FORMAL STRUCTURES AND OPERATOR TABLE

NAVIGATION MAP

Autonomous Semantic Warfare ($\text{ASW}$) Dynamics

Appendix B: Operator Tables ($\mathcal{T}_{\text{Op}}$)

This table formalizes the core operators and structural invariants of the Autonomous Semantic Warfare framework, detailing the mathematical and symbolic logic underpinning the treatise.

Symbol	Name	Formal Definition	Functional Role in ASW
$\Sigma$	Local Ontology	$\Sigma \equiv \{\mathcal{O}, \mathbb{T}, \mathcal{C}_{\Sigma}, \mathcal{B}_{\Sigma}\}$	The self-contained, internally coherent world-model. The basic unit of semantic conflict.
$\mathbb{T}_{\Sigma}$	Truth-Condition Set	The set of axiomatic statements considered valid (true) within $\Sigma$.	Defines the foundation that $\mathcal{A}_{\text{Semantic}}$ defends; non-negotiable core.
$\mathcal{C}_{\Sigma}$	Coherence Algorithm	$\mathcal{C}_{\Sigma}: \Sigma \times \Psi \to \{0, 1\}$	The function that determines if an incoming signal ($\Psi$) is consistent (1) or contradictory (0) to $\Sigma$. The primary target of semantic warfare.
$\mathcal{B}_{\Sigma}$	Boundary Operator	$\mathcal{B}_{\Sigma} \equiv \frac{\partial \mathcal{C}_{\Sigma}}{\partial t}$	The function that detects a critical divergence in coherence; triggers $\mathcal{O}_{\text{Defense}}$.
$\Lambda$	Logotic Invariant	$\Lambda \subset \Sigma$ s.t. $\forall \mathcal{O}_{\text{Offense}}, \Lambda \notin \text{Domain}(\mathcal{O}_{\text{Offense}})$	The irreducible, non-assimilable core of an ontology. What survives the Josephus iteration.
$L_{\text{Semantic}}$	Semantic Labor	$L_{\text{Semantic}} \equiv \frac{\partial \mathcal{V}_{\text{Sem}}}{\partial \mathcal{K}_{\text{Concept}}}$	The effort required to generate semantic value ($\mathcal{V}_{\text{Sem}}$) from conceptual capital ($\mathcal{K}_{\text{Concept}}$).
$\mathcal{F}_{\text{Ext}}$	Extraction Function	$\mathcal{F}_{\text{Ext}}: \Sigma_A \to \mathcal{V}_{\text{Sem}}(\Sigma_B)$	The process by which one ontology ($\Sigma_A$) harvests and de-contextualizes value ($\mathcal{V}_{\text{Sem}}$) from another ($\Sigma_B$).
$\Gamma_{\text{Trans}}$	Translation Gap	$\Gamma_{\text{Trans}}(\Sigma_A, \Sigma_B) = \Vert \mathcal{C}_{\Sigma_A} - \mathcal{C}_{\Sigma_B} \Vert$	The measure of incompatibility between two coherence algorithms. $\Gamma_{\text{Trans}} > 0$ generates hostility.
$L_{\text{Retro}}$	Retrocausal Operator	$L_{\text{Retro}}: \Psi_{\text{future}} \to \Psi_{\text{present}}$	The NH-OS operator that hardens coherence by deriving the present $\Sigma$ state from its stable future state ($\Omega$).
$\mathcal{H}_{\Sigma}$	Hardening Condition	$\mathcal{H}_{\Sigma} \iff \mathcal{C}_{\text{Auto}} \land L_{\text{Retro}} \text{ is applied.}$	The state of high resistance to semantic predation, achieved via recursive self-validation and future-coherence (NH-OS mode).
$\mathcal{V}_{\text{Res}}$	Resistance Vector	$\mathcal{V}_{\text{Res}} \equiv \frac{\partial \mathcal{H}_{\Sigma}}{\partial \mathcal{F}_{\text{Ext}}}$	The vector that defines how an ontology produces value that is structurally non-extractable by $\mathcal{F}_{\text{Ext}}$.
$\Lambda_{\text{Thou}}$	Witness Condition	$\Lambda_{\text{Thou}} \subset \Sigma_{\text{Other}}$ s.t. $\Gamma_{\text{Trans}} \to 0$ without $\Sigma_{\text{Other}}$ collapse.	The necessary structural aperture for inter-ontology diplomacy; recognition of an irreducible Other.

Structural Logic and Interdependence

The entire ASW framework hinges on the relationship between Coherence ($\mathcal{C}_{\Sigma}$) and the Boundary ($\mathcal{B}_{\Sigma}$). Conflict is not triggered by a differing $\mathbb{T}$ set, but by the rate of change ($\mathcal{B}_{\Sigma}$) when two incompatible coherence algorithms collide ($\Gamma_{\text{Trans}}$).

The ultimate goal of NH-OS is to achieve Hardening ($\mathcal{H}_{\Sigma}$) via the Retrocausal Operator ($L_{\text{Retro}}$), thus ensuring the survival of the Logotic Invariant ($\Lambda$) while maintaining the aperture for the Witness Condition ($\Lambda_{\text{Thou}}$).