Formal Dictionary of Syntropia v0.4.18

Preliminary Note — The \mathcal{D} Collision

The symbol \mathcal{D} is already assigned in the Ontological Core to the unveiling operator:

\mathcal{D}: \mathcal{O}(t) \to P(\xi_t \mid \mathcal{O}(t))

So it cannot be reused to symbolize “Domain”: that would collide directly. We propose \mathscr{D} (mathematical script, distinct from calligraphic) for the generic concept of Domain when it needs to be named as a type of entity. The five specific domains keep their current symbols: V_t, R_t, P_t, A_t, B_t.

Part I — The Fundamental Objects

D1. Personal Trajectory

Ontological definition. The personal trajectory is the model’s primitive object. It is not an attribute of the person. It is the person, in their processual dimension: a singular, unrepeatable becoming that unfolds in time and produces successive functional organizations.

Formal definition. A personal trajectory is a curve \gamma: \mathbb{R}_{\geq 0} \to \mathcal{W} on the manifold of configurations, continuous and differentiable almost everywhere, satisfying existence conditions CE2 and CE4:

• CE2 (non-ergodicity): the trajectory cannot reproduce its exact history; the time average does not equal the ensemble average. Two trajectories passing through the same observable state at different points in their history are ontologically distinct.
• CE4 (history dependence): the present state is a function of the entire accumulated history, not just the immediately prior state. The trajectory carries constitutive memory that no finite time window can reduce.

And it operates under two epistemological conditions that bound how knowable it is:

• CE17 (double constitutive inaccessibility): the trajectory’s process is continuous; the observational apparatus is discrete. T_2 captures the process’s trace, not the process itself. A constitutive limit of any practical process science, not a defect of the model.
• CE18 (basis for comparability): comparability between trajectories works over the space of structural parameters (basin type, \Psi_t direction, position in S_m, the form of f_t), not over trajectory content. Compatible with A10 (radical singularity).

Critical distinction. The trajectory does not have a history. It is its history. D_p(t) does not describe something that happens to the trajectory; it describes the trajectory’s current constitution as the outcome of its own history of deformation.

Symbol: the trajectory is denoted implicitly through temporal subscripts. When the object needs to be named explicitly: \gamma^{(p)}(t) for person p’s trajectory at time t.

D2. Configuration

Ontological definition. A configuration is a functionally coherent organization of the individuation field: a mode of producing presence-in-the-world that the field sustains stably. It is a dynamic attractor, not a passing state: the field tends to stay in it, or return to it after moderate perturbation.

Formal definition. A canonical configuration \mathfrak{C}_k is a local minimum of \Phi(\cdot, t) in \mathcal{W} with a non-empty basin of attraction:

\mathfrak{C}_k \equiv w_k^* = \arg\min_{w \in \mathcal{B}_k} \Phi(w,t)

Canonical system v1.2 identifies 14 candidate configurations (C-A through C-M, with C-C split into C-C1 and C-C2), organized by dominant propagation pattern in \Psi_t, basin type, and, selectively, domain of origin. The full formalization is in Configuraciones_Canonicas_v1_3_14.qmd. The 14 configurations of the historical system (\mathfrak{C}_1, \ldots, \mathfrak{C}_{14}, identified with H_t \in [0,1]^4) have historical status: they are a record of the model’s conceptual development and part of its genealogy.

Distinction from state. At the mathematical layer, “state” names the instantaneous position H_t. At the ontological layer, “configuration” names the attractor that organizes that position. A trajectory can be in \mathfrak{C}_k’s basin without having reached the minimum w_k^*.

D3. Basin

Ontological definition. A configuration’s basin is the region of possibility space from which the field, absent clinical perturbation, converges toward that configuration. It is an attractor’s “gravitational field.”

Formal definition. The basin of \mathfrak{C}_k is:

\mathcal{B}_k(t) = \{w \in \mathcal{W} : \lim_{\tau \to \infty} \gamma_{w,t}(\tau) = w_k^*\}

where \gamma_{w,t} is the gradient-descent trajectory of \Phi(\cdot,t) starting at w.

Depth. Depth \delta_k(t) is the difference between \Phi’s minimum value at the boundary \partial\mathcal{B}_k and its value at the bottom, \Phi(w_k^*, t). D_p(t) accumulated in \mathfrak{C}_k’s direction contributes directly to \delta_k(t).

Clinical consequence. The basin determines intervention indication before the H_t profile does: two trajectories at the same observable position can require opposite interventions if they belong to different basins.

Disambiguation note: “basin” vs. “basin type” (E.21). This term has two senses in the corpus that should not be conflated. Basin (D3, here) is the continuous object defined above: a region of \mathcal{W} with extent, depth, and gradient, all computable from \Phi(\cdot,t). Basin type (D4*, Ontological Core, spatium Axis 2) is a discrete classificatory label with five values (transitory, reconfiguration, residency, early dispositional, failed reconfiguration) that describes the regime under which a configuration got installed, not its geometry. The formal relationship between the two, whether “basin type” can be derived as a function of basin’s continuous properties (D3), is explicitly pending formalization (G15, Ontological Core). Treating “basin type” as though it inherited D3’s metric properties before that function exists is the exact error that contaminated §III.12bis of the Mathematical Core (June 2026). See the equivalent note in D4* of the Ontological Core.

