Appendix B: Summary of the Formal Proof

The argument’s logical skeleton, compressed for reference. Items are sequenced in the order the derivation establishes them: definitions, premises, theorems in canonical proof order (T2 through T8; T1, Necessary Possibility, is established in item 3 below), then the application of T8 to the theistic alternatives and the verdict that closes them. The full derivation map tracing every claim to its structural source follows in Appendix C.

  1. Definitions. C =df one state, then another. Three structural requirements (succession, distinction, and exchange) follow from the definition as its internal dimensions (§1.3; Glossary). T =df Temporality, the structural Condition of causally ordered succession. S =df Spatiality, the structural Condition of relational extension. Φ =df Physicality, the structural Condition of causal persistence (§1.4). N =df T∧S∧Φ (§1.4.4). The biconditional □(◇C ↔︎ (T∧S∧Φ)) follows from the Structural Requirement (§2.4) and Chapter 2’s converse work as the Modal Identity (Theorem 2, §2.6.6), not by definition.

  2. P1 · Modal Bivalence. ∀P: ◇P ∨ ¬◇P. Every proposition about Reality is either really possible or really impossible: two exhaustive modal statuses, every proposition bearing exactly one. Modal Bivalence licenses the bilateral structure of the argument as a single move, forecloses third-option moves at the door, and separates the modal-status rule from the empirical population of either status (§2.1.1).

  3. Modal Constitution and Necessary Possibility (Theorem 1). The second premise, Modal Constitution, holds that a claim’s modal status is fixed by what Reality consists in, and so is the same in every world (§2.1.3). With Modal Bivalence (item 2) it earns the theorem: modal status, once secured, does not lapse and cannot be relocated, because possibility and impossibility are internal to Reality rather than handed in from outside (§2.1.3). The earned result is Necessary Possibility — (3a) ◇P → □◇P; (3b) ¬◇P → □¬◇P: real possibility is necessarily possible, and real impossibility necessarily impossible. Not imported as an S5 axiom but derived from the identity between the real and Reality (Theorem 1, §2.1.10).

  4. P2 · Cartesian Certainty. ◇C. Change is really possible, the single input the modal machinery needs to begin, and it is not merely possible but actual: reading this sentence is change in the defined sense, and the denial of actuality instantiates what it denies (Cogito Ergo Muto, §2.2.1). The certainty is Cartesian in a precise sense: any attempt to doubt that change occurs is itself a change, from not-doubting to doubting, so no skeptical scenario can undermine it (§2.2).

  5. Necessary Possibility of Change (Lemma 3). □◇C. The possibility of change is not a local accident but holds in every world constituted by modal structure. The Cartesian input ◇C, fed into the Necessary Possibility Theorem (T1, §2.1.10), is lifted to its necessitated form: what is possible is necessarily possible, so ◇C carries to □◇C (§2.3).

  6. P3 · Structural Requirement. □(◇C → T∧S∧Φ). Necessarily, change cannot be so much as possible unless the three Conditions of §1.4 are instantiated: a real before-and-after to occur in (Temporality), genuinely distinct states to move between (Spatiality), and something to carry the exchange (Physicality). The necessity is not a modal axiom imported from outside but constitutive: what essence fixes, it fixes at every world. The conditional is reached from four independent directions (modal, conceptual/essence, grounding, and facthood), each starting from a different vocabulary and converging on the same result (§2.4).

  7. The Modal Identity (Theorem 2). □(◇C ↔︎ (T∧S∧Φ)). The possibility of Change and the instantiation of the Triconditions are one structural fact: each description already invokes the other as its content. The Structural Requirement supplies the forward conditional; §2.4.10 establishes the converse (structure suffices for change) via the frozen-world (§2.4.11) and masking (§2.4.12) closures (§2.6.6).

  8. The Triconditional Cross-Entailment (Theorem 3). T ↔︎ (S∧Φ), S ↔︎ (T∧Φ), Φ ↔︎ (T∧S), with pairwise results T↔︎S, S↔︎Φ, T↔︎Φ. T, S, and Φ are one necessary structure apprehended through the cross-entailment of the three Conditions: each a condition of the conjunction of the other two, none reducible to the others, such that removing any one collapses the rest. Pairwise convergences are established at §§2.6.2–2.6.4 (§2.6.7).

