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The mark cost is exactly \(\hbar/2\), and the worst-case theorem survives as a statistical one

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A mark made by coupling the body’s position to a probe has canonically conjugate error and recoil, so its cost is \(\kappa=\delta\Delta\ge\hbar/2\) for every probe state (Theorem A). In the Gaussian statistical model every protocol of marks with \(\delta_j\Delta_j\ge\kappa\) needs \(\tau\Delta E\ge48z_{1-\epsilon}^2\kappa\), with a sharp constant (Theorem B), hence \(\tau\Delta E\ge24z_{1-\epsilon}^2\hbar\) for Gaussian probes whose error and recoil are uncorrelated, about \(65\hbar\) at five per cent error. A correlation \(\rho\) between error and recoil multiplies the floor by \(\sqrt{(1-\rho)/(1+\rho)}\) (Theorem C), and three marks attain that dependence within a factor \(4/3\) (Proposition D), deciding at every force as \(\rho\to1\): the mechanism of Yuen’s contractive states, applied to the meter. These floors concern non-adaptive Gaussian marks read by tests invariant under the body’s unknown initial position and velocity. Invariance is what carries them: the body’s state cancels from every invariant statistic, and a protocol that knows the preparation escapes them even with Gaussian states (§4). For probes in arbitrary states and for every instrument, the statements that hold are the recoil bound of the recoil note and the disturbance bound of the disturbance note.

Revisions, 2026-09-22. The first version applied Theorem B to every Gaussian probe. Theorem B assumes independent error and recoil, and a probe with correlated position and momentum delivers correlated ones, which Theorem C now prices. The constant of Theorem B improved from \(\frac29\) to \(\frac1{24}\) in \(d^2\,m\kappa/F^2\tau^3\): the invariance under \(v_0\) makes the function \(T\) of the proof vanish at both ends, so the Poincaré constant \(\tau^3/12\) replaces the Cauchy–Schwarz bound \(\frac49\tau^3\). The same squeezing mechanism corrects the aperture corollary of the probabilistic note: a floor of order \(\hbar\) needs the shape of the uncertainty ellipse bounded as well as its area.

Reopening the question of what was wrong with the derivation note: almost nothing. Its Proposition 6 claimed a positive quantum value for the mark cost \(\kappa=\delta\Delta\), and that claim is correct. The proof was invalid and the constant was too large by a factor of four, and the worst-case formulation of \(\delta\) and \(\Delta\) as support widths is what made the quantum instance empty. Replacing support widths by standard deviations repairs everything at once, and two theorems then close the gap the derivation note left open.

Theorem A. For a mark realized by the impulsive coupling \(\lambda\hat y\otimes\hat P_A\) of the body’s position to a probe, with the probe’s pointer read afterwards, the mark’s error operator and the impulse it delivers are canonically conjugate,

\[\hat N=\frac{\hat Q_A}{\lambda}-\hat y,\qquad \hat D=-\lambda\hat P_A, \qquad [\hat N,\hat D]=-i\hbar,\qquad\text{hence}\qquad \delta\Delta=\Delta_\psi\hat N\cdot\Delta_\psi\hat D\ \ge\ \frac\hbar2\]

for every probe state whatever. So \(\kappa\ge\hbar/2\), exactly, with no error–disturbance relation and no calibration argument.

Theorem B. In the Gaussian statistical model, for every non-adaptive protocol of marks with \(\delta_j\Delta_j\ge\kappa\), of any number and at any times, with the initial position and velocity unknown and the tests invariant under them, the optimal test’s deflection obeys

\[d^2\ \le\ \