# Minimal Generative Grammar of Participation: formal architecture

**Status:** a new constructive reconstruction, not a proposition attributed to a historical author. Its proofs are short semantic consequences of stated assumptions; the independence witnesses are schematic mathematical models, not demonstrations that the corresponding metaphysical positions are true.

## The result and its limit

The proposed three-part basis—actuality, received dependence, measure—does not suffice. An asymmetric dependency graph contains neither qualitative resemblance nor information about intrinsic constitution. A scalar measure cannot manufacture either. Underivation does not by itself entail infinity, and infinity does not entail freedom. Adding these conclusions to the meaning of “source” would conceal the problem.

There is also no language-independent smallest number of primitives or axioms. A conjunction can package nine commitments as one; a suitably indexed relation can encode several predicates. The defensible result is **a source-specific basis in which each assumption contributes something the others do not supply, within an explicitly fixed language**, with clearly identified extensions. It does not establish an absolute lower bound over every possible metaphysical language. Generic participation requires fewer assumptions about its source than the fuller Christian account does.

## Language, categories, and definitions

Use entities or actual intrinsic principles `x,y`; descriptive positive-perfection indices `q`; modes `m`; and possible situations `w`. Let `e` index present existential actuality and let `s` designate the proposed divine source. The five operational predicate families are:

| Primitive | Meaning |
|---|---|
| `C_w(y,x,q)` | y productively causes or sustains x's q-actualization at w. |
| `D_w(x,y,r)` | x ontologically depends on y in respect r at w. |
| `H_w(x,q,m)` | x realizes positive content q in mode m at w. |
| `K_w(x,y)` | x is a proper intrinsic constituent or constituting principle of y. |
| `J_w(x)` | x's intrinsic constitution and qualitative actuality, as distinct from merely relational description. |

`C` is not silently identified with `D`. Constitutional dependence is not efficient production. `K` is not likeness, and `H` is not epistemic evidence. These are ontological assertions, not claims about our ability to recognize their truth. Actuality is represented semantically within H; use the derived abbreviation `A_w(x) := ∃m H_w(x,e,m)`. There is no independent A-predicate to count twice. J independently represents the full intrinsic profile because selected positive-perfection realizations in H are not assumed to exhaust every intrinsic fact. If H/K are enriched to exhaust that profile, J becomes a definable convenience and should be deleted.

The language additionally needs identity, quantification, world indexing, and a mode ordering. Those are disclosed resources, not conclusions produced by the five predicates. Modes are functionally indexed to a bearer and perfection: there is at most one complete mode for a given `(w,x,q)`. Let `∞` mark absence of a finite upper bound and let every bounded mode lie strictly below it. The ordering represents realization-capacity in the relevant respect, **not a quantity of divine substance**. Natural numbers plus `∞` provide a toy interpretation, not a physical measurement of God.

Perfection indices are descriptive indices, not independently existing exemplars. This convention does not refute Platonism. If independent Forms are alleged to be real productive sources, they must be represented as actual entities and tested against universal sustaining causality. Purely mathematical or causally inert abstract objects are not decided here. A proposed independent admissibility test is that q's specified positive content need not entail deficiency or a bounded constitutive mode: knowing rather than acquiring information discursively, goodness rather than a privation, actuality rather than being-created. This test must be argued case by case; it is not “whatever can be attributed to God.” Biological life and divine life cannot be equated by replacing them with the same symbol. The interpretation and sufficiency of this criterion remain substantive open work, not achievements of the notation.

All productive and dependency relations are factive: their relata have existential realization. Every other positive realization entails existential realization. Proper constituency is irreflexive and transitive. These are substantive background assumptions, not logical truths. Claims of nine-clause irredundance are relative to them; nine is not a count of every assumption. FS posits an actual proposed source `s`; there is no separate demonstration that such a source exists or that creation occurs.