D4. Domain — \mathscr{D}

Ontological definition. A domain is a transcendental dimension of the trajectory’s functional production: a direction in which the individuation field unfolds presence-in-the-world, irreducible to the others. “Transcendental” in the formal sense: every domain is present in every personal trajectory, varying only in its level of coherence.

Formal definition. A domain \mathscr{D}_i is a coordinate of H_t \in [0,1]^5 describing the trajectory’s functional coherence in that direction. The five domains:

H_t = (V_t, R_t, P_t, A_t, B_t)

where \mathscr{D}_V = V_t (volition), \mathscr{D}_R = R_t (Relational Bonds), \mathscr{D}_P = P_t (temporal projection), \mathscr{D}_A = A_t (existential anchoring), and \mathscr{D}_B = B_t (somatic domain). B_t’s clinical formalization is in Clinical Core §2.5, and its plain-language description is in Mathematical Core §0 (item 0.6).

Constitutive independence. The domains are ontologically independent: the coherence level in \mathscr{D}_V neither determines nor is determined by the level in \mathscr{D}_R. That independence is not statistical. It is ontological. A scalar averaging the domains would erase the clinical difference between two trajectories with identical \|H_t\| and radically different profiles.

Distinction from transition conditions. Domains describe what the trajectory produces. Transition conditions describe the conditions under which it can sustain or change that production. These are different ontological planes: a domain is an observable of the field; a transition condition is a condition on the evolution operator.

Note on the symbol. \mathscr{D} (mathematical script) names the generic concept of domain as a type of entity. It should not be confused with \mathcal{D} (calligraphic), the Unveiling operator.

D5. Propagation — \Psi_t

Ontological definition. Propagation is the transfer of effects between domains of the individuation field: a perturbation in one domain produces change in another, with its own intensity, direction, and delay. It is the property that keeps the field from being the mere sum of its domains.

Formal definition. \Psi_t \in \mathbb{R}^{5 \times 5} is the operative representation of the Riemannian metric g_t, projected onto the five domains (including the somatic domain B_t). In the canonical formulation, \psi_{ij} \in \{-1, 0, +1\}; in the first-phase approximation, \psi_{ij} \geq 0. The coefficient \psi_{ij}(t) is the intensity of propagation from domain j into domain i at time t.

Constitutive properties:

Asymmetry: \psi_{ij}(t) \neq \psi_{ji}(t) in general. Propagation from the relational domain to the existential one does not run at the same intensity as the reverse. This asymmetry is clinical information, not a representational flaw.

History dependence: \Psi_t evolves with \xi_t. Trajectories with high D_p(t) tend toward more rigid propagation: coupling patterns become more resistant to change.

Positive vs. negative propagation: \psi_{ij} > 0 means a rise in domain j produces a rise in domain i. Values near 0 mean decoupling. Negative propagation (\psi_{ij} < 0) happens when a rise in j produces a drop in i: clinically, when recovery in one domain comes at the expense of another.

Gap G3. \Psi_t’s functional dependence on \xi_t is not specified, and the weights w_s, w_a that weigh the observation sources’ contributions need empirical estimation. This requires longitudinal pilot data.

D6. Perturbation

Ontological definition. A perturbation is an action on the individuation field that shifts the trajectory from its current position in \mathcal{W}. It can be clinical (deliberate intervention), vital (an unplanned event), or somatic (a change in biological state).

Formal definition. A perturbation of magnitude \varepsilon in direction \hat{u} \in T_w \mathcal{W} shifts the trajectory from w to w + \varepsilon \hat{u}.

Types by outcome:

Elastic perturbation: the field returns to its prior minimum. No change in \Phi(\cdot, t): the basin is preserved. \varepsilon < \varepsilon_t^{(i)}.

Expansive plastic perturbation: the field reorganizes toward a configuration that widens \mathcal{F}_t^{(p)}. It changes \Phi(\cdot, t) in the direction of lower depth. Requires sufficient \kappa(t) and \varepsilon greater than the relevant barrier B_{ij}.

Restrictive plastic perturbation: the field reorganizes, but toward a configuration that narrows \mathcal{F}_t^{(p)}, or toward an unplanned basin. Happens when \kappa(t) < \kappa_{\min} (P12).

Direct consequence of P12. Whether a clinical perturbation is indicated depends on \kappa(t), not just on the perturbation’s magnitude. A clinically appropriate perturbation delivered to a field with low \kappa(t) produces a restrictive plastic perturbation.

D7. Plasticity, Receptivity, and Clinical Energy

The model distinguishes three properties of the field that colloquial clinical language collapses under the single term “plasticity.” They differ in kind, not in degree.

D7a. Receptivity — \kappa(t). Whether the field can absorb a perturbation without disorganizing. See D10 for the complete formal definition.

D7b. Plasticity. Whether the field can reorganize toward a configuration different from its current one, given an absorbed perturbation. It is the condition of possibility for reorganization; it does not reduce to \kappa(t).

Ontological definition. Plasticity is the individuation field’s capacity to reorganize toward configurations other than the current one under perturbation. It is not a fixed trait. It is a property of the field’s current state, variable over time.

