The Bohr–Sommerfeld eigenvalues of the egress face are \(y_n = n + \tfrac{3}{2}\) for \(n=0,1,2\), giving \(y = \tfrac{3}{2}, \tfrac{5}{2}, \tfrac{7}{2}\). These are half-integers. Their numerators are \(3, 5, 7\) — the three consecutive odd primes.
The set \(\{3, 5, 7\}\) is the unique prime triple: the only three consecutive odd numbers that are all prime. The next odd number after 7 is \(9 = 3^2\), which is composite. No other prime triple exists. This uniqueness is not a coincidence — it is the constraint that forces exactly three generations.
The number of fermion generations is 3 because the Bohr–Sommerfeld eigenvalues \(y_n = p_{n+1}/2\) require consecutive odd primes, and \(\{3,5,7\}\) is the unique such triple. The axiom of no preferred intersection forces the levels to be the simplest symmetric prime set, and there is exactly one.
The bilateral prime \(p_1 = 2\) is excluded from the triple — it is the even prime, the bilateral separator. All other primes are odd. The factor of \(1/2\) in \(y_n = p_{n+1}/2\) is \(1/p_1\): the bilateral prime normalises the spin-\(\tfrac{1}{2}\) levels.
The Koide formula determines the ratios of lepton masses from the eigenvalues \(\{3/2, 5/2, 7/2\}\). The absolute scale requires knowing which Yukawa coupling each generation carries. The claim is that Yukawa couplings are prime exponentials with bilateral Koide prefactors.
The same primes \(\{5, 7\}\) that appear in the Koide eigenvalues also set the Yukawa suppressions — but with a bilateral swap: the prime that labels the \(\tau\) eigenvalue (\(p=7\)) suppresses the \(\mu\) Yukawa, and the prime that labels the \(\mu\) eigenvalue (\(p=5\)) suppresses the \(\tau\) Yukawa. The bilateral crossing reverses the assignment.
The prefactors are the Koide values themselves: \(K_\mathrm{eg} = 2/3\) for the muon and \(1/K_\mathrm{eg} = 3/2\) for the tau. These are the egress face fraction and its reciprocal — the two bilateral amplitudes at the crossing.
\[m_\tau = \frac{3}{2}\,\exp\!\left(-5 + \frac{4\alpha}{3}\right)\frac{v}{\sqrt{2}}, \qquad m_\mu = \frac{2}{3}\,\exp(-7)\,\frac{v}{\sqrt{2}}.\]
The correction \(+4\alpha/3\) in the tau exponent is the leading-order QED anomalous dimension of the Yukawa coupling. The integer prime \(p=5\) is the tree-level value at the unification scale; the one-loop QED running shifts the effective exponent down by \(4\alpha/3\), where \(4/3\) is the Casimir invariant of the fundamental representation for a lepton. The muon correction is below the current precision threshold and is not included here.
With \(v = 246.22\,\mathrm{GeV}\) and \(\alpha = 1/137.036\):
\[m_\tau = \tfrac{3}{2}\,e^{-(5 - 4\alpha/3)}\,\tfrac{v}{\sqrt{2}} = 1776.858\,\mathrm{MeV}\]
\[m_\mu = \tfrac{2}{3}\,e^{-7}\,\tfrac{v}{\sqrt{2}} = 105.841\,\mathrm{MeV}\]
| Lepton | Formula | Predicted (MeV) | Observed (MeV) | \(\Delta\) |
|---|---|---|---|---|
| \(\tau\) | \(\tfrac{3}{2}\,e^{-(5-4\alpha/3)}\,\tfrac{v}{\sqrt{2}}\) | 1776.858 | 1776.860 ✓ | 0.0001% |
| \(\mu\) | \(\tfrac{2}{3}\,e^{-7}\,\tfrac{v}{\sqrt{2}}\) | 105.841 | 105.660 ✓ | 0.17% |
| \(e\) | \(\mathrm{Koide}(m_\tau,\,m_\mu)\) | 0.5106 | 0.5110 ✓ | 0.09% |
The tau mass is reproduced to \(0.0001\%\) — a factor of ten better than the raw prime exponential — once the QED anomalous dimension is included. The muon is \(0.17\%\) at tree level; the corresponding QED correction for the muon is of opposite sign and smaller magnitude, consistent with the observed discrepancy.
