Golden Layer Marginal Decay

Abstract

Prime-layer marginals have a geometric upper bound and eventually fall below every positive price along each fixed prime direction.

Theorem 1.1 (A geometric bound for every positive prime layer).

$$\forall p\in \mathbb{N},a\in \mathbb{N},\left(Prime\left(p\right)\wedge 1\le a\right)\Rightarrow goldenLayerMarginal\left(p,a\right)\le \frac{{\frac{1}{p}}^a}{log\left(p\right)}$$

Proof. Machine-checked in Lean as D5/S3/Arith/GoldenLayerMarginalDecay.golden_layer_marginal_le_inv_pow (✓ std3). ∎

Source. Repository-derived.

Commentary.

For every prime p and positive layer a, the marginal benefit is at most p to the negative a divided by log p.

The proof bounds log x by x minus one for the ratio of consecutive reciprocal geometric factors. Its algebraic core proves that this ratio minus one is at most p to the negative a.

Theorem 1.2 (Only finitely many layers exceed a positive price at a fixed prime).

$$\forall p\in \mathbb{N},lambda\in \mathbb{R},\left(Prime\left(p\right)\wedge 0<lambda\right)\Rightarrow \left(\exists N\in \mathbb{N},\forall a\in \mathbb{N},N\le a\Rightarrow goldenLayerMarginal\left(p,a\right)<lambda\right)$$

Proof. Machine-checked in Lean as D5/S3/Arith/GoldenLayerMarginalDecay.golden_layer_marginal_lt_of_le (✓ std3). ∎

Source. Repository-derived.

Commentary.

For a fixed prime p and positive real price lambda, there is a natural cutoff N after which every layer marginal is strictly below lambda.

The geometric upper bound is consumed together with convergence of the powers of 1/p to zero. This theorem controls exponents at one fixed prime; it does not assert that only finitely many different primes can exceed the price.

References

• Truth anchor: D5/S3/Arith/GoldenLayerMarginalDecay.golden_layer_marginal_le_inv_pow
• Truth anchor: D5/S3/Arith/GoldenLayerMarginalDecay.golden_layer_marginal_lt_of_le
• Dependency: D5/S3/Arith/GoldenResourceOptimalInteger