eml-skill

Verify whether a claimed EML tree really computes a stated elementary function. Use when a user presents an EML expression (nested eml(...) form or RPN) and asks "does this really equal sin(x)?" / "is this a valid witness for log10?", when auditing proof-engine witness trees, when checking a compiler's output against a reference formula, or when someone needs a branch-cut / removable-singularity audit with an interior-domain sampler. Produces a structured audit report (audit.json, audit.md, audit.blog.md) covering leaf set, shape stats, numerical agreement, branch-cut flags, and removable-singularity caveats. The `--format blog` option emits a self-contained README/blog-friendly markdown artifact with embedded Mermaid, K-context table, witness provenance, and a probe table. Handles complex arithmetic via principal-branch cmath.

Deterministic library-first regression — fit a CSV against the calculator-primitive witness library (unary, affine `a·w(x)+b` with constant snapping, depth-2 composite `w(v(x))`, binary `w(x,y)`) and emit a machine-checkable JSON verdict ranked by max |residual| and R². Use when you need a reproducible, audit-able answer to "which elementary law generated this data?" — the JSON output is downstream-consumable (no LLM in the loop), exit codes encode the verdict, complex-plane evaluation via cmath catches branch-cut hazards. Snaps to π, e, 1/ln(10), Catalan G, ζ(3), Khinchin K, log₂(e), e^π, γ, etc. Optional `--noise-sigma σ` for measured data; reports SE(a)/SE(b).

Compile ordinary formulas (exp(x+y), x**y, ln(x*y), sin(x)+cos(x), sqrt(x*y), x/y, asin(x), atan(x), log10(x)) into EML trees, look up calculator-primitive witnesses (exp, ln, add, mult, sub, pow, neg, inv, div, pi, i, sin, cos, tan, sqrt, asin, acos, atan, log10), inspect arbitrary EML trees, or run one-shot compile-render to emit tree + diagram + audit + summary from a sympy expression. Use when a user wants to lower a sympy-parseable expression into the EML IR, ask "how many tokens does the mult witness take?", visualize a tree as Graphviz/Mermaid, convert between nested / RPN / JSON forms, read shape stats (K, depth, leaf histogram), or produce a shareable artifact bundle in one command. Every named elementary primitive has a stored tree; the only `needs_tree` entry is `apex` (the closure proof itself, not a callable primitive).

Verify numerical equivalence of two EML trees (interior samples + branch-cut probes), search for shorter trees via witness-swap peephole, or enumerate shortest trees bottom-up via beam search with function-hash deduplication, meet-in-the-middle complement lookup, backward goal-propagation priority population, and optional library-witness seeding. Use when a user wants to confirm two EML trees compute the same function, audit whether a subtree collapses to a known library entry (exp, ln, e, add, mult, sub, pow, neg, inv), or rediscover the shortest EML witness for a named claim (exp, ln, e, mult, sub, neg, inv, simple composites) within a K budget up to K=17. Produces delta-K, equivalence verdict with branch flags, or best-K tree with per-K candidate counts.

Numerically check whether two elementary-function expressions are equal. Use when someone asks "is sin(x)^2 + cos(x)^2 = 1?", "does log(x*y) equal log(x)+log(y)?", "verify this identity", "is this trig/log/algebraic identity true?", or when reviewing an LLM-generated proof, textbook answer, or student submission that asserts two closed-form expressions are equal. Handles sympy-parseable Python-style expressions and LaTeX (`\frac`, `\sqrt`, etc.). Produces a `verified` / `refuted` / `branch-dependent` / `cannot-verify` verdict with a concrete counterexample when the identity fails. Backs onto the EML proof engine when both sides compile to its witness library; falls back to sympy lambdify otherwise. NOT a symbolic proof — for that use sympy.simplify or a CAS.

Verify whether a claimed EML tree really computes a stated elementary function. Use when a user presents an EML expression (nested eml(...) form or RPN) and asks "does this really equal sin(x)?" / "is this a valid witness for log10?", when auditing proof-engine witness trees, when checking a compiler's output against a reference formula, or when someone needs a branch-cut / removable-singularity audit with an interior-domain sampler. Produces a structured audit report (audit.json, audit.md, audit.blog.md) covering leaf set, shape stats, numerical agreement, branch-cut flags, and removable-singularity caveats. The `--format blog` option emits a self-contained README/blog-friendly markdown artifact with embedded Mermaid, K-context table, witness provenance, and a probe table. Handles complex arithmetic via principal-branch cmath.

Deterministic library-first regression — fit a CSV against the calculator-primitive witness library (unary, affine `a·w(x)+b` with constant snapping, depth-2 composite `w(v(x))`, binary `w(x,y)`) and emit a machine-checkable JSON verdict ranked by max |residual| and R². Use when you need a reproducible, audit-able answer to "which elementary law generated this data?" — the JSON output is downstream-consumable (no LLM in the loop), exit codes encode the verdict, complex-plane evaluation via cmath catches branch-cut hazards. Snaps to π, e, 1/ln(10), Catalan G, ζ(3), Khinchin K, log₂(e), e^π, γ, etc. Optional `--noise-sigma σ` for measured data; reports SE(a)/SE(b).

Compile ordinary formulas (exp(x+y), x**y, ln(x*y), sin(x)+cos(x), sqrt(x*y), x/y, asin(x), atan(x), log10(x)) into EML trees, look up calculator-primitive witnesses (exp, ln, add, mult, sub, pow, neg, inv, div, pi, i, sin, cos, tan, sqrt, asin, acos, atan, log10), inspect arbitrary EML trees, or run one-shot compile-render to emit tree + diagram + audit + summary from a sympy expression. Use when a user wants to lower a sympy-parseable expression into the EML IR, ask "how many tokens does the mult witness take?", visualize a tree as Graphviz/Mermaid, convert between nested / RPN / JSON forms, read shape stats (K, depth, leaf histogram), or produce a shareable artifact bundle in one command. Every named elementary primitive has a stored tree; the only `needs_tree` entry is `apex` (the closure proof itself, not a callable primitive).

Verify numerical equivalence of two EML trees (interior samples + branch-cut probes), search for shorter trees via witness-swap peephole, or enumerate shortest trees bottom-up via beam search with function-hash deduplication, meet-in-the-middle complement lookup, backward goal-propagation priority population, and optional library-witness seeding. Use when a user wants to confirm two EML trees compute the same function, audit whether a subtree collapses to a known library entry (exp, ln, e, add, mult, sub, pow, neg, inv), or rediscover the shortest EML witness for a named claim (exp, ln, e, mult, sub, neg, inv, simple composites) within a K budget up to K=17. Produces delta-K, equivalence verdict with branch flags, or best-K tree with per-K candidate counts.

Numerically check whether two elementary-function expressions are equal. Use when someone asks "is sin(x)^2 + cos(x)^2 = 1?", "does log(x*y) equal log(x)+log(y)?", "verify this identity", "is this trig/log/algebraic identity true?", or when reviewing an LLM-generated proof, textbook answer, or student submission that asserts two closed-form expressions are equal. Handles sympy-parseable Python-style expressions and LaTeX (`\frac`, `\sqrt`, etc.). Produces a `verified` / `refuted` / `branch-dependent` / `cannot-verify` verdict with a concrete counterexample when the identity fails. Backs onto the EML proof engine when both sides compile to its witness library; falls back to sympy lambdify otherwise. NOT a symbolic proof — for that use sympy.simplify or a CAS.