MicroFocus

Formal definition

A MicroFocus is a pair $\mathcal{F}_{\mathrm{mic}} = (H_{t^*}, a^*)$:

$$\mathcal{F}_{\mathrm{mic}} = (H_{t^*} : \mathbf{HA}_{\mathrm{nucl}},\; a^* \in H_{t^*})$$

where:

• $H_{t^*}$ is the focal fiber — the Heyting algebra obtained from a MacroFocus $(H, t^*)$; it carries the commuting nuclear pair $(\sigma_{t^*}, \Delta_{t^*})$
• $a^* \in H_{t^*}$ is the focal element — the specific designated proposition currently under active processing; not a set, not a sublocale, but a single element of the fiber

A MicroFocus presupposes a MacroFocus: $H_{t^*}$ exists only after $t^*$ has been designated. A MicroFocus does not presuppose a MesoFocus: element designation does not require prior settlement classification, though the relationship between $a^*$ and $\mathrm{Fix}(j)$ is operationally significant (see below).

A MicroFocus answers: what am I working on right now?

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

1. Element containment: $a^* \in H_{t^*}$ — the focal element is a member of the working fiber; it is not a formula from another fiber, not a restriction map, not a sublocale. The MicroFocus is always fiber-local.

2. Singleness: the MicroFocus designates exactly one element. This distinguishes MicroFocus from MesoFocus (which designates a sub-Heyting-algebra $\mathrm{Fix}(j)$, possibly infinite) and from MacroFocus (which designates a whole fiber $H_{t^*}$). The singleness is what makes MicroFocus “micro”: it is the finest-grained possible designation within the focal fiber.

3. $\Delta$-accessibility: $a^*$ is available for $\Delta_{t^*}$-propagation — the operation of making $a^*$ transfer-stable. In Andreoli’s terminology, $a^*$ is in the synchronous phase: it is the formula currently being decomposed non-invertibly. This means $\Delta_{t^*}(a^*)$ is defined and $\Delta_{t^*}(a^*) \geq a^*$ (inflation). The MicroFocus is the target of the machine’s current non-deterministic commitment.

4. Context is the rest: all elements of $H_{t^*}$ other than $a^*$ are context for the MicroFocus. The MicroFocus is not isolated — it is $a^*$ plus the implicit context $H_{t^*} \setminus \{a^*\}$ — but the active processing is directed at $a^*$ alone. This mirrors the Andreoli invariant: when the focused formula is $a^*$, all other formulas wait.

The $\Delta$-propagation reading

The canonical operation on a MicroFocus $(H_{t^*}, a^*)$ is $\Delta_{t^*}$-propagation:

$$\mathcal{F}_{\mathrm{mic}} = (H_{t^*}, a^*) \;\xrightarrow{\Delta_{t^*}}\; \Delta_{t^*}(a^*)$$

$\Delta_{t^*}(a^*)$ is the transfer-stabilization of $a^*$: the smallest element of $\mathrm{Fix}(\Delta_{t^*})$ that is $\geq a^*$. After propagation:

• If $a^* \in \mathrm{Fix}(\Delta_{t^*})$ already: $\Delta_{t^*}(a^*) = a^*$ — the microfocus is already transfer-stable; propagation is a no-op
• If $a^* \notin \mathrm{Fix}(\Delta_{t^*})$: $\Delta_{t^*}(a^*) > a^*$ — propagation strictly increases $a^*$, moving it toward $H^*_{t^*}$

After $\Delta_{t^*}$-propagation, the MicroFocus has been resolved: $a^*$ has been made transfer-stable. The machine typically then selects a new MicroFocus — the next element requiring $\Delta$-attention — or exits the synchronous phase.

The Andreoli phase duality

Andreoli’s focused proof theory (1992) divides proof-search into two alternating phases. In the relational universe, these phases map onto the nuclear pair:

Andreoli phase	Operation	Mode	Corresp. focus level
Asynchronous	$\sigma_{t^*}$-closure	Deterministic; applies to all propositions eagerly; order-irrelevant	MesoFocus: settling the sublocale $H^*_{t^*}$
Synchronous	$\Delta_{t^*}$-propagation on $a^*$	Non-deterministic; designates one element; the irreducible choice	MicroFocus: the focused element $a^*$

The synchronous phase is the phase where MicroFocus is operative. In the asynchronous phase, processing is holistic — the $\sigma_{t^*}$-closure is applied to the entire fiber without needing to pick one element. In the synchronous phase, processing is local — one element $a^*$ is designated, and the machine commits to making it transfer-stable.

