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I003: finite propagation, internal state and resolution

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Finite propagation remains a promising route to the structure of the state at a cut. This revised draft preserves that question, the exact two-state commutator comparison and the measurement-resolution test. It separates them from the stronger claims superseded by the repository review.

Origin: the user’s I003 remarks and the independent Fable draft of 2026-09-08. Revised 2026-09-09 after coordinator review. The earlier version is preserved in Git at commit 0632c7d. No claim ID is assigned; R02 owns the remaining theorem and prior-art audit.

1. Finite-propagation classification to audit

Use Fourier phase \(e^{-ikx}\) and write

\[M_t(k)=e^{t(B-ikA)},\qquad \partial_t\psi=-A\partial_x\psi+B\psi.\]

A candidate theorem is that every translation-invariant strongly continuous \(L^2(\mathbb R;\mathbb C^n)\) contraction semigroup whose convolution kernels have support in \([-ut,ut]\) has this first-order form, with \(A=A^\dagger\), \(\|A\|\le u\) and \(B+B^\dagger\le0\). In the unitary case \(B=-iB_H\) with \(B_H\) Hermitian.

This is the corrected proposed formulation, not a newly accepted classification. The scalar unitary argument remains in the two-faces note. The matrix proof must justify pointwise continuity in time from strong operator continuity, the local matrix-logarithm construction, and differentiation of the generator.

A useful route is to apply Bernstein bounds to every matrix entry: finite propagation gives exponential type at most \(ut\), while contraction bounds the multiplier on the real axis. If a differentiable generator is established, the second derivative estimate should force its affine dependence on \(k\). Dissipativity for all real \(k\) then constrains the derivative coefficient. These are proof obligations for R02.

Positivity preservation adds a diagonal real velocity matrix in the physical component basis. To conclude that \(B\) is a conservative Markov switching generator one must also impose conservation of total probability. For example, \(T_tf=e^{-t}f\) has positivity, contraction and zero propagation but loses mass. Choose row/column conventions explicitly, and retain the \(L^2\) contraction hypothesis or its appropriate invariant-weight norm.

2. Exact two-state scale and its scope

For symmetric velocities \(V=u\sigma_z\) and switching \(Q=\lambda(\sigma_x-I)\), the accepted C020 comparison gives

\[\|[V,Q]\|=2u\lambda,\qquad H_*=\frac{mu^2}{\lambda} =\frac{2mu^3}{\|[V,Q]\|}.\]

For the coherent specialization \(A=c\sigma_z\), \(B_H=\omega_0\sigma_x\), the same matrix calculation gives \(\|[A,B_H]\|=2c\omega_0\). Assigning rest energy through a supplied action unit \(K\) gives \(m=K\omega_0/c^2\). C039 provides the resulting Dirac model. This keeps the source of \(K\) separate from the frequency.

The two-state equalities are useful. A general multicomponent system carries more than one rate, and a commutator norm need not detect every slow mode. C041’s independent hidden sign changes the full gap while preserving the velocity commutator and plateau.

The commuting case also needs the preparation: a random constant velocity \(V=\pm u\) with \(Q=0\) gives

\[\operatorname{Var}(\Delta X)=u^2\Delta^2,\qquad \mathsf h(\Delta)=mu^2\Delta.\]

A deterministic velocity gives zero variance instead. Thus commuting matrices describe unchanged velocity sectors; their mixture can be ballistic. The old general zero-response conclusion is superseded by this distinction. Within C019’s finite irreducible reversible class, its spectral proof—not a commutator alone—establishes positivity.

3. A specified measurement-noise double limit

Suppose a finite-speed process is measured with independent additive errors, independent also of the motion, each centered with variance \(\varepsilon^2/12\). For two measurements the error in the increment has variance \(\varepsilon^2/6\), so

\[\mathsf h_\varepsilon(\Delta) =\mathsf h(\Delta)+\frac{m\varepsilon^2}{6\Delta}.\]

This is an additive-noise or suitably dithered readout model. Deterministic rounding to a fixed lattice does not automatically supply independent uniform errors. In this specified model the two iterated limits are zero and infinity; a path \(\varepsilon^2=6\kappa_0\Delta/m\) gives the selected value \(\kappa_0\). It tests readout dependence, not a restriction on real-number subdivision.

For a uniform partition with \(N-1\) interior nodes, pin endpoint errors to zero and give independent interior errors that same variance. The expected added kinetic action is \(m(N-1)\varepsilon^2/(12\Delta)\); normalizing by \(2/(N-1)\) gives the same bias. Other endpoint/readout conventions need their own calculation. The algebra and general classification remain under R02 review.

4. Preserved research questions

R02 should start with the scalar theorem and the corrected matrix convention, then obtain a bounded prior-art audit. The earlier scripts remain historical artifacts under the hard rule and are not an acceptance gate.