Overview
This source note fixes the phenomenological target of the charged-lepton mass sector of the fermionic matter sub-programme as a single scale-free invariant. Writing the square-root mass vector $r_\ell = (\sqrt{m_e}, \sqrt{m_\mu}, \sqrt{m_\tau})$, the Koide quotient is exactly $Q_\ell = (1 + \mathrm{CV}(r_\ell)^2)/3$ with $\mathrm{CV} = \sigma_r/\bar r$, so that $Q_\ell = 2/3 \iff \mathrm{CV}(r_\ell) = 1$: the square-root masses have standard deviation equal to their mean.
This is sharper than “why $3477$” or “why $45^\circ$”, and it separates the intrinsic dispersion content of Koide from the hierarchy. The note does not derive the masses; it isolates a target invariant, and records where the projective cascade under-shoots it and what an entropic derivation would require.
Core result
- Unit-dispersion identity. With mean $\bar r$ and standard deviation $\sigma_r$ of the three components of $r_\ell$, $Q_\ell = (1 + \mathrm{CV}(r_\ell)^2)/3$ exactly, so $Q_\ell = 2/3 \iff \mathrm{CV}(r_\ell) = 1 \iff \sigma_r = \bar r$.
- Data. For the pole masses, $\bar r = 17.716$ and $\sigma_r = 17.715$, so $\mathrm{CV}(r_\ell) = 0.99999 \pm 0.00001$ and $Q_\ell = 0.666661$: unit relative dispersion at the $10^{-5}$ level, consistent with exact at $0.9\sigma$ (limited by $m_\tau$) — consistent with exact, not established as exact.
- Robustness and scope. $\mathrm{CV} = 1$ is an infrared (pole-mass) statement, not an RG invariant (at $M_Z$, $\mathrm{CV} \simeq 1.002$), though it is invariant under any flavour-universal rescaling of $\sqrt m$, so the leading QED running leaves it fixed. It is specific to the charged leptons: the quark sectors sit away from it (indicatively $\mathrm{CV} \simeq 1.09$ down-type, $1.24$ up-type), and the neutrino value is undefined without the absolute mass scale.
- One-number dictionary. $Q$ is monotone in $\mathrm{CV}$: $\mathrm{CV} = 0$ is degenerate ($Q = 1/3$), $\mathrm{CV} = 1/\sqrt2$ is the $Z_2$-ladder value ($Q = 1/2$), and $\mathrm{CV} = 1$ is Koide ($Q = 2/3$).
Dispersion versus hierarchy
The relation fixes only the relative dispersion; it does not say which generation is light. In the cyclic parametrisation $r_g = \bar r\,(1 + \sqrt2\,\cos(\delta + 2\pi g/3))$, the condition $Q_\ell = 2/3$ fixes the amplitude $\sqrt2$ for every phase $\delta$, while $\delta$ places the generations relative to the nodes of the square-root profile (zeros of $1 + \sqrt2\cos\phi$ at $135^\circ, 225^\circ$). The large hierarchy arises when one generation lies near a node — for the observed masses the electron sits about $2.6^\circ$ from the $135^\circ$ node, so $\sqrt{m_e}$ is nearly annulled by interference. The two targets are therefore distinct: $\mathrm{CV}(\sqrt m) = 1$ (Koide, scale-free), and the node placement of $\delta$ (the concrete hierarchy), the latter being the harder, exponentially sensitive residue.
Cascade-ladder obstruction
The projected generation levels form the pinned ladder $(1, \tfrac12 + u, \tfrac12 - u)$, with dispersion bounded by $\mathrm{CV} \le 1/\sqrt2$. But this bounds the static levels (their square roots give $\mathrm{CV}(r_\ell) \in [0.09, 0.37]$ at $u = 1/10$, with $\lambda_Y$ cancelling), not the physical masses: the physical charged-lepton mass carries, beyond the levels, the depth amplification of the cascade, $m_g = \lambda_Y\,w_g\,S_g\,A(n_g)$ with $A(n_g) = e^{\beta^\ast n_g}$, so its coefficient of variation is set by the cascade depths, not capped by the ladder. Presenting the ladder bound as the obstruction to $\mathrm{CV} = 1$ would conflate the levels with the mass. Within the corpus's present (depth) realisation the forced $O(1)$ depths give neither unit dispersion nor the hierarchy — but this is a limitation of the depth realisation, not a foundational obstruction. Unit dispersion would instead be carried by a saturation acting on the resolved observable-type capacities (a singlet–doublet coincidence of the A4 projection lock), an arena not yet constructed but not forbidden.
