Time is not the engine: it is read from the slope of persistence

The most faithful image is the one used in the monograph: the sieve is like a clock that has already finished ticking. Primes are not produced one after another; the whole arithmetic hierarchy is there at once.

Still, we can read it layer by layer: parity, then $p=3$, then $p=5$, then $p=7$, and so on. That succession is not physical time yet. It is a logical order of dependence, like pages in a book that is already written: they exist together, even if we read them one after another.

Time appears when this hierarchy receives a geometry. The Fisher metric of the sieve selects an internal coordinate, $\mu$, and when its component $g_{00}$ becomes negative, that coordinate becomes timelike in the relativistic sense.

Why? Because a metric classifies directions. The three spatial directions of the sieve contribute like ordinary distances, with a positive sign. If the $\mu$ direction contributes with the opposite sign, it is no longer one more spatial distance: it becomes the direction that separates states by proper duration. That is exactly the difference between geometry of space and geometry of spacetime.

In plain language: the sieve does not unfold in time; our reading of the sieve, made metric by Fisher geometry, becomes time. The measurable duration is not bare $\mu$, but proper time $d\tau=\sqrt{|g_{00}|}\,d\mu$.

So “before time” is not a chronological before. Asking what came before this reading is like asking what lies north of the North Pole: the question uses a coordinate beyond the domain where that coordinate has meaning.

The monograph identifies the spacetime metric with the Fisher metric of the sieve. In that metric, the component $g_{00}$ becomes negative beyond a threshold: this is the appearance of Lorentzian signature $(-,+,+,+)$.

The decisive point is the relative sign. The Fisher blocks associated with the active primes $3,5,7$ give three positive spatial contributions. The curvature of the $\mu$ direction gives $g_{00}=-|g_{00}|$. The interval then becomes $ds^2=-|g_{00}|d\mu^2+\sum_p a_p^2dx_p^2$: displacements in $\mu$ no longer add as spatial length, they define proper duration.

Physical time is therefore not postulated. It is defined as proper time: $\tau = \int\sqrt{|g_{00}|}\,d\mu$. The variable $\mu$ gives a coordinate, but observables depend on $\tau$, not on the chosen coordinate name.

$\mu$ is a sieve-depth coordinate; $\tau$ is what a physical clock measures. PT is not merely saying “call $\mu$ time”; it says the metric forces a direction in which proper durations become defined.

The arrow of time then receives an informational reading. It is not a mysterious flow added to the world; it is the stable orientation along which persistence structure becomes readable, measurable, and irreversible for internal observables.

The relativity chapter treats $\mu \leftrightarrow \tau$ structurally: the spacetime metric is a spectral invariant of the transfer matrices $\{T_m\}$, and $\mu$ is the unique coordinate producing the time component.

Time Rigidity then shows that $\tilde\mu=f(\mu)$, for any smooth strictly monotone reparametrization, preserves signature, proper time, and the observables $H_0$, $\Omega_\Lambda$, $q_0$, and $G_{\mu\nu}$.

Technically, the persistence potential $S(\mu)=-\ln\alpha_{EM}(\mu)$ gives a metric of the form $ds^2=-|S^{\prime\prime}|d\mu^2+\sum_p(S^{\prime}_p)^2dx_p^2$. The negative sign on the $d\mu^2$ term is not decorative: it turns a persistence scale into a Lorentzian time coordinate.

The physical content is therefore invariant under time-gauge changes. One may rename the coordinate, but not the proper durations, directional Hubble rates, or Einstein equations derived from it.