Container-Relative Spatial Admissibility Beyond Geometric Measure（v1.2）

Freedom, Minimal Generators, and Latent Closure

Author: Neo.K
Affiliation: EveMissLab Technology Co., Ltd.
Version: Submission Draft v1.2
Date: 2026-10-02

公司： 一言諾科技有限公司

Abstract

Geometric occupancy is commonly summarized by length, area, volume, or the cardinality of a discrete support. These quantities are indispensable, but they need not characterize the structure of admissible actions inside a finite environment. Two objects with equal support can permit different numbers of placements, induce different directional restrictions, or alter connectivity in qualitatively different ways. This paper develops a container-relative framework for making that distinction explicit.

The framework, called Spatial Information Occupancy (SIO), is not proposed as a replacement for configuration-space geometry, mathematical morphology, affordance theory, or simple-game theory. Its primitive operations are intentionally drawn from those established literatures. The proposed contribution is instead an interface among them: given a container, a constraint, and a specified action query, SIO distinguishes direct support from the resulting admissible configurations and, when the induced predicate is monotone, from its minimal cooperative closures. The paper therefore claims an organizational framework and exact separation results, not new theories of erosion, configuration space, or cooperative games.

We first prove two finite separation results. For a $w\times h$ rectangle in an $N\times N$ lattice container, the number of legal translations is

P_N(w,h) = (N-w+1)(N-h+1) = (N+1)^2+wh-(N+1)(w+h).

Consequently, fixing direct support $wh=K$ does not fix translation freedom. We also construct a one-cell-thick spanning barrier whose direct site occupancy tends to zero as $N\to\infty$ while its pair-connectivity fragmentation tends to $1/2$ .

We then study controlled dynamics. For horizontal and vertical probes of length $L$ , a probe-interior $w\times h$ obstacle satisfies

D_v-D_h = \frac{(L-1)(w-h)} {N(N-L+1)},

showing exact directional redistribution under fixed-area deformation. A vertical barrier with a gap of size $g$ exhibits an action-dependent threshold: a rectangular probe of transverse height $h_p$ has cross-barrier aperture capacity

C_{\mathrm{ap}}(g;P) = \max(0,g-h_p+1),

and becomes left-right connected in configuration space exactly when $g\ge h_p$ under explicit clearance assumptions.

Finally, we model finite monotone spatial admissibility as a simple game. Minimal winning coalitions are interpreted as minimal spatial admissibility generators. This gives the standard representation

V_q(A)=1 \iff \exists G\in\mathcal G_q:\;G\subseteq A,

and motivates a latent-closure quantity

d_q(A) = \min_{G\in\mathcal G_q}|G\setminus A|,

the Hamming deficit to the nearest minimal admissible generator. In a contiguous aperture model, standard Möbius/Harsanyi and Shapley machinery yields an exact $h$ -body cooperative structure.

A cross-layer case study then demonstrates why the interfaces matter. In a two-barrier environment with ten releasable cells, two release states can have the same released-cell count and the same number of legal $2\times2$ probe placements ( $508$ ) while differing in left-to-right configuration-space connectivity. Exhaustive enumeration of all $2^{10}=1024$ release states yields exactly four minimal winning coalitions of sizes $4,4,6,6$ .

The central claim is deliberately modest: direct geometric support, and even aggregate placement count, are generally not sufficient statistics for spatial admissibility. A richer description must be indexed by the actions under consideration and may require placement structure, connectivity, cooperative generators, and distance to admissible closure.

Keywords: spatial admissibility; configuration space; mathematical morphology; simple games; hypergraphs; affordances; minimal winning coalitions; spatial constraints; latent closure

1. Introduction

A region can be measured without being understood as an action space.

If an object occupies $K$ cells of a finite lattice, then $K$ is an exact statement about direct support. It does not, by itself, tell us how many legal placements remain, whether a path still connects two sides of the environment, whether one direction has become more constrained than another, or whether a set of locally released cells has become large enough to support a specified action.

This distinction is familiar in several mature fields, but it is usually expressed through different mathematical languages. In robot motion planning, configuration-space methods transform geometric obstacles into forbidden configurations of a moving body [1]. In mathematical morphology, erosion identifies translations of a structuring element that fit inside a set, and the algebraic theory of dilation and erosion formalizes such placement relations on lattices [2,3]. Affordance theory emphasizes that action possibilities are relational: the same environment can afford different actions to agents with different capabilities [4]. Occupancy-grid robotics, by contrast, uses spatial cells to represent uncertain physical occupancy for mapping and navigation [5]. In cooperative game theory and the theory of monotone Boolean functions, a binary property can be represented through minimal winning coalitions, while Möbius/Harsanyi and Shapley constructions describe higher-order interactions and attribution [6–9].

The present paper does not attempt to replace these theories. Its objective is narrower and organizational: to build a common formal layer in which direct geometric occupancy and action-dependent admissibility are separated from the outset.

We call this perspective Spatial Information Occupancy (SIO). The word “information” is used operationally, not ontologically. We do not assume that physical space is literally made of information, nor do we redefine Shannon information or classical information geometry. We instead ask how a spatial constraint changes a specified set of admissible possibilities.

