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Replicating pi(x) from Era Ordering with a Script

A numerical probe of generational admissibility

An M. Rodriguez

2026-03-25

One-Sentence Summary: A small script computes era-bounded integers, sorts the cumulative generated set, and bins the recovered primes to compare causal ordering with the usual prime-counting picture.

Summary: This note records a numerical experiment for the era-ordering idea. Integers are bounded to the earliest era in which their generators are available, with power formation treated as primitive and coprime recombination admitted afterward. A script computes the era function tau(n) era by era, caches prior runs, and plots both a dense local staircase and the global truncated frontier. The point is not to recover a smooth density but to watch a jagged prime-counting profile emerge from discrete generational admission.

Keywords: prime counting, pi(x), era ordering, natural numbers, generational admissibility, scripts

# Replicating pi(x) from Era Ordering with a Script The numerical question is simple: if integers are bounded to the earliest era in which their generators are available, what does the cumulative admitted set look like when it is sorted back into the usual order on `N`? The script for this experiment now lives at [`C:\Users\an\src\siran\writing\.scripts\tools\era_order.py`](C:\Users\an\src\siran\writing\.scripts\tools\era_order.py). It now works era by era rather than through a fixed numerical cutoff. On each run it: 1. loads the cached era state, if any; 2. saves a rolling `latest` image of the current era-truncated counting plots; 3. reports the known eras and the size of the admitted set; 4. asks how many more eras to compute, together with a timing estimate based on the most recent era; 5. computes those eras, updates the cache, and saves both per-era snapshots and a refreshed `latest` plot; 6. can optionally save checkpoint plots during materialized eras. When the admitted right endpoint becomes too large to read comfortably, the script automatically switches the global horizontal axis to `log10(N)` so late eras remain visible and very large integer endpoints do not break plotting. The plotted function is not the classical prime-counting function `pi(N)`. It is the era-truncated version $$ \pi_E(N)=\#\{p\le N : p \text{ is prime and has been admitted through era } E\}. $$ That distinction matters. The point of the experiment is to watch `\pi_E(N)` approach the familiar jagged prime frontier as more eras are added, not to pretend that a low-era truncation is already the full classical `\pi(N)`. Each saved image now contains two complementary panels: 1. a dense local staircase, with one plotted point for each successive integer in a finite window; 2. the global compressed frontier of `\pi_E(N)` across the whole admitted range. By default the dense local window runs from `1` through the current era, so the upper panel shows the recovered staircase in the region where it is already classical. A larger dense window can be requested explicitly with `--dense-xmax`, in which case the plot plateaus after the currently admitted primes. It uses the current era rule: $$ \tau(1)=1, $$ and for $n>1$, $$ \tau(n) = \min\left( \kappa(n), \min_{\substack{ab=n\\1 --- - [Preferred Frame Writing on GitHub.com](https://github.com/siran/writing) (built: 2026-03-25 18:30 EDT UTC-4)