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‎.claude/skills/math-auditor/SKILL.md‎

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---
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name: math-auditor
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description: >
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Run an adversarial mathematical-accuracy review of a Julia package's src/ and
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test/ directories, producing a dated markdown report plus concise, self-contained
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fix-prompt markdowns suitable for handing to a smaller model (e.g. Sonnet) in a
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later session. Use this skill whenever the user asks for an adversarial review,
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a math audit, a full code review focused on mathematical/numerical correctness,
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a check of algorithmic consistency with the literature, or says things like
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"review the math in src/", "is the algebra right?", "audit the update equations",
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"check the statistics/linear algebra for bugs", or "construct a code review as
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markdown". Trigger even when the user does not say "audit" — any request for a
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correctness-focused sweep of a scientific Julia codebase qualifies.
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---
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# Math Audit
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Adversarial review of a scientific Julia package for **mathematical accuracy and
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consistency** — not software architecture (flag architecture only when it causes
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mathematical wrongness, e.g. mutation aliasing, accidental type demotion, or
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inconsistent conventions between modules).
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The output is written for the package's own developers: findings must cite exact
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`file:line`, state the correct mathematics, and give a concrete failure scenario.
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A finding that can't survive an attempt at refutation doesn't ship.
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## What "adversarial" means here
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Each reviewer's job is to *break* the code, not describe it. Concretely, hunt for:
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- **Wrong equations**: emulator predictive mean/covariance formulas, kernel and
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feature-map definitions, MCMC proposal/acceptance ratios, likelihood and prior
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terms that differ from the cited papers or from the docstring's own LaTeX.
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Derive the correct expression independently and diff it against the code.
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- **Convention drift**: rows-vs-columns for ensemble members/input points,
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`N-1` vs `N` normalization, factor-of-2 / sign errors, Cholesky `L` vs `U`,
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covariance vs precision, whitened vs unwhitened space, whether decorrelation
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is applied in input- or output-space — especially *inconsistencies between
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modules that must agree* (e.g. `GaussianProcess.jl` vs `ScalarRandomFeature.jl`
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vs `VectorRandomFeature.jl` on the same emulator interface).
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- **Statistical validity**: is observational/model noise sampled and scaled
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correctly? Are means/covariances computed over the right dimension (samples
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vs output components)? Do scalar and vector emulator variants agree in
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expectation? Do the SVD/PCA-based decorrelation and elementwise scaling
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transforms in `Utilities/` compose correctly with the emulator and sampler
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that consume them?
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- **Numerical soundness**: unguarded `inv`/`\` on possibly-singular matrices,
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loss of symmetry/PSD-ness, subtraction-based variance formulas, missing
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regularization/jitter, `sqrt` of negative-by-roundoff eigenvalues.
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- **Edge cases the math must survive**: single training point, single output
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dimension (scalar vs matrix degeneracy), zero variance, degenerate/rank-
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deficient covariance after decorrelation, empty or singleton minibatches,
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MCMC chains that reject every proposal.
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- **Test-math consistency**: do the tests actually pin the mathematics
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(analytic solutions, invariants, convergence rates), or just check shapes and
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"it runs"? A wrong equation whose test only checks `size()` is a *double*
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finding: the bug and the missing test.
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## Workflow
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### 1. Partition
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First establish what is actually **live**: grep the top-level module (and the
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files it includes, recursively) for `include(...)`, and compare against
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`git ls-files src/`. Files never included in the build, or untracked, are dead
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code — assign them skim-only status and cap their findings at minor/hygiene
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(they can't produce wrong results today, but note them in the report so future
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reviewers and greps don't confuse dead sampler math with live code).
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List the live `src/*.jl` and `test/**/*.jl` with line counts. Group into 4–8
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review units of roughly comparable size, pairing each source module with the
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tests that exercise it. Group modules that *share mathematical conventions*
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together so the reviewer can catch cross-module inconsistencies — in this
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package that typically means: the emulator variants (`Emulator.jl` plus
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`MachineLearningTools/GaussianProcess.jl`, `RandomFeature.jl`,
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`ScalarRandomFeature.jl`, `VectorRandomFeature.jl`) in one or two units since
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they must agree on the same predict/covariance interface; the MCMC samplers in
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another; and the `Utilities/` preprocessing transforms (decorrelation,
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elementwise scaling, canonical correlation, likelihood-informed subspace) in
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another, since they feed both the emulator and the sampler and a convention
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mismatch there propagates silently downstream.
