Hotelling's rule predicts a price shape structurally distinct from both the vault's hold-then-collapse and monotonic-decline commodity shapes

The vault already holds two shapes for what happens to a commodity's price when a cheaper production or extraction method becomes available: claim-asteroid-pgm-price-holds-then-collapses (price holds near the terrestrial-anchored level, then collapses toward a new cost floor once cheaper off-world supply displaces the old) and claim-cheaper-extraction-disruptions-fall-monotonically-not-hold-then-collapse (price falls continuously as cumulative production of the cheaper method scales, per claim-wrights-law-cost-falls-per-cumulative-production-doubling). Hotelling's rule describes neither: it assumes no new cheaper substitute enters the model at all, and its baseline prediction — net price rising continuously at the discount rate — points in the opposite direction from both existing shapes.

The rule's own extensions complicate a clean "rising" label further. Slade and Thille show the rate of rise slows once extraction cost depends on remaining reserves: "Since CR < 0, the shadow price increases at a slower rate... This is true because extraction today leads to higher costs tomorrow, and the owner internalizes this externality." And the observed market price (distinct from the theoretical shadow price) can even trace a U-shape under some extensions — falling while a fixed resource meets new extraction technology, then rising once scarcity dominates. In no variant discussed does the model produce the vault cluster's "flat, then a cliff" or "continuously falling" shapes: the three models are answering different questions. The vault cluster asks what happens to price when production or extraction gets cheaper; Hotelling's rule asks what happens to price when a fixed stock gets scarcer with no cheaper alternative in the model at all.