no_global_recognition_work_scale_composition_model
plain-language theorem explainer
No cost function on a configuration space can represent every real number as a recognition-work value under a scale-composition model. Anyone building the T5–T6 self-similarity bridge cites this to rule out an all-real work domain. The proof is a one-line contradiction: work_value at −1 collides with nonnegativity of cost.
Claim. Let $E$ be a configuration space and $\kappa$ a cost function on $E$ (nonnegative, with dichotomy and independent additivity). There do not exist maps $\mathrm{workEvent}:\mathbb{R}\to E$, $\mathrm{compose}:E\times E\to E$, and $\mathrm{op}:\mathbb{R}\times\mathbb{R}\to\mathbb{R}$ such that $\kappa(\mathrm{workEvent}(a))=a$ for every real $a$ and event composition represents the real scale operation $\mathrm{op}$.
background
In the Unified Forcing Chain, T0–T8 are forced from the cost foundation (Recognition Composition Law, normalization, calibration). Cost lives on a configuration space: empty config, binary join, consistency, and independence. A cost function $\kappa$ assigns a real to each config, is nonnegative, vanishes exactly on consistent configs (dichotomy), and is additive on independent joins.
A recognition-work scale-composition model would embed every real work value $a$ as an event of cost $a$, with event composition representing a real binary operation on those values. If composition were independent join, additivity would force that operation to be ordinary addition. The model is the natural candidate for treating recognition work as an unrestricted real scale.
The module’s stronger claim is complete inevitability of the chain from the absolute floor through $\varphi$ and $D=3$. Ruling out an all-real work representation is a local obstruction used when bridging unique $J$ (T5) to self-similar $\varphi$ (T6).
proof idea
Assume a recognition-work scale-composition model. Instantiate the work-value axiom at $a=-1$: the cost of $\mathrm{workEvent}(-1)$ equals $-1$. Cost-function nonnegativity gives $0\le\kappa(\mathrm{workEvent}(-1))$. Linear arithmetic yields the contradiction. No composition or operation axioms are needed; nonnegativity alone kills global real representation.
why it matters
Feeds t5_to_t6_bridge_holds, which packages the T5-to-T6 self-similarity bridge as theorem-backed once $J$-uniqueness is available. In the forcing chain, T5 pins $J(x)=(x+x^{-1})/2-1$; T6 forces $\varphi$ as the self-similar fixed point on the discrete ledger. An all-real work scale would blur the nonnegative, discrete cost geometry that makes self-similarity and the $\varphi$-ladder well-posed.
The result is a cheap but sharp domain constraint: recognition work cannot be a free real coordinate if cost is a CostFunction. It closes a modeling dead-end rather than advancing a numbered T-step, and supports the module’s “no gaps” claim between unique $J$ and forced $\varphi$.
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