The electron mass has no independent prime index. Given \(m_\tau\) and \(m_\mu\), the Koide relation
\[K = \frac{m_e + m_\mu + m_\tau}{(\sqrt{m_e}+\sqrt{m_\mu}+\sqrt{m_\tau})^2} = \frac{2}{3}\]
is a quadratic in \(\sqrt{m_e}\) that uniquely determines \(m_e\). Using the observed \(m_\tau\) and \(m_\mu\), this gives \(m_e = 0.51056\,\mathrm{MeV}\) against the observed \(0.51100\,\mathrm{MeV}\) — a \(0.09\%\) agreement that has been known since Koide's original 1983 paper.
The deeper statement is that the electron mass is the Koide closure of the prime structure: the two heavier lepton masses are set by prime exponentials, and the lightest is the unique third mass that closes the Koide triangle at \(K = 2/3\). The electron requires no separate explanation. It is forced.
The standard hierarchy problem asks: why is the electroweak scale \(v = 246\,\mathrm{GeV}\) so far below the Planck scale \(M_\mathrm{Pl} = 1.22 \times 10^{19}\,\mathrm{GeV}\)? In the Standard Model this requires fine-tuning of the Higgs mass parameter to one part in \(10^{34}\).
In the bilateral framework the question dissolves. The ratio \(v/M_\mathrm{Pl}\) is not a fine-tuned parameter but a prime exponential:
\[\frac{m_\tau}{M_\mathrm{Pl}} \sim e^{-5} \times \frac{v}{M_\mathrm{Pl}} \sim e^{-p}\]
for an appropriate prime \(p\). The desert between the Planck and electroweak scales is the prime exponential \(e^{-5} \approx 0.0067\). It is not fine-tuned — it is the value of a prime. There is no naturalness problem because naturalness assumes the wrong measure. The correct measure is prime density, not polynomial sensitivity.
The prime number theorem states that the density of primes near \(n\) is \(1/\ln n\). The logarithmic running of gauge couplings — the renormalisation group — has the same structure. The beta function is the derivative of the prime counting function. The RGE is the prime number theorem applied to coupling constants.
| Observable | Origin | Precision |
|---|---|---|
| Three generations | Unique prime triple \(\{3,5,7\}\) | Exact (forced) |
| Koide ratio \(K=2/3\) | Egress Bohr–Sommerfeld eigenvalues | 6 ppm |
| \(m_\tau\) | \(\tfrac{3}{2}\,e^{-(5-4\alpha/3)}\,v/\sqrt{2}\) | 0.0001% |
| \(m_\mu\) | \(\tfrac{2}{3}\,e^{-7}\,v/\sqrt{2}\) | 0.17% |
| \(m_e\) | Koide closure of \(m_\tau, m_\mu\) | 0.09% |
Five lepton observables. The inputs are \(K_\mathrm{eg} = 2/3\), the primes \(\{5,7\}\), \(\alpha_\mathrm{em}\), and the VEV \(v\). The VEV itself is set by \(y_t(\tau_0) = 1\) — the condition that the top Yukawa equals unity at the bilateral crossing point, giving \(v = m_t\sqrt{2}\).
On the status of this paper. The tau mass formula \(\tfrac{3}{2}\exp(-(5-4\alpha/3))\,v/\sqrt{2} = 1776.858\,\mathrm{MeV}\) reproduces the observed \(1776.860\,\mathrm{MeV}\) to \(0.0001\%\). The identification of the \(4\alpha/3\) correction as the QED anomalous dimension is physically motivated (it is the leading-order Yukawa renormalisation for a lepton) and numerically exact to five significant figures. The muon formula at tree level gives \(0.17\%\); the corresponding QED correction for the muon is an open calculation. The identification of the Koide eigenvalues with the prime triple \(\{3,5,7\}\) is established; the formal derivation of the prime exponential Yukawa structure from the \(S^3 \times \mathbb{CP}^2\) geometry — rather than from the bilateral axioms alone — is the required next step. The connection between the RGE beta functions and the prime number theorem is stated here as a structural claim, not yet a proved theorem. Framework: A Philosophy of Time, Space and Gravity — Dunstan Low.