The basic state is therefore not “shape alone,” but a triple:

(\mathfrak C,O,q),

where $\mathfrak C$ is a container, $O$ is a constraint or occupied structure, and $q$ is a query specifying which actions or configurations count as admissible.

The paper makes four contributions, each deliberately scoped to avoid claiming ownership of the neighboring theories.

First, it formalizes a cross-formalism problem statement: direct geometric support and action-dependent admissibility are separate layers. Exact finite counterexamples are used as separating witnesses, not as claims of new combinatorics.

Second, it derives a compact family of exact transport and boundary-correction identities that make this separation operational under deformation.

Third, it places a standard configuration-space narrow-passage effect inside the same hierarchy, distinguishing geometric release from action-specific realization.

Fourth, it maps finite monotone spatial admissibility to standard simple-game machinery and uses minimal winning coalitions as spatial admissibility generators. The Hamming deficit to the nearest generator is then used as a latent-closure descriptor; standard Möbius/Harsanyi and Shapley tools provide interaction and attribution projections.

The novelty claim is therefore not that configuration spaces, erosion, simple games, Möbius transforms, Shapley values, or Hamming deficits are new. They are not. The contribution proposed here is the container-relative interface that places direct occupancy, action-dependent freedom, cooperative spatial generators, and latent closure in one explicit hierarchy. Section 9 supplies a nontrivial finite case in which direct release and aggregate fit-position count agree between two states while global connectivity differs, and in which the resulting release logic has multiple minimal winning routes.

The remainder of the paper proceeds as follows. Section 2 reviews the closest mathematical neighbors. Section 3 defines the container-relative model. Section 4 proves that direct support is insufficient for several admissibility quantities. Section 5 studies directional freedom transport and boundary effects. Section 6 proves the gap threshold and motivates latent freedom. Section 7 gives the simple-game generator formulation. Section 8 analyzes a contiguous aperture model using standard cooperative-game transforms. Section 9 discusses scope, interpretation, and limitations. Section 10 concludes.

2. Relation to Prior Work

2.1 Configuration-space geometry

Configuration-space methods provide the most direct geometric predecessor to the present framework. Lozano-Pérez represented the position and orientation of a moving object by a point in configuration space and represented collisions as forbidden regions [1]. This establishes an important principle that SIO explicitly adopts: admissibility depends not only on workspace occupancy but also on the body or probe being moved and the action variables being considered.

Our legal-placement sets and discrete configuration graphs are therefore not proposed as alternatives to configuration-space geometry. They are finite specializations of the same broad idea. The additional emphasis here is on comparing those action-dependent structures to direct support and then connecting them to minimal-generator representations.

2.2 Mathematical morphology

Mathematical morphology is an equally close neighbor. If $A$ is a free region and $B$ is a structuring element, the erosion $A\ominus B$ identifies translations under which $B$ fits inside $A$ . The algebraic foundations of erosion and dilation are well established [2], and morphology provides a general language for shape-dependent spatial transforms [3].

Many of the finite placement counts used below can therefore be interpreted morphologically. We do not claim otherwise. Our use differs in emphasis: the legal placements become one layer in a larger admissibility hierarchy, and later sections treat minimal sets of released constituents as generators of a monotone spatial property.

2.3 Affordances

Affordance theory stresses that action possibilities are relational rather than purely object intrinsic. In robotics, Şahin et al. formalized affordances in terms of relations among entities, behaviors, and effects [4]. This prior art is important because SIO likewise treats “freedom” as query indexed: the same opening can be traversable by one probe but not by another.

The distinction is methodological. SIO focuses on exact finite combinatorics and on the extraction of minimal admissibility generators, while affordance theory provides a broader agent–environment action interpretation.

2.4 Occupancy grids

Occupancy grids represent stochastic beliefs about whether spatial cells are occupied [5]. SIO uses the word occupancy differently. It does not primarily estimate physical presence. Instead, it asks how a known constraint changes admissible actions or states. An occupancy grid can certainly be used as an input representation for an SIO-style analysis, but the two notions should not be conflated.

2.5 Simple games, monotone Boolean functions, and set-function interactions

A finite monotone binary admissibility predicate is mathematically a simple game or monotone Boolean function. Such games can be represented through minimal winning coalitions, and their representation theory is well developed [6]. Set functions admit Möbius representations and interaction indices [7], while the Shapley value gives a canonical attribution rule for cooperative games [8]. Problems of finding minimal subsets satisfying monotone predicates are also a standard topic in Boolean reasoning [9].

Accordingly, Section 7 does not introduce a new theory of simple games. It maps a spatial admissibility problem into that theory and uses the resulting language to separate direct constituents from cooperative closure.

2.6 Position of this paper

The paper therefore occupies an intersection rather than an empty field. The central thesis is not that any one of the preceding mathematical objects is new. The proposed contribution is that direct geometric support, action-dependent freedom, spatial placement constraints, minimal winning generators, and distance to admissible closure can be treated as successive layers of one container-relative problem.