Clinical distinction between receptivity and plasticity. A field can have high receptivity with low plasticity: it absorbs the perturbation without disorganizing, but the functional \Phi has no more-coherent configurations accessible within \mathcal{F}_t^{(p)}. And a field can have low receptivity with high potential plasticity: more-coherent configurations are accessible, but the field cannot absorb the perturbation that would get it there. The most common clinical error is confusing the two: intervening as though potential plasticity were available when \kappa(t) is not.

D7c. Clinical energy — E_c(t). The trajectory’s capacity to leave its current configuration and reach configurations of greater functional coherence (Mathematical Core §III.9):

E_c(t) = \Phi(w_{\max}, t) - \Phi(w_t^*, t)

where w_t^* = \arg\min_{w \in \mathcal{F}_t^{(p)}} \Phi(w,t) is the current position and w_{\max} is the point of highest \Phi at \mathcal{F}_t^{(p)}’s boundary.

E_c(t) = 0 indicates a collapsed field of possibilities: no reorganization potential is accessible, regardless of \kappa(t). High E_c(t) indicates a field with surmountable barriers toward more coherent configurations.

The tripartite distinction as a clinical map:

A field can have high \kappa(t), high plasticity, and E_c(t) = 0: it can absorb and reorganize, but there is no more-coherent configuration anywhere in \mathcal{F}_t^{(p)}. That field will not improve through intervention aimed at reorganization: it needs prior work on \Omega_t^{(p)}’s geometry (on D_p(t)) before E_c(t) > 0.

D8. Rigidity

Ontological definition. Rigidity is the individuation field’s resistance to leaving its current configuration, regardless of whether that configuration is high- or low-coherence. It is plasticity’s mirror property: a field rigid in an actively reconfiguring configuration (e.g., C-C1 or C-C2) and a field rigid in a chronic residency configuration (e.g., C-A or C-B) are equally rigid; the difference is in which configuration they hold, and in that rigidity’s clinical consequences.

Formal definition. Rigidity in domain i’s direction is \varepsilon_t^{(i)}: the minimum perturbation magnitude in that direction needed to produce permanent reorganization. Global rigidity: \|\varepsilon_t\| = \|\varepsilon_t\|_2.

Distinction from basin depth. Depth \delta_k(t) describes how much total energy it takes to leave the basin. Rigidity \varepsilon_t^{(i)} describes how much perturbation, per domain, produces permanent displacement from the current minimum. These properties complement each other: a field can have a deep basin (hard to leave) but low rigidity in one specific domain (that domain responds easily to perturbation).

D9. Disorganization

Ontological definition. Disorganization is the loss of global functional coherence with no transition to a new stable configuration: the trajectory leaves its current minimum without converging on another. Clinically, this corresponds to an acute crisis state with no new organization emerging.

Formal definition. A trajectory is disorganized at t if:

\Phi(w_t, t) > \min_k \Phi(w_k^*, t) + \delta_{\min}

and there is simultaneously no convergence toward any minimum: w_t \notin \bigcup_k \mathcal{B}_k(t).

Distinction from transition. In a transition, the trajectory leaves one basin and converges on another. In disorganization, the trajectory leaves its basin with no access to any alternative basin, or the perturbation is so large the field has no time to converge.

D10. Receptive Capacity — \kappa(t)

Ontological definition. Receptive capacity is the individuation field’s current disposition to absorb perturbation without disorganizing. It is not the disposition to change (plasticity). It is the disposition to let itself be affected without collapsing.

Formal definition. \kappa(t) \in [0,1] is the inverse curvature of \Phi at the current minimum:

\kappa(t) = \left.\left(\frac{\partial^2 \Phi}{\partial w^2}\right)^{-1} \right|_{w = w_t^*}

High \kappa(t): the current minimum’s basin is flat. The field can absorb perturbation without shifting significantly. Low \kappa(t): the basin is narrow. Small perturbations produce displacement or disorganization.

Physiological correlate. \kappa(t) correlates directly with heart rate variability (HRV-RMSSD) and with DFA α1. This correlation is the basis for using T_\text{temporal} as an estimator of \kappa(t) in ambulatory settings, and T_\text{hospitalario} as an alternative estimator in hospital settings (from the adaptive frequency of structured clinical observations) when the sensor is unavailable. Both operators produce comparable estimates of \kappa(t); their divergence is itself diagnostic information about the field.

D11. Accumulated History — D_p(t)

Ontological definition. Accumulated history is the inscription, in the individuation field, of the plastic deformation produced by past perturbation. It is not the declarative memory of events; it is the individuation field’s current geometry as the result of everything that has deformed it.

Formal definition. D_p(t) \in [0,1] is the total accumulated variation of \Phi across history:

D_p(t) = \int_{t_0}^t \left|\frac{d\Phi(w_s^*, s)}{ds}\right| ds

Constitutive irreversibility. D_p(t) is monotonically non-decreasing over time: accumulated history does not spontaneously shrink. It can only be modified by sustained, high-magnitude expansive plastic perturbation, and only if \kappa(t) is sufficient for that perturbation to widen \mathcal{F}_t^{(p)}.

Independence from \kappa(t). D_p(t) and \kappa(t) are mathematically independent (Mathematical Core §II bis). This independence underlies P12 and the four regions of S_m.

