Golden Resource Endpoint Comparisons at 5040

Abstract

A single prime layer determines the two adjacent objective comparisons at 5040.

Theorem 1.1 (The exact objective change from adding one prime layer).

$$\forall lambda\in \mathbb{R},n\in \mathbb{N},p\in \mathbb{N},\left(1\le n\wedge Prime\left(p\right)\right)\Rightarrow goldenResourceObjective\left(lambda,n\cdot p\right)-goldenResourceObjective\left(lambda,n\right)=(goldenLayerMarginal\left(p,factorization\left(n,p\right)+1\right)-lambda)\cdot log\left(p\right)$$

Proof. Machine-checked in Lean as D5/S3/Arith/GoldenResource/GoldenResource5040EndpointComparison.golden_resource_objective_single_layer_delta (✓ std3). ∎

Source. Repository-derived.

Commentary.

For every real price lambda, positive natural number n, and prime p, multiplication of n by p changes the objective by the next layer marginal minus lambda, multiplied by log p. Positivity of n is an explicit hypothesis.

The proof applies the frozen finite-support decomposition to n and n times p on one common support. Prime factorization changes only at p; all other local differences vanish. The remaining difference is evaluated using exact logarithm identities.

Theorem 1.2 (The two adjacent comparisons beyond the price interval).

$$\forall lambda\in \mathbb{R},\left(\frac{log\left(\frac{31}{30}\right)}{log\left(2\right)}\le lambda\Rightarrow goldenResourceObjective\left(lambda,5040\right)\le goldenResourceObjective\left(lambda,2520\right)\right)\wedge \left(lambda\le \frac{log\left(\frac{12}{11}\right)}{log\left(11\right)}\Rightarrow goldenResourceObjective\left(lambda,5040\right)\le goldenResourceObjective\left(lambda,55440\right)\right)$$

Proof. Machine-checked in Lean as D5/S3/Arith/GoldenResource/GoldenResource5040EndpointComparison.golden_resource_5040_endpoint_comparisons (✓ std3). ∎

Source. Repository-derived.

Commentary.

At a price at least log(31/30) divided by log 2, the objective at 2520 is at least the objective at 5040. At a price at most log(12/11) divided by log 11, the objective at 55440 is at least the objective at 5040. The comparisons include their endpoints and are non-strict.

This companion consumes the single-layer delta at n = 2520, p = 2 and n = 5040, p = 11. The exact boundary marginals are those of layer (2,4) and layer (11,1). The dependency direction is endpoint comparisons to single-layer delta.

The statement covers only these adjacent comparisons. The previously frozen strict-interval sufficiency result remains upstream. Global endpoint maximality, numerical decimal bounds, and comparisons with the continuous allocation model are not asserted here.

References

• Truth anchor: D5/S3/Arith/GoldenResource/GoldenResource5040EndpointComparison.golden_resource_5040_endpoint_comparisons
• Truth anchor: D5/S3/Arith/GoldenResource/GoldenResource5040EndpointComparison.golden_resource_objective_single_layer_delta
• Dependency: D5/S3/Arith/GoldenResource5040PriceInterval