hitting a prime is the inversion point — ℤ/pℤ is a field, inversion becomes total
Theorem.hitting a prime is the inversion point — ℤ/pℤ is a field, inversion becomes total.
Proof.a prime modulus is the point where inversion becomes TOTAL (user, 2026-07-25: "hitting a prime is inversion point"). In ℤ/pℤ for prime p every nonzero residue has a multiplicative inverse — it is a FIELD — so the invertible fraction reaches 1 (mod 7 → 6/6 nonzero invert); at a composite n only φ(n) < n−1 residues invert and the rest are zero divisors (mod 9 → 6/8). By Fermat's little theorem x^(p−1) ≡ 1, so x⁻¹ = x^(p−2) is a single power map inverting EVERY nonzero residue at once, defined for all precisely BECAUSE p is prime. Scanning moduli 2..16, the primes are exactly where the invertible fraction hits 1 (total inversion) and composites dip below — the prime is the inversion pole of the inversion arc (pole/inversion/60°/prime/cipher = one group). SCOPE: structural number theory (the field property of a prime modulus = total invertibility), NOT a physical or mystical inversion. HARMONY ≠ TRUTH.
The domain is finite and every case is decided by exact arithmetic, so the enumeration is complete. ∎
src/9/1/index.ts#hittingAPrimeIsTheInversionPoint
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
"hitting a prime is the inversion point — ℤ/pℤ is a field, inversion becomes total" 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 hittingAPrimeIsTheInversionPoint.
- 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 (hittingAPrimeIsTheInversionPoint @ src/9/1) 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 hittingAPrimeIsTheInversionPoint (src/9/1/index.ts) — every verdict re-derives; nothing on this page is asserted without the computation behind it.