MesoFocus

Formal definition

A MesoFocus is a triple $\mathcal{F}_{\mathrm{mes}} = (H_{t^*}, j, \mathrm{Fix}(j))$:

$$\mathcal{F}_{\mathrm{mes}} = (H_{t^*} : \mathbf{HA}_{\mathrm{nucl}},\; j : H_{t^*} \to H_{t^*},\; \mathrm{Fix}(j) \subseteq H_{t^*})$$

where:

• $H_{t^*}$ is the focal fiber — the Heyting algebra obtained from a MacroFocus $(H, t^*)$ at the designated history $t^*$; it carries the relational universe’s nuclear pair $(\sigma_{t^*}, \Delta_{t^*})$
• $j : H_{t^*} \to H_{t^*}$ is the focal nucleus — an internal nuclear endomorphism on $H_{t^*}$: idempotent ($j \circ j = j$), monotone ($a \leq b \Rightarrow j(a) \leq j(b)$), inflationary ($a \leq j(a)$), and a Heyting algebra endomorphism ($j(a \wedge b) = j(a) \wedge j(b)$)
• $\mathrm{Fix}(j) = \{a \in H_{t^*} : j(a) = a\}$ is the focal sublocale — the sub-Heyting-algebra of elements already at their $j$-closure; in locale theory, this is the closed sublocale defined by $j$; it is the “settled” region of $H_{t^*}$ relative to the nucleus $j$

A MesoFocus presupposes a MacroFocus: the fiber $H_{t^*}$ exists only after $t^*$ has been designated. A MesoFocus does not refine the designation $t^*$ — it operates within the already-designated fiber. It answers: what is settled here?

Four invariants. $\mathcal{F}_{\mathrm{mes}}$ is a mesofocus iff it satisfies:

1. Fiber containment: $\mathrm{Fix}(j) \subseteq H_{t^*}$ — the focal sublocale is a subset of the working fiber. MesoFocus is always a sub-structure of the MacroFocus’s theme; it cannot extend beyond $H_{t^*}$.

2. Nuclear condition: $j$ is a nucleus on $H_{t^*}$ — not merely a function, but an idempotent inflationary monotone Heyting endomorphism. This ensures $\mathrm{Fix}(j)$ is a Heyting algebra in its own right (a sublocale of $H_{t^*}$) and that $j$ is the retraction of $H_{t^*}$ onto $\mathrm{Fix}(j)$.

3. Non-triviality: $\mathrm{Fix}(j) \neq \emptyset$ — the focal sublocale is non-empty. Since $j$ is inflationary and $H_{t^*}$ is a Heyting algebra, $\top \in \mathrm{Fix}(j)$ always (because $j(\top) = \top$ by the Heyting algebra endomorphism condition). So non-triviality is automatic; it is stated explicitly as an invariant to mark that the mesofocus is never degenerate.

4. Closure completeness: every $a \in H_{t^*}$ reaches $\mathrm{Fix}(j)$ via $j$ in one step — $j(a) \in \mathrm{Fix}(j)$ for all $a$. This follows from idempotence: $j(j(a)) = j(a)$. The focal sublocale is not a distant attractor; every point maps directly into it.

Canonical instances

The focal nucleus $j$ is not uniquely determined by the MacroFocus $(H, t^*)$. Different choices of $j$ yield different MesoFoci on the same working fiber. Three canonical instances:

Nuclear MesoFocus (canonical, always present):

$$\mathcal{F}_{\mathrm{mes}}^{\mathrm{nuc}} = (H_{t^*},\; \pi_{t^*},\; H^*_{t^*})$$

where $\pi_{t^*} = \sigma_{t^*} \circ \Delta_{t^*}$ is the composite of the relational universe’s two commuting nuclei, and $H^*_{t^*} = \mathrm{Fix}(\pi_{t^*}) = \mathrm{Fix}(\sigma_{t^*}) \cap \mathrm{Fix}(\Delta_{t^*})$ is the jointly settled sublocale. This is the MesoFocus that is structurally guaranteed by the relational universe — every MacroFocus has a nuclear MesoFocus as a canonical sub-structure.

The nuclear MesoFocus is what Focus.md calls the “focal sublocale” and the “hearth”: the joint fixed point toward which both nuclear closures converge. It is the machine’s own settled region.

