Finite propagation: scalar rigidity, internal states and record limits
Two scalar rigidity statements hold with exact ballistic support. A probability convolution semigroup on the line supported in \([-ut,ut]\) is deterministic drift. A strongly continuous translation-invariant unitary group on \(L^2(\mathbb R)\) with kernels supported in \([-u|t|,u|t|]\) is translation at some \(|a|\le u\), up to a constant phase. The proofs below use variance additivity and the imported Paley–Wiener–Schwartz theorem, respectively. These are internal written derivations, not independent peer review or a claim of literature novelty.
Nontrivial bounded-speed dispersion can be carried by internal states. The two-state identities, hidden slow-mode counterexample and specified readout-noise limit below do not classify all matrix semigroups or force an action scale. The historical cone/arrow comparison is maintained in source evidence and interfaces.
Scalar probability kernels
C032. For probability measures, if \(\mu_0=\delta_0\), \(\mu_{s+t}=\mu_s*\mu_t\) and \(\operatorname{supp}\mu_t\subset[-ut,ut]\), then \(\operatorname{Var}\mu_T=n\operatorname{Var}\mu_{T/n}\le u^2T^2/n\to0\) and \(\mu_t=\delta_{bt}\) with \(|b|\le u\). Adjacent sections of such a law are translates of one another. The telegraph process escapes the theorem because its position alone is not Markov: its kernel’s Fourier transform \(e^{-\lambda t}[\cosh(st)+(\lambda/s)\sinh(st)]\) with \(s=\sqrt{\lambda^2-u^2k^2}\) is an even function of \(s\), hence entire in \(k\), but it is not a convolution semigroup in \(t\).
Scalar unitary kernels
Proposition. Let \((U_t)_{t\in\mathbb R}\) be a strongly continuous one-parameter unitary group on \(L^2(\mathbb R)\) that commutes with translations, and suppose that for every \(t\) the kernel \(K_t\) with \(U_t\psi=K_t*\psi\) has support in \([-u|t|,u|t|]\). Then there are real constants \(a,b\) with \(|a|\le u\) such that \(U_t\psi(x)=e^{-ibt}\psi(x-at)\).
Proof. Commutation with translations makes \(U_t\) a Fourier multiplier by a unimodular measurable \(m_t(k)\), and Stone’s theorem gives \(m_t(k)=e^{-it\omega(k)}\) for a real measurable \(\omega\). The support hypothesis and the Paley–Wiener–Schwartz theorem extend each \(m_t\) to an entire function with \(|m_t(k)|\le C_t(1+|k|)^{N}e^{u|t||\operatorname{Im}k|}\). The identity \(m_tm_{-t}=1\) holds on the real axis, hence on \(\mathbb C\), so \(m_t\) has no zeros and \(G_t=\log m_t\) is entire. For fixed \(t\), the bounds for \(m_t\) and \(m_{-t}\) give \(|\operatorname{Re}G_t(z)|\le u|t||\operatorname{Im}z|+N_t\log(1+|z|)+C_t\). For its harmonic real part, second-derivative estimates on a disk of radius \(R\) are \(O((R+\log R)/R^2)\) and vanish as \(R\to\infty\). Thus the real part, and hence \(G_t\), are affine; its value on the real axis is zero. Thus \(m_t(k)=e^{-i(a_tk+b_t)}\), with real \(a_t,b_t\) and \(|a_t|\le u|t|\). The group law makes \(a_t\) additive; the bound implies \(a_t=at\). Strong continuity makes the remaining phase a continuous character, hence \(e^{-ibt}\). This proves the form without selecting logarithms continuously in time. \(\square\)
Reading. Scalar evolutions in this class are translations at any \(|a|\le u\), with an arbitrary constant phase rate. The massless chiral choices \(a=\pm u\) saturate the speed bound. Interpreting phase rate as energy requires an action conversion factor. The Dirac evolution keeps light-cone support and unitarity by carrying two components; its scalar reduction has \(\omega(k)=\sqrt{u^2k^2+\omega_0^2}\), which is real on the axis but has branch points at \(k=\pm i\omega_0/u\) and so cannot be the dispersion of a scalar kernel of compact support. At \(u=c\) and \(\omega_0=mc^2/\hbar\) the branch points sit at the inverse reduced Compton length. Matching a telegraph rate to \(\omega_0\) makes their reciprocal scales coincide algebraically. This is a chosen model identification; analytic continuation alone identifies neither probabilities nor physical dynamics.