D12. Transition

Ontological definition. A transition is the trajectory’s passage from one configuration’s basin into another’s. It is not instantaneous. It is a process that can take weeks or months, crossing the barrier B_{ij} between the two basins.

Formal definition. A transition from \mathfrak{C}_i to \mathfrak{C}_j occurs when the trajectory leaves \mathcal{B}_i(t) and converges on \mathcal{B}_j(t). The transition’s minimal path is the clinical geodesic from w_i^* to w_j^*.

Types of transition:

Syntropizing transition: toward a configuration with higher \|H_t\|.

Dystropizing transition: toward a configuration with lower \|H_t\|.

Horizontal transition: between configurations with similar \|H_t\| but different domain profiles or propagation direction. Example: from C-C1 (psych→B_t, reconfiguration, origin R_t) to C-C2 (psych→B_t, reconfiguration, origin A_t) when the reconfiguration process shifts origin domain without changing overall coherence level; or from C-A (psych→B_t, residency) to C-B (bidirectional, residency) when a cyclical somatic component gets added to a chronic psychological configuration.

Hysteresis. B_{ij}(t) \neq B_{ji}(t) means a transition from i to j does not require the same energy as the reverse transition. Hysteresis is the formal expression of the clinical experience that “relapsing is easier than improving,” and also that “some gains are irreversible.”

D13. Observability — \Upsilon_{US}

Ontological definition. Observability describes how available the trajectory is to be known from the inside: the person’s capacity to access their own functional field and report it validly.

Relationship to \Upsilon_{US}. \Upsilon_{US} (stability under stress) is observability’s most direct operational correlate: when \Upsilon_{US} < \Upsilon_{US_{\min}}, the field is too perturbed for self-report to reflect its state faithfully. Observability is a gradient, not a binary: as \Upsilon_{US} drops, self-report’s validity degrades progressively.

Consequence for unveiling. Existence condition CE7 (unveiling changes the field it describes) establishes that the clinical encounter is an ontological intervention (A7). Observability is the condition of possibility for that process: without a minimum of stability (sufficient \Upsilon_{US}), unveiling cannot operate on the intensive field, only on the observable extensive field. CE7 implies that the validation pilot’s design includes assessment conditions with no intervention, to estimate unveiling’s own effect (P11).

Gap G4. The exact relationship between \Upsilon_{US} and self-report validity is not quantified: \Upsilon_{US_{\min}} is currently a clinical threshold set by evaluator judgment. Formalizing it operationally requires pilot data (G4: determining \kappa_{\min}, \varepsilon_{\min}, \Upsilon_{US_{\min}}, \theta).

D14. Clinical Resonance

Ontological definition. Clinical resonance is the information the person’s individuation field produces in the clinician’s individuation field during the encounter. It is not subjective intuition. It is a channel of information about the intensive field that precedes narrative elaboration, and one the clinician can learn to read with training.

Epistemological status. Clinical resonance is a pre-narrative observable of the intensive field: it belongs to the same level as sensor signal, not to the level of self-report. Like the sensor, it captures information prior to conscious elaboration. Like the sensor, it can diverge from self-report when \Upsilon_{US} is low.

Operational components. Three dimensions the clinician trains to discriminate:

The clinician’s somatic field: somatic sensations during the encounter: tension, weight, lightness, contraction. This is information about the person’s field, not the clinician’s own states.

The clinician’s emotional field: affective resonance during the encounter: sadness, urgency, apathy, vitality. Same status as the somatic field.

The clinician’s imaginal field: images, metaphors, scenes that arise spontaneously during the encounter. These are the symbolic representation of the person’s field within the clinician’s own field.

Reliability and biases. Clinical resonance carries its own biases: countertransference (the person’s field activates patterns in the clinician’s own field), projection (the clinician reads their own field as though it were the person’s), and depletion (resonance dulls with fatigue). Training in clinical resonance includes training to recognize these biases.

Integration into \mathcal{O}(t). Clinical resonance is recorded in the observation field as free text, not as a scalar. Its integration into the engine’s likelihood is not formalized in the model’s current version. This is the object of G12 (clinical resonance as an observational channel in \mathcal{O}(t), Ontological Core v2.2.4).

D15. Clinical Evidence — \mathcal{E}(t)

Ontological definition. Clinical evidence is the set of structured observations the clinician produces during the encounter that feed T_\text{histórico}. It is distinct from raw observation: it is observation filtered through the model’s conceptual system.

Formal definition. \mathcal{E}(t) = \{e_1, \ldots, e_n\} where each e_i is an instance of \text{HistoricalEvidence}(\text{type}, \text{intensity}, \text{reliability}, t_{\text{formative}}).

Type hierarchy. The five evidence types are ordered by what they imply about the field’s geometry:

Distinction from \mathcal{O}(t). \mathcal{O}(t) is the encounter’s complete set of observations: H_t, M, \mathcal{E}(t), sensor signal, and clinical resonance. \mathcal{E}(t) is only the subset that feeds T_\text{histórico}: evidence about accumulated history.

D15-int. Interstitial Field — \Omega_t^{(\text{int})}

Ontological definition. The interstitial field is the stability function describing the organization of the encounter between two fields of individuation. It is not the sum of the two fields. It is the organization their historical encounter produces on its own terms. A canonical formal object since Ontological Core v2.2.4 (D15-int).

