work_extensive_scale_composition_unique
plain-language theorem explainer
Any two binary operations on the reals that are work-extensive on scale values coincide pointwise: both must be ordinary addition. Citation target for uniqueness of ledger-style scale composition once extensivity is assumed. The proof is a two-line rewrite from the structure field that forces op(a,b)=a+b.
Claim. If $\mathrm{op},\mathrm{op}':\mathbb{R}\times\mathbb{R}\to\mathbb{R}$ are both work-extensive scale compositions (each satisfies $\mathrm{op}(a,b)=a+b$ and likewise for $\mathrm{op}'$), then $\mathrm{op}(a,b)=\mathrm{op}'(a,b)$ for all real $a,b$.
background
In the Unified Forcing Chain, scale and work values on the ledger are composed by a binary operation on $\mathbb{R}$. The structure WorkExtensiveScaleComposition packages the physical requirement that composition be extensive in the scale-as-work observable: composing two work values yields their ordinary sum. It is the theorem-facing stand-in for silently fixing addition as ledger composition.
The module's program is the complete inevitability chain T-1 through T8 from the cost foundation (Recognition Composition Law, normalization, calibration). Uniqueness of the work-extensive composition is a local algebraic fact used when certifying that a chosen scale-composition law is the canonical ledger law, not an arbitrary alternative binary operation.
Upstream, the same extensivity field is the content of the structure; related "scale" definitions elsewhere (phi-powers, action scaling) are not needed for the equality argument itself.
proof idea
Term-mode proof by introduction of $a,b$, then simultaneous rewrite of both sides with the structure field work_extensive. That field states $\mathrm{op},a,b=a+b$ and $\mathrm{op}',a,b=a+b$, so the two operations agree. No further lemmas are required.
why it matters
Feeds directly into canonical_scale_composition, whose doc-comment states that any work-extensive scale composition is canonical: the uniqueness result (together with equality to ledger composition) supplies the certificate that the operation is the forced ledger law rather than a free choice.
In the forcing-chain narrative this closes a small but sharp gap: once work extensivity is the physical constraint on scale composition, addition is unique, so downstream ledger and recognition constructions cannot hide a different binary law. It supports the module claim that composition structure is forced from cost/ledger axioms rather than postulated ad hoc. Not itself a T0–T8 step, but scaffolding uniqueness for the ledger-compose layer those steps rely on.
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