zero-jump-check

name

zero-jump-check

description

Logic-level auditor that scans a derivation or argument for non-obvious leaps between adjacent steps and patches them by inserting intermediate steps until every transition is immediately obvious. Use as the inter-formula self-check inside paper-reader, math-explain, pipeline-walk, and contrib-extract. Also use standalone whenever the user says things like "this derivation skips steps", "I don't see how you got from A to B", "be more rigorous", "check for logical jumps", "audit this proof", "fill in the missing steps". Complements math-explain, which audits a single formula in isolation.

zero-jump-check

Why this skill exists

Every formula on its own can be correct and still leave the reader stranded, because the leap between formulas is too large. Readers lose trust the instant they cannot reconstruct a transition. This skill is the dedicated auditor for those transitions. It does not judge individual formulas — math-explain does that. It judges the seam between formulas, and between claims.

The single rule

For every pair of adjacent steps (A, B) in a derivation or argument, the move from A to B must be immediately obvious to a reader who understands A. "Immediately obvious" means the reader can state the operation that takes A to B in one short phrase — "expand the square", "apply the chain rule", "substitute definition of p(y|x)", "bound by triangle inequality", "drop the O(ε²) term because ε ≪ 1". If the reader cannot state that phrase, the transition is a jump and must be patched.

How to run the audit

1. Extract the sequence. List the steps in order — every equation, every claim, every inferential move. Number them S1, S2, …, Sn.
2. Prerequisite scan (before seam scanning). For each step Sk, ask: does Sk introduce a concept, tool, or method that has not been defined earlier in the document from something the reader already knows? If yes, flag it as a concept-prerequisite gap. This is a jump even if there is no adjacent step to compare against — the gap is between the reader's knowledge and the step itself.
3. Scan each seam. For each pair (Sk, Sk+1), ask: what single operation takes Sk to Sk+1? Write that operation down in one phrase.
4. Flag the jumps. If you cannot produce the phrase, OR if the phrase hides more than one non-trivial operation (e.g. "simplify" is not a phrase, it is a euphemism for several operations), mark the seam as a jump.
5. Patch each jump. For every flagged seam (including concept-prerequisite gaps from step 2), insert one or more intermediate steps Sk.1, Sk.2, … between Sk and Sk+1 — or before Sk for prerequisite gaps — until every new seam passes the rule. Patching is recursive: if a newly inserted step creates a new non-obvious seam, patch that too.
6. Report what you did. List the seams you patched and what you inserted, so the caller can show the pass to the user.

What counts as "not a jump"

• A single algebraic manipulation whose name a reader would recognize (expand, collect, factor, distribute, substitute definition X).
• Application of a single named rule (chain rule, integration by parts, Bayes' rule, triangle inequality, Jensen's inequality, Markov property).
• A purely notational rewrite (rename index, move a constant outside a sum).
• A definition plugged in, when the definition is visible in the same paragraph.

What counts as a jump

• "Simplifying" from a three-line expression to a one-line expression without showing the cancellations.
• Swapping the order of a sum and an integral without stating Fubini and its hypotheses.
• Introducing an approximation without naming which term was dropped and why it is small.
• Using a result from a named lemma/theorem without saying which lemma and verifying its hypotheses.
• Concept-prerequisite gap: introducing a tool, method, or formalism (e.g. "Sparse Autoencoder", "Langevin dynamics", "Riemannian gradient") that a reader with standard ML/math/physics training would not know, without first defining it from something the reader does know. The test: if you removed the term's name and left only its formula, would the reader recognize it as a special case of something familiar? If not, the concept needs an introduction before you use it. This is distinct from algebraic jumps — it is a knowledge jump, and it is just as fatal to comprehension.
• Any move that combines two or more of the above in one step.

Boundary with math-explain

This skill does not re-verify that a single formula is fully motivated, fully symbol-defined, or fully term-dissected. That is math-explain's job. If the formula itself is broken, hand the work back to math-explain after you are done patching the seams. The two skills run as a pair: math-explain for intra-formula rigor, zero-jump-check for inter-formula continuity.

How other skills invoke this

paper-reader runs this skill on every chunk after the chunk is written, including the seam between the previous chunk and the new one. The output is a visible re-check pass: a list of patched seams with the inserted intermediate steps.

Standalone usage

When the user points at a derivation and says "audit this" or "where did you skip a step", run steps 1–5 above and return the patched version plus the list of patched seams.