Decay Function of Marketability
M(Xₙ) = M(X₀)·e^(−λ·n·σ(t)) — the master construct. Every transformation between a direct claim and its paper substitute imposes a measurable saleability haircut, exponential in the hop count and amplified by current stress.
Formula
M(Xₙ) = M(X₀) · e^(−λ · n · σ(t)) M(X₀) initial marketability (normalized to 1.0) n number of substitution hops from the underlying λ asset-class-specific decay constant σ(t) current substitute-layer stress (Z-score scale)
Inputs
Higher λ → faster saleability decay per hop. Stress-sensitive asset classes (long-duration paper, exotic derivatives) carry higher λ than substrate-grade collateral.
Aggregate substitute-layer stress on a Z-score scale. Calm: 0.5. Mild stress: 1.5–2.0. 2008 / 2020 acute: 3.0+.
Implied hop half-life at current parameters:
6.93 hops
Number of substitution layers at which marketability falls to 50% of the underlying.
Marketability by hop
Each row is M(Xₙ) at the current λ and σ. The bar height collapses exponentially as substitution hops accumulate.
Worked hop examples
What "n" means in practice, for a precious-metals reference asset:
Physical gold (bullion)
n = 0 · Direct claim. No substitute layer between the holder and the asset.
100.0%
Allocated ETF (e.g., physically backed)
n = 1 · One layer of substitution: custodian relationship + creation/redemption mechanism.
90.5%
Synthetic ETF / unallocated paper claim
n = 2 · Two layers: paper claim on a custodian who holds derivatives, not metal.
81.9%
Tokenized synthetic on a chain
n = 3 · Three layers: chain layer over paper claim over derivative over physical.
74.1%
Lent / rehypothecated tokenized synthetic
n = 4 · Four layers: counterparty exposure stacked on top of all of the above.
67.0%
The decay function is the framework's master construct. The Mengerian Stress Index is its empirical implementation: each MSI component measures the observable spread at a specific n-and-asset combination, and the composite reconstructs the unobservable σ(t).