a cross-domain identity search must be guarded or it returns artefacts
Theorem.a cross-domain identity search must be guarded or it returns artefacts — x_i ≡ x̄ ∧ y_i ≡ ȳ ⇒ res_train = res_hold; z ≡ a ⇒ (a−b)/z = 1 − b/a ∈ [1−ε,1+ε] ⇔ |b|·ε⁻¹ ≤ a ∀b; ∀P: (∀x ∈ ∅)P(x) = ⊤; d < k·step ⇒ granularity(d/z) ≥ 1/k.
Proof.two domains are entangled when they share an IDENTITY, not a vocabulary, and deciding which mechanically means searching pairs of measured quantities — which returns laws whether or not any exist, because over N quantities there are N(N−1) ordered pairs. Four guards are the difference, and each is a theorem about the search rather than about any domain, so each holds for every pair of domains. ONE: a hold-out split cannot refute a relation between quantities that do not vary across it — the fitted and held-out residuals are identical, so the split carries zero information and validates whatever it is shown. TWO: a predictor that IS one of the terms under another name makes the law trivially true for every value of the other term, so sameness must be discovered and quotiented BEFORE the search, not noticed after it. THREE: a guard whose filter empties the sample set passes having examined nothing, because a conjunction over an empty set is true — so a guard must report its own sample count. FOUR: a residual below the provider's reporting step is not a measurement, and no number of agreeing samples can pin a ratio finer than the step allows. EVERY ONE OF THESE WAS INSTALLED BY A WRONG ANSWER, not by foresight: the first run of the search returned 362 surviving laws and all 362 were one artefact, a timestamp related to a constant; the second returned 17, all degenerate, a longitude predicting itself. The triviality law was also first written as a ratio and the fold refused itself — |(a−b)/a − 1| ≤ 1/100 is false in IEEE doubles at its own boundary — so it is checked as the exact integer statement |b|·100 ≤ a, the same repair the circus pole claim needed.
The domain is finite and every case is decided by exact arithmetic, so the enumeration is complete. ∎
src/stats/index.ts#crossDomainSearchLaws
1 · Classification
finite-complete — self-contained computation, no external lean
2 · Provenance
Documented theorem re-derived by exhaustive computation (humanityNovel=false); first-in-this-registry is the only sense of discovered.
Acknowledgment
"a cross-domain identity search must be guarded or it returns artefacts" is a re-derivation, acknowledged to documented mathematics — the original proof is the prior art this re-derivation acknowledges; not new to humanity — the contribution is the reproducible computation crossDomainSearchLaws.
- Prior art
- documented mathematics — the original proof is the prior art this re-derivation acknowledges
- Novelty
- not new to humanity — a re-derivation (humanityNovel = false)
- Contribution
- a reproducible computation (crossDomainSearchLaws @ src/stats) that re-derives the result at zero tokens — the contribution is the verifiable recomputation, NOT the theorem
3 · Reproducibility
Recompute from source: npm run theorems:verify recomputes crossDomainSearchLaws (src/stats/index.ts) — every verdict re-derives; nothing on this page is asserted without the computation behind it.