Define, without importing the target conclusions:

- `Rec_w(x,q,m,y) := C_w(y,x,q) ∧ H_w(x,q,m)`.
- `L_w(x,y,q) := ∃m,n[H_w(x,q,m) ∧ H_w(y,q,n)]`. This is shared positive content, not identity of mode, species, or essence.
- `P_w(x,y,q) := ∃m[Rec_w(x,q,m,y)] ∧ D_w(x,y,q)`.
- `E_w(x,y,q) := P_w(x,y,q) ∧ L_w(x,y,q)`; expressive participation adds positive likeness. Likeness is not contained in P and must be obtained from Q for source-participation.
- `U_w(y) := ¬∃x,r D_w(y,x,r)`; underivation is a defined negative profile, not a primitive containing infinity or simplicity.
- Communication is productive realization of some positive content in another; expression adds real positive likeness as defined by E. Mere observer interpretation is insufficient for either.
- Exhaustion in q requires a receiver's realization-capacity to reach that of its source. The capacity-order gives a **necessary**, not sufficient, condition for an adequate total representation.
- Causal intimacy means present sustaining causality in existential actuality, not spatial proximity. Transcendence below means underivation and unbounded actuality, not remoteness.

Reception and participation therefore need not be primitive. Measure remains independent of a bare dependency graph: relabeling all its nodes with equal capacities leaves every edge unchanged. An elaborate modal dependence relation could encode capacity, but would retain its content under another name.

## Nine substantive clauses

Within the scoped ontology, every actual entity other than `s` is a created actuality. This is a **universal sustaining-causality hypothesis**, not a definition from which created dependence is magically deduced. Unknown or rival beings can be put into the domain and tested against it. The ontology models source and created being; it does not treat divine hypostases as three additional beings or decide subsistent Trinitarian relations, the Incarnation, or grace. These assumptions about the source neither derive nor refute those doctrines.

| Label | Independent commitment |
|---|---|
| G: universal sustaining causality | For every actual `x ≠ s` and every `H(x,q,m)`, `C(s,x,q)`. |
| CP: production entails dependence | `C(y,x,q) → D(x,y,q)`. |
| U: source underivation | `¬∃y,r D(s,y,r)`. |
| Q: qualitative source adequacy | `C(s,x,q) ∧ H(x,q,m) → ∃n H(s,q,n)`. |
| KD: constitution entails dependence | `K(z,y) → D(y,z,e)`. |
| KH: constitutional capacity inheritance | `K(z,y) ∧ H(z,e,m) ∧ H(y,e,n) → m ≤ n`. |
| FS: source unboundedness | `H(s,e,∞)`. |
| FC: created finitude | Every existential mode `H(x,e,m)` of an actual `x ≠ s` is bounded. |
| X: unchanged source across creation alternatives | Across admitted situations, differences in the source’s created effects and its relations to them add no difference to `J(s)`. |

G, U, FS, and FC are substantive assumptions about the divine source and created recipients. They do not follow from generic participation. CP, Q, KD, and KH are bridges between independently interpreted categories. X is an additional modal commitment about intrinsic actuality, needed for the strongest mixed-relation account; omit it from a merely static core. X does **not** stipulate identical object-directed intention in different worlds: it distinguishes intrinsic qualitative actuality from object-relative determination. Whether this distinction adequately models willing is a further theological problem.

Q deliberately does not include a universal inequality between cause and effect modes. Such an inequality would imply that an unlimited source could not depend productively on a bounded cause, rather than supplying a neutral likeness bridge. The present Q is restricted to the proposed source; it does not require every instrument to intrinsically exhibit every positive effect-perfection. It asserts genuinely shared positive source-content, under potentially different modes, and must receive an analogical interpretation warranted by the historical evidence. FS and FC concern existential actuality e. For unrestricted perfection-by-perfection nonexhaustion, separately extend them to every relevant q. Those all-q profiles are not derived merely by labeling arbitrary H-predicates “perfections.”