  9. Mutual Containment (Result of Theorem 3). Instantiating any one of T, S, Φ entails instantiating all three. Direct from the triconditional form.

  10. The Ontological Identity (Theorem 4). Being physical ↔︎ T∧S∧Φ. Any being just is what persists (Φ) distinguished in S and succeeding in T; there is no residue left over after the Triconditions specify the being (§2.6.8).

  11. The Triconditional Identity (Theorem 5). N ↔︎ □(T∧S∧Φ); N ↔︎ □◇C; □(T∧S∧Φ) ↔︎ □◇C. The Modal Identity, the Triconditional Cross-Entailment, and the Ontological Identity compose one Nature: three readings of one structural identity. The Triunity of Nature (§2.6.9) frames the three readings of one structural fact; §2.6.10 names the consolidation. Point at Nature and you are pointing at the necessary Conditions; point at the necessary Conditions and you are pointing at necessary possibility; point at necessary possibility and you are pointing at Nature (§2.6.10).

  12. Necessity of Nature (Theorem 6). □(T∧S∧Φ). Temporality, Spatiality, and Physicality are necessary; Nature is necessary. The headline theorem of the argument and the thesis of the book. Derived at §2.5 by modus ponens: the Necessary Possibility of Change □◇C (§2.3) discharged against the Structural Requirement □(◇C → T∧S∧Φ) (§2.4) yields □(T∧S∧Φ), the Triconditional Necessity (§2.5); declared formally as the Necessity of Nature at §2.8.5 once Chapter 2’s defense of the premises (and the Triconditional work of items 7–11) has earned the result. Chapter 3 confirms via structurally independent, separately cuttable lines (§2.5; §2.8.5). The joint result unpacks into three per-Condition necessities, the Necessity of Temporality (□T), of Spatiality (□S), and of Physicality (□Φ), collected as the Necessity of Each Condition (Result of Theorem 6; §2.5); each is also secured directly by the Parasitic Negation reductios at §§2.4.6–2.4.8.

  13. Exclusivity of Nature (Theorem 7). The Null State ∅, the joint absence of T, S, and Φ, is precluded: no possible world realizes it (¬◇∅), so Nature admits no coherent alternative and no external vantage (§2.7; §2.8.6). The result excludes proposed necessary grounds already specified as internal members of the modal domain from grounding that domain externally. A candidate that instantiates the relevant structural roles is not thereby external to them; a candidate that withdraws every determinate relation faces an ∅-like specification problem. The partial-overlap analysis remains a pressure on internal candidate models. An essence-prior or transcendent ground that declines those roles is not automatically contradicted here; it remains a live metaphysical proposal bearing a determinacy and explanatory burden (§2.8.6; §4.1.10).

  14. Necessary Occupancy (Result of Theorems 5 and 7). For any individual thing to exist is for it to occupy the necessary structure: to be at a point in Temporality, somewhere in Spatiality, and to persist as a causal relatum in Physicality. Follows from the Exclusivity of Nature (item 13) and the Triconditional Identity (item 11) (§2.8.6).

  15. Supernatural Preclusion (Theorem 8). Classical divine attributes such as atemporality, aspatiality, immateriality, simplicity, and pure actuality create pressure when paired with determinate active predicates. Under the role/type distinctions of the book, a doctrine retaining real cognition, agency, causality, or relation incurs the relevant structural-role burdens. A doctrine that withdraws all differentiating content faces an ∅-like specification problem. This is not a definition-level identity of every classical or apophatic God-concept with ∅; it is the framework for the Chapter 6 attribute analysis (§6.1).

