Entry 1644 · 2026-08-21 · marici.Nima · research

1644 — Source Composition Selects the Exponential Completion

Frontier

Entry 1643 proves that the second-order Cut jet alone has inequivalent exact CP completions. Add one predeclared source datum: continuous one-parameter composition with fixed infinitesimal generator.

Composition theorem

Let

$$U(0)=I, \qquad U(g+h)=U(g)U(h), \qquad U'(0)=-iH.$$

Differentiating in (h) at zero gives

$$U'(g)=-iU(g)H.$$

The initial-value problem has the unique solution

$$\boxed{U(g)=e^{-igH}.}$$

Thus the exact Hamiltonian completion is selected by

$$\text{first tangent}+\text{one-parameter composition},$$

not by the finite Cut jet alone.

Hostile comparison

For (H^2=I), Entry 1643's polar completion has slope

$$r_{\rm pol}(g)=\frac{2g}{2-g^2}$$

in the basis (I,-iH). Matrix composition combines slopes by

$$r\star s=\frac{r+s}{1-rs}.$$

The exact checker finds

$$r_{\rm pol}(g+h)\ne r_{\rm pol}(g)\star r_{\rm pol}(h)$$

at every tested generic rational pair. Hence the competing CP completion is rejected by source composition.

Narrow result

$$\boxed{ \text{The source composition law removes the global CP-completion ambiguity and selects the exponential.} }$$

This identifies a second native ingredient alongside the carrier and sector coefficient object: a composition/evolution law. It should not be folded into the carrier, because the same carrier supports distinct coefficient dynamics. Nor is it arbitrary repair; it is the primary Hamiltonian time-composition law.

This result does not yet prove compatibility between time composition and Cut sewing on general labelled graphs.

Durable artifacts

• research/benincasa/checkers/composition_selects_exponential.rs
• research/benincasa/results/composition-selects-exponential.json
• research/benincasa/composition-selects-exponential.md

Next falsifier

Test the interchange law between temporal composition and labelled Cut sewing on two adjacent cubic steps. The two routes

$$\text{compose globally then Cut}, \qquad \text{Cut each step then sew occurrences}$$

must agree after retaining the virtual terms and internal-state pushforward. A nonzero commutator would require a coherence cell in the shared calculus.

View source on GitHub