Maximum-entropy reading (open test)
Unit coefficient of variation, $\sigma_r = \bar r$, is the signature of the exponential — the maximum-entropy distribution on $\mathbb{R}_+$ at fixed mean. The crux is not the existence of an entropy but the constraint: only a first-moment condition on $\sqrt m$ gives $\mathrm{CV} = 1$. Fixing instead the natural quadratic norm $\langle m \rangle$ gives the half-normal value $\mathrm{CV} = 0.756$; fixing the support gives the uniform value $\mathrm{CV} = 0.577$. Reaching $\mathrm{CV} = 1$ requires fixing the linear sum $\langle \sqrt m \rangle$ alone. A structural hook would make this natural: the non-injective pushforward combines fibre content linearly, $\nu(o) = \sum_{\text{fibre}} \mu$, so if $\sqrt m$ is that linearly summed fibre content, its total is the conserved datum and the first-moment constraint is forced.
A lock of the continuous route only. On the continuous max-entropy route, even granting the constraint, $\mathrm{CV} = 1$ is a property of the continuous exponential and three discrete masses do not inherit it (maximising the entropy of the three discrete weights at fixed sum gives the uniform distribution — equal square-root masses, $\mathrm{CV} = 0$; natural three-point realisations are recipe-dependent, none pinned at unity: tercile means $\approx 0.81$, quantiles $\approx 0.76$, sample CV of three draws $\approx 0.65$). This is not a universal obstruction, however. A route that imposes unit dispersion directly on the discrete triple — the singlet–doublet equipartition of the cyclic form, where three phases $120^\circ$ apart inherit $\mathrm{CV} = A/\sqrt2$ exactly and independently of the phase (the cyclic form is the exact real DFT of any triple) — dissolves the bridge: the discrete triple has $\mathrm{CV} = 1 \iff A = \sqrt2$ directly. This is how the ternary phase carrier addresses it (speculatively), relocating the lock to the amplitude condition $A = \sqrt2$.
Status
Exact. The identity $Q_\ell = (1 + \mathrm{CV}(r_\ell)^2)/3$ and the equivalence $Q_\ell = 2/3 \iff \mathrm{CV}(r_\ell) = 1$.
Obstruction. The pinned cascade ladder has $\mathrm{CV} \le 1/\sqrt2$, and the committed Yukawa carrier gives $\mathrm{CV}(r_\ell) \in [0.09, 0.37]$; the observed unit dispersion is reached by no presently named mechanism of the projective cascade.
No-go (determinant realisation). One concrete realisation of the discrete singlet–doublet route is now excluded: no real-linear amplitude map that is self-adjoint in the inherited positive Hermitian structure can make the Koide radicand $s^2 - d^2$ the square of a determinant operator on a non-zero projected block — for a doublet vector the required square would be negative, contradicting the positivity of the square of a self-adjoint operator. This closes the linear self-adjoint determinant realisation only; the discrete equipartition route itself, and its non-linear, Krein, or enlarged-tower extensions, remain open.
Open. The maximum-entropy route, whose crux is a first-moment constraint on $\sqrt m$ and a forced discrete-to-continuous bridge; and, harder still, the concrete hierarchy — the node placement fixed by the phase $\delta$ — which is where the genuine residue of the charged-lepton hierarchy remains.
Relation to the Cosmochrony programme
This note belongs to the fermionic matter sub-programme. It is downstream of the projected Yukawa line and operator notes, whose level ladder it shows to under-disperse, and it points the entropic reading at the projection entropy of the non-injective projection. The mass target it isolates is not fixed by the projected Yukawa line itself; it is recorded here separately as the unit-dispersion Koide target.
References
Jérôme Beau. The Charged-Lepton Koide Relation as a Unit-Dispersion Constraint. Working paper, 2026. 10.5281/zenodo.21109812