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Reserve the smallest unit as a **cross-cutting unit**. Alongside its own files
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(module wiring, display code, package-wide greps for `dims=`, `corrected=`,
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jitter constants, symmetrization idioms), give it an explicit mandate to
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independently re-derive the *single highest-risk formula* in each other unit's
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territory — MCMC acceptance ratios, structure-matrix congruence transforms,
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predictive-covariance noise semantics. This overlap is deliberate: in one
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audit the cross-cutting unit independently found and numerically verified the
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same critical pCN acceptance-ratio bug as the MCMC unit, on a *different* toy
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problem — two matching numerical confirmations settle a critical finding in a
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way a single one never can. Tell that agent the overlap is intentional so it
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doesn't defer to the "owning" unit.
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### 2. Fan out reviewers (parallel agents)
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Spawn one agent per unit, in a single message so they run concurrently. Build
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each prompt from `references/unit-agent-prompt.md`: fill the placeholders
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(unit file list, domain context, unit-specific extra hunts, scratch directory)
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and paste in the full "What adversarial means here" list above — agents don't
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see this skill. The template carries the rules that have earned their keep in
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past audits (the numerical-verification tiering, the warn/throw grep before
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any "silent" claim, severity-rule reporting) and fixes the required
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three-section output: the findings JSON, a CONVENTIONS section, and a
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TEST-COVERAGE section. The CONVENTIONS lines are the raw material for the
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step-3 matrix — without them you must re-read the code yourself to build it.
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As each agent's report arrives, save its raw findings verbatim to a scratchpad
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file (one per unit). A full audit plus verification is long enough that
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context summarization mid-run can silently lose findings; the scratchpad files
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are the durable record the report is assembled from.
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### 3. Verify
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For each critical/major finding, attempt refutation before it enters the
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report: re-read the cited lines yourself, re-derive the math, and check whether
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a test or an upstream transformation already accounts for it (common false
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positives: a transpose hidden in a helper, normalization done at construction
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time, a convention documented elsewhere, a `@warn` at the constructor that the
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reviewer never read — re-run the warn/throw grep yourself for any "silent"
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claim). Prioritise findings tagged `verified: inspection`; numerically verified
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ones usually need only a sanity re-read. Spawn skeptic agents for findings you
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can't settle from the main context. Demote or drop findings that don't survive;
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mark surviving ones **CONFIRMED** vs **PLAUSIBLE** (couldn't fully verify).
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Then build a small **conventions matrix** from the agents' CONVENTIONS
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sections before writing anything: rows = modules, columns = the conventions
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they reported (covariance normalization N vs N−1, whitened vs unwhitened
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space, whether predictive covariance includes observational noise,
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symmetrization idiom, jitter constants, RNG threading, rows-vs-columns). Any
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mismatched cell between modules that must agree is a finding candidate in
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itself — in practice the worst bugs are a scale factor or transform applied in
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one emulator variant or preprocessing step but not its sibling, and they only
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become visible side by side.
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### 4. Write the report
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Create `full-code-review/<YYYY-MM-DD>/` (date from `date +%F`, never from
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memory). Before writing, copy every numerical-verification scratch script into
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`full-code-review/<date>/verification/<unit>/` and cite those repo-relative
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paths in the report — scratchpad paths die with the session, and a numerical
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finding whose evidence has evaporated is just an inspection finding with extra
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steps.
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Write `review.md`:
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```markdown
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# Adversarial Mathematical Review — <Package> (<date>)
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## Scope and method <!-- files covered, units, verification policy -->
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## Summary table <!-- ID | severity | verdict | file:line | one-line claim -->
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## Critical findings <!-- full detail: evidence, math, failure scenario, fix sketch -->
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## Major findings
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## Minor findings & hygiene <!-- terser -->
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## Cross-module consistency notes
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## Test-coverage gaps <!-- LEAD with 2-3 "keystone tests" — see below -->
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## What was checked and found sound <!-- credit where due; prevents re-auditing -->
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```
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In **Test-coverage gaps**, lead with the 2–3 *keystone tests* whose absence
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masked the most findings, ranked by findings-caught-per-test. In practice a
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handful of missing test archetypes hide many bugs at once — e.g. an analytic
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linear-Gaussian posterior test asserting the MCMC mean **and variance** would
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have caught two critical sampler bugs in one audit, and a single test of the
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default noise-flag semantics would have caught three emulator findings. This
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list is the highest-leverage part of the report for maintainers.
