Golden Resource Price Interval

Abstract

The best unadopted prime layer has an attained positive price. Comparing it with the worst adopted layer characterizes existence of a common optimal price.

Definition 1.1 (Upper layer price).

$$\forall n\in \mathbb{N},goldenUpperPrice\left(n\right)=sSup\left(nextPrimeLayerValues\left(n\right)\right)$$

Formalization. D5/S3/Arith/GoldenResource/GoldenResourcePriceInterval.goldenUpperPrice (✓ std3).

Source. Repository-derived.

Commentary.

L(n) is the real supremum of the set of next-layer marginals over all primes, with exponent factorization n p plus one. For positive n the supremum is attained, as proved below. The Lean API is named goldenUpperPrice.

Definition 1.2 (Lower layer price).

$$\forall n\in \mathbb{N},1<n\Rightarrow goldenLowerPrice\left(n\right)=min\left(adoptedPrimeLayerValues\left(n\right)\right)$$

Formalization. D5/S3/Arith/GoldenResource/GoldenResourcePriceInterval.goldenLowerPrice (✓ std3).

Source. Repository-derived.

Commentary.

For n greater than one, U(n) is the minimum of the adopted-layer marginals over the finite nonempty set of prime divisors of n. For empty prime support, namely n equal to zero or one, goldenLowerPrice is defined to equal goldenUpperPrice. This real-valued boundary convention extends the existential criterion to one; it does not make the full set of prices at one a bounded interval.

Definition 1.3 (Colossal abundance).

$$\forall n\in \mathbb{N},IsColossallyAbundant\left(n\right)\iff \left(\exists lambda\in \mathbb{R},0<lambda\wedge IsGoldenResourceOptimal\left(lambda,n\right)\right)$$

Formalization. D5/S3/Arith/GoldenResource/GoldenResourcePriceInterval.IsColossallyAbundant (✓ std3).

Source. Repository-derived.

Commentary.

The predicate asserts existence of a strictly positive real price at which n maximizes the frozen resource objective among all positive integers.

Theorem 1.4 (A prime attains the best next-layer price).

$$\forall n\in \mathbb{N},1\le n\Rightarrow \left(\exists p\in \mathbb{N},Prime\left(p\right)\wedge \left(\forall r\in \mathbb{N},Prime\left(r\right)\Rightarrow goldenLayerMarginal\left(r,factorization\left(n,r\right)+1\right)\le goldenLayerMarginal\left(p,factorization\left(n,p\right)+1\right)\right)\right)$$

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

Source. Repository-derived.

Commentary.

The next layer at prime two has positive marginal. The frozen uniform cutoff at that value leaves finitely many candidate primes. A maximum among these dominates the omitted tail, since every tail value is below the candidate at two. This constructs the preregistered witness for an arbitrary positive integer, without enumerating a fixed instance.

Theorem 1.5 (The supremum is an attained maximum).

$$\forall n\in \mathbb{N},1\le n\Rightarrow \left(\exists p\in \mathbb{N},Prime\left(p\right)\wedge \left(goldenUpperPrice\left(n\right)=goldenLayerMarginal\left(p,factorization\left(n,p\right)+1\right)\wedge \left(\forall r\in \mathbb{N},Prime\left(r\right)\Rightarrow goldenLayerMarginal\left(r,factorization\left(n,r\right)+1\right)\le goldenUpperPrice\left(n\right)\right)\right)\right)$$

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

Source. Repository-derived.

Commentary.

The attained maximum identifies the real supremum and supplies the upper bound for every next-layer marginal. The final criterion consumes this specification, so attainment is on its proof dependency path.

Theorem 1.6 (The upper price is positive).

$$\forall n\in \mathbb{N},1\le n\Rightarrow 0<goldenUpperPrice\left(n\right)$$

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

Source. Repository-derived.

Commentary.

The upper price equals a positive prime-layer marginal. This gives the positive price selected in the sufficient direction of the criterion.

Theorem 1.7 (Existence of a common price).

$$\forall n\in \mathbb{N},1\le n\Rightarrow \left(IsColossallyAbundant\left(n\right)\iff goldenUpperPrice\left(n\right)\le goldenLowerPrice\left(n\right)\right)$$

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

Source. Repository-derived.

Commentary.

For n at least one, colossal abundance is equivalent to L(n) at most U(n). Necessity bounds L by an optimal price and bounds that price by every adopted-layer marginal. Sufficiency chooses the positive price L and applies the frozen fixed-price threshold criterion. Equality is kept. This slice does not separately state the characterization of every admissible price, reduce absent-prime comparisons to the smallest missing prime, or classify ties at critical parameters.

References

• Truth anchor: D5/S3/Arith/GoldenResource/GoldenResourcePriceInterval.IsColossallyAbundant
• Truth anchor: D5/S3/Arith/GoldenResource/GoldenResourcePriceInterval.colossally_abundant_iff_price_interval_nonempty
• Truth anchor: D5/S3/Arith/GoldenResource/GoldenResourcePriceInterval.goldenLowerPrice
• Truth anchor: D5/S3/Arith/GoldenResource/GoldenResourcePriceInterval.goldenUpperPrice
• Truth anchor: D5/S3/Arith/GoldenResource/GoldenResourcePriceInterval.golden_upper_price_attained
• Truth anchor: D5/S3/Arith/GoldenResource/GoldenResourcePriceInterval.golden_upper_price_pos
• Truth anchor: D5/S3/Arith/GoldenResource/GoldenResourcePriceInterval.golden_upper_price_spec
• Dependency: D5/S3/Arith/GoldenPrimeLayerCofinite
• Dependency: D5/S3/Arith/GoldenResource/GoldenResourceThresholdCriterion