Newton’s record as a parallel move: what forgetting a record leaves
Result, 2026-09-27 (atlas of halving, open cell 6). Inserting an unobserved time into Newton’s cell is a series move and closes exactly (refinement note, §2). Inserting a record at that time attaches a new variable, the pointer, to the body; forgetting it is the Newtonian counterpart of the parallel move of the series/parallel note. For a position record with Gaussian pointer of width \(\sigma\) this move is exact (Proposition 1): it leaves the body’s position distribution unchanged and convolves its momentum distribution with a Gaussian of variance
\[v=\frac{\hbar^2}{4\sigma^2}.\]
Three consequences fill the Newton row of the atlas.
- Records at one instant combine in parallel: precisions add, \(\sigma^{-2}=\sigma_1^{-2}+\sigma_2^{-2}\), and so do the momentum variances, as the heat times of parallel faces combine by conductances in the gauge case.
- Refining with a record at every inserted time converges iff \(\sum_j\sigma_j^{-2}<\infty\): the total momentum diffusion is \(\frac{\hbar^2}4\sum_j\sigma_j^{-2}\), and the same sum measures the information gathered about the place. The continuous-measurement limit of Caves and Milburn is the case \(\sigma_j^{-2}=\kappa\tau_j\) (Corollary 2).
- Information and disturbance are locked at \(\hbar^2/4\) per unit: \(\sigma^2v=\hbar^2/4\) for every Gaussian record and every mesh (for other pointers the product is bounded below by the Planck paper’s Theorem 4). For \(\hbar=0\) the move is the identity for every \(\sigma\) on the classical phase-space distribution, since averaging the posterior over a forgotten outcome returns the prior (the factor \(e^{-s^2/(8\sigma^2)}\) contains no \(\hbar\); its effect on \(W\) is a momentum convolution of variance proportional to \(\hbar^2\)): records are series-exact, the Newtonian counterpart of a theory with no parallel planes, and exactly the joint determinacy of the fifth-postulate note.
The Gaussian record channel and its continuous limit are standard (Caves and Milburn 1987, metadata, cited in the Planck paper); the reading as the parallel move of the atlas is this note’s. For non-Gaussian records the general floor is Theorem 6 of the Planck paper.
1. The record channel
A position record with Gaussian pointer of width \(\sigma\) and outcome \(y\) has Kraus operator \(M_y\) acting on wavefunctions by multiplication with \((2\pi\sigma^2)^{-1/4}e^{-(x-y)^2/(4\sigma^2)}\); forgetting the outcome gives the channel \(\rho\mapsto\int M_y\rho M_y^\dagger\,dy\). Write \(W(q,p)\) for the Wigner function of \(\rho\).
Proposition 1. The forgotten record acts as \(\rho(x,x')\mapsto\rho(x,x')\,e^{-(x-x')^2/(8\sigma^2)}\), that is,
\[W(q,p)\ \mapsto\ \int W(q,p-p')\,\frac{e^{-p'^2/(2v)}}{\sqrt{2\pi v}}\,dp', \qquad v=\frac{\hbar^2}{4\sigma^2}.\]
The position marginal is unchanged; the momentum marginal is convolved with a centred Gaussian of variance \(v\).
Proof. \(\int M_y(x)M_y(x')\,dy=(2\pi\sigma^2)^{-1/2}\int e^{-[(x-y)^2+(x'-y)^2]/(4\sigma^2)}dy=e^{-(x-x')^2/(8\sigma^2)}\). With \(s=x-x'\) and \(W(q,p)=(2\pi\hbar)^{-1}\int\rho(q+\frac s2,q-\frac s2)e^{-ips/\hbar}ds\), multiplication by \(e^{-s^2/(8\sigma^2)}\) is convolution in \(p\) with the Gaussian whose characteristic function in \(s/\hbar\) is \(e^{-s^2/(8\sigma^2)}\), of variance \(v\) with \(v/(2\hbar^2)=1/(8\sigma^2)\). The factor equals one on the diagonal \(s=0\), so the position marginal is unchanged. \(\square\)
The record’s error is \(\sigma\) and its delivered impulse has spread \(\sqrt v\); their product is \(\hbar/2\), the saturated mark cost of the Planck paper’s Theorem 4.
2. Composition
Parallel. Two forgotten records at one instant multiply the density matrix by \(e^{-s^2/(8\sigma_1^2)}e^{-s^2/(8\sigma_2^2)}=e^{-s^2(\sigma_1^{-2}+\sigma_2^{-2})/8}\): one record with \(\sigma^{-2}=\sigma_1^{-2}+\sigma_2^{-2}\). Precisions add like conductances, and the momentum variances add. This is the Newtonian counterpart of two parallel faces at heat time \(2t\) combining into one at \(t\) (the series/parallel note, §3).
Series. Between records at different times the body evolves freely or under constant force, an affine symplectic map on \(W\); a momentum convolution at one time becomes a sheared convolution in \((q,p)\) at a later time. The channels compose exactly, because Gaussian convolutions and affine symplectic maps form a closed family.
Corollary 2 (refinement with records). Put a record of width \(\sigma_j\) at each vertex of a partition of \([0,T]\). The forgotten records convolve the final momentum distribution with a Gaussian of variance \(\frac{\hbar^2}4\sum_j\sigma_j^{-2}\); the shear adds the position spread \(\sum_jv_j(T-t_j)^2/M^2\le(T/M)^2\sum_jv_j\). The noise covariance of the forgotten-record channel converges along the refinement iff \(\sum_j\sigma_j^{-2}\) converges; for a divergent sum the momentum marginal flattens and the channel has no limit. The records’ Fisher information about the initial place, for free motion with known momentum, is \(\sum_j\sigma_j^{-2}\) and converges on the same condition. With \(\sigma_j^{-2}=\kappa\tau_j\) the sum is \(\kappa T\): individual records become coarse as the mesh shrinks, and the rate \(\kappa\) and the momentum diffusion rate \(\hbar^2\kappa/4\) survive the limit (the Lindblad term \(-\frac\kappa8[x,[x,\rho]]\)). This is a continuous limit of the kind Caves and Milburn construct; the Gaussian-pointer parametrization is stated here, their abstract being the part checked.
Proof. The multiplicative factors of Proposition 1 at one instant multiply; free evolution between vertices is an affine symplectic map, which carries Gaussian convolutions to Gaussian convolutions without changing the momentum variance of a convolution applied before it. The momentum variances therefore add (a harmonic force would rotate the kicks, which is why the corollary is stated for free and constant-force motion). \(\square\)
3. What the Newton row of the atlas now says
- Series move (unobserved insertion): exact after one scalar counterterm, at every \(\hbar\).
- Parallel move (forgotten record): a momentum convolution of variance \(\hbar^2/(4\sigma^2)\). It is the identity for \(\hbar=0\), as a gauge theory with no transverse planes has no parallel defect.
- What survives refinement with records: the information rate \(\kappa\) and the diffusion rate \(\hbar^2\kappa/4\), with fixed product per record. This is the Newtonian form of the atlas’s error budget: summable records, like summable parallel defects, give a limit, and the ratio that survives is set by the one action constant.
4. Consequence for STATE
Atlas cell 6 is filled for Gaussian records: forgetting a record is an exact parallel move whose defect is proportional to \(\hbar^2\), which makes joint determinacy the statement that Newton’s refinement has no parallel defect. The general, non-Gaussian record is covered by the Planck paper’s Theorem 6 and by Theorem E of the fifth-postulate note.