Constants

constants

Every parameter is $K$ at a different address.

K

$$\dot{\theta}_i = \omega_i + \frac{K}{N}\sum_j \sin(\theta_j - \theta_i)$$

K ≪ 1

K < 1

K = 1

K > 1

The coupling strength. The only free parameter in the framework, and it is not free — it is determined self-consistently by the population it generates. $K$ is the dial that selects which physics applies.

$K \ll 1$: free phases, no structure, quantum vacuum. $K < 1$: partial locking — Schrödinger governs. $K = 1$: critical coupling — Einstein takes over. $K > 1$: rigidity, cascading failure.

Every other constant is $K$ evaluated at a specific rational address $p/q$ in the Stern-Brocot tree. The standard model does not have 19 free parameters. It has one parameter at 19 addresses.

C

$$c = \lim_{\Delta t \to 0} \frac{\Delta(\text{phase information})}{\Delta t}$$

phase information propagating at finite speed

The maximum rate at which one oscillator can learn about another’s phase. Not a property of light — a property of the coupling topology. The causal bound exists before photons do.

In the Kuramoto model, coupling propagates through the mean field. At $K = 1$, the coherence tensor $C_{ij}$ becomes the spatial metric $\gamma_{ij}$, and the lapse function $N$ (local clock rate) is the Kuramoto order parameter $|r|$. The speed at which the lapse can change — the speed at which synchronization updates propagate through the metric — is $c$.

Newton assumed infinite propagation: every body knows where every other body is. Reality: each body only knows where the others were. This finite propagation is not imposed. It emerges from the topology of the self-consistency loop — information cannot outrun the iteration that generates it.

III

E

EULER’S NUMBER

$$\delta x_n = \delta x_0 \cdot (f')^n = \delta x_0 \cdot e^{\,n \ln f'(x^*)}$$

iterates spiraling to x*

The third primitive generates it. Near a fixed point $x^* = f(x^*)$, small perturbations evolve as $(f')^n$. The moment you have iteration + fixed points, you have $e$. It is not imported — it is the natural base of the map’s linearization. The Floquet multipliers governing mode-lock stability go as $e^{-\lambda}$. Convergence rates, Lyapunov exponents, decay constants — all expressions of $e$ emerging from the fixed-point primitive.

ELEMENTARY CHARGE

$$\alpha = \frac{e^2}{4\pi\varepsilon_0 \hbar c} \approx \frac{1}{137}$$

tongue width at the EM address

The fine structure constant $\alpha$ is the Arnold tongue width at the electromagnetic address in the tree. The elementary charge $e$ is the coupling quantum for the U(1) mode-locking — the minimum coupling strength that locks two oscillators at the photon frequency ratio. That $\alpha \approx 1/137$ is not a coincidence to be explained; it is a tree depth.

M

$$C(p/q) = \frac{1}{q^2 \sin^2(\pi\, p/q)}$$

Mass is not substance. Mass is how much coupling budget a frequency ratio consumes. High-denominator rationals are expensive to lock. A proton is not heavy because it contains stuff — it is heavy because its internal frequency ratios sit at costly addresses in the tree.

The particle mass spectrum is a map of the Stern-Brocot tree weighted by synchronization cost. Light particles live at simple rationals (low $q$). Heavy particles live at complex rationals (high $q$). The mass hierarchy is not a mystery — it is the denominator hierarchy.

THE DECOMPOSITION

The standard model’s 19 parameters, decomposed. Each is $K$ at an address, filtered through the four primitives.

Parameter	Tree expression	Primitive
$g_1, g_2, g_3$	Arnold tongue widths at U(1), SU(2), SU(3) addresses	Parabola
$m_u, m_c, m_t$	$C(p/q)$ at up-type quark frequency ratios	Fixed point
$m_d, m_s, m_b$	$C(p/q)$ at down-type quark frequency ratios	Fixed point
$m_e, m_\mu, m_\tau$	$C(p/q)$ at charged lepton frequency ratios	Fixed point
$\theta_{12}, \theta_{13}, \theta_{23}$	Rotation angles between mass eigenstates on the tree	Mediant
$\delta_{\text{CP}}$	Phase offset at the CP-violating node	Integers
$v$ (Higgs VEV)	Critical coupling threshold: $K = 1$ energy scale	Fixed point
$\lambda_H$	Self-coupling curvature at the $K = 1$ fixed point	Parabola
$\theta_{\text{QCD}}$	Winding number of the strong coupling path	Integers

Three gauge couplings: parabola (Arnold tongue boundaries are saddle-node bifurcations). Nine masses: fixed point (synchronization cost at self-consistent addresses). Three mixing angles: mediant (rotation between Farey neighbors). Two phases: integers (winding numbers on the circle). Two Higgs parameters: the fixed point that is $K = 1$ and the parabola that is its local curvature.

THE STAIRCASE

$$W(\Omega) = \lim_{n\to\infty}\frac{\theta_n}{n}$$

devil’s staircase — $W(\Omega)$ at $K = 1$

The winding number $W$ as a function of driving frequency $\Omega$ is a devil’s staircase: a continuous, monotone function that is locally flat on every rational plateau and yet climbs from 0 to 1. The flat steps are mode-locked regions — Arnold tongues seen from the side. At $K = 1$ the steps touch and the staircase becomes a singular function: all rise, no slope. Every constant in the table above lives on one of these steps.

VII

ℤ × K

everything falls out of $\mathbb{Z}$ and $K$

The integers give the addresses. The coupling gives the physics. Together they generate the four primitives — parabola, fixed point, mediant, integers — and from those, every constant in the standard model. The loop above traces this: $\mathbb{Z}$ and $K$ on the left, the four primitives branching outward, the 19 parameters orbiting as their consequences. There is nothing else.