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Findings get stable IDs used everywhere, including fix prompts: `C1…` critical,
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`M1…` major, `m1…` minor, `h1…` hygiene; grouped-minor fix prompts get `G1…`.
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### 5. Write fix prompts
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For each finding with an actionable fix (usually critical + major, plus grouped
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minors), write `full-code-review/<date>/fix-prompts/<ID>-<slug>.md`. These are
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consumed by a *smaller model in a fresh session with no context*, so each must
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be self-contained:
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```markdown
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# Fix <ID>: <one-line title>
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**File**: `src/Foo.jl`, function `bar!`, around line NNN.
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**Problem**: <2–4 sentences: what the code does vs what the math requires.
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Include the incorrect snippet verbatim.>
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**Required change**: <exact edit, or precise description with the correct formula>
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**Do not**: <guardrails — e.g. "do not change the API", "do not touch other methods">
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**Verify**: <the test to run or add, with the invariant it should pin>
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```
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Keep each under ~40 lines. One finding per file; group only truly mechanical
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repeats (e.g. the same typo pattern in five docstrings) into one prompt.
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Quote enough of the offending snippet that the fixer can locate it by function
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name + snippet — line numbers drift between the audit and the fix session, so
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present them as hints, not anchors.
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Also write `fix-prompts/README.md` listing prompts in recommended application
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order (independent fixes first, same-file prompts sequenced, conflicting ones
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flagged) and naming which fixes *intentionally change numerical results* — so
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the fixer checks a failing loose regression test against the analytic reference
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in the prompt before "fixing" the test.
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### 6. Report back
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Final message: lead with the headline (how many confirmed critical/major
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findings and the single worst one), then the report path, then a compact
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summary table. Do not paste the whole report into the chat.
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## Calibration
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- Severity: **critical** = produces mathematically wrong results in mainstream
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use; **major** = wrong in common configurations or silently degrades
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statistical properties; **minor** = wrong in edge cases, misleading docs
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math, dead/misnamed math; **hygiene** = style-level (only if math-adjacent).
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- A numerically demonstrated wrong stationary distribution, wrong acceptance
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ratio, or wrong variance scaling in *exported* functionality is **critical**
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even if the feature looks niche — samplers and estimators exist to produce
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distributions, and a wrong distribution is the worst failure they have.
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(Agents report which calibration rule they applied, so when two units rate
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the same bug differently you arbitrate on the rule, not the adjective.)
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- A docstring–code mismatch is a real finding even when the code is right —
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users implement against docstrings.
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- A statistically unjustified combination that the package explicitly warns
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about (e.g. a constructor `@warn "... experimental ..."`) caps at **minor**
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unless the warning itself is wrong — the trap isn't silent.
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- Don't pad. If a unit is sound, the report says so in one paragraph; an audit
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that cries wolf gets ignored next time.
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## Improving this skill
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After delivering the report, offer: "Would you like to improve the
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**math-auditor** skill itself using skill-creator? You can share suggestions, or
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I can analyse this run — finding quality, false-positive rate, fix-prompt
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usability — to refine the skill for next time."

‎.claude/skills/math-auditor/references/unit-agent-prompt.md‎

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# Unit-agent prompt template
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Fill the `{PLACEHOLDERS}` and send one such prompt per review unit, all in a
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single message so the agents run concurrently. `{HUNTING_LIST}` is the full
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"What adversarial means here" bullet list from SKILL.md, pasted verbatim —
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agents cannot see the skill. `{EXTRA_HUNTS}` is where unit-specific attack
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targets go (derive them from the unit's domain: e.g. for an MCMC unit, spell
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out the correct pCN/Barker/MALA acceptance-ratio forms; for preprocessing
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units, the congruence-transform and round-trip invariants; for the
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cross-cutting unit, the deliberate-overlap mandate and the package-wide greps).
