The cut paradox in two signatures: sections of a solid and instants of a motion
Test 1 of I003, drafted 2026-09-08 by the independent Claude Fable 5.1 session. Earlier scripts are historical artifacts; current verification uses written proof and source review. The proposition below is a working derivation awaiting review and a prior-art audit; no claim ID is assigned. The 2026-09-09 review preserves the scalar rigidity candidate and routes proof and source review to R02. The cone/arrow connection below is a modern heuristic; analytic continuation relates the specified evolution equations, rather than establishing a historical identity.
Result. The Euclidean statement C032, that a position-only law with ballistic support and independent stationary increments is a deterministic drift, has a Lorentzian counterpart: a scalar, translation-invariant, strongly continuous unitary evolution on the line whose kernels are supported in the light cone \([-ut,ut]\) is a uniform translation at some speed \(|a|\le u\), up to a constant phase. In both signatures a bounded-speed law with nontrivial fluctuation or dispersion needs a second state variable, the velocity sign of the telegraph process or the second component of the Dirac evolution, and the continuation \(\lambda\to i\omega_0\) of A09 carries the one situation into the other. The obstruction sits at the Compton wave number: the scalar reduction of the massive evolution has dispersion \(\omega(k)=\sqrt{u^2k^2+\omega_0^2}\) with branch points at \(k=\pm i\omega_0/u\), which at \(u=c\), \(\omega_0=mc^2/\hbar\) is \(k=\pm imc/\hbar\). This is the reading of the user’s remark 2 in I003 that the present calculation supports; remark 3, an \(h\) that controls convergence, is the crossover \(g(z)\) of the coordinator’s exact bridge note.
1. The two paradoxes as slicing laws
Cone. A solid is a family of sections \(A(z)\), \(0\le z\le h\). The cone \(A(z)=\pi r^2(1-z/h)^2\) has \(A'(z)=-2\pi r^2(1-z/h)/h\ne0\) below the apex; the cylinder has \(A'=0\). Democritus asks whether adjacent sections are equal or unequal, Chrysippus answers neither, and the continuum answer is that \(A(z+\delta)-A(z)=A'(z)\delta+O(\delta^2)\): adjacent sections differ by a quantity that vanishes with the cut spacing while their ratio of differences to spacing does not. The Mohist Canon stops the halving at an endpoint; Liu Hui and Archimedes cut “until it cannot be cut” and keep the limit. Cutting the solid at more places preserves the old sections; this is the restriction consistency of a fixed solid, the analogue of C034 for a fixed bridge.
Arrow. A motion is a family \(X(t)\). Zeno asks whether at each instant the arrow is where it is, and Aristotle answers that the argument takes time to be composed of nows (Physics VI.9). In the project’s models the state at a cut is \((X,V)\): the finite-speed bridge of C033 retains the velocity sign at every cut, and its midpoint atom is exactly the arrow at its turning instant, where the position is \(uT/2\) with probability \(1/I_0(\lambda T)\) and the velocity is fixed only by the right-continuous convention. Al-Naẓẓām’s leap is the discrete crossing of the same instant.
Under the exchange of the slicing variable \(z\) with \(t\) and of the slope \(A'\) with the velocity, the two dilemmas are one question: what an infinitesimally adjacent slice carries. The continuation of A09 makes the exchange precise for the telegraph and checkerboard dynamics, and the rest of this note states what survives on each side.
2. The Euclidean face
C032. 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 \(t\).
3. The Lorentzian face
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 \(|\operatorname{Re}G_t(z)|\le u|t||\operatorname{Im}z|+N_t\log(1+|z|)+C_t\). Harmonic derivative estimates make this function 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. The analytic obstruction and the physical crossover \(\Delta_*\) of the P02 note are the same scale seen from the two sides of the continuation.
4. The double limit
Refining cuts in time and resolving positions together is the double limit of I003’s remark 2. At finite speed and fixed \(\lambda\), the window observable obeys \(\mathsf h(\Delta)\le H_*\Delta/\Delta_*\) (P02), and the conditioned midpoint obeys the exact law \(\kappa_{\rm mid}/H_*=g(\lambda T)=zR(z)[1-R(z)]\) of the bridge-crossover note, with cubic onset. A window-independent coefficient therefore exists only above \(\Delta_*=K/(mu^2)\), and the joint limit of mesh and resolution to zero leaves no coefficient unless the speed bound is removed, which is the Gaussian class. The surviving constant is the plateau, and \(g\) is the rate it controls. Which premise makes that plateau positive and common to all bodies is A08’s question, and this note adds nothing to it.
5. Review and next step
R02 will audit the scalar proof and its Paley–Wiener/harmonic-growth ingredients, then examine the matrix extension in the revised I003 note. The historical analogy remains available for idea generation. Its mathematical content must be established model by model.