# RRLS v0.29 — Semantic Canonicalization Probe

## Status

**P = NP is not proved.**

v0.29 attacks the wall exposed by v0.28: many syntactically different residual CNFs can represent only a handful of Boolean futures.

## Main result

A new direct CNF→ROBDD compiler canonicalizes residual functions symbolically, without first enumerating complete truth tables. For one fixed variable order and one shared ROBDD manager, equal roots mean equal Boolean functions.

Hard snapshots:

- n=36: 3,258 residual CNF syntaxes → 5 exact ROBDD semantic roots.
- n=40: 38,191 residual CNF syntaxes → 1 exact ROBDD semantic root → terminal FALSE.

These class counts exactly match the previous exponential truth-table audit.

## Semantic collapse appears early

For the special n=40 formula:

- tick 5: 67 live states, 67 syntactic futures, 33 future variables, 1 ROBDD root.
- tick 30: 6,860 live states, 6,860 syntactic futures, 22 future variables, 1 root.
- tick 63: 84,394 live states, 38,191 syntactic futures, 11 future variables, 1 root.

The suffix was already semantically FALSE very early.

## Fresh peak controls

Across four fresh phase-transition random-3SAT formulas at each tested n, median peak results were:

- n=20: 353 live, 178 syntactic, 2 semantic roots.
- n=28: 3,321 live, 746 syntactic, 5 semantic roots.
- n=36: 27,980 live, 7,447 syntactic, 3.5 semantic roots.
- n=40: 98,910 live, 36,714 syntactic, 4.5 semantic roots, about 40 reachable canonical ROBDD nodes across those roots.

Two n=44 peak controls also compiled to 4 and 1 semantic roots. At n=48/n=52 the upstream frontier hit its 300,000-state cap before the semantic audit could run.

These are finite observations, not asymptotic bounds.

## Actual early jump

v0.29 adds an exact semantic checkpoint solver:

    frontier pressure → residual futures → shared direct ROBDD canonicalization.

If every live future canonicalizes to FALSE, the engine terminates UNSAT immediately.

For the special hard n=40 formula, the ordinary frontier reaches 84,394 states. The semantic checkpoint proves UNSAT at tick 19 with 3,172 live states, 2,211 residual syntaxes, and one semantic root = FALSE. No truth-table oracle is used.

This is an exact computational shortcut, although current Python ROBDD construction is slower in wall-clock time on this small instance.

## The new wall

The final canonical representation can be tiny while discovering it is expensive. For the hard n=40 suffix, the final canonical function is simply FALSE — zero nonterminal ROBDD nodes — but direct compilation still constructs temporary nodes.

Full-formula direct ROBDD compilation with exact garbage collection still hits a 100,000-live-node cap on the tested generic n>=36 random-3SAT controls.

Therefore:

    small semantic future != cheap semantic canonicalization.

## Variable order is a real coordinate problem

For the equality family x_i ↔ y_i:

- 24 variables: interleaved order 36 reachable nodes; separated order 12,285.
- 28 variables: interleaved 42; separated 49,149.
- 32 variables: interleaved 48; separated exceeds a 300,000-live-node cap.

So a coordinate/order change can produce exponential representation differences. Discovering a universally good order is itself part of the complexity problem.

## Proof-safe semantic witnesses

v0.29 adds conservative polynomial semantic witnesses for recognized languages. Any polynomially certified UNSAT residual has semantic key FALSE. Recognized GF(2) systems can be normalized to exact RREF witnesses; 2-SAT has an implication-closure witness.

The witness layer passed 96/96 invariance checks; 94 cases produced certified semantic keys. Coverage remains partial and unrecognized residuals stay UNKNOWN.

## What v0.29 establishes

It establishes on tested controls that direct symbolic canonicalization recovers the same semantic classes as the exponential oracle, semantic diversity can be tiny long before syntactic diversity peaks, and semantic FALSE checkpoints can skip later exact computation.

It does not establish polynomial ROBDD size, polynomial good-order discovery, polynomial arbitrary-CNF equivalence, polynomial temporary compilation size, or P=NP.

## Next theorem target

The target is now a local quotient update:

    Q_(t+1) = Update(Q_t, event)

where Q_t is the canonical quotient of all live futures.

A P=NP result along this route would require proving for arbitrary CNF that quotient initialization, every update, every intermediate quotient, and the final decision all have polynomial time and polynomial description size, with no exponential truth table, unrestricted branching, hidden advice, or exponential order search.

Current proof gate: **P = NP PROVED: FALSE.**