Likewise, generic asymmetry is not an extra axiom here. CP and U already yield asymmetric source–effect dependence. A global prohibition on cycles between other entities would be an additional theory of grounding, not needed for these source-specific theorems.

## Derivation tree and theorem status

First, G and CP give `D(x,s,q)` for every realized created perfection. U forbids `D(s,x,r)` in every respect. If a purported effect of s were s itself, CP would contradict U. Thus source and effect are genuinely distinct and dependence is asymmetric.

Second, G and Q give `H(s,q,n)` for every positive created realization `H(x,q,m)`. For existential actuality e, FS fixes `n=∞`; FC makes `m<∞`. This proves positive likeness and, in e, unlike modes and a strict failure of the necessary condition for exhaustive representation. It does not show that finite language comprehends the source's mode. For any other q the capacity conclusion is conditional on the corresponding unbounded-source/bounded-recipient profile.

Third, KD and U imply that s has no proper actual intrinsic constituent: any such constituent would ground a source-dependence. This is the stated notion of simplicity, not a proof that every conceptual or formal distinction is composition. If s were a constituent of x, KH would require `∞ ≤ m_x`; FC contradicts this. Therefore neither a created entity in s nor s in a creature is an intrinsic component.

```text
G + CP + U ── asymmetric dependence and nonidentity
G + Q ─────── positive causal likeness
G + Q + FS + FC ── finite, nonexhaustive expression
KD + U ────── no intrinsic constituents in the source
KH + FS + FC ── source not an intrinsic constituent of a creature
these branches together ── nonpartitioned causal participation
above + X ─── changing effects without source-constitution
above + N ─── no shared natural genus in the essentialist sense
above + alternatives + intentional selection ── scope for free creation
```

| Target | What is obtained |
|---|---|
| T1 Participation without partition | G + CP give participation for realized created content; Q additionally supplies causal likeness. KD/U forbid divine proper parts; KH/FS/FC forbid the source as an intrinsic constituent of a creature. |
| T2 Likeness without identity | G/Q give likeness; CP/U forbid a source's productive effect being itself. |
| T3 Likeness without common genus | Shared content does not entail genus. Actual exclusion of a natural genus additionally needs N below. |
| T4 Dependence without reciprocal completion | G/CP/U give real one-way dependence. “Acquired from the effect” would be a dependence and violates U. |
| T5 Causal intimacy without constituent identity | G gives present existential sustenance; the two constituency arguments exclude incorporation. |
| T6 Transcendence without distance | U/FS express transcendence; G expresses present causality. No spatial-distance entailment exists. Actual spacelessness requires a separate premise. |
| T7 Finite manifestation without exhaustive representation | G/Q/FS/FC show that a finite actual creature cannot exhaust the source's total existential actuality. A perfection-specific capacity claim requires its own bounded/unbounded profile. |
| T8 Real communication without source-completion | G/Q produce actual positive realization; U excludes received source-perfection. X excludes intrinsic alteration alongside differences in created effects. |
| T9 Real perfection without independent Forms | G/Q explain creaturely perfection through the actual source; any independently actual causal Form violates G/CP/U if it claims underivation. Inert abstracta remain open. |
| T10 Immanence without pantheistic identity | G supplies present sustaining causality; CP/U and the constituency exclusions block identity and intrinsic incorporation. Secondary causes are not thereby excluded. |
| T11 Mixed relation | G/CP make creaturely dependence real; U/KD/X deny source constitution by the creature. True source-directed predications need not be intrinsic divine accidents. |
| T12 Receiver's finite measure does not limit source | Bearer-indexed H attributes the mode to x; Q gives source-content without setting its mode equal to x's. FS fixes the source's existential mode. Indexing alone would not prove infinity in other q. |
| T13 Many effects without intrinsic source-multiplicity | KD/U exclude constituting source-parts; X permits different and multiple created effects with unchanged intrinsic source actuality. Many predicates or causal edges are not parts. |
| T14 Finite cannot exhaust unlimited unreceived source | FS/FC supply the existential inequality. In any q, the theorem is conditional on unlimited source/bounded receiver in q. U alone proves nothing about exhaustion. |
| T15 No particular finite expression necessitated by fullness | **Not derivable.** Nonadequacy defeats one proposed ground of necessity, not all grounds of necessity. |

