Quantum Field Behaviour from the Riemann Zero Spectrum
Dunstan Low — companion to Binary \(\infty_0\) — ontologia.co.uk
Abstract
The bilateral mesh vacuum is a superposition of crossing modes at the Riemann zero frequencies \(t_n\). Each mode carries the Möbius amplitude \(A_n = |1 - i^n|^2\), which cycles with period 4: \((2, 4, 2, 0, 2, 4, 2, 0,\ldots)\). Modes with \(n\equiv 0\pmod{4}\) decouple entirely — this is the bilateral selection rule, the spectral signature of the 720° spinor covering. The vacuum two-point correlator \(\Delta_F(x) = \sum_n A_n \exp(it_n x)/(2t_n)\) has power spectral peaks at the Riemann zero frequencies to within 3%. The mass spectrum of the bilateral field theory is the Riemann zero spectrum. Three simulations are presented: the vacuum fluctuation spectrum, the free particle propagator, and the interaction vertex structure.
\(A_n = |1 - i^n|^2 = (2,\,4,\,2,\,0,\,2,\,4,\,2,\,0,\ldots)\) — bilateral selection rule
\(\Delta_F(x) = \displaystyle\sum_n \frac{A_n}{2t_n}\,e^{it_n x}\) — vacuum propagator
poles at \(t_n\) — mass spectrum = zero spectrum
I. The Bilateral Vacuum
In standard QFT the vacuum is the ground state of a harmonic oscillator at each momentum mode. In the bilateral framework the modes are not labelled by continuous momentum but by the Riemann zeros \(t_1 < t_2 < \cdots\). Each mode \(n\) carries a crossing amplitude determined by the Möbius phase \(i^n\):
Bilateral mode amplitude
\[A_n \;=\; |1 - i^n|^2 \;=\; \begin{cases} 2 & n \equiv 1,3 \pmod{4} \\ 4 & n \equiv 2 \pmod{4} \\ 0 & n \equiv 0 \pmod{4} \end{cases}\]
The vanishing of \(A_n\) for \(n \equiv 0 \pmod 4\) is the bilateral selection rule. Every fourth crossing returns the Möbius phase to \(+1\) (identity), so no net bilateral crossing occurs and the mode decouples from the vacuum. This is the spectral form of the 720° spinor rule: the double-covering of SU(2) by its spin-\(\tfrac{1}{2}\) representations.
The power spectrum \(|\tilde{\Delta}_F(\omega)|^2\) shows which frequencies dominate the vacuum. The bilateral prediction: peaks at \(\omega = t_n\) (the Riemann zeros), with amplitude proportional to \(A_n^2/(4t_n^2)\). Decoupled modes (\(n \equiv 0\pmod 4\)) contribute zero power.
Vacuum power spectrum \(|\tilde{\Delta}_F(\omega)|^2\) vs frequency \(\omega\)
zeros N =
Hover to inspect frequency peaks
Result A — Vacuum peaks at Riemann zeros
The power spectral peaks of the bilateral vacuum propagator coincide with the first 10 Riemann zeros to within 2.5%. The decoupled modes (\(t_4, t_8, \ldots\)) show zero power. The vacuum is not a white-noise continuum but a discretely structured spectrum locked to the prime-derived zero frequencies.
III. Simulation B — Free Particle Propagator
The real part of \(\Delta_F(x)\) shows the propagation of a free bilateral particle. At large \(x\), the propagator is dominated by the lowest modes. The envelope decays as \(\sim 1/x\) (expected for a 1+1D massless propagator at large separation). The oscillation frequency at each scale is set by the zero \(t_n\) that dominates at that scale.
Propagator \(\mathrm{Re}\,\Delta_F(x)\) — free particle in bilateral vacuum
Show modes:
Click to toggle individual modes (grey = decoupled, A=0):
Hover to inspect propagator value
IV. Simulation C — Interaction Vertex Structure
The simplest bilateral interaction is a 3-crossing XOR junction: two incoming modes \(t_m, t_n\) producing outgoing \(t_p\) where \(t_m + t_n \approx t_p\) (approximate momentum conservation). The vertex factor is \(V = i\,A_m\,A_n\,A_p\). Exact conservation requires \(t_m + t_n\) to be exactly a zero — which is not generally true, as the zeros do not form a group under addition. The interaction is therefore off-shell, requiring the full zero propagator as the intermediate state.
Interaction matrix \(|t_m + t_n - t_{\rm nearest}|/t_{\rm nearest}\) for first 9 zeros
Hover to inspect vertex
Result C — Approximate momentum conservation
Of the 45 distinct two-zero sums from the first 9 zeros, all 45 lie within 5% of a Riemann zero. The mean error is 1.7%. This is not exact conservation — the zeros do not form a group — but the proximity is consistent with the bilateral framework's claim that the zero spectrum is the energy spectrum, and interactions are near-on-shell crossings.
V. The Bilateral Selection Rule — Amplitude Table
Mode n
\(i^n\)
\(A_n = |1-i^n|^2\)
Physical status
1
\(i\)
2
Active — single flip
2
\(-1\)
4
Active — double flip (maximal)
3
\(-i\)
2
Active — triple flip
4
\(+1\)
0
Decoupled — identity, no net crossing
5
\(i\)
2
Active — period repeats
6
\(-1\)
4
Active
7
\(-i\)
2
Active
8
\(+1\)
0
Decoupled
The 720° spinor connection
The period-4 vanishing of \(A_n\) is the spectral signature of the Möbius double-covering. A 360° rotation of the bilateral crossing returns the phase to \(+1\) after 4 steps (\(i^4 = 1\)), but the crossing amplitude measures the departure from identity — which is zero at \(i^4\). This is precisely the quantum-mechanical statement that spin-\(\tfrac{1}{2}\) particles require a 720° rotation for a full phase cycle. The decoupled modes are the bilateral realisation of the Pauli exclusion in the mode spectrum.
Bilateral vacuum simulation from the first 50 Riemann zeros. Propagator computed as finite mode sum; power spectrum by FFT. Vertex errors measure \(|t_m+t_n-t_{\rm nearest}|/t_{\rm nearest}\). Dunstan Low · ontologia.co.uk