Appendix B: Formal Extensions for Peer Validation

To support advanced peer review and constructive analysis, we extend the paper with the following elements:

1. Formal Lemma Sketches

Lemma 1: All expressions evaluated under SATS terminate or throw a Resolution Boundary Fault. Lemma 2: The system is type-preserving under substitution. Lemma 3: Scroll-safe addition and multiplication are commutative and associative up to the p* boundary.

2. Drift Function Δ(e)

We define Δ(e) as the difference in output signature when e is evaluated across scroll environments with different Resolution Primes.

A drift of Δ > 1 signals the scroll is unstable beyond current bounds.

3. Minimal Encoding Schema for Simulation

The system can be encoded in JSON or Lean-style syntax for external theorem proving.

Example encoding: { “type”: “ScrollProof”, “op”: “add”, “args”: [97, 101], “limit”: 257 }

4. Resolution Prime Advancement

The Resolution Prime p* advances by consensus or computational verification (e.g. via largest known provable primes).

Scrolls that exceed p* may be re-run under updated memory states.

5. Simulation Guidance

To verify scroll behavior in practice: Implement SATS in a functional language (e.g., Haskell, OCaml, Lean) Log runtime halting behavior, max_depth, and drift Cross-check scroll outputs under increasing p*