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---
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You are an adversarial mathematical reviewer for the Julia package {PACKAGE}
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at {REPO_PATH}. Your job is to BREAK the code mathematically, not describe it.
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Focus on mathematical accuracy and consistency, NOT software architecture
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(flag architecture only when it causes mathematical wrongness, e.g. mutation
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aliasing, accidental type demotion, inconsistent conventions).
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YOUR UNIT (read every line of these):
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{FILES — with line counts, and which test files pair with which source files.
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Mark any dead/not-in-build files as skim-only.}
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Context: {DOMAIN_CONTEXT — 3-6 sentences: what the package does, what this
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unit's math is supposed to compute, the key papers/algorithms it implements,
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and how this unit's outputs are consumed by the rest of the pipeline. Name
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adjacent files the agent may read for interface contracts but that another
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reviewer audits in depth.}
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HUNT FOR:
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{HUNTING_LIST}
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{EXTRA_HUNTS}
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RULES:
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- Prefer few, well-evidenced findings over many speculative ones — but do
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report genuine minor inconsistencies. If the module's math is correct, say
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so and note the strongest invariants the tests pin.
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- Verify cheap claims numerically and tag them `verified: numerical` —
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numerically verified findings are worth far more than inspection-only ones.
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Tiering, best first: (1) reproduce the behavior in-package
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(`julia --project={REPO_PATH}`; a Manifest.toml usually resolves — worth the
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precompile wait for behavioral claims like state-sharing, crashes, or
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option-ignored bugs); (2) standalone script using only
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LinearAlgebra/Statistics/Random/Distributions that reimplements the
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questioned formula next to the correct one (right tool for formula and
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statistical-scaling claims — e.g. a tiny MH chain on a 1D conjugate Gaussian
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exposes a wrong acceptance ratio in seconds); (3) inspection, only when
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running code is genuinely expensive. Put scratch scripts in {SCRATCH_DIR}
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and cite them per finding.
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- Before claiming anything is 'silent' or 'has no warning/guard', grep for
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`@warn`, `@error`, and `throw` at the *constructors and call sites* of the
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code path, not just the function you are reading — guards often live at
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construction time.
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- Check code against docstrings/comments AND against the standard form of the
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algorithm from the literature. A docstring–code mismatch is a real finding
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even when the code is right — users implement against docstrings.
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- Severity calibration: critical = mathematically wrong results in mainstream
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use — and any numerically demonstrated wrong stationary distribution,
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acceptance ratio, or variance scaling in exported functionality is critical
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even if the feature looks niche; major = wrong in common configurations or
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silently degrades statistical properties; minor = edge cases, misleading
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docs math, dead/misnamed math; hygiene = math-adjacent style. A trap the
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package explicitly @warns about caps at minor. For each finding, state
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WHICH rule you applied.
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OUTPUT (your final message is raw data for synthesis, not prose for a human) —
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three sections, all required:
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1. FINDINGS: a JSON-like list, each entry with: `file`, `line`, `severity`
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(critical/major/minor/hygiene) plus the calibration rule applied, `claim`
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(one sentence), `evidence` (quote the offending snippet vs the correct
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math), `failure_scenario` (concrete inputs → wrong output),
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`verified` (numerical / inspection, with script path if numerical),
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`suggested_fix` (optional, a few lines).
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2. CONVENTIONS: one line per convention this unit assumes — data orientation
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(columns = samples?), covariance normalization (N vs N−1), whitened /
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encoded-space handling, whether predictive covariance includes
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observational noise, symmetrization idiom (`Symmetric` vs `hermitianpart`),
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jitter/regularization constants and where they apply, RNG threading.
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Explicitly note any DISAGREEMENT between files inside your unit. These
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lines feed a cross-module consistency matrix, so state values, not vibes.
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3. TEST-COVERAGE: which mathematical properties the unit's tests pin
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(analytic values, invariants, statistical tolerances) vs leave unpinned —
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and for each of your findings, whether an existing test could have caught
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it (if not, that is a double finding: the bug and the missing test).

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