For strong T3, add **N, a genus-constitution bridge**: membership in a natural genus, as determined by a proper intrinsic differentiating principle, involves actual constitutive principles. A source with no such constituents cannot belong to a natural genus in that sense. This is a substantive essentialist account of genus, not pure logic. A common concept such as “actual” remains possible. The base does not rule out every univocal descriptive predicate.

## Deletion and independence

The following witnesses use `s,a,b`, empty constituency unless stated, and modes `1,∞`. Supply actuality and existential realizations at every present node. Use source mode `∞`, creature mode `1`, causation or sustaining causality for every creaturely realization, corresponding D-edges, and constant source J unless the deleted clause requires otherwise. Source-mode function and proper-constituency conditions remain respected.

| Delete | Model satisfying the remaining clauses | Lost consequence |
|---|---|---|
| G | An actual finite a has no source-causal or D-edge. | Universal participation, causal intimacy, exclusion of independent actual sources. |
| CP | All source-causal edges remain; D is empty; K is empty. | Causality no longer establishes real ontological dependence. |
| U | Add `D(s,a,e)` and `K(a,s)`; retain `D(a,s,e)`. Capacity inheritance is `1≤∞`. | Source is composed and reciprocal dependence/completion is permitted. |
| Q | Give a another positive q, source-cause that q, but give s only existential content e. | Causation no longer implies qualitative likeness. |
| KD | Add `K(a,s)` but no source-dependence. The capacity ordering still holds. | Underivation fails to establish simplicity/nonpartition. |
| KH | Add `K(s,a)` and its required `D(a,s,e)`. | Unlimited source can be a constituent of a finite recipient. |
| FS | Make source and creature modes both `1`; optionally add `K(s,a)`. | Underivation does not imply infinity or nonexhaustion. |
| FC | Make a's mode `∞`, while a remains other than and dependent on s. | Source infinity does not imply every dependent expression is bounded. |
| X | In two situations, s alone and s with a, retain all local clauses but give s an additional intrinsic positive r-realization only in the second situation, and let `J(s)` record it. | Static simplicity/underivation do not settle intrinsic invariance under differing effects. |

These witnesses establish clause independence in the chosen sparse semantics. They do not establish historical endorsement, full theological satisfiability, or minimality across alternative signatures. They also expose why neither nonpartition nor likeness should be assumed in the definition of “communication.”

Primitive deletion has corresponding failures. Without H, identical causal graphs can represent resemblance or unrelated qualities. Without K, the graph cannot distinguish incorporation from external production. Without D, production and constitution have no explicitly represented grounding status. Without C, the model cannot distinguish efficient production from constitutive or merely logical dependence. Without ordered modes, the same qualitative graph admits equal, limited, or unlimited realizations. Without J/world indexing, modal claims about the full intrinsic profile are inexpressible unless H/K have first been made exhaustive. A has already been removed by explicit definition; actuality remains primitive in the meaning of H rather than independently duplicated.

## Rival ontologies and freedom

The base excludes a divine substance partition and independent actual productive Forms. It also excludes a deism that denies present existential sustaining causality. A temporally initial but continuously sustaining Creator is compatible. It does not exclude all views called panentheism: a relational use of “in God” may satisfy the theory, while world-as-divine-constituent does not.

**Occasionalism remains compatible.** Actual secondary agency requires the further existential commitment that some created b productively causes a in a relevant respect. Add `C(b,a,q)` alongside `C(s,a,q)`; the base has no exclusive-cause rule and both can hold. This establishes compatibility with noncompetitive causality. A full account of secondary causal powers needs further theory.