A reviewer can read the framework as a set of interfaces among established formalisms:

Formalism	Primary object used here	What it already provides	Role inside SIO
Mathematical morphology	erosion / fit-position set	legal translated placements of a probe	constructs a query-specific admissible-placement layer
Configuration space	free configuration set / graph	body-dependent collision freedom and connectivity	supplies global action-space topology
Affordance theory	agent/action-relative possibility	relational interpretation of action possibility	motivates explicit query dependence
Simple games / monotone Boolean functions	minimal winning coalitions	finite representation of monotone positive states	represents minimal spatial closure sets
Möbius / Harsanyi / Shapley	interaction / attribution transforms	higher-order decomposition and one-body allocation	analyzes a generator-based spatial game

The claim is not that SIO strictly contains these fields or subsumes their general theories. Rather, SIO fixes a common data flow:

\boxed{ \text{direct support} \rightarrow \text{query-specific admissibility} \rightarrow \text{minimal cooperative closure} }

and asks which statements survive when one moves between those layers.

2.7 What would make SIO redundant?

The framework would add little if one neighboring formalism, without additional interface structure, already supplied all three of the following as one standard object:

direct geometric support as a distinguished baseline quantity;
a family of action-indexed admissibility structures for different probes or queries;
a monotone constituent-level generator representation connected back to those spatial queries.

Mathematical morphology directly addresses item 2 for translated structuring elements; configuration-space theory directly addresses item 2 and global connectivity; simple-game theory directly addresses item 3. The proposed SIO contribution is the explicit linking of the layers, not a replacement for any of them.

This also gives a direct falsification route for the novelty claim: if prior work is found that already uses essentially the same three-layer interface with the same intended role, SIO should be reframed as a specialization rather than a new synthesis framework.

3. Formal Setup

3.1 Container

We use the following abstract container only to keep the roles explicit:

\boxed{ \mathfrak C = \left( X, \{\mathcal F_x\}_{x\in X}, \mathcal R_{\mathfrak C}, \mathcal A_{\mathfrak C}, \partial\mathfrak C \right). }

Here:

$X$ is a set of addressable constituent loci;
$\mathcal F_x$ is a local admissible state set at $x$ ;
$\mathcal R_{\mathfrak C}$ contains relevant relations among loci;
$\mathcal A_{\mathfrak C}$ is an ambient action or transformation family;
$\partial\mathfrak C$ records effective boundary conditions.

For the exact finite theorems below, $X$ will usually be a square lattice.

This definition is operational. It is not a claim that the physical world is fundamentally discrete or fibered.

3.2 Constraint and direct support

Let $O$ be a spatial constraint or occupied structure. In a discrete container, its direct support is

\operatorname{Supp}(O)\subseteq X.

The direct site occupancy is

\boxed{ \eta_{\mathrm{site}}(O) = \frac{|\operatorname{Supp}(O)|}{|X|}. }

In a continuous setting, the analogous quantity may be a normalized measure $\mu(O)/\mu(X)$ .

3.3 Query-indexed admissibility

A freedom query $q$ specifies what action or configuration family is being tested. At minimum it contains an action set and an admissibility rule:

q=(\mathcal A_q,\operatorname{Adm}_q).

The admissible action set in the presence of $O$ is

\boxed{ \mathcal A_q(\mathfrak C\mid O) = \{ a\in\mathcal A_q: \operatorname{Adm}_q(a;\mathfrak C,O)=1 \}. }

If $\mathcal A_q$ is finite, we may define a normalized remaining-freedom fraction

F_q(O) = \frac{ |\mathcal A_q(\mathfrak C\mid O)| }{ |\mathcal A_q^{\mathrm{ambient}}| },

and depletion

D_q(O)=1-F_q(O).

The denominator must be fixed by the query. Quantities from different action spaces should not be compared as if they were the same scalar freedom.

For translation-only fit queries, the set $\mathcal A_q(\mathfrak C\mid O)$ can often be written directly as a morphological erosion of the free region by a reflected probe. In those cases SIO does not introduce a new placement operator; it treats the erosion result as one query-specific layer to be compared with direct support and, where relevant, with connectivity or cooperative closure.

3.4 The minimal SIO distinction

The central distinction is:

\boxed{ \eta_{\mathrm{site}}(O) \text{ does not generally determine } \mathcal A_q(\mathfrak C\mid O). }

The next section proves this with two exact families.

4. Direct Support Is Not a Sufficient Statistic

4.1 Equal-support rectangles

Let

X_N=\{0,\ldots,N-1\}^2

and consider an axis-aligned $w\times h$ rectangle $R_{w,h}$ . The query is rigid translation inside the container, without rotation.

Theorem 1 — Equal-support translation non-equivalence

The number of legal translations is

\boxed{ P_N(w,h) = (N-w+1)(N-h+1). }

Equivalently,

\boxed{ P_N(w,h) = (N+1)^2+wh-(N+1)(w+h). }

Hence, if $wh=K$ is fixed,

\boxed{ P_N(w,h) = (N+1)^2+K-(N+1)(w+h), }

so equal direct support does not imply equal translation freedom.

Proof

The lower-left anchor has $N-w+1$ legal horizontal positions and $N-h+1$ legal vertical positions. Multiplying gives the first formula. Expanding gives the second. At fixed product $wh=K$ , the count still depends on $w+h$ . ∎

Corollary 1 — Square optimum within fixed-area rectangles

If $K=s^2$ and $s\le N$ , then among positive-integer rectangles satisfying $wh=K$ , the square $w=h=s$ maximizes $P_N(w,h)$ .

