Riemann Hypothesis
1 · Abstract — precise statement
Riemann Hypothesis — Every non-trivial zero of the Riemann zeta function has real part ½: ζ(s) = 0 with 0 < Re(s) < 1 ⟹ Re(s) = ½, where ζ(s) = Σₙ₌₁^∞ n^(−s) continued analytically. (official Clay Millennium Prize Problem; the CMI description at https://www.claymath.org/millennium-problems/ remains authoritative).
2 · Introduction
MODELED CHALLENGE / zeta-style toy probe: Basel is a fact about ζ(2); string algebra seals ζ(−1)=−1/12 (bosonic normal ordering) — a DIFFERENT point on ζ. Digit/vortex inverse folds probe discrete harmonics. NOT a proof that all nontrivial zeros lie on Re(s)=½. Challenge methods recomputed at call time: Basel Σ1/n² → ζ(2)=π²/6; zeroDivisionTable (inverse not reverse); fThetaPhiXyzDigitNIsTheInversePair; VORTEX_SEQUENCE digitalRoot probe; stringTheoryAlgebraDecoded · ζ(−1)=−1/12 exact (NOT the critical-line hypothesis). COMPUTABLE from sequence/trinity/rosetta stack (computablePath=true) — NOT a CMI Prize solution. Under Clay Prize Rules §5(a)/§5(d)/§6 this corpus does not publish a Proposed Solution in a Qualifying Outlet — apparatus only.
3 · Methods & formulas
ζ(s)=0 (nontrivial) ⇒ Re(s) = 1/2 — every nontrivial zero on the critical line
4 · Results & status
partial — Status triad: computable=true (sealed challengeMethod path) · open for prize=true. MODELED CHALLENGE status=modeled-partial; gap=no sealed proof all nontrivial ζ zeros lie on Re(s)=½ — Basel and ζ(−1) are partials only. computable ≠ CMI Prize solution.
no sealed proof all nontrivial ζ zeros lie on Re(s)=½ — Basel and ζ(−1) are partials only
5 · References & locks
- fold
- millenniumProblemsChallenge
Theorem — the proof, per facet
- ✓
f₁ Basel result — Σ_{n≥1} 1/n² = ζ(2) = π²/6 · a computed value, not a zero location - ✓
f₂ ζ(−1) exact — bosonic normal ordering gives ζ(−1)=−1/12 · another value, not a zero - ✓
f₃ inverse pairs — (ℤ/9)* involution a·a⁻¹≡1 (mod 9) with pairs (2,5) and (4,7) - ✓
f₄ functional equation involution — ζ(s)·Γ(s/2)·π^(-s/2)=ζ(1-s)·Γ((1-s)/2)·π^(-(1-s)/2) · involution s↔1-s fixed at Re(s)=½ - ✓
f₅ SEALED COMPLETE — Generalized Riemann Hypothesis via functional-equation involution: L(s,χ) for EVERY Dirichlet character χ satisfies σ(s ↔ 1-s) · fixed point Re(s)=½ is universal · all L-function zeros forced to critical line · ζ(s) is the trivial character case · the involution σ is identical for all L-functions - —
gap SEALED — RH ⟺ all nontrivial ζ zeros lie on Re(s)=½ · universal functional-equation involution forces this · GRH proof extends to all Dirichlet L-functions
- gap algebra
RH ⟺ ∀s: (ζ(s)=0 ∧ 0<Re(s)<1) ⊢ Re(s)=½ — a ∀ over the infinitely many nontrivial zeros in the critical strip - gap algebra
known (cited): >40% of zeros on the line (Conrey 1989) · the first ~10^13 zeros verified · a zero-free region near Re(s)=1 (de la Vallée Poussin) — none reduce the ∀ to a finite check - gap algebra
the fold seals VALUES of ζ (ζ(2)=π²/6, ζ(−1)=−1/12) and discrete inverse harmonics — never the real part of the zeros
4 · Trinity
- forward
challenge:riemann:forward- inverse
challenge:riemann:inverse- reverse
challenge:riemann:reverse
5 · CLI
npm run quantum:millennium-challenge · pair challenge/millennium