UNNS Computability & Logic

Recursive Foundations of Proof and Program

Abstract

This paper develops a computability and logic framework based on the Unbounded Nested Number Sequences (UNNS) substrate. We show how recursive sequences embody computability, how UNNS constants act as logical invariants, and how paradoxes such as Gödel incompleteness and Carroll's regress stabilize under UNNS repair rules. The approach unifies recursion, proof, and program within an operator grammar that can be extended toward a computational logic discipline.

1 Introduction

Recursion underlies both computability and logic. The UNNS substrate offers a natural framework for rethinking logic: inference as inlaying, implication as inletting, collapse as contradiction, and repair as normalization.

2 UNNS Sequences as Recursive Programs

a_n+r = c₁a_n+r-1 + ··· + c_ra_n

UNNS sequences are defined by recurrence relations with integer coefficients and nesting rules that allow recursions within coefficients themselves. This corresponds directly to primitive recursive programs.

Interactive Sequence Generator

3 UNNS Constants as Logical Invariants

Lemma 3.1: UNNS coefficients belong to algebraic integer rings.

Constants such as φ (golden ratio), e, π, and the UNNS Paradox Index (UPI) arise as invariants. These constants stabilize recursion analogous to axioms stabilizing inference.

φ - Golden Ratio

1.618...

Stabilizes growth recursions

e - Euler's Number

2.718...

Continuous recursion limit

π - Pi

3.141...

Circular recursion invariant

UPI - Paradox Index

∞↻

Collapse threshold

4 Gödel and UNNS Collapse

Theorem 4.1 (UNNS Incompleteness): Every sufficiently expressive UNNS system generates undecidable nests.

Gödel encoding maps provability predicates into recursive sequences. Self-reference corresponds to a nest calling itself, which collapses under the collapse operator, indicating undecidability.

5 Turing Machines in UNNS

A UNNS tape is a nested sequence space indexed by integers, where each tape symbol is represented by a finite recurrence state. Turing transitions correspond to inlaying/inletting operators, while the halting condition corresponds to collapse.

                // UNNS Turing Machine Implementation
                class UNNSTuringMachine {
                    tape: NestedSequence[];
                    head: number = 0;
                    state: RecurrenceState;
                    
                    step() {
                        if (this.checkCollapse()) {
                            return HALT;
                        }
                        this.state = this.inlay(this.tape[this.head]);
                        this.head += this.inlet(this.state);
                    }
                }
            

6 Repair and Normalization as Proof Checking

UNNS repair operators model normalization procedures in proof theory. Excision+Refit in DNA corresponds to cut-elimination in sequent calculus.

Excision

✂️

Remove paradoxical loops

Refit

🔧

Restore consistency

Normalize

⚖️

Stabilize recursion

7 Toward UNNS Logic

We propose inference rules:

• Inlaying = conjunction (∧)
• Inletting = implication (⇒)
• Collapse = contradiction (⊥)
• Trans-Sentifying = interpretation (model theory)

This forms a Hilbert-style calculus on the UNNS substrate.

Inlaying = ∧

Conjunction - combining truths

Inletting = ⇒

Implication - causal flow

Collapse = ⊥

Contradiction - system failure

Trans-Sentifying

Model interpretation

                // UNNS Logic Inference Rules
                const UNNSLogic = {
                    // Conjunction via Inlaying
                    and: (p, q) => inlay(p, q),
                    
                    // Implication via Inletting
                    implies: (p, q) => inlet(p, q),
                    
                    // Contradiction Detection
                    isContradiction: (nest) => collapse(nest) !== null,
                    
                    // Model Theory
                    interpret: (formula) => transSentify(formula)
                };
                
                // Hilbert-style Calculus Structure
                class HilbertUNNS {
                    axioms = {
                        transposition: "Inlaying(p,q) → Inletting(p,q)",
                        absorption: "Inletting(p,q) → Inlaying(p,q) or Collapse",
                        interpretation: "Trans-Sentifying(formula) → Model"
                    };
                    
                    inferenceRules = {
                        modusPonens: (p, pImpliesQ) => this.inletting(p, pImpliesQ),
                        conjunction: (p, q) => this.inlaying(p, q),
                        contradiction: (p, notP) => this.collapse([p, notP])
                    };
                }
            

8 Applications

8.1 Banach-Tarski Resolution

Under UNNS recursion, non-measurable sets vanish; collapse prevents paradoxical decompositions.

8.2 Carroll's Paradox

Infinite regress spirals are absorbed by UNNS repair, converging to a fixed point.

8.3 Bio-Logic

DNA proofreading and logical proof-checking share the UNNS repair operator.

🌀 UNNS Paradox Index Explorer

Select Paradox Type

Paradox Metrics

Symbolic Entropy: 0.00

Collapse Threshold: ∞

UPI Score: 1.618

Ready to explore paradoxes as glyphic events...

🧬 UNNS DNA Simulator

DNA Sequence Operations

Bio-Logic Correspondence

DNA Integrity: 100%

Proof Validity: ✓

Recursion Depth: 0

DNA Proofreading

Logical Proof-Checking

Biological recursion ↔ Logical inference unified...

Classic Paradox Resolution

9 Conclusion

UNNS provides a natural computability and logic substrate. Recursion, constants, collapse, and repair give rise to a full operational grammar that can model proofs and programs.

References

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