Aluminium's Hall-Héroult price decline and low-background steel's fallout decay are the same functional form — exponential decay in calendar time — for unrelated mechanisms

A vault bridge-check run against the seed pairing claim-low-background-steel-became-a-metaphor-for-pre-2022-ai-free-web-text and claim-aluminium-price-fell-monotonically-after-hall-heroult (unlinked, 0.75 cosine) found the metaphor note is not actually the aluminium note's nearest unlinked neighbor. claim-low-background-steel-need-is-fading-as-atmospheric-fallout-decays is (0.824 vs. 0.803) — and that pair is itself unlinked, despite the closer score. The bridge the seed pairing pointed at is real, but it runs through the wrong pair of notes.

Both describe an exponential decline in calendar time, for causally unrelated reasons. Aluminium's price fell continuously after the Hall-Héroult process — "the price fell to $2 per pound in 1889 and to $0.5 per pound in 1894" — a learning-by-doing decline: Sahal's identity shows a Wright's-law cost curve (falling with cumulative production) becomes exponential in calendar time whenever production itself scales exponentially, which 1880s–90s electrolytic aluminium plausibly did. Low-background steel's falling necessity follows a different exponential: short-to-medium-lived fallout isotopes physically decaying since atmospheric testing ended in 1963. One is a market learning to produce more cheaply; the other is nuclei losing energy on a fixed schedule. The convergent shape (exponential decay in calendar time) and divergent mechanism (economic scaling vs. radioactive decay) make this a form-only bridge, not a shared-mechanism one.

[unverified — inherits flags]. Both curve shapes rest on Tier 4 (Wikipedia) figures already flagged [unverified-quant] in their own notes — see question-verify-aluminium-price-history-primary-source and question-verify-low-background-steel-fallout-decline-figures. This note stays seedling until those clear.