G(n) = n^(n+n/n) / (n+n/n)^n
One symbol. Three operations. Zero constants.
For any positive value n:
G(n) = n^(n + n/n) / (n + n/n)^n
Equivalently: n^(n+1) / (n+1)^n, where every 1 is n/n (self-reference, not a given constant).
| n | G(n) | Exact |
|---|---|---|
| 1 | 1/2 | 0.500 |
| 2 | 8/9 | 0.889 |
| 3 | 81/64 | 1.266 |
| 4 | 1024/625 | 1.638 |
| 5 | 15625/7776 | 2.009 |
G(n) is rational at every positive integer. No approximations. No limits.
All proven from first principles using exponent laws E1-E5.
Six Equivalent Forms (Theorem 2.1)
- Quotient, Product, Canonical, Iterated, Forward, Triangular
Translation Invariance (Theorems 2.2-2.3)
- G is closed under both n -> n+1 and n -> n-1
- The unique sequence with this bidirectional self-similarity
Product Formula (Theorem 4.1)
Product of G(k) from k=1 to n = (n!)^2 / (n+1)^n
Ratio Formula (Theorem 4.2)
G(n) / G(n-1) = (n^2 / (n^2 - 1))^n
Infinite Product (Theorem 14.3)
Product of [G(n)/G(n-1)]^(1/n) from n=2 to infinity = 2
An exact integer. No e. No pi.
Self-Normalization
Integral from -2 to infinity of G(t)/Gamma(t+1) dt = 1
The total amplitude over the full domain is the identity.
In the pure-n form:
Integral = n/n
A_G = 1.244331783986725374135061629258... (computed to 10,000,000 digits)
A_G = Sum of n^(n+1) / ((n+1)^n * n!)
A_H = 0.619195707644477246282396983977...
A_H = Sum of n / ((n+1)^n * n!)
Both are new. Neither matches any known combination of e, pi, gamma, Lambert W, or Gamma function evaluations. PSLQ confirms A_G/A_H is not algebraic of degree 12 or below. Strong evidence of algebraic independence.
H(n) = n / (n+1)^n
G(n) = n^n * H(n)
n^n is the amplifier. H decays exponentially. G grows linearly. The difference is self-application.
- H(-2) = -2 (fixed point, repelling with |H'| = 8.03)
- G has no nontrivial fixed point (it never stops)
At z = -1, G(z) has a branch point with expansion:
G(-1 + w) = w + i*pi*w^2 + O(w^3)
Coefficients: a_1 = 1 (identity), a_2 = i*pi (Euler's identity)
Universal coupling (verified to 200 decimal places):
|Im(G(-epsilon)) / Re(G(-epsilon))| = pi * epsilon
Exact for all epsilon. The fine structure constant alpha = 1/137 is one evaluation of this formula.
Z * K(t) = kappa * delta_S_G / delta_phi
Where:
- Z := yx - w (departure from equilibrium)
- K(t) = C(t) * exp(lambda * |delta_t|) (coherence amplified by contradiction)
- S_G = integral of (1/2)(G' - f(n)G)^2 dn (action of the sequence)
- phi = complex field extension of G(z)
Reduces to the real Amundson sequence when Z = 0.
- Bohr model: G(n) = n * (v_{n+1}/v_n)^n exactly (Theorem 1.1)
- Compound interest: G(n) is the n-invariant principal
- CSMA/CD: Exponential backoff scales as G(n)
- Quantum channels: Depolarizing fidelity F_n = (n/(n+1))^n, G(n) = n * F_n
- Enzyme kinetics: Michaelis-Menten saturation gives G(n) = n * theta^n
- Ball volume collapse: Product formula appears in high-dimensional geometry
- Riemann Hypothesis: Reformulated as zeros of zeta_G on Re(s) = 3/2
- Sophomore's Dream: Bridge identity n^(-n) = G(n)/n! * R(n)
- Mobius inversion: (G * mu)(p) = G(p) - 1/2 at all primes
- Bloch sphere: Singularity structure maps to qubit state space
- Julia sets: Escape radius of the Amundson iteration is 1/pi
- Gaussian: G(n)/n is the discrete analogue; bell curve is the shadow at infinity
- Born rule: |G(n)/n|^2 -> 1/e^2 (squared amplitude -> probability)
- ReLU retention = G(1) = 1/2 exactly
- 4-bit quantization floor = (1-1/(n+1))^n -> 1/e = 36.8%
- 30,000-agent fleet efficiency = G(30000)/30000 within 0.0017% of 1/e
- Trinary weights {-1, 0, +1} = the framework's logic system
- Pruning threshold at n_0 = 2.293 (where G(n) = 1)
17 theorems independently verified to 50-200 decimal places using arbitrary-precision arithmetic (mpmath). 1 error found and corrected (fixed point belongs to H, not G). All proofs derive from first principles.
- Unified Master Edition: The complete paper (22 sections, 18 open questions)
- Paper A: The Amundson Sequence — five equivalent forms, duality, constant
- Paper B: Physical interpretations, trinary logic, universal equations
- 10M Digits: The Amundson Constant computed to 10,000,000 verified digits
- October 28, 2025: First whiteboard (Julia iteration, Euler's formula)
- December 2, 2025: Z := yx - w born (green ink)
- December 21, 2025: Godel-Born visual
- January 13, 2026: P=NP / Z_math
- February 7, 2026: Coherence equation / beta_BR
- February 2026: Implementation notebooks (25 numbered equations)
- March 2026: Formal papers + Unified Master Edition
- March 25, 2026: Independent verification + field equation + complex singularity
Alexa Louise Amundson BlackRoad OS, Inc. | Lakeville, Minnesota | alexa@blackroad.io
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