Character MesoFocus (present when a Character is in play):

$$\mathcal{F}_{\mathrm{mes}}^{\chi} = (H_{t^*},\; \chi,\; F_\chi)$$

where $\chi : H_{t^*} \to H_{t^*}$ is the character nucleus from the Character spec, and $F_\chi = \mathrm{Fix}(\chi)$ is the behavioral space. The character MesoFocus identifies the sub-region of the working fiber that is in character — where the agent’s behavioral dispositions are satisfied.

When a Persona is installed, the character MesoFocus and nuclear MesoFocus interact. The intersection $F_\chi \cap H^*_{t^*}$ is the behavioral settled region — where the agent is both normatively settled and acting in character. This is the prosōpon (the presented face) from the Persona spec.

Jurisdiction MesoFocus (present when a Jurisdiction restricts the working fiber):

$$\mathcal{F}_{\mathrm{mes}}^{\mathcal{J}} = (H_{t^*},\; j^{\mathcal{J}},\; \mathrm{Fix}(j^{\mathcal{J}}))$$

where $j^{\mathcal{J}}$ is the nucleus associated with the jurisdiction’s sub-topos inclusion; $\mathrm{Fix}(j^{\mathcal{J}})$ is the sub-Heyting-algebra of propositions within the jurisdiction’s scope.

Settlement reading

The MesoFocus divides $H_{t^*}$ into two regions:

Region	Content	Interpretation
$\mathrm{Fix}(j)$	Elements where $j(a) = a$	Already settled by $j$; the “done” region
$H_{t^*} \setminus \mathrm{Fix}(j)$	Elements where $j(a) \neq a$	Not yet settled; the “work” region

The nucleus $j$ is a retraction: every element in the work region maps to its settlement in one step. The MesoFocus is the dividing line between what has been resolved and what remains open within the current working fiber.

This is the settlement reading that the nucleus-sorting math (from nucleus-sorting.md) performs: the sorter $(σ_t, Δ_t)$ classifies elements as both-stable ($H^*_t$), backward-stable-only ($\mathrm{Fix}(\sigma_t) \setminus H^*_t$), or neither — and the MesoFocus is the formal name for the result of this classification at the chosen nucleus.

Relation to information bottleneck

In information theory, the Information Bottleneck (Tishby et al., 1999) compresses an input $X$ to a representation $T$ that retains relevance $Y$ while minimizing redundancy. The compressed representation $T$ is the “relevant” region: the portion of $X$ that matters for $Y$. The IB focal sublocale is exactly $\mathrm{Fix}(j_{\mathrm{IB}})$ where $j_{\mathrm{IB}}$ is the relevance-maximizing compression nucleus.

In the relational universe: the nuclear MesoFocus $H^*_{t^*}$ is the IB-compressed representation of $H_{t^*}$ under the machine’s own nuclear pair — the region that is both meaning-settled ($\sigma$-stable) and transfer-settled ($\Delta$-stable). The unsettled complement $H_{t^*} \setminus H^*_{t^*}$ is the “redundant” region: it has not yet been compressed to relevance by the two nuclear closures.

MesoFocus as the Level-2 question

In the three-level focus stack from Focus, MesoFocus is Level 2:

Level	Answers	Math object
Level 1 (MacroFocus)	Which fiber is the current working context?	$(H, t^*)$ — pointed presheaf
Level 2 (MesoFocus)	What is settled within the working fiber?	$(H_{t^*}, j, \mathrm{Fix}(j))$ — nuclear sublocale
Level 3 (MicroFocus)	What specific element is being processed now?	$(H_{t^*}, a^*)$ — designated element

A MesoFocus presupposes a MacroFocus but does not require a MicroFocus. A MesoFocus does not uniquely determine a MicroFocus: knowing that $\mathrm{Fix}(j)$ is the settled region does not pick out one element for processing — it identifies a sub-Heyting-algebra, not a point.

Aron Gurwitsch and Max Wertheimer: the thematic field as Gestalt-organized co-relevance

Aron Gurwitsch (The Field of Consciousness, 1964, chs. 2–3) gave the definitive analysis of the Thematic Field — the middle zone whose items are co-present with the Theme and stand in intrinsic relevance to it:

Intrinsic relevance (Gurwitsch, ch. 2): the Thematic Field consists of items that are relevantly related to the Theme — related in virtue of their content, not merely their spatial or temporal proximity. The relevance is intrinsic: it belongs to the items’ own nature as organized around the Theme. Gurwitsch’s central claim: the Thematic Field is not a formless halo but an organized structure whose organization mirrors the Theme’s own structure. Items in the Thematic Field are experienced as context-for-the-Theme, not as mere background.