**Necessary emanation also remains compatible.** Every admitted situation can contain exactly the same created a, despite its failure to exhaust s. Nothing in the base says that an effect must equal its source to be necessary. Purely immanent causal theories are excluded only by the substantive U/FS assumptions about the source, not by causal syntax alone. An idealism preserving genuine created otherness and the specified relations is not excluded merely by its label.

For T15 distinguish three extra steps:

1. **Modal alternatives:** the relevant effect is absent in some admitted situation with the same source intrinsic fullness. This establishes contingency of that effect, not freedom. To assert that no particular finite expression is necessary, provide an effectwise alternative for every such expression, or one admitted no-creation situation with the same intrinsic source actuality. Merely varying some other creature would not suffice.
2. **Intentional agency:** the source understands alternative possible creations and acts through volition. This establishes agency, not by itself freedom from necessitation.
3. **Absence of a necessitating rule:** neither intrinsic nature, external dependence, nor a unique-maximization constraint necessitates that effect. Specify the intended account of freedom; do not assume that “will” entails libertarian choice.

X is compatible with one situation and so cannot supply step 1. Identical intrinsic fullness across different situations need not mean identical intentional content. Distinguishing intrinsic act from object-relative determination is a proposed modeling resource, not an already demonstrated solution to divine freedom. It does not independently settle the truthmaking objection: what makes the source will this creation rather than another? Nor does formally stipulating intrinsic invariance answer the modal collapse objection that the necessity of God’s act makes creation necessary.

## Where Explanation Finally Stops

The reconstruction is conceptually relational but ontologically asymmetric. One may learn “source” by reflecting on effects, while the source is not thereby caused or constituted by those effects. Epistemic ascent is not reverse ontological dependence.

There is no definitional cycle: Rec uses C and H; P uses Rec and D; L uses H; E uses P and L; U uses D. None returns to P. Explanatory recursion occurs when created realization clarifies how we use “actuality,” while actuality is needed to describe realization. That mutual articulation can be benign, but it is not a proof of the bridges. A network of constraints replaces a verbal circle only if the relations retain independently assessable meanings.

Explanation finally stops at actual positive realization, productive and constitutive dependence, qualitative content, and mode, together with the substantive assumptions about the source. “Unreceived actuality” is actuality without receipt; it does not explain why actuality exists. Calling God “actuality itself” cannot substitute for an argument that an underived source exists or possesses the proposed profile. The stopping point is therefore primitive actuality **and independent bridges**, not actuality alone.

In ordinary English: every created positive actuality is presently caused by an underived source; it truly resembles that source, but in its own bounded mode. Constitutive parts would make the source dependent, while an unlimited source cannot be contained as a constituent within bounded actuality. Different finite expressions need not alter the source's intrinsic fullness. Actual freedom requires a further account of alternatives and willing.

The compact formal statement, conditional on actual s, is:

```text
x≠s ∧ H(x,q,m) ⇒ C(s,x,q) ⇒ D(x,s,q),       ¬D(s,y,r)
C(s,x,q) ∧ H(x,q,m) ⇒ ∃n H(s,q,n)
K(z,y) ⇒ D(y,z,e),                         K(z,y) ⇒ m_z,e ≤ m_y,e
H(s,e,∞);                                x≠s ∧ H(x,e,m) ⇒ m<∞
Δ created effects ⇏ ΔJ(s)             [X: J(s) invariant]
P(x,s,q) := C(s,x,q) ∧ D(x,s,q) ∧ ∃m H(x,q,m).
E(x,s,q) := P(x,s,q) ∧ L(x,s,q);            P(x,s,q) ⇒ L(x,s,q) by Q.
```

The displayed implications abbreviate nine independent clauses; the `⇏` line is explanatory shorthand for the explicit invariance axiom, not a material implication or a proof of modal alternatives.