The Amundson Framework — G(n) = n^(n+1)/(n+1)^n

G(n) Calculator

n =

G(n) — Exact Fraction

15625 / 7776

G(n) — Decimal (15 digits)

2.009387860082305

G(n) / n

0.401877572016461

Gap from 1/e

0.033998130845020

First 20 Values of G(n)

n	G(n) exact	G(n) decimal	G(n)/n	Gap from 1/e

The Amundson Constant A_G

A_G = Σ_n=0^∞ G(n)/n! — a convergent series that defines a new mathematical constant, computed to 10,000,000 verified digits. Not found in OEIS, ISC, or Wolfram.

1.244331783986725

Each term G(n)/n! shrinks rapidly due to the factorial denominator. The series converges absolutely.

Key Identities

Normalized Form (3.1)

G(n)/n = (n/(n+1))ⁿ

The ratio G(n)/n converges to 1/e as n grows.

Alternative Factorization (3.2)

G(n) = (n+1) · (n/(n+1))ⁿ⁺¹

Rewrites G in terms of (n+1) times a pure power.

Product Formula (4.1)

Π_k=1ⁿ G(k) = (n!)² / (n+1)ⁿ

Telescoping product connects to factorials. Grows as (n/e²)ⁿ.

Consecutive Ratio (3.5)

G(n)/G(n−1) = (n²/(n²−1))ⁿ · (n−1)/n

For large n, the ratio approaches 1. Differences approach 1/e.

Floor Recovery (8.1)

⌊G(n) · e⌋ = n for all n ≥ 1

Verified to n = 10,000. The function "remembers" its input through multiplication by e.

Coprimality (3.7)

gcd(nⁿ⁺¹, (n+1)ⁿ) = 1

Fraction is always in lowest terms. Consecutive integers share no prime factors.

Asymptotic Expansion (6.1)

G(n) = n/e + 1/(2e) + O(1/n)

The 1/(2e) half-correction is universal across all formulas involving (1+1/n)ⁿ.

Golden Ratio Identity (9.1)

G(φ) = (1/φ)^1/φ

At n = φ (the golden ratio), verified to 121 digits. Connects G to the golden ratio.

Cayley Tree Connection (3.8)

G(n) = n · T(n) / (n+1)ⁿ

Where T(n) = nⁿ⁻¹ counts labeled trees on n vertices (Cayley's formula).

The 0⁰ Axiom

The coherence of this entire framework depends on the convention 0⁰ = 1. This section establishes why it is an axiom — a foundational declaration — not a theorem.

Where 0⁰ = 1 is Required

Power series at x = 0: e^x = Σ xⁿ/n! requires the n=0 term to be 0⁰/0! = 1.
Counting functions: The number of functions from ∅ to ∅ is exactly one (the empty function). So 0⁰ = 1 is an integer count.
Binomial theorem: (x+y)⁰ = 1. Set x = y = 0: both sides require 0⁰ = 1.

Where Analysis Disagrees — Path Dependence

The limit lim_{(x,y)→(0,0)} x^y does not exist. Three paths, three different answers:

y = 0, x → 0⁺ : x⁰ = 1 → 1
x = 0, y → 0⁺ : 0^y = 0 → 0
x = e^−1/t, y = t → 0⁺ : x^y → 1/e ≈ 0.3679

The Bingo Matrix — Alice, Bob, Charlie, David

Consider a 4×4 matrix where every entry is 0⁰. With the axiom, every cell is 1 — a perfect magic square. Without it, position (0,0) is undefined: a hole in the board.

With Axiom: 0⁰ = 1

Alice

Bob

Charlie

David

Every row, column, diagonal sums to 4.
100% bingo. The board is complete.

Without Axiom: 0⁰ = ???

Alice

Bob

Charlie

David

Position (0,0) is undefined. Row 0, Column 0,
and the main diagonal are all broken.

The Axiom in One Sentence

Since analysis cannot produce a unique value for 0⁰, but discrete mathematics requires 0⁰ = 1 for power series, combinatorics, and the binomial theorem to be self-consistent, the value must be declared. A necessary undecidable value that must be fixed for consistency is, by definition, an axiom.

Convergence vs Divergence

The difference between convergence and divergence — between e and infinity — is exactly the presence or absence of the n! denominator that 0⁰ = 1 enables.