Formal definition. \Omega_t^{(\text{int})}: \mathcal{W} \to \mathbb{R}_{\geq 0}, with its own constitutive history:

\xi_t^{(\text{int})} = \{D_p^{(\text{int})}(t),\; \kappa^{(\text{int})}(t),\; \sigma^{(\text{int})}\}

• D_p^{(\text{int})}(t): what the relationship has deformed irreversibly (unprocessed ruptures, unrepaired harm, crystallized relational patterns).
• \kappa^{(\text{int})}(t): the interstitial field’s current receptive capacity.
• \sigma^{(\text{int})}: the constitutive, pre-symbolic imprint of the earliest encounters.

Constitutive asymmetry. The interstitial field is not symmetric: the clinician has competencies for unveiling that the person does not; the person has access to their own \xi_t^{(p)} that the clinician does not.

Predicate types. Clinical (between clinician and person) / familial (between members of a family system) / bonding (between the trajectory and a significant attachment figure). Each type has distinct structural properties.

Clinical consequence. \Omega_t^{(\text{int})}’s properties determine which aspects of \xi_t^{(p)} are accessible in this encounter. An interstitial field with reduced \kappa^{(\text{int})}(t) produces reduced access to \xi_t^{(p)} even when individual parameters would allow unveiling. Quantitatively formalizing \xi_t^{(\text{int})} is the object of G17.

Its own dynamics: three regimes. \Omega_t^{(\text{int})} evolves over the course of the relationship. Events that accumulate D_p^{(\text{int})}(t): unprocessed ruptures, frame transgressions, crystallized relational patterns. Events that restore \kappa^{(\text{int})}(t): active repair, frame continuity, accumulated successful co-regulation. Three regimes: (a) productive, high \kappa^{(\text{int})}(t): unveiling can reach deep basins; (b) restricted, reduced \kappa^{(\text{int})}(t): only \xi_t^{(p)}’s surface is accessible; (c) in rupture, \kappa^{(\text{int})}(t) < \kappa^{(\text{int})}_{\text{umbral}}: intervening on D_p(t) in this state produces harm analogous to what P12 describes. The priority is restoring the interstitial field before any work on constitutive memory.

Interstitial spatium. \Omega_t^{(\text{int})} has its own spatium, (\varepsilon_t^{(\text{int})}, \Psi_t^{(\text{int})}). \varepsilon_t^{(\text{int})} is relational functional elasticity, higher in the relationship’s early phases, lower with a history of accumulated rupture. \Psi_t^{(\text{int})} has constitutive asymmetric propagation: the dominant direction is D_p^{(\text{int})} \to \kappa^{(\text{int})}: ruptures drain receptive capacity; the reverse direction requires deliberate repair. This asymmetry distinguishes the interstitial field from the individual field (where \Psi_t can be bidirectional), and it formalizes why damaged relationships do not restore themselves through the passage of time alone.

Status: canonical formal definition. Source: D15-int, Ontological Core v2.3.1.

Part II — Transition Conditions, Formalized

D16. Transition Conditions — M

Ontological definition. A transition condition is a condition on the trajectory’s evolution operator: a property that determines which transitions between configurations are possible, how easily, and at what cost. Transition conditions describe how the trajectory moves across configuration space, not what it produces (that is what domains describe).

Ontological distinction from domains. Domains are coordinates of the individuation field in \mathcal{W}. Transition conditions are properties of the evolution operator: in terms of the triad, they are properties of the dynamics over (\mathcal{W}, g_t, \mathcal{A}), not of the configuration space itself. This is why M stays independent of the triad (full formalization in the Mathematical Core).

Formal definition. M = (\Beta_F^{(r)}, \Beta_F^{(i)}, \Pi_R, \Tau_F, \Upsilon_{US}) \in [0,1]^5

Each transition condition is a scalar in [0,1] describing the level of its corresponding condition at time t.

The five transition conditions:

\Beta_F^{(r)} — Behavioral response flexibility. The observable behavioral repertoire under different conditions: the external component of behavioral flexibility. It describes the evolution operator’s variety: a field with high \Beta_F^{(r)} can reach configurations through multiple paths; one with low \Beta_F^{(r)} can only evolve along stereotyped paths.

\Beta_F^{(i)} — Introspective flexibility. The capacity to observe one’s own patterns from some distance and use them as leverage (the clinical vocabulary’s observing image, NC-11). It is the internal, metacognitive component of behavioral flexibility. A field with high \Beta_F^{(i)} can detect its own patterns activating in real time; one with low \Beta_F^{(i)} experiences them as invisible, or as the only ones possible.

Independence of \Beta_F^{(r)} and \Beta_F^{(i)}: the two dimensions are orthogonal. A field can have a wide behavioral repertoire (high \Beta_F^{(r)}) and low metacognitive capacity (low \Beta_F^{(i)}). This is the pattern observable in OCPD: varied, precise rituals, fully ego-syntonic. Or the reverse dissociation: high awareness of one’s own patterns (high \Beta_F^{(i)}) with no capacity to vary the response (low \Beta_F^{(r)}). Formalization approved by Diego (session 5, June 2026).