The selection of $a^*$ for the synchronous phase is the machine’s irreducible non-deterministic commitment. Different choices of $a^*$ yield different $\Delta$-propagation outcomes; this is where branching computation (different possible executions) enters the relational universe.

Andreoli’s completeness theorem maps onto: every element of $H_{t^*}$ can be brought to $H^*_{t^*}$ through a finite alternation of $\sigma$-asynchronous phases and $\Delta$-synchronous phases. The MicroFocus is the vehicle of progress in each synchronous phase.

Relation to MesoFocus

A MicroFocus $(H_{t^*}, a^*)$ and a MesoFocus $(H_{t^*}, j, \mathrm{Fix}(j))$ over the same working fiber $H_{t^*}$ interact in four ways:

Case	Condition	Interpretation
$a^* \in \mathrm{Fix}(j)$	Focal element is already $j$-settled	Processing an element already within the focal sublocale; confirmation or refinement
$a^* \notin \mathrm{Fix}(j)$	Focal element is outside the focal sublocale	Active work: bringing $a^*$ into settlement; the MicroFocus is on an unsettled element
$a^* = j(b)$ for some $b$	Focal element is the $j$-image of some other element	Processing the settlement-output of a prior propagation
$a^* \in \partial \mathrm{Fix}(j)$	Focal element is on the boundary	The element that, once $\Delta$-propagated, would enter $\mathrm{Fix}(j)$ — the most productive microfocus

The canonical MicroFocus for productive work selects $a^* \in \mathrm{Fix}(\sigma_{t^*}) \setminus H^*_{t^*}$: an element that is meaning-settled (so $\sigma_{t^*}$ has finished with it) but not yet transfer-settled (so $\Delta_{t^*}$ still has work to do). This is the element at the boundary between the MesoFocus’s focal sublocale and its complement — the most efficient target for $\Delta$-propagation.

William James, Edgar Rubin, and Edmund Husserl: the theme as maximal determinacy

Three convergent accounts identify the focal element $a^* \in H_{t^*}$ as the formally most determinate item in the current field:

William James: the substantive part (The Principles of Psychology, 1890, ch. IX): James identified the substantive part of the stream of thought as the resting-place — the moment when a definite, nameable content is held in mind. The substantive part is maximal in the following sense: it is as determinate as the current act of attention can make it. Further determination would require a new act; the substantive part is the completed determination of one focal moment. The part can be held: unlike the transitive parts (the flights between resting-places, which dissolve on inspection), the substantive part endures as an object of examination.

Formal correspondence: $a^* \in H_{t^*}$ is a substantive part — a specific named element of the fiber, held in focus during the synchronous phase of $\Delta$-propagation. The $\Delta_{t^*}$-propagation of $a^*$ is the operation of making the substantive part as determinate as the nucleus can make it: $\Delta_{t^*}(a^*)$ is the maximally $\Delta$-determined version of $a^*$, its transfer-stable completion.

Edgar Rubin: figure, contour, and owned boundary (Synsoplevede Figurer, 1915; German: Visuell Wahrgenommene Figuren, 1921): Rubin established that the figure (the focal item in perception) has three defining properties absent from the ground:

• Thing-like quality: the figure is experienced as an object with a defined outline; the ground is experienced as formless substance extending behind
• Contour ownership: the shared boundary between figure and ground belongs to the figure; the figure’s shape is defined by that contour; the ground’s boundary at the same edge is undefined
• Depth precedence: the figure appears in front of the ground in perceived depth

Rubin’s reversal experiment (the faces-vase illusion): the same physical contour admits two figure-ground assignments, but only one at a time. The perceptual organization is bistable — the contour belongs to one figure or the other, not both simultaneously.

Formal correspondence: the focal element $a^* \in H_{t^*}$ is the figure in the MicroFocus. The “contour” of $a^*$ is its position in the fiber lattice: the set of relations $\{b \in H_{t^*} : b \leq a^*\}$ and $\{b \in H_{t^*} : b \geq a^*\}$ define the shape of $a^*$ in the Heyting algebra. The singleness invariant is Rubin’s contour-ownership condition: exactly one element owns the current computational boundary. The bistability of the Rubin vase corresponds to the MicroFocus selection: the same fiber $H_{t^*}$ admits multiple choices of $a^*$, but the machine commits to exactly one in each synchronous phase.