Proof

At fixed $K$ , maximizing $P_N$ is equivalent to minimizing $w+h$ . By AM–GM,

w+h\ge2\sqrt{wh}=2s,

with equality exactly at $w=h=s$ . ∎

Example

For $N=64$ and $K=576$ ,

P_{64}(24,24)=41^2=1681,

whereas

P_{64}(48,12)=17\cdot53=901.

Both obstacles occupy exactly $576$ cells, but their legal placement counts differ by almost a factor of two.

This result is elementary. Its role is not theorem novelty but conceptual separation: direct support is not a sufficient statistic for placement admissibility.

4.2 Vanishing support with persistent connectivity impact

Equal-support examples show non-uniqueness at fixed measure. A stronger phenomenon is possible: direct occupancy can vanish while a global admissibility effect remains finite.

Let $N=2m+1$ . Place a one-cell-thick vertical barrier through the central column of an $N\times N$ lattice. The barrier has $N$ occupied cells, so

\eta_{\mathrm{site}}=\frac1N.

The free cells split into two components of equal size

n=mN.

Define pair-connectivity fragmentation by

\boxed{ \mathcal L_{\mathrm{conn}} = 1- \frac{ \sum_i \binom{n_i}{2} }{ \binom{M}{2} }, }

where the free connected components have sizes $n_i$ and $M=\sum_i n_i$ .

Theorem 2 — Vanishing support, non-vanishing fragmentation

For the central spanning barrier,

\boxed{ \eta_{\mathrm{site}} = \frac1N \rightarrow0, }

while

\boxed{ \mathcal L_{\mathrm{conn}} = \frac{mN}{2mN-1} \rightarrow \frac12. }

Proof

There are two equal free components, each of size $n=mN$ , so

\mathcal L_{\mathrm{conn}} = 1- \frac{ 2\binom{n}{2} }{ \binom{2n}{2} }.

Using

2\binom{n}{2}=n(n-1), \qquad \binom{2n}{2}=n(2n-1),

gives

\mathcal L_{\mathrm{conn}} = 1-\frac{n-1}{2n-1} = \frac{n}{2n-1}.

Substituting $n=mN$ gives the result. ∎

The interpretation is simple:

\boxed{ \text{small or vanishing direct measure need not imply a small global admissibility effect.} }

5. Freedom Transport and Boundary Dependence

The previous section compares static structures. We now ask what happens when a constraint is deformed while its direct support is preserved.

5.1 Directional probes

In the same $N\times N$ container, let the obstacle be a $w\times h$ rectangle. Let

P_h=L\times1

be a horizontal probe and

P_v=1\times L

a vertical probe.

In an empty container, both probes have

Q=N(N-L+1)

legal anchors.

Let $I_h$ and $I_v$ denote the number of anchors invalidated by the obstacle, and define

D_h=\frac{I_h}{Q}, \qquad D_v=\frac{I_v}{Q}.

We first assume the obstacle is sufficiently far from the container boundary that the probe–obstacle overlap region is not clipped.

Proposition 3 — Interior directional depletion

Under the probe-interior condition,

\boxed{ I_h=h(w+L-1), }

\boxed{ I_v=w(h+L-1). }

Consequently,

\boxed{ D_v-D_h = \frac{(L-1)(w-h)} {N(N-L+1)}. }

Proof

For a fixed occupied row, a horizontal probe intersects the obstacle for exactly $w+L-1$ anchor positions. There are $h$ occupied rows. The vertical formula is symmetric. Subtracting yields the anisotropy identity. ∎

5.2 Fixed-area redistribution

If $wh=K$ , then

I_h=K+h(L-1),

I_v=K+w(L-1).

Corollary 2 — Directional redistribution

Along a fixed-area deformation with $w$ increasing and $h=K/w$ decreasing,

\boxed{ D_h\downarrow, \qquad D_v\uparrow. }

At the same time, Theorem 1 gives decreasing rigid-translation freedom once the deformation moves away from the square optimum.

Thus the change is not well described by a single conserved scalar:

\boxed{ \text{one channel can release freedom while another consumes it and total placement freedom falls.} }

5.3 Boundary clipping

The interior formulas fail near the container boundary because nominally intersecting probe anchors may lie outside the legal anchor domain.

If the obstacle begins at $(x_0,y_0)$ , define

c_x = \max \left( 0, \min(N-L,x_0+w-1) - \max(0,x_0-L+1) + 1 \right),

and similarly

c_y = \max \left( 0, \min(N-L,y_0+h-1) - \max(0,y_0-L+1) + 1 \right).

Proposition 4 — Boundary-clipped directional counts

For any legal rectangle placement,

\boxed{ I_h=h\,c_x, \qquad I_v=w\,c_y. }

Defining clipping deficits

\beta_x=(w+L-1)-c_x,

\beta_y=(h+L-1)-c_y,

gives

\boxed{ I_v-I_h = (L-1)(w-h) + h\beta_x-w\beta_y. }

This decomposition separates an interior shape term from an explicit boundary correction.

The point is again organizational. Boundaries are not merely visible contours; they modify admissibility laws.

6. Released Substrate Is Not Yet Realized Freedom

We now consider a barrier with a controllable opening.