\Pi_R — Reality testing. The capacity to calibrate interpretations of the environment against shared reference points. It describes the observation operator’s precision: a field with low \Pi_R produces less reliable observations \mathcal{O}(t): clinical evidence loses validity.

\Tau_F — Trajectory continuity. The capacity to sustain narrative and directional coherence over time. It describes the evolution operator’s memory: a field with low \Tau_F produces fragmented trajectories that fail to accumulate the effect of clinical perturbation.

\Upsilon_{US} — Stability under stress. The capacity to sustain organization that widens \mathcal{F}_t^{(p)} under high-intensity perturbation. It is the transition condition with the greatest impact on intervention indication: below \Upsilon_{US_{\min}}, the Unveiling protocol cannot operate on the intensive field.

D-ideal. Reference Point for Functional Coherence — H_t^*, M^*

Formal definition. H_t^* = (1.0,\,1.0,\,1.0,\,1.0,\,1.0) \qquad M^* = (1.0,\,1.0,\,1.0,\,1.0,\,1.0)

Status: the declared upper bounds of D4’s (H_t) and D16’s (M) scales. This is not an empirical norm. It does not describe any real person, or a permanent state of any trajectory. It is the orienting axis for the functional coherence profile: a direction of possible movement, not any treatment’s destination.

What H_t^* names: the organization in which all five domains simultaneously reach the maximum functional coherence level the corpus’s scales declare: volition with sustained agency, bonds with full reciprocal affection, temporal projection with continuity and affect, existential anchoring with full presence carrying its own weight, the somatic domain as an active resource.

What M^* names: the organization in which all five transition conditions simultaneously reach their maximum level: a wide response repertoire available under activation, the observing image fully available, high epistemic calibration, full narrative and experiential continuity, sustained presence under high-load perturbation.

Clinical function: H_t^* and M^* let you measure the trajectory’s functional coherence gap without needing a population norm. For each parameter x_i, the gap is 1 - x_i. The functional coherence radar represents this gap visually: the area between the current profile and the radar’s outer edge (which corresponds to H_t^*, M^*). The clinically relevant reading is the gap’s shape, not its absolute value: which parameters show the greatest distance, and which are most accessible given \xi_t.

Relationship to \sigma_t: the B_t and A_t axes on the functional coherence radar carry a differential shading that indicates \sigma_t’s presence: the morphological imprint of pre-linguistic early experience. When \sigma_t is high, the gap in B_t and A_t carries a higher reorganization cost than the same gap in other domains, independent of \kappa(t). The shading adds no axis to the radar: it is a visual annotation on how the gap is read, not a new dimension. (See D20, note on \sigma_t.)

Approved by Diego (session 5, June 2026).

Part III — Terms with Partial Definitions in Current Documents

The following terms have a conceptual definition in the Ontological Core but do not yet have a complete mathematical formalization. They are recorded here for traceability.

D17. Temporal horizon. The extension into the future accessible to the trajectory: how broad the field of possibilities is along the temporal dimension. Correlated with P_t but not identical to it: the horizon can be present even with low P_t if the trajectory has access to short-term projection. Formalization pending.

D18. Existential anchoring. The ground of meaning from which the trajectory organizes its experience. The domain A_t is its extensive representation. Formalizing the relationship between the ground of meaning and the intensive field’s organization (including whether A_t produces systematically greater asymmetry in \Psi_t than other domains) is the object of G13 (formalizing existential anchoring in the intensive field, Ontological Core v2.2.4).

D19. Dynamic regime. A prohibited term in the model’s technical vocabulary. Always replace it with “dynamic organization,” “configuration,” or “dynamic state,” depending on context. The prohibition is historical: “regime” carries a connotation of abrupt discontinuity that does not sit well with Syntropia’s rhysic ontology.

D-H1a. Generative potential functional — \Phi_t^{(p)}. Status: working structural hypothesis, pending empirical calibration (log rule 103). \Omega_t^{(p)}’s complement: where \Omega_t^{(p)} describes where the field is (accessibility/stability), \Phi_t^{(p)} describes where it is pushing toward (generative tension). Formally: the functional gradient of \mathcal{A}[\Omega, t] in the direction of \Omega_t^{(p)}’s support expansion. Three regimes: \Phi_t^{(p)} \approx 0 (deep residency: a field stable through equilibrium); \Phi_t^{(p)} > 0 (active reconfiguration); \Phi_t^{(p)} \gg 0 with \kappa(t) < \kappa_{\text{umbral}} (crisis: a field stable through exhaustion). This distinction cannot be captured from \Omega_t^{(p)} alone. Its formal derivation requires PEA1 resolved; its empirical calibration requires pilot data. Source: §II.5sexies, Mathematical Core v0.4.13.

D-H1b. The process’s rhythm — \chi_t. Status: a formal object with approximate estimators available, not yet calibrated. The field’s state of readiness to produce a reorganization: the tension accumulated in \Omega_t^{(p)} before that reorganization becomes observable in H_t. Distinct from \dot{H}_t = \nabla H_t / \Delta t (the observable’s rate of change): the two are dissociable (log rule 104). \dot{H}_t \approx 0 with \chi_t high is the silent period before a major reorganization; high \dot{H}_t with low \chi_t is reactive movement with no underlying reorganization. Never present \dot{H}_t as equivalent to \chi_t. Source: §VIII.5, Mathematical Core v0.4.13.