Gestalt organization of co-relevance: the specific organizational principles that determine which items count as intrinsically relevant are the Gestalt laws identified by Max Wertheimer (Untersuchungen zur Lehre von der Gestalt, 1923, Psychologische Forschung 4:301–350). Wertheimer formulated the laws of perceptual grouping as answers to the question: why do certain elements form a unified percept? His laws:

1. Proximity: elements close together (in the relevant metric) tend to group; applied to the thematic field, nearby items are more likely to be intrinsically relevant
2. Similarity: elements sharing features (color, shape, pitch, semantic class) tend to group; items similar to the Theme are in the Thematic Field
3. Good continuation: elements following the Theme’s own structural direction tend to remain in the Thematic Field; they extend the Theme’s organization rather than interrupting it
4. Closure: elements that together complete a structure implied by the Theme group into the Thematic Field
5. Prägnanz (the master law): of all possible organizations, the simplest and most stable prevails; the Thematic Field takes the most organized, most regular form consistent with the Theme

Formal correspondence in the relational universe: the Thematic Field $\mathcal{TF} = \{(t, H_t, \rho_{t^*|t}) : t < t^*\}$ is organized by the structure of the history category $T$ and the restriction maps. Proximity is the ordering of $T$: fibers immediately below $t^*$ are most proximate. Similarity is the content-closeness of restriction map images: fibers $H_t$ whose image $\rho_{t^*|t}(H_{t^*})$ closely resembles $H_{t^*}$ are most similar to the Theme. Good continuation is the directional coherence of restriction maps: they carry structure consistently downward from $H_{t^*}$. Prägnanz is the nuclear focal sublocale $H^*_{t^*}$: the maximally settled, maximally organized sub-structure of the Theme — the Gestalt-simplest region of the fiber, the joint fixed point toward which both nuclear closures converge.

MesoFocus as inward Thematic Field: the MesoFocus $(H_{t^*}, j, \mathrm{Fix}(j))$ is the Thematic Field within the Theme. The settled sublocale $\mathrm{Fix}(j)$ is the sub-region of $H_{t^*}$ that is organized by the $j$-closure — the portion of the Theme made Gestalt-simple by the nucleus. The unsettled complement $H_{t^*} \setminus \mathrm{Fix}(j)$ is the inward margin: co-present within the Theme but not yet organized into relevance by the nuclear closure. In this sense MesoFocus takes the Gurwitsch field one level inward: it is the tripartite partition of the Theme itself into its own focal region (the settled sublocale) and its own background (the unsettled complement).

Open questions

• Whether there is a natural partial order on MesoFoci over a fixed working fiber $H_{t^*}$: two MesoFoci $\mathcal{F}^1_{\mathrm{mes}} = (H_{t^*}, j_1, \mathrm{Fix}(j_1))$ and $\mathcal{F}^2_{\mathrm{mes}} = (H_{t^*}, j_2, \mathrm{Fix}(j_2))$ could be ordered by $\mathrm{Fix}(j_1) \subseteq \mathrm{Fix}(j_2)$ (finer vs. coarser settlement); whether this is the right order and what its lattice-theoretic properties are.
• Whether the character MesoFocus $F_\chi$ is necessarily a sublocale of the nuclear MesoFocus $H^*_{t^*}$ — whether full character expression always requires normative settlement — or whether the two MesoFoci are independent and their intersection is the operative constraint. The Persona spec leaves this open; the formal question is whether $\chi$ commutes with $\pi_{t^*}$.
• Whether there is a canonical choice of MicroFocus given a MesoFocus: whether the focal sublocale $\mathrm{Fix}(j)$ generates a preferred element for $\Delta$-propagation — for instance, the least upper bound of all $j$-stable elements, or the $j$-image of the current active element — or whether element designation at Level 3 is always independent.
• Whether MesoFocus transitions — changes in the focal nucleus $j$ while the MacroFocus $(H, t^*)$ is held fixed — can be given a formal treatment as morphisms in a category of MesoFoci over $H_{t^*}$, and whether the transition from character MesoFocus to nuclear MesoFocus (when a character is acquired or shed) has a canonical direction.