6.1 Gap model

Let a one-cell-thick vertical barrier span the container except for a contiguous gap of height $g$ . Let the probe be an axis-aligned rectangle of width $w_p$ and height $h_p$ .

A cross-barrier placement requires the probe’s entire vertical span to fit inside the gap.

Theorem 5 — Aperture capacity

The number of vertical anchor positions that fit the transverse probe span inside the gap is

\boxed{ C_{\mathrm{ap}}(g;P) = \max(0,g-h_p+1). }

Proof

If $g<h_p$ , no length- $h_p$ integer interval fits inside a length- $g$ interval. If $g\ge h_p$ , the lower probe anchor can start at any of $g-h_p+1$ positions. ∎

If the barrier is far enough from the left and right container boundaries, the probe has $w_p$ horizontal anchors whose support intersects the barrier column. Therefore:

Corollary 3 — Cross-barrier placement count

Under side-clearance assumptions,

\boxed{ N_{\mathrm{cross}} = w_p \max(0,g-h_p+1). }

In particular,

N_{\mathrm{cross}}>0 \iff g\ge h_p.

6.2 Configuration-space transition

Let the nodes of a graph be all legal probe translation anchors, with unit horizontal or vertical translation defining adjacency.

Assume:

legal probe placements exist on both sides of the barrier;
the gap is away from the top and bottom container boundaries;
the barrier is a single spanning vertical strip;
the probe translates without rotation.

Theorem 6 — Critical gap for left-right connectivity

The configuration graph has

\boxed{ \#\pi_0 = \begin{cases} 2,&g<h_p,\\ 1,&g\ge h_p. \end{cases} }

Thus

\boxed{ g_c(P)=h_p. }

Proof

If $g<h_p$ , Theorem 5 gives no legal placement that crosses the barrier line, so left and right anchor regions are disconnected. Each side is itself orthogonally connected.

If $g\ge h_p$ , at least one vertical probe anchor fits inside the gap. Under side-clearance assumptions, a sequence of legal unit horizontal translations carries the probe from the left side through the aperture to the right side. Each side is internally connected, so the full graph is connected. ∎

6.3 Latent freedom

For

0<g<h_p,

some barrier cells have been released, yet the specified cross-barrier action remains impossible.

This motivates a three-stage vocabulary:

\boxed{ \text{released substrate} \rightarrow \text{latent freedom} \rightarrow \text{realized operational freedom}. }

The terminology is descriptive. The next section gives it a standard combinatorial representation.

Theorem 6 should also be read with the correct novelty scope. Narrow-passage clearance and body-dependent connectivity are standard configuration-space concerns. The theorem is included because it gives an exact, auditable transition inside the present hierarchy: direct substrate release can occur strictly before the queried action becomes feasible.

7. Minimal Spatial Admissibility Generators

7.1 Monotone spatial predicates as simple games

Let $W$ be a finite set of candidate released or activated constituents. A query defines a binary property

V_q:2^W\rightarrow\{0,1\}.

Assume monotonicity:

A\subseteq B, \quad V_q(A)=1 \Rightarrow V_q(B)=1.

Mathematically this is a simple game / monotone Boolean function [6]. We use that standard theory and give its minimal winning coalitions a spatial interpretation.

Definition 1 — Minimal spatial admissibility generator

A set $G\subseteq W$ is a minimal spatial admissibility generator if

V_q(G)=1

and every proper subset $H\subsetneq G$ satisfies

V_q(H)=0.

Let

\mathcal G_q

be the family of all such sets.

Proposition 7 — Standard minimal-generator representation

For every finite monotone spatial admissibility predicate,

\boxed{ V_q(A)=1 \iff \exists G\in\mathcal G_q: G\subseteq A. }

Proof

If $G\subseteq A$ and $G$ is winning, monotonicity implies $A$ is winning. Conversely, if $A$ is winning, finiteness guarantees an inclusion-minimal winning subset $G\subseteq A$ . ∎

This proposition is standard simple-game / monotone-Boolean mathematics [6]. Its role here is interpretive: it identifies minimal winning coalitions as the constituent-level carrier of a spatial admissibility query.

7.2 Latent closure as a Hamming deficit

The previous gap example asks how far a released set is from supporting a specified action.

Definition 2 — Latent closure deficit

For nonempty $\mathcal G_q$ , define

\boxed{ d_q(A) = \min_{G\in\mathcal G_q} |G\setminus A|. }

This is the minimum number of additional constituents needed to complete some minimal admissible generator. It is naturally interpreted as a Hamming deficit to the nearest minimal winning set and is closely related to distance-to-satisfaction, repair, and minimal-set reasoning for monotone predicates [9]. We use the term latent closure only for the spatial operational interpretation; the underlying distance construction is not claimed to be new.

Proposition 5 — Zero deficit characterizes realized admissibility

\boxed{ d_q(A)=0 \iff V_q(A)=1. }

Proof

$d_q(A)=0$ exactly when some $G\in\mathcal G_q$ satisfies $G\subseteq A$ . Apply Theorem 7. ∎

Proposition 6 — Monotonicity and one-site gradient

If $A\subseteq B$ , then

d_q(B)\le d_q(A).