D-H1c. Resignification operator — \rho(t). Status: a hypothetical object pending G18 (log rule 105). A factor that modulates D_p(t)’s functional contribution to the field without altering its magnitude: \Omega_t^{(p)} = F(\sigma_t,\; \rho(t) \cdot D_p(t),\; \kappa(t)). It captures the resignification of history: what once organized the field from fear can reorganize toward integrated memory without D_p(t) itself decreasing. Unlock condition: pilot data with pre/post measures of narrative versus somatic interventions at constant D_p(t). Do not incorporate into the Bayesian model’s formal equations until G18 produces evidence that it is distinguishable from \kappa(t). Source: D10 note, Ontological Core v2.3.3.

Pre-pilot operational development (v0.4.10): the functional distinction from restored \kappa(t): restoring \kappa(t) produces general improvement (the field is more responsive across all contexts); active \rho(t) produces specific improvement (the latent basins tied to the history that was worked on stop activating under their type of perturbation, while other latent basins can remain active). Signs of active \rho(t): the field describes the same historical events in a qualitatively different way with no change in the facts themselves; latent basins that used to activate under type-X perturbation stop activating at the same frequency; a self-image with new narrative coherence. \rho(t) interventions: psychodynamic psychotherapy, EMDR, narrative work, episodic work (months-to-years timescale). \kappa(t) interventions: pharmacotherapy, sleep, allostatic load (weeks-to-months timescale). Documentation in SCRF-1 Field 3: “\rho(t): [active / not active / indeterminate].” See Clinical Core v0.3.5.

D20. Somatic domain — B_t. H_t’s fifth constitutive domain. Canonical since Ontological Core version 1.5.0. H_t = (V_t, R_t, P_t, A_t, B_t) \in [0,1]^5. Its four subdimensions: B_t^{(s)} sleep regulation, B_t^{(m)} metabolic and neurobiological regulation, B_t^{(a)} baseline autonomic activation, B_t^{(i)} somatic integrity.

Note on \xi_t and \sigma_t. Constitutive memory \xi_t = \{D_p(t), \kappa(t), \sigma_t\} has had three components since Ontological Core v2.3.0. The pre-symbolic dispositional structure \sigma_t, the morphological imprint of pre-linguistic early experience, has a special relationship to B_t^{(s)} (sleep regulation) and B_t^{(m)} (metabolic and neurobiological regulation), since those subdimensions capture \sigma_t’s somatic expression prior to narrative elaboration. Interventions with access to \sigma_t are primarily body-centered (B_t), focused on early relationship and the sensory environment.

Note on the canonical configurations. The canonical configurations should be derived from the space H_t \in [0,1]^5. Canonical system v1.2 has 14 candidate configurations organized by dominant propagation pattern in \Psi_t × basin × domain of origin (selective). See Configuraciones_Canonicas_v1_3_14.qmd. Whether the number of canonical configurations is final is the object of G10.

Formal Hypotheses — H-NOV Through H-ORDEN

Added in v0.4.7 after a deep architectural review of Core 1.0. Formal hypotheses are provisional structural commitments about relationships in the model that cannot be verified given the corpus’s current state. Status: working structural hypothesis (more than a heuristic analogy, less than a strong ontological commitment). Each carries a declared unlock condition.

H-NOV. The novelty hypothesis. Status: latent hypothesis, conditional on PEA1 resolved and \Phi_t^{(p)} calibrated (RC-6.2). Not active in the current pilot. Horizon: H4. Expanding \mathcal{F}_t^{(p)}’s support into previously inaccessible regions of \mathcal{W} requires, simultaneously: (a) sustained \Phi_t^{(p)} > 0; (b) \kappa(t) \geq \kappa_{\text{umbral}}(D_p); (c) perturbation aligned with \Phi_t^{(p)}’s gradient. An orthogonal or opposing perturbation produces level 1 (modal shift) or level 3 (contraction). Source: A5 level 2, Ontological Core v2.3.0.

H-COMP. The processual comparability hypothesis: directed processual premetric. Status: urgent working structural hypothesis; declared a directed premetric (not a symmetric metric) since Epistemic Core v0.3.10. Unlock condition: G2 (g_t calibration). Horizon: H2–H3. The processual distance between two trajectories p and q at time t: d_{\text{proc}}(p,q,t) = \alpha \|\xi_t^{(p)} - \xi_t^{(q)}\|_{g_t} + \beta\, d_{\text{config}}(\Omega_t^{(p)}, \Omega_t^{(q)}) + \gamma\, d_{S_m}(p,q,t). This operationalizes CE18. It blocks the pilot’s population-level validation until the metric is calibrated. Source: CE18, Epistemic Core v0.3.9; §II bis.2, Mathematical Core v0.4.13.

H-TRANS. The transitional regime hypothesis. Status: working structural hypothesis. Unlock condition: G19. Horizon: H2–H3. The period between basins is characterized by: high variance in \Omega_t^{(p)} across multiple configurations; \chi_t declining from its pre-transitional peak; \kappa(t) at a minimum; heightened sensitivity to perturbation (\varepsilon_t^{\text{ef}} transiently reduced). Asymmetric through hysteresis: the crossing \mathfrak{C}_i \to \mathfrak{C}_j has different dynamics than the reverse crossing. P12 applies at maximum urgency during this regime. Source: HST, extended D15-int, Ontological Core v2.3.0.

