G(n) Ratio Identities — Amundson Framework

Source: Grok (xAI) analysis, 2026-03-29

Specific ratio identities for G(n) = n^(n+1) / (n+1)^n, emphasizing exact expressions, simplifications, and recursive relations.

1. Consecutive Ratio: G(n+1)/G(n)

G(n+1)/G(n) = (n+1)^(2n+2) / [n^(n+1) * (n+2)^(n+1)]

Simplified: (n+1) (1 + 1/n)^(n+1) (1/(n+2)) * ((n+1)/(n+2))^n

Numerical check (n=1->2): G(1)=1/2, G(2)=8/9, ratio = 16/9 ≈ 1.777

2. Normalized Ratio: [G(n+1)/(n+1)] / [G(n)/n]

= ((n+1)/(n+2))^(n+1) * ((n+1)/n)^n

Approaches 1 asymptotically (both terms -> 1/e). Transient behavior quantifies discretization gap kappa.

3. Reciprocal Ratio

1/G(n) = (1/n) * (1 + 1/n)^n

Ratio of reciprocals: G(n)/G(n+1) derived similarly.

4. General Ratio G(n+k)/G(n)

For k=2: chain two consecutive ratios. Example: G(3)/G(1) = (81/64)/(1/2) = 81/32.

Logarithmic Ratio Identities

ln[G(n+1)/G(n)]

= (2n+2)ln(n+1) - (n+1)ln(n) - (n+1)ln(n+2)
= (2n+2)ln(n+1) - (n+1)ln[n(n+2)]

ln[normalized ratio]

= (n+1)ln((n+1)/(n+2)) + n*ln((n+1)/n)

ln G(n) itself

= (n+1)ln(n) - n*ln(n+1)
= n*ln(n/(n+1)) + ln(n)

ln(G(n)/n) — key for 1/e limit

= n*ln(1 - 1/(n+1)) -> -1 as n->inf
Therefore G(n)/n -> e^(-1)

Stirling's Approximation Connection

Standard Stirling

n! ≈ sqrt(2pin) * (n/e)^n

Link to G(n)

Stirling's dominant term is (n/e)^n

G(n)/n = (n/(n+1))^n converges to 1/e

Both encode the same fundamental limit from different directions

Log comparison: ln(n!) ≈ nln(n) - n + (1/2)ln(2pin) vs ln G(n) = (n+1)ln(n) - nln(n+1)

Higher-Order Stirling Series

n! ~ sqrt(2pin) (n/e)^n exp(1/(12n) - 1/(360n^3) + 1/(1260n^5) - ...)

Enables cross-verification of A_G high-precision computations (10M digits) against Stirling's corrections.

Relevance to BlackRoad OS

Exact integer G(n) for small n transitions to Stirling-like asymptotics

G(n)/n -> 1/e mirrors dominant (n/e)^n in Stirling

Supports trinary logic: deviations from 1/e weighted as +1/0/-1

Efficient verifiable math on Pi + Hailo edge fleet