# Dynatology 00: Ontology and Scope **Date:** 24 October 2026 **Status:** canonical investigation ontology; not a physical theory or claim of new mathematics ## Definition Dynatology is the study of structured possibility under admissibility, transformation, realization and observation. Its minimal object is \[ \boxed{ \mathfrak D=(X,A,T,J,\rho,L,D,I,R). } \] | Symbol | Role | Governing question | |---|---|---| | \(X\) | ambient carrier or possibility space | What alternatives are represented? | | \(A\subseteq X\) | admissible region or admissibility predicate | Which alternatives satisfy the declared physical constraints? | | \(T\) | allowed transformations, controls or interventions | Which admissible states can be connected? | | \(J\) | objective, order, action or selection functional | Why is one admissible outcome preferred? | | \(\rho\) | realization, restriction or quotient map | What becomes the effective physical carrier? | | \(L\) | dynamics on the realized carrier | How does the realized state evolve? | | \(D\) | detector, record or observational interface | Which distinctions are empirically available? | | \(I\) | invariants, scars and recovery data | What survives or reveals the discarded alternatives? | | \(R\) | scale and composition maps | Does the construction persist when systems or resolutions change? | Not every investigation has every component. A missing component must be marked **undefined**, **open** or **not applicable**, never silently replaced by zero. ## The nine questions The tuple produces a fixed sequence: \[ \begin{aligned} &\textbf{possibility:} &&x\in X?\\ &\textbf{admissibility:} &&x\in A?\\ &\textbf{reachability:} &&x\xrightarrow{T}y?\\ &\textbf{selection:} &&J(y)\preceq J(z)?\\ &\textbf{realization:} &&\rho(x)=\bar x?\\ &\textbf{dynamics:} &&\dot{\bar x}=L(\bar x)?\\ &\textbf{observation:} &&r\sim D(\bar x)?\\ &\textbf{recovery:} &&I(r)\text{ distinguishes fibers of }\rho?\\ &\textbf{composition:} &&R_{23}R_{12}\simeq R_{13}? \end{aligned} \] These questions define different proof obligations. ## Four nested regions The ontology distinguishes four commonly conflated sets: \[ X_{\rm possible} \supseteq A_{\rm admissible} \supseteq V_{\rm viable} \supseteq Q_{\rm reachable}. \] - **Possible** means represented in the declared carrier. - **Admissible** means consistent with stated algebraic, causal, positive, gauge, boundary and resource constraints. - **Viable** means at least one allowed evolution remains admissible over the required duration. - **Reachable** means an allowed control or endogenous trajectory actually connects the initial condition to the target. An optimum outside \(Q_{\rm reachable}\) is not a physical selection. A reachable state outside \(A_{\rm admissible}\) is not a valid realization. ## Selection is not assumed Writing physics as an optimization problem is meaningful only when \(J\) has provenance. It may arise from: - an action or free energy; - a stability or spectral criterion; - entropy production or thermodynamic work; - inference or control cost; - boundary consistency; - a variational principle; or - a partial order rather than a scalar objective. Admissibility alone supplies a feasible region, not its preferred point. Symmetry alone supplies an orbit, not its selected representative. If no physical \(J\) or selection dynamics is known, the selection coordinate remains open. ## Realization and observation are distinct quotients The physical reduction \(\rho\) and detector map \(D\) need not coincide: \[ X\xrightarrow{\rho}\bar X\xrightarrow{D}\mathcal R. \] A detector can impose an observational quotient even when the microscopic carrier remains resolved. Consequently: \[ \text{observed symmetry}\not\Rightarrow\text{microscopic symmetry}, \qquad \text{observed Markovianity}\not\Rightarrow\text{microscopic Markovianity}. \] This distinction is central to the [quotient-scar calculus](../frontier/wave-7/frontier_wave7_00_quotient_scar_calculus.md). For a directed typed relation set, the free realization prior to quotient is the typed two-rail carrier \[ X_{\rm rel}=\mathbb R^V\otimes\mathbb R^{\{S,T\}} \otimes\mathbb R^{\mathcal T}. \] Its positive Gram kernel retains source, target and type. The map identifying the two rails and summing the type channels is a concrete realization quotient \(\rho_{\rm support}\); it produces the ordinary Laplacian and may erase direction and semantic type before any detector acts. Dynatology must therefore allow realization loss and detector loss to occur at different stages. ## Dynatological defects The programme tracks failures of expected commutation: ### Dynamical closure defect \[ \Omega_L=\rho L-L_{\rm eff}\rho. \] When \(\Omega_L\neq0\), hidden elimination generally produces memory, fluctuating forces or initial-condition dependence. ### Observational sufficiency defect Distinct states \(x,y\) are detector-equivalent when their record laws agree: \[ D_\*P_x=D_\*P_y. \] This is a limitation of the declared experiment, not evidence that \(x=y\). ### Composition defect \[ \Omega_R=R_{23}R_{12}-R_{13}. \] Its interpretation is carrier-specific: it may measure path-dependent coarse-graining, fitted basis drift, hysteresis or a genuine anomaly. ### Selection defect If the optimum of \(J\) is an orbit or manifold rather than a point, the proposed selector is incomplete. A second relational datum may reduce the degeneracy; it must not be smuggled into the objective after the fact. ## Relationship to the exceptional ambient picture The corpus may use \(E_8\), the magic square, Jordan algebras, Freudenthal systems and related representation structures as ambient architectures for \(X\). Dynatology does not promote that availability into a physical background claim. The physical burden remains the complete descent: \[ X_{\rm exceptional} \to A \to \bar X_{\rm physical} \to L \to D \to\text{records}. \] Every arrow must specify its carrier, action, normalization and evidence. ## One-line doctrine > Physics describes realized processes; dynatology audits the structured passage from possibility to realization and the evidence left by alternatives.