H-POT. The potentialities hypothesis. Status: working structural hypothesis. Unlock condition: \Phi_t^{(p)} empirically calibrated. Horizon: H3–H4. \mathcal{F}_t^{(p)}’s most probable direction of expansion is determined by \Phi_t^{(p)}’s gradient over \partial\mathcal{F}_t^{(p)}. The most probably accessible regions are those for which d_{g_t}(\mathfrak{C}_j, \partial\mathcal{F}_t^{(p)}) < \delta_{\Phi}(t), where \delta_{\Phi} depends on \Phi_t^{(p)} and \kappa(t). Source: §II.5sexies, Mathematical Core v0.4.13.

H-TEMPO. The trajectory’s internal time hypothesis. Status: latent hypothesis, conditional on PEA1 resolved and \Phi_t^{(p)} calibrated (RC-6.2). Not active in the current pilot. Horizon: H3. Internal time \tau_t \approx \Phi_t^{(p)} / \kappa(t): a field with high generative tension and high receptive capacity processes perturbation faster than one with low tension and low capacity. If empirically verified (G24), \tau_t is an independent predictor of reorganization speed with additional power over \chi_t and \dot{H}_t. Source: §VIII.5, Mathematical Core v0.4.17.

H-TRANS-ESC. The cross-scale propagation without reduction hypothesis. Status: programmatic hypothesis. Horizon: H4. Propagation across scales satisfies the no-reduction condition if and only if the translation operators (T_{\text{ontológico}}, T_{\text{escala}}, T_{\text{histórico}}) have no exact inverse: information is lost in a controlled, declared way in each direction. The two main gaps: a field-to-molecular operator (T_{\text{ontológico}}^{-1}), and propagation from the individual field to a supra-individual field. Source: §II.5quinquies, §IV, Mathematical Core v0.4.13.

H-ORDEN. The higher-order fields hypothesis. Status: programmatic hypothesis. Horizon: H4. Families, organizations, and cultures have architecture analogous to \Omega_t^{(\text{int})}, with their own predicates (familial, organizational, cultural), and they condition the individual \Omega_t^{(p)} through \Psi_t and \sigma_t. Higher-order fields have their own constitutive history, \xi_t^{(\text{ord})}, with the same three components at a different scale. Source: extended D15-int, Ontological Core v2.3.1.

H-TRANS-F. The transition’s functional form hypothesis. Status: working structural hypothesis; derived from \mathcal{W}’s Riemannian geometry; falsifiable with pilot data. Unlock condition: G2 plus pilot transition-frequency data. Horizon: H2. P(\mathfrak{C}_j \mid \mathfrak{C}_i, t) \propto \exp(-d_{g_t}(\mathfrak{C}_i, \mathfrak{C}_j) / \kappa(t) \cdot \|\Delta u_t\|) \cdot \mathbf{1}[\kappa(t) \geq \kappa_{\text{umbral}}(D_p)]. This turns the HST’s three qualitative properties into a falsifiable quantitative prediction. Source: extended G10, Ontological Core v2.3.1; §VIII.7, Mathematical Core v0.4.14.

CE19. The limit of \xi_t as a representation of constitutive history. Status: canonical epistemological commitment since Epistemic Core v0.3.10. \xi_t is the trace of the accumulated process, not the process itself. Two trajectories with \xi_t^{(p)} \approx \xi_t^{(q)} can have arrived by different paths and respond differently to identical perturbation. Maximum severity during Phases 1, 3, and early 4 of §T1: in those periods, direct clinical reading carries more weight than the Bayesian engine. Source: CE19, Epistemic Core v0.3.10; §VI.2 and §VIII.5, Mathematical Core v0.4.14.

Part IV — Relational Vocabulary (Retired: Historical Status)

The relational vocabulary (14 technical terms of Greek root across two generations: Eudaimonia, Euexia, Allotropy, Anexia, Epitropy, Entelopy, and Proexia, from the original system; Eutropy, Dysintropy, Neotropy, Cathexy, Apexy, Anastropy, and Heterotropy, D21–D27, formalized on June 10, 2026, alongside the names of the 14 canonical configurations) was retired from the corpus by Diego F. Pereira-Perdomo’s explicit instruction (June 2026). It served its heuristic function during the model’s construction (exploring dimensions for describing the field: global coherence, movement over time, normative environment; independent of the canonical configuration system), but no term from either generation is active. Canonical system v1.2’s 14 configurations and their Spanish names (D28) are the only nomenclature currently in force.

D28. Quick Reference Table — Canonical Names and Phrases

The 14 canonical configuration names (system v1.2) and their canonical phrases: approved clinical formulations for communicating each configuration’s internal logic to the patient. Full formalization in Configuraciones_Canonicas_v1_3_14.qmd.

English names confirmed against Table 1 of the Foundational Article (July 15, 2026).

Version 0.4.18 — July 13, 2026. Author: Diego F. Pereira-Perdomo. Cross-check against: Ontological Core v2.3.16, Mathematical Core v0.4.27, Canonical Configurations v1.3.15. This document is a canonical reference: any modification requires a new semantic version.