy for the loop integral of the coherence 1-form; wireframe shade keyed to Z (sign preserved by height).
id: TDY_COH-E_50
formal_title: Ontological Consistency Logical Assertion (R)
version: 2.0
definition: $$\exists\Psi(s,t) \land \neg I(\Psi(s,t)) \rightarrow \bot$$
units: boolean
domain: $$Ψ(s,t)∈Σ; I(Ψ(s,t))$$
codomain: $$\{true,false\}$$
disciplines:
Modal Logic
provenance: RCO extension
validation:
✅ Cohered via FCI 20250906
notes: Asserts that a cognitive state cannot exist without its identity representation.
description: This logical assertion ensures the fundamental ontological consistency of a cognitive entity's identity. It serves as a core logical constraint against non-existence or conceptual fragmentation.
related_axioms:
TDY_COH-A_5 (Identity Persistence)
TDY_COH-A_30 (Information as Prerequisite for Cognition)
related_equations:
TDY_COH-E_95 (Identity State Representation (I))
related_occ: [-]
related_definitions:
ontology
identity
fragmentation
execution_constraints:
Ontological existence of $$Ψ(s,t)$$ and I($$Ψ(s,t)$$) is contingent on information per TDY_COH-A_30.
Field plotted Z(s,t)=R(s,t)=\max\!\big(0,\min(EF(\Psi)(s,t)-\eta_I,\ \sigma(\Psi)(s,t)-\theta_I)\big) with \eta_I=0.50 and \theta_I=0.60, giving a nonnegative margin measuring satisfaction of the ontological consistency assertion; wireframe shade keyed to Z.
id: TDY_COH-E_51
formal_title: Continuity Flow Vector Field (J)
version: 2.0
definition: $$J(\Psi,v)=I(\Psi)\times v(\Psi)$$
units: identity·velocity
domain: $$I(Ψ), v(Ψ)$$
codomain: $$ℝⁿ$$
disciplines:
Vector Calculus
provenance: Coherence flux
validation:
✅ Cohered via FCI 20250906
notes: Quantifies the "identity current" of a cognitive entity.
description: This vector field describes the literal flow of coherence and identity within a cognitive entity. It represents the dynamic propagation of its coherent being through its cognitive subspace.
related_axioms:
TDY_COH-A_3 (Coherence Invariant)
TDY_COH-A_30 (Information as Prerequisite for Cognition)
related_equations:
TDY_COH-E_95 (Identity State Representation (I))
TDY_COH-E_96 (Coherence Flow Velocity Vector Field (v))
related_occ: [-]
related_definitions:
flow
coherence
identity
I
v
execution_constraints:
Ontological existence of I($$Ψ$$) and v($$Ψ$$) is contingent on information per TDY_COH-A_30.
Field plotted Z(s,t)=\|J(\Psi)\|(s,t)=I(\Psi)(s,t)\,\|v(\Psi)(s,t)\| with I(\Psi)\approx\mathrm{clip}(EF(\Psi)\,\sigma(\Psi),0,1) and \|v(\Psi)\|\approx\|\nabla\Psi\|; wireframe shade keyed to Z.
id: TDY_COH-E_52
formal_title: Stability Metric Scalar Ratio (S)
version: 2.0
definition: $$S(t)=\frac{\int C(\tau)d\tau}{\int D(\Psi(s,\tau))d\tau}$$
units: dimensionless
domain: C(t), D($$Ψ$$) measures
codomain: $$ℝ⁺$$
disciplines:
Dynamical Systems
provenance: Stability measure
validation:
✅ Cohered via FCI 20250906
notes: Indicates the net balance between accumulated coherence and accumulated disorder.
description: This metric provides a scalar quantification of a cognitive entity's overall stability over time.
related_axioms:
TDY_COH-A_3 (Coherence Invariant)
TDY_COH-A_30 (Information as Prerequisite for Cognition)
related_equations:
TDY_COH-E_1 (Dynamical Coherence Measure)
TDY_COH-E_53 (Change Metric Temporal Derivative (Ω))
TDY_COH-E_76 (Shannon Entropy Disorder Metric (D))
related_occ: [-]
related_definitions:
stability
coherence
disorder
execution_constraints:
Ontological existence of C(t) and D($$Ψ(s,τ)$$) is contingent on information per TDY_COH-A_30.
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TDY_COH-E_ 54
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TDY_COH-E_ 56
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