Exact named statement
∀ x. ∀ y. ∀ z. ∀ n. ∀ m. ∀ k. ∀ i. ∀ j. ∀ u. ∀ v. ∀ w. ∀ x0. ∀ x1. ∀ x2. ∀ x3. ∀ x4. ∀ x5. ∀ x6. ∀ x7. ∀ x8. ¬x7 = 0 → ¬x8 = 0 → x · x + y · y = x1 + (x · y + y · x) → z · z + n · n = x2 + (z · n + n · z) → m · m + k · k = x3 + (m · k + k · m) → i · i + j · j = x4 + (i · j + j · i) → u · u + v · v = x5 + (u · v + v · u) → w · w + x0 · x0 = x6 + (w · x0 + x0 · w) → u + (x · k + y · m + 2 · (z · j + n · i)) = x · m + y · k + 2 · (z · i + n · j) + v → w + (x · j + y · i + (z · k + n · m)) = x · i + y · j + (z · m + n · k) + x0 → ¬x7 · x7 = 0 ∧ (¬x8 · x8 = 0 ∧ (¬x7 · x8 · (x7 · x8) = 0 ∧ (¬x7 · x7 · (x8 · x8) = 0 ∧ x5 · (x7 · x7 · (x8 · x8)) + (x1 · (2 · x4) + 2 · x2 · x3) · (x7 · x8 · (x7 · x8)) = 2 · x6 · (x7 · x7 · (x8 · x8)) + (x1 · x3 + 2 · x2 · (2 · x4)) · (x7 · x8 · (x7 · x8)))))No displayed named abbreviation.
Complete certificate and independent checks
46 actual local proof nodes. Each node carries its complete ordinary HA proof body.
Full canonical proof bundle · Original HA receipt · Independent Lean receipt
Target SHA-256: ab88f7b597011f85f10d543e582abb37db52852d940b4b7a8c4c2cf7ae22c335
Bundle SHA-256: 70c033412b498cb8e180a800c45773610adb27aedad70716b6cc0e332edd9c68
Checked statement containment
These exact targets occur inside this certificate. These links summarize containment, not direct dependency arrows; the downloadable bundle records every private lemma and ordered edge.
Scope
rational norm product denominator not full IR031
None of these finite arithmetic statements closes an IR campaign parent. The analytic identification, interpolation and final positive irrationality certificate remain open.