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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 safely even for enormous late-era endpoints, and plots a dense local staircase. 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). A companion experiment now lives at [`C:\Users\an\src\siran\writing\.scripts\tools\era_factor_heuristic.py`](C:\Users\an\src\siran\writing\.scripts\tools\era_factor_heuristic.py). It computes exact `\tau(n)` values up to a numerical cutoff, then compares naive prime-divisor search up to `\sqrt{n}` against an era-filtered search up to `\min(\tau(n),\sqrt{n})`. That gives a direct way to test whether era depth actually shrinks the candidate-prime set for factoring. 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. Late eras produce endpoints with thousands of decimal digits. The cache now stores those very large integers in a safe encoded form, so extending the run does not fail when JSON tries to serialize a huge decimal expansion. 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 is a dense local staircase, with one plotted point for each successive integer in a finite window. By default that window runs from `1` through the current era, so the saved figure stays close to the familiar staircase shape of `\pi(x)`. A larger dense window can be requested explicitly with `--dense-xmax`, in which case the staircase 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 19:00 EDT UTC-4)