nonnegativeWork_universal_independence
plain-language theorem explainer
Any two nonnegative scalar work values are independent as configurations in the cost-from-distinction ConfigSpace. The result is cited when assembling the canonical scalar work carrier (join = addition, empty support). The proof is a one-line definitional discharge: scalar values carry no internal predicates, so independence holds by construction.
Claim. For all nonnegative real work values $a,b \geq 0$, the pair $(a,b)$ satisfies the ConfigSpace independence relation of CostFromDistinction: the two configurations share no predicates.
background
The Unified Forcing Chain module derives T-1 through T8 as inevitabilities from the Recognition Composition Law plus normalization and calibration. Early in that chain one needs a concrete carrier for recognition work: nonnegative reals, abbreviated NonnegativeWork as ${x:\mathbb{R}\mid 0\le x}$.
CostFromDistinction.ConfigSpace is the abstract structure used throughout: empty configuration, binary join, consistency, and an independence relation meaning the two configurations share no predicates. Join is a commutative monoid with empty as identity; independence is symmetric and empty is independent of everything.
Scalar work is the realizable replacement for an impossible all-real model. Costs are nonnegative and geometric scales are positive, so this domain is exactly what the scale-closure bridge needs. Independence of pure scalars is the structural fact that their internal support is empty.
proof idea
Term-mode proof consisting of a single trivial. On the NonnegativeWork instance of ConfigSpace, Independent is definitionally true for every pair: scalar values have empty predicate support, so the independence obligation reduces to True and discharges without lemmas.
why it matters
Feeds directly into canonical_scalar_work_carrier, which packages the commutative additive work carrier: join is addition, zero is empty, every pair of scalars is independent (this theorem), and cost is the scalar value itself. That certificate is part of the foundation layer of the Complete Inevitability Chain (T-1 absolute floor through T8, D=3), where work and cost must sit on a clean monoidal carrier before J-uniqueness (T5), $\varphi$ forcing (T6), and the eight-tick octave (T7) are stated. Without universal scalar independence the aggregate carrier would not be a lawful ConfigSpace instance.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.