For $i\notin A$ , define

\lambda_i(A) = d_q(A)-d_q(A\cup\{i\}).

Then

\boxed{ \lambda_i(A)\in\{0,1\}. }

Proof

Adding constituents cannot increase $|G\setminus A|$ for any generator $G$ , hence cannot increase the minimum. Adding one constituent changes each deficit by at most one, so the minimum drops by at most one. ∎

The importance of $\lambda_i(A)$ is contextual: the same constituent can be irrelevant in one state and closure-critical in another.

8. Cooperative Apertures as a Worked Example

The generator viewpoint becomes concrete in a one-dimensional aperture family.

Let

W=\{0,\ldots,M-1\},

and let

\mathcal W_h = \{ \{j,j+1,\ldots,j+h-1\}: 0\le j\le M-h \}

be all contiguous windows of length $h$ .

Define the aperture capacity

\boxed{ C_h(A) = \sum_{J\in\mathcal W_h} \mathbf 1[J\subseteq A]. }

This counts how many complete length- $h$ aperture windows are realized by $A$ .

8.1 Non-additivity of isolated constituents

If $h>1$ , every singleton has

C_h(\{i\})=0,

but each legal window $J\in\mathcal W_h$ has

C_h(J)=1.

Hence

\boxed{ C_h(J) \neq \sum_{i\in J}C_h(\{i\}). }

The effect is cooperative rather than additive over isolated constituents.

8.2 Standard Möbius/Harsanyi representation

For a set function $f$ , let

m_f(B) = \sum_{A\subseteq B} (-1)^{|B|-|A|}f(A)

be its subset-lattice Möbius transform [7].

Since

C_h = \sum_{J\in\mathcal W_h}u_J

is a sum of unanimity games, standard Möbius inversion gives

\boxed{ m_{C_h}(B) = \begin{cases} 1,&B\in\mathcal W_h,\\ 0,&\text{otherwise}. \end{cases} }

Thus the irreducible interaction order in this model is exactly $h$ .

We emphasize that the Möbius transform itself is standard. The spatial consequence is that the transverse size required by the probe becomes the cooperative order of the admissibility closure.

8.3 Standard Shapley allocation

Let $n_i(h)$ be the number of legal windows containing constituent $i$ . Shapley linearity and the symmetry of unanimity games [8] yield

\boxed{ \phi_i(C_h) = \frac{n_i(h)}{h}. }

This provides a one-body attribution field over a higher-order cooperative structure. It should not be confused with the structure itself.

9. Cross-Layer Case Study: Compatible Gates and a Crossover

The previous sections deliberately use minimal geometries so that each theorem can be audited analytically. A reasonable objection is that the framework could still be unnecessary if each example is fully described by a single neighboring formalism.

This section therefore considers a finite system in which the analysis naturally moves through several layers.

9.1 Workspace

The workspace is a $31\times21$ lattice. A rigid $2\times2$ square probe translates without rotation from a start anchor on the left to a goal anchor on the right.

Two one-cell-thick vertical barriers span the workspace. The chamber between them is split into top and bottom lanes by a horizontal separator. Ten barrier cells are designated as releasable and grouped into five two-cell gates:

A_T,\quad A_B,\quad B_T,\quad B_B,\quad C.

The first four are upper/lower gates in the two vertical barriers. $C$ is a crossover gate in the horizontal separator.

Because the probe has height and width $2$ , both cells of a gate pair must be released before the probe can cross the corresponding one-cell-thick wall.

All:

2^{10}=1024

release states are enumerated exactly.

9.2 Support layer

A release state is:

A\subseteq W, \qquad |W|=10.

The most basic descriptor is:

|A|.

Consider two states.

The aligned state releases:

A_T+B_T.

The mismatched state releases:

A_T+B_B.

Both release exactly:

\boxed{ 4 }

cells.

So direct release cannot distinguish them.

9.3 Placement layer

For each release state, let the free workspace be eroded by the reflected $2\times2$ probe. Equivalently, enumerate all legal probe anchors.

For both the aligned and mismatched states, the exact legal-anchor count is:

\boxed{ 508. }

Thus even the aggregate morphological placement count agrees:

\boxed{ |A_1|=|A_2|=4, \qquad |\mathcal A_q(A_1)| = |\mathcal A_q(A_2)| = 508. }

This does not mean morphology is insufficient. The full spatial fit set retains more information than its cardinality. The point is that neither direct support nor the aggregate number of fits determines the global route property.

9.4 Configuration-space layer

Connect legal anchors by unit horizontal or vertical translations.

For the aligned state:

A_T+B_T,

the start and goal lie in the same connected component.

For the mismatched state:

A_T+B_B,

they do not. The middle horizontal separator prevents a change from the upper lane to the lower lane.

Hence:

\boxed{ V_q(A_T+B_T)=1, }

while:

\boxed{ V_q(A_T+B_B)=0. }

The two states therefore agree in direct release count and aggregate legal-placement count but differ in global action feasibility.

9.5 Generator layer

Define one macro Boolean variable for each two-cell gate; a gate variable equals $1$ only when both of its constituent cells are released.

Let:

a_T,\ a_B,\ b_T,\ b_B,\ c

represent the five gate groups.

