# PrimeMind Factor–Power Certificate & Gap Half-Balance Lab v1

## Purpose

This standalone lab separates two ideas that are easy to confuse:

1. **Exact vertical correspondence:** inside every integer, each prime-factor occurrence has one canonical partner among the active nested prime-power coordinates.
2. **Open horizontal balance:** inside a prime-to-prime interval, the exact structural units may fall more heavily in the earlier half, the later half, or equally in both.

The first statement is a consequence of unique factorization. The second is a measurable question and is not forced by the first.

## Central exact certificates

For

\[
n=\prod_p p^{e_p},
\]

copy number \(k\) of prime factor \(p\) is paired with coordinate \(p^k\), for \(1\le k\le e_p\). Therefore

\[
\#\{\text{factor occurrences}\}
=
\#\{\text{active nested coordinates}\}
=
\Omega(n).
\]

After compressing each prime column to its maximum active exponent,

\[
\#\{\text{distinct prime bases}\}
=
\#\{\text{exponent entries}\}
=
\omega(n).
\]

Both equalities hold row by row, and therefore after summing any finite interval or prime gap.

The integer 1 is included as the empty multiplicative origin:

\[
\Omega(1)=\omega(1)=0.
\]

It has no factor occurrences, no active prime-power coordinates, and no largest prime factor.

## Why a prime gap does not have to be centered

For consecutive odd primes \(P<Q\), the interior contains an odd number of integers, so there is one central integer and equally many rows on either side. For a selected mass \(w(n)\), the lab defines

\[
L=\sum_{\text{earlier half}}w(n),\qquad
C=w(\text{center}),\qquad
R=\sum_{\text{later half}}w(n).
\]

The directional statistic is

\[
\Delta=R-L.
\]

Splitting the center equally between the two halves changes both half totals by \(C/2\), so it does **not** change \(\Delta\). A gap may therefore be earlier-heavy, later-heavy, or exactly equal.

The total structural sequence can grow even when \(R\le L\), because every new integer contributes nonnegative mass and new prime or power coordinates are introduced as the atlas expands. Local later-heavy imbalance is not required for global growth.

For a monotone size metric such as \(\log n\), the later member of every mirrored pair is automatically larger. A later-heavy result under that metric is therefore partly construction-driven. For \(\Omega\), no such monotone rule exists.

## Preliminary independent scan through 10,000,000

Across 664,577 completed odd-prime gaps:

| Metric | Later-heavy | Earlier-heavy | Equal | Mean \(R-L\) |
|---|---:|---:|---:|---:|
| \(\Omega\) active cells | 268,628 | 269,838 | 126,111 | -0.009471 |
| \(\omega\) distinct bases | 247,006 | 247,732 | 169,839 | -0.008109 |
| \(\Omega-\omega\) higher-power cells | 263,413 | 264,001 | 137,163 | -0.001362 |
| \(\log n\) | 605,597 | 0 | 58,980 twin-gap equalities | +0.000065 |

Thus the strong statement “the later half is always heavier” is false for the combinatorial factor metrics. The distribution is very close to directionally balanced in aggregate, which is a better and more interesting question to study.

## Tabs

### 1 · Integer certificate

- canonical factor-occurrence ↔ nested-coordinate links;
- expanded and compressed balance checks;
- product reconstruction;
- weighted logarithmic identity;
- sequence-step check;
- controlled tamper demonstrations.

### 2 · Gap certificate

- one row per integer from the lower prime through the upper prime;
- exact row-level differences;
- inclusive visual totals;
- non-overlapping half-open totals \([P,Q)\).

### 3 · Half balance & growth

- earlier / center / later decomposition;
- split-center shares;
- raw and normalized half imbalance;
- mass center;
- normalized cumulative bridge;
- mirrored row-pair table;
- interchangeable metrics.

### 4 · Coordinate decomposition

For \(\Omega\)-mass,

\[
R-L
=
\sum_{p^a}
\left(
\#\text{later appearances of }p^a
-
\#\text{earlier appearances of }p^a
\right).
\]

The page splits the result into short- and long-period coordinates. A large period means a small frequency.

### 5 · Bulk scan

- scans every completed odd-prime gap through the selected endpoint;
- counts later-heavy, earlier-heavy, and equal gaps;
- plots the running mean and imbalance distribution;
- summarizes results by global layer regime;
- opens counterexample gaps directly;
- exports one compact row per gap.

### 6 · Definitions & exports

Exports current integer, current gap, half balance, coordinate decomposition, and bulk-scan data.

## Validation

The browser build was tested through 10,000,000:

- 664,579 primes generated;
- 664,577 completed odd-prime gaps scanned;
- the \(\Omega\) bulk counts exactly matched an independent C++ implementation;
- n=72 produced 5 factor occurrences and 5 active nested coordinates;
- n=1 produced the empty 0 ↔ 0 certificate;
- tampering with one occurrence produced an immediate certificate failure;
- no JavaScript errors appeared in the tested tabs and exports.

## Claim boundary

The factor–power correspondence is exact. The direction and scaling of half-gap imbalance are empirical. A later predictive use must freeze its metric and signature before revealing the target gap.