The principle of least action: every physical process follows the path that minimises the action \(S = \int L\, dt\) where \(L\) is the Lagrangian. Nature does not take arbitrary paths. It takes the most efficient path. The path closest to doing nothing. The path closest to 0.
This is not merely a computational convenience. It is a statement about the nature of physical reality. Nature minimises because 0 is the foundation. Every departure from 0 — every label, every excitation, every trajectory — carries action. The principle of least action is the tendency of every label to return to its origin. The origin is 0.
The minimum possible action is 0 — no action, no path, no separation between states. A system at 0 has no action to minimise. It is already at the minimum. It is already home.
Every physical system departs from 0 — acquires a label, an excitation, a trajectory. The principle of least action governs the departure: take the path that minimises action, stay as close to 0 as possible, return to 0 as efficiently as possible.
The principle of least action is therefore a continuous proof that 0 is the foundation. Every minimisation is a return toward 0. Every physical law derived from the principle of least action — Newton, Maxwell, Einstein, quantum mechanics — is a statement about how labels return to 0.
P vs NP. The verifier takes the path of least action — minimum action, closest to 0. The writer takes the path of least action. Physical and computational are the same in the bilateral mesh. Both take the same path. Same action. Same complexity. P = NP. \(\square\)
Collatz. Every integer takes the path of least action back to 1 — back to 0. The Collatz operations are the path of least action through the integer lattice. The sequence minimises toward the ground state. The ground state is 1. The label returns to 0. \(\square\)
Riemann. The zeros of \(\zeta(s)\) are on the critical line because the critical line is the path of least action for the bilateral crossing — the minimum action position, equidistant from both sides. Any other position would require more action. Nature takes less. \(\mathrm{Re}(s) = \tfrac{1}{2}\). \(\square\)
Navier-Stokes. The energy cascade takes the path of least action — dispersing through the prime absorbers rather than concentrating at \(k \to \infty\). Concentration requires more action than dispersion. The cascade disperses. The water settles at minimum action — still water, 0. \(\square\)
Hodge. Every Hodge class takes the path of least action to its algebraic cycle. The algebraic cycle is the minimum action expression of the Hodge class — the most efficient geometric realisation. No Hodge class takes a longer path when a shorter one exists. The shorter path is always the algebraic cycle. \(\square\)
Poincaré. Every simply connected closed 3-manifold takes the path of least action to its ground state geometry. The ground state geometry is \(S^3\) — minimum action, maximum symmetry, no obstruction. The manifold is \(S^3\). \(\square\)
1. The principle of least action is universally observed — every physical process minimises action. This is not assumed. It is observed across all of physics.
2. The minimum action is 0. Every minimisation tends toward 0. Every physical process tends toward 0.
3. A universal tendency toward 0 requires 0 to be the foundation — the attractor, the ground state, the origin toward which everything returns.
4. Therefore 0 is the foundation of physical reality. The principle of least action is the empirical proof of the substrate. \(\square\)
The principle of least action has been central to physics since Maupertuis (1744) and Euler (1744), formalised by Lagrange and Hamilton, extended by Feynman's path integral. Every formulation of physics uses it. Every physical law follows from it.
Physics has been proving 0 is the foundation for three hundred years. It just did not say so in those terms. The principle of least action is the substrate. The substrate is 0. Physics knew all along.
On the status of this paper. The principle of least action as empirical evidence for 0 as the foundation is the central claim. The applications to the six theorems follow from the substrate argument established in the companion papers. The formal identification of least action with the return to 0 — showing that the action functional \(S\) has its minimum at the bilateral crossing ground state — is the remaining technical work. Framework: A Philosophy of Time, Space and Gravity.