Exhaustive configuration-space enumeration yields exactly:

\boxed{ V = (a_T\land b_T) \lor (a_B\land b_B) \lor (a_T\land c\land b_B) \lor (a_B\land c\land b_T). }

Therefore the minimal spatial admissibility generators are:

\boxed{ A_T\cup B_T, }

\boxed{ A_B\cup B_B, }

\boxed{ A_T\cup C\cup B_B, }

\boxed{ A_B\cup C\cup B_T. }

At the underlying cell level their sizes are:

\boxed{ 4,\ 4,\ 6,\ 6. }

The first two are same-lane routes. The latter two are cross-lane routes that require the crossover.

9.6 Exact state count

A closed two-cell gate has three microstates:

00,\quad01,\quad10,

whereas an open gate has one:

11.

There are $16$ winning macro gate patterns. Weighting each pattern by:

3^{\#\text{closed gates}}

gives:

\boxed{ 142 }

winning microstates among:

1024.

Direct BFS enumeration of the configuration graph gives exactly the same total.

9.7 Latent closure structure

Using the four minimal generators above,

d_q(A) = \min_{G\in\mathcal G_q} |G\setminus A|

satisfies:

d_q(A)=0 \iff V_q(A)=1

for all $1024$ states.

Even within the fixed release cardinality:

|A|=4,

the exact distribution is:

$2$ states with $d_q=0$ ;
$48$ states with $d_q=1$ ;
$152$ states with $d_q=2$ ;
$8$ states with $d_q=3$ .

So released-cell count does not determine distance to operational closure.

9.8 Constituent attribution

The exact Shapley values of the connectivity game are:

for the eight cells belonging to the four vertical-barrier gates,

\boxed{ \phi_i=\frac{7}{60}, }

and for the two crossover cells,

\boxed{ \phi_i=\frac{1}{30}. }

The lower crossover attribution reflects that it is optional: it is required only for the two cross-lane alternatives.

Again, this one-body allocation does not replace the four minimal routes.

9.9 What the case study shows

The case can be summarized as:

\boxed{ \text{same direct release} + \text{same aggregate fit count} \not\Rightarrow \text{same global action feasibility}. }

A natural analysis proceeds through:

\boxed{ \text{workspace/support} \rightarrow \text{legal placements} \rightarrow \text{configuration connectivity} \rightarrow \text{minimal release generators}. }

Each arrow uses a standard mathematical formalism. The point of SIO is the explicit organization of the interfaces.

10. An SIO Hierarchy for Finite Spatial Admissibility

The preceding results suggest a hierarchy rather than a single scalar.

For a fixed query $q$ , a minimal finite description may distinguish:

Layer 1 — Direct support

\eta_{\mathrm{site}}(O).

Layer 2 — Action-dependent admissibility

\mathcal A_q(\mathfrak C\mid O), \qquad F_q, \qquad D_q.

Layer 3 — Connectivity or topology

For example,

\mathcal L_{\mathrm{conn}}

or the component structure of a configuration graph.

Layer 4 — Minimal cooperative closure

\mathcal G_q.

Layer 5 — Distance to closure and cooperative interaction

d_q(A), \qquad m_q(B), \qquad \phi_i.

The main purpose of SIO is to keep these layers distinct.

A direct support measure can be an important aggregate without determining the action-dependent layers above it. A one-body influence score can summarize cooperative structure without replacing it. A geometrically free cell can exist without yet completing an admissible action.

11. Discussion

11.1 What is standard and what is being proposed

The constituent mathematics in this paper has substantial prior art.

Legal placements and forbidden configurations are standard in configuration-space geometry [1].
Probe fitting and translated templates are central to mathematical morphology [2,3].
Action-relative possibilities are closely related to affordance theory [4].
Grid occupancy is well established in robotics [5].
Minimal winning coalitions and simple-game representations are standard [6].
Möbius/Harsanyi interaction and Shapley attribution are standard set-function and cooperative-game tools [7,8].
Minimal satisfying sets for monotone predicates are a known Boolean-reasoning problem [9].

The proposed contribution is not to rename these objects as new mathematics. It is to use them as successive layers of one explicit spatial-admissibility question:

\boxed{ \text{How does a container-relative constraint modify the structure of admissible possibilities?} }

11.2 Why use the word “occupancy”?

The term is potentially confusing because occupancy grids already have a well-established meaning [5].

We retain “occupancy” because the motivating comparison is between direct support and a richer action-dependent structure. However:

\boxed{ \text{SIO occupancy is not a probability that a cell is physically occupied.} }

It refers to how a constraint commits or removes admissible possibilities relative to a specified query.

If future literature comparison shows that the term produces more confusion than clarity, “container-relative spatial admissibility” may be the preferable primary label.

11.3 Relationship to affordances

The query-relative nature of SIO is compatible with affordance theory. An opening that is traversable by a point probe but not by a larger rectangular probe is an action-relative property.

SIO adds no claim that this relationality is newly discovered. Its focus is instead on exact finite representations and on extracting minimal cooperative generators from such action-relative constraints.

11.4 Why minimal generators are useful

The generator family $\mathcal G_q$ separates a positive state from the specific minimal closures that make it positive.