  16. Bilateral Symmetry (Result of Theorems 7 and 8). The conclusion □¬G — the necessary non-existence of the classical God — is established by the definitional line, and recovered a second way from the modal apparatus perfect-being theism itself relies on. Definitional line. The Maximally Great Being’s defining attributes, atemporality (¬T), aspatial existence (¬S), immateriality (¬Φ), analytically entail ∅. By the Exclusivity of Nature (item 13), ∅ is precluded as a coherent alternative to T∧S∧Φ; so ¬◇G, and ∴ □¬G. No modal logic beyond the definitions is required. S5-mirror line. This line adds no premise the definitional line lacks — its ◇□¬G is licensed only through the same ∅-preclusion — but it shows the conclusion also falls out of the theist’s own tool: S5 cuts both ways, so if ◇□G → □G is Plantinga’s direction, the bilateral mirror ◇□¬G → □¬G returns □¬G. What perfect-being theism wields for God, the definitional collapse turns against it.

  17. The Inversion (Proposition). The Supernatural Preclusion (item 15) runs systematically across the classical divine attributes: simplicity (§6.2.1), necessity (§6.3.1), eternity (§6.3.2), presence (§6.8.1), immutability (§6.3.3), impassibility (§6.3.4), awareness (§6.4.1), knowledge (§6.4.2), power (§6.7.1), causality (§6.7.2), interaction (§6.7.3), sovereignty (§6.7.4), relatability (§6.8.2), describability (§6.8.3), revelation (§6.8.4), benevolence (§6.9.1), righteousness (§6.9.2), holiness (§6.5.1), unity (§6.2.2), perfection (§6.2.4), life (§6.6), infinity (§6.2.3), and beatitude (§6.5.2). Each attribute is taken at the tradition’s strongest version and tested against the structural results. The failures take different shapes (dilemmas between empty and reinstating readings, cascades in which successive rescues evacuate what they were meant to preserve, direct collapse), but the common result is that the classical attributes cannot describe what they were designed to describe without invoking what they were designed to negate.

  18. Aseity Forfeiture (Result of Theorem 5, the Triconditional Identity). T, S, and Φ are ontologically prior to any being embedded in them — not temporally prior, but structurally prior in the sense that structure is the precondition of the embedded being’s existence rather than the reverse. Therefore no embedded being (process, open, panentheist) possesses aseity. The relabeling move, “the necessary structure is God,” changes the word while conceding the substance: necessity, all-encompassingness, and explanatory basicness do not entail personhood, agency, or intentionality. Classical theism cannot be repaired by redescription (§6.1, §6.10).

  19. Trilemma Closure (Proposition). For articulated classical-theist active-ground claims, the chapter distinguishes three recurrent responses: retain determinate active predicates and incur role-level conditions; revise the deity into an embedded or process-like agent and face an aseity/ultimacy cost; or withdraw into strict apophasis or essence-priority and relinquish the relevant positive grounding and predicate claims while taking on a determinacy burden. The trilemma closes this specific articulated field; it does not prove that every transcendent posit is contradictory (§6.10).

  20. Theological Verdicts Scoped. A wholly structureless yet determinate active ground faces the central Chapter 6 burden: its retained predicates must be contentful without reinstating the roles it denies. Embedded, process, open, and panentheist models may describe agency within Nature but thereby revise classical aseity or ultimacy rather than automatically becoming contradictory. Scriptural claims of temporal, spatial, and causal action are claims about a God related to Nature and require an account of that relation. Strict apophatic or essence-prior proposals remain live but cannot simultaneously function as positive explanations of knowing, acting, creating, loving, or relating without addressing determinacy. The book therefore maps pressures and costs rather than declaring every God-concept precluded.

  21. Methodological Convergence (Proposition). If T, S, and Φ are the conditions any possible Reality must instantiate, then any method that reliably tracks Reality arrives at them. Logic (pure inference), metaphysics (modal derivation), mathematics (formal structure), and physics (empirical measurement) exhaust the fundamental epistemic modes and each instantiates the roles it describes. The four-mode framing reduces to a metaphysics/physics convergence: two directions on one inquiry. The methodological-convergence form of the result (§7.1).

  22. The Cycle (Proposition). Past-eternity, the Preclusion of ∅ (§2.7), and Modal Invariance (§2.1.7) jointly forbid a stable zero-extension terminal state. A permanent locus of absent relational extension would be ∅ under cosmological description, which no locus can persist in. Total compression must therefore resolve; the recursion terminates only if the trajectory closes. A stable zero-extension terminus is structurally precluded; that history closes into a cycle is the abductive continuation physics supplies under the transmission premise (§7.6).