Expanded correspondence
p copy k ↔ active coordinate pᵏ
#occurrences = #active cells = Ω(n)
#occurrences = #active cells = Ω(n)
Canonical and exact for every integer n≥2.
PrimeMind · exact correspondence and balance diagnostics
Factor–Power Certificate & Gap Half-Balance Lab
Verify the exact one-to-one correspondence between factor occurrences and active prime-power coordinates, then keep that vertical identity separate from the open question of how structural mass is distributed between the earlier and later halves of a prime gap.
Large builds run in a background worker.
Ready.
Current integer——
Factorization——
Expanded correspondence—factor occurrences ↔ active cells
Compressed correspondence—prime bases ↔ exponent entries
Current prime gap——
Later-minus-earlier half—interior Ω-mass
Exact factor–power correspondence at one integer
Each copy of a prime factor is paired canonically with one nested coordinate in the same prime column: copy 1 with p¹, copy 2 with p², and so on. The compact factorization pairs each distinct base with exactly one exponent entry.
factor occurrenceactive nested coordinatecanonical one-to-one link
Exact certificate across a complete prime gap
The local equality holds row by row, so it also holds after summing any finite set. The visual gap includes both prime boundaries; the non-overlapping global partition uses [P,Q), so adjacent gaps do not double-count their shared prime.
| offset | n | class | factorization | factor occurrences | active cells | difference | distinct bases | exponent entries | difference |
|---|
—
Earlier half, center row, and later half
A consecutive odd-prime gap has an odd number of interior integers, so it has one central row and equal numbers of rows before and after it. Nothing in the exact factor–power correspondence forces the two sides to contain equal mass. This page tests that separate question directly.
Per-integer mass profile
Earlier rows, the unique center, and later rows are separated explicitly.
Three-part mass
Compare earlier half L, center C, and later half R without forcing C onto either side.
Normalized cumulative bridge
The bridge is internal shape after total mass is normalized; it need not be zero at the midpoint.
Mirrored row pairs
| distance from center | earlier n | earlier mass | later n | later mass | later−earlier |
|---|
—
Which recurrence coordinates create the half difference?
For Ω-mass, every active coordinate contributes one unit. The later-minus-earlier gap difference is therefore the sum of the later-minus-earlier appearance count of every pᵃ coordinate. Long period means low frequency, and a long-period coordinate usually appears zero or one time in a short gap.
Coordinate contributions by period
Each bar is #later appearances minus #earlier appearances for one coordinate.
Short versus long period totals
The two groups must add exactly to the observed Ω half difference.
| coordinate | period | frequency | earlier | center | later | later−earlier | first interior hit | last interior hit |
|---|
Bulk test of the “later half is always larger” idea
Scan every complete odd-prime gap through a chosen endpoint. This separates exact local correspondence from a statistical directional claim and immediately exposes counterexamples to any overly strong version.
Running mean later−earlier imbalance
Downsampled for display; exact values remain in the export.
Distribution of normalized half imbalance
Zero means equal half mass after the center row is excluded.
Results by global layer regime
| regime L | prime range | gaps | later | earlier | equal | mean difference | mean normalized |
|---|
Run a scan.
Definitions, claim boundaries, and exports
The exact theorem is vertical: factor occurrences correspond one-to-one with nested power coordinates. The half-balance question is horizontal: where those exact units fall inside an ordered prime gap. These must not be conflated.
Compressed correspondence
pᵉ ↔ pair (p,e)
#bases = #exponent entries = ω(n)
#bases = #exponent entries = ω(n)
One compact pair per distinct prime factor.
Gap halves
L = Σ earlier rows
C = center row
R = Σ later rows
Δ = R−L
C = center row
R = Σ later rows
Δ = R−L
No theorem forces Δ=0 or Δ>0 for Ω.
Coordinate decomposition
ΔΩ = Σₚ Σₐ (#R hits of pᵃ − #L hits of pᵃ)
The exact total can be separated into short- and long-period coordinates.
Empty origin
n=1: Ω(1)=ω(1)=0
One has no factor occurrences and no active prime-power coordinates. It is not a prime-gap boundary.
Growth caution
Σ positive row masses grows automatically
local R−L need not be positive
local R−L need not be positive
The number line expands by adding integers and coordinate births, not because every gap must be later-heavy.
Research boundary: completed gaps and factorizations are descriptive truth. Any predictive use of a half-balance signature, coordinate tail, or gap certificate must freeze its definition before the target interval is revealed.