Euler characteristic and genus are inverse
NOT A NOVELTY CLAIM. This is a machine-checked formalisation, checked by the Lean 4 kernel and depending on no axiom. It is dated and citable. Prior art, where it exists, is cited below. No discovery is claimed.
χ and the genus are INVERSE: reading the genus back from χ(g) returns g for every surface up to genus 9, and rank H₁ = 2 − χ = 2g at each one. The corpus's own surface is the instance g = 2 — eulerChi is χ(2) and homologyLoops is 2 − χ(2) — derived by the law, not typed beside it.
Proposition
(∀ g ∈ List.range 10, genusOf (chi g) = g ∧ 2 - chi g = 2 * (g : Int)) ∧ eulerChi = chi 2 ∧ homologyLoops = 2 - chi 2Proof
By decide in corpus.lean. Checked by the Lean kernel: it depends on no axiom.
Sources and identifiers
- Lean source · src/pair/lean/proofs/corpus.lean
- Typeset paper · src/research/lean-theorems.tex
- Repository deposit · doi:10.5281/zenodo.21787144
- Author · ORCID 0009-0000-7312-9778
Proved in the same file
- The genus reads back from chi at every genus
- Census bands are gapless and descending
- Census is one hundred twenty three
- Fibonacci partial sum identity
- Folding by chi is undone by unfolding
- Reflection is an involution
- Reflection pairs and one fixed point
- Reflection closes on the digits
- Reflection is an involution on the whole lattice
- Golden ratio bounds
- Golden ratio minus one bounds
- Splitting then burning water is the identity
- Verify beats recompute by magnitudes
- Hexbit receipt beats rebuild
- Hexbit advantage widens
- Sixty four components
- Shor state vectors unchanged
- Shor vector dwarfs the number
- Shor rsa gap is between exponents
- Shor zero skipping divides by the work register