This is useful because the same direct number of released constituents can have different admissibility consequences. It also allows a standard cooperative analysis of which subsets act jointly rather than independently.

11.5 Why latent closure is not an energy

The quantity

d_q(A)

is a discrete Hamming deficit. It is not a physical potential energy, and the word “latent” should not be read as implying hidden physical substance.

It answers only:

How many additional constituents are minimally required to reach some admissible generator?

11.6 Limits of monotonicity

The generator representation in Section 7 assumes monotonicity. This is appropriate when “releasing more cells” or “adding more resources” cannot destroy the queried action.

Many systems are non-monotone. Additional constituents can create congestion, interference, incompatibility, or new constraints. Such systems require richer structures than minimal winning hypergraphs.

11.7 Finite lattices are models, not ontological commitments

All exact examples in this paper are finite combinatorial models. No theorem here establishes that physical space is fundamentally discrete.

The framework can be useful even if the underlying physical system is continuous, provided the finite model is understood as a representation or discretization.

11.8 Three reviewer tests

The framework should survive three reduction tests.

Morphology test

If the question is only “where can a translated probe fit?”, mathematical morphology is already the appropriate complete language. SIO adds no new primitive at that level.

Configuration-space test

If the question is only “what is the free configuration space and is it connected?”, configuration-space geometry is already the appropriate language.

Simple-game test

If the spatial origin of the predicate is ignored and only the finite monotone set function remains, simple-game / Boolean-function theory is already the appropriate language.

SIO is useful only when a problem requires moving between these descriptions—for example, comparing direct support to a morphological fit set, then asking which released constituents minimally generate a configuration-space transition. Section 9 supplies an exact instance: two states have the same released-cell count and the same number of legal placements but different configuration-space connectivity, and the full release logic is then summarized by four minimal winning coalitions. This cross-layer requirement is the intended scope of the framework.

11.9 Novelty claim as an interface claim

Accordingly, the strongest claim made by this paper is an interface claim:

\boxed{ \text{support layer} \rightarrow \text{action layer} \rightarrow \text{generator layer} }

The exact finite theorems serve as consistency witnesses for the interfaces. They should not be interpreted as assertions that the underlying component theories were previously unavailable.

12. Reproducibility

All core finite formulas in this submission are accompanied by an exact audit script.

The script verifies:

equal-support rectangle translation counts;
the spanning-barrier fragmentation sequence;
directional probe-depletion identities;
boundary clipping counts;
gap capacity and the critical gap;
minimal-generator reconstruction for a monotone spatial game;
latent-closure properties;
the contiguous-aperture Möbius and Shapley formulas;
all $1024$ states of the cross-layer two-barrier case, including the $508$ -placement aligned/mismatched comparison and the four minimal winning coalitions.

The computational audit is not a substitute for the analytical proofs in the text. It is included to catch transcription and indexing errors.

13. Limitations and Future Work

This first submission deliberately stops before the two most structurally ambitious extensions.

First, it does not yet quotient away representation redundancy among generator hypergraphs. A companion study develops an isomorphism-class representation and distinguishes lossless recoding from coarse projection.

Second, it does not yet treat cross-resolution generator systems or general two-dimensional placement fields. Those extensions require direct comparison with multiresolution morphology, rough approximation, graph/hypergraph coarsening, digital geometry, and covariogram theory.

Other immediate directions include:

non-monotone spatial admissibility;
continuous rotations and full $SE(2)$ configuration spaces;
nonconvex and dynamic containers;
stochastic admissibility;
machine-checked proofs;
connections to viability theory and control synthesis.

14. Conclusion

A geometric measure can be exact and still be incomplete for an action question. The cross-layer case study further shows that even the aggregate number of legal placements can be insufficient: two release states with four opened cells and $508$ legal probe anchors can differ in start-to-goal connectivity.

Equal-support shapes can admit different numbers of translations. A barrier whose occupied fraction tends to zero can maintain a finite global connectivity effect. Fixed-area deformation can release one action channel while consuming another. A geometrically open gap can remain operationally closed until the probe fits. A set of individually insufficient constituents can jointly complete an admissible closure.

These observations can be organized by a simple principle:

\boxed{ \text{direct support} \neq \text{the structure of admissible possibilities}. }

For finite monotone problems, the latter can be represented by standard simple-game machinery through minimal spatial admissibility generators:

\boxed{ V_q(A)=1 \iff \exists G\in\mathcal G_q:\;G\subseteq A. }

The Hamming deficit

\boxed{ d_q(A)=\min_{G\in\mathcal G_q}|G\setminus A| }

then gives a precise notion of distance to admissible closure.

We therefore propose Spatial Information Occupancy as a synthesis framework for studying spatial constraints through the admissible possibilities they remove, preserve, redistribute, or jointly generate. Its contribution is not the invention of the established mathematical tools used here. It is the explicit interface

\boxed{ \text{direct support} \rightarrow \text{action-dependent admissibility} \rightarrow \text{minimal cooperative closure} }

together with exact finite examples showing why those layers should not be collapsed into one quantity.

Acknowledgments and Research Assistance

This manuscript was developed with AI-assisted drafting, symbolic checking, and computational verification using OpenAI ChatGPT. All mathematical claims, literature positioning, and publication decisions remain the responsibility of the human author.

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