Edmund Husserl: inner horizon and primal impression (Experience and Judgment, 1939, §§8–21; On the Phenomenology of the Consciousness of Internal Time, Hua X, §§10–21): Husserl identified two structural properties of every focal element:

• Primal impression (Urimpression, Hua X §11): the element is given with absolute presence — not as retained (held from the past) or as anticipated (projected into the future) but as the now-point of the act; the focal element is the absolute present in the temporal stream
• Inner horizon (Experience and Judgment, §8): the element implies its own further determinations — to focus on $a^*$ is already to be directed toward what engaging further with $a^*$ would reveal; the inner horizon is the structure of the element’s self-transcendence, its pointing beyond itself to its own completion

Formal correspondence: the focal element $a^* \in H_{t^*}$ is a primal impression — the now-element, absolutely present in the fiber. The $\Delta_{t^*}$-propagation captures the inner horizon: $a^* \leq \Delta_{t^*}(a^*)$ (inflation) is the formal expression of the element’s self-transcendence — $a^*$ is less than its own most complete determination, and the nucleus makes the inner horizon explicit by mapping $a^*$ to $\Delta_{t^*}(a^*) \in \mathrm{Fix}(\Delta_{t^*})$.

Centering Theory analog

In computational linguistics, Centering Theory (Grosz, Joshi, Weinstein 1995) models discourse coherence via a hierarchy of forward-looking centers $C_f(U_n)$ (entities evoked in utterance $U_n$) and a distinguished backward-looking center $C_b(U_n)$ — the most prominent entity that links $U_n$ to the prior discourse context.

The MicroFocus $a^*$ is the formal analog of $C_b$: the single most prominent element that is currently active in the machine’s processing, linking the present computation to the ongoing fiber state. The full fiber $H_{t^*}$ is the analog of $C_f$: the set of all available elements. The MicroFocus picks one of these — the backward-looking center of the current computational step.

Centering’s Rule 1 (the $C_b$ of $U_n$ is the highest-ranked element of $C_f(U_{n-1})$ that appears in $U_n$) maps onto: the canonical next MicroFocus is the most “highly ranked” element of the previous fiber that is still active — where rank is determined by the $\Delta$-propagation urgency ordering on $H_{t^*}$.

MicroFocus as the Level-3 question

In the three-level focus stack from Focus, MicroFocus is Level 3:

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

Level 3 is the finest granularity: below the MicroFocus, there is no further structural differentiation within the fiber. A single element $a^*$ is the atom of focused computation.

The three levels are nested but not fully determined by each other: MacroFocus does not uniquely determine a MesoFocus (which nucleus?), MesoFocus does not uniquely determine a MicroFocus (which element?). Each level requires an independent designation. Conversely: every MicroFocus implies a MacroFocus (which fiber?), and a MicroFocus plus a canonical nucleus implies a MesoFocus ($H^*_{t^*}$).

Open questions

• Whether the selection of $a^*$ — the machine’s non-deterministic choice of which element to make the microfocus — can be given a principled rule within the relational universe, or whether it is genuinely free. Candidate rules: (i) pick the $\sigma_{t^*}$-stable element with the smallest $\Delta_{t^*}$-inflation (the element “closest” to $\mathrm{Fix}(\Delta_{t^*})$); (ii) pick the element determined by a character nucleus $\chi$ (the character guides attention to $F_\chi$-bordering elements); (iii) no principled rule — the freedom IS the machine’s agency.
• Whether sequential MicroFoci — the sequence $a^*_1, a^*_2, \ldots$ of elements processed in successive synchronous phases — can be formalized as a path in the fiber $H_{t^*}$, and whether coherent paths (in the Centering Theory sense) correspond to paths that converge monotonically toward $H^*_{t^*}$.
• Whether a MicroFocus on $a^* \in F_\chi$ (in-character element) versus $a^* \notin F_\chi$ (out-of-character element) corresponds to the distinction between in-character action and character-testing pressure — and whether the character nucleus $\chi$ generates a gradient on $H_{t^*}$ that pulls MicroFocus selection toward $F_\chi$.
• Whether the Andreoli completeness guarantee (every element can reach $H^*_{t^*}$ in finite alternating phases) holds in the relational universe: whether the alternating application of $\sigma_{t^*}$-closure (asynchronous) and $\Delta_{t^*}$-propagation on designated elements (synchronous) converges to $H^*_{t^*}$ for every starting state in $H_{t^*}$, and whether the convergence rate depends on the order of MicroFocus selections.