Bounded-speed fluctuations and the double limit
Refining time cuts and position resolution are distinct limits. For the symmetric telegraph model with mass \(m\), speed \(u\) and switching rate \(\lambda>0\), put \(H_*=mu^2/\lambda\) and \(\Delta_*=1/\lambda\). The window observable obeys \(\mathsf h(\Delta)\le H_*\Delta/\Delta_*\) (finite-speed response), and the conditioned midpoint obeys the exact law \(\kappa_{\rm mid}/H_*=g(\lambda T)=zR(z)[1-R(z)]\) of the return-bridge calculation, with cubic onset. The fixed-preparation short-window intrinsic coefficient vanishes under a speed ceiling; the specified crossover window \(\Delta_*=H_*/(mu^2)=1/\lambda\) concerns a long-window plateau, and \(g\) describes its approach. This does not assert a limit for noisy records whose error scale changes with the cut. Which premise makes the plateau positive and common to all bodies is A08’s question, not a consequence of scalar finite propagation.
Internal states, a matrix conjecture and readout noise
The scalar proposition is not a classification of matrix-valued finite-speed semigroups. A candidate, still unproved, says that a translation-invariant, strongly continuous contraction semigroup on \(L^2(\mathbb R;\mathbb C^n)\) whose convolution kernels have support in \([-ut,ut]\) has Fourier multipliers \(e^{t(B-ikA)}\), where \(A=A^\dagger\), \(\|A\|\le u\) and \(B+B^\dagger\le0\). The missing steps include pointwise time continuity from strong operator continuity, a justified matrix logarithm and differentiation of its generator. Positivity alone does not make \(B\) a conservative Markov switching generator: \(T_tf=e^{-t}f\) is positive and has zero propagation but loses mass. Conservation must be required separately. The scalar proof above does not discharge these obligations.
For the symmetric two-state telegraph generator \(Q=\lambda(\sigma_x-I)\) and \(V=u\sigma_z\), the exact matrix identity is \(\|[V,Q]\|=2u\lambda\), while the stationary velocity plateau is \(H_*=mu^2/\lambda=2mu^3/\|[V,Q]\|\). A coherent two-component model with \(A=c\sigma_z\), \(B_H=\omega_0\sigma_x\) similarly has \(\|[A,B_H]\|=2c\omega_0\). Converting frequency into rest energy uses a supplied action \(K\), \(m=K\omega_0/c^2\); physical \(\hbar\) enters only if separately identified with \(K\). A commutator norm does not see every slow mode: the four-state counterexample keeps the plateau fixed as an independent hidden label’s relaxation gap closes. For \(Q=0\), a random constant velocity \(\pm u\) instead has \(\operatorname{Var}(X_\Delta-X_0)=u^2\Delta^2\); a deterministic velocity has zero variance. Commuting velocity sectors therefore do not justify the old general zero-response claim.
There is also an exact, model-dependent readout effect. Define \(\mathsf h(\Delta)=m\operatorname{Var}(X_\Delta-X_0)/\Delta\) for a fixed bounded-speed motion. At the two endpoints add independent centered errors, independent of the motion, each of variance \(\varepsilon^2/12\). The observed increment then satisfies
\[\mathsf h_\varepsilon(\Delta) =\mathsf h(\Delta)+\frac{m\varepsilon^2}{6\Delta}.\]
For fixed \(\varepsilon>0\), the \(\Delta\downarrow0\) limit is infinite; for fixed \(\Delta>0\), removing the readout error first and then refining gives zero by finite speed. Along \(\varepsilon^2=6\kappa_0\Delta/m\), the observed coefficient tends to any chosen \(\kappa_0\ge0\). Independent dither or additive-noise measurement is a premise here; fixed-lattice deterministic rounding need not satisfy it. On a uniform \(N\)-cell partition, pin endpoint errors to zero and give the \(N-1\) interior errors this variance. Expanding the kinetic-action squares, the expected added action is \(m(N-1)\varepsilon^2/(12\Delta)\), with correlated neighboring increment errors accounted for in the expectation. This scaling selects a readout bias, not a lower bound on intrinsic action and not physical \(\hbar\).
Exact free-kernel composition
The independently useful time-refinement manuscript and PDF retain the normalized free-kernel proof, Rivero’s 1998 intermediate-coordinate proposal and the separate Brouder/Cariñena source leads. Their hypotheses are not those of the ballistic-support theorems above.
In that proof a positive Gaussian kernel of variance \(\kappa t/m\) composes exactly under integration over an inserted position for every \(\kappa>0\). Weak continuity and Gaussian time homogeneity permit every \(\kappa=ma\ge0\); \(a=0\) is the deterministic Dirac member. Requiring a positive density rules out zero by premise. Passing from positive kernels to oscillatory unitary propagation is another chosen step, and neither step identifies \(\kappa\) with physical \(\hbar\). The Newton synthesis distinguishes this free fixed-line result from the two-regulator RG question.
Consequence for STATE
Scalar finite-speed rigidity, the open matrix classification and the specified readout double limit remain separate. At fixed preparation, bounded speed makes the intrinsic short-window coefficient vanish; independent readout errors can create any prescribed resolution-dependent coefficient. Exact Gaussian composition also permits a deterministic zero member. None of these statements supplies a universal positive physical action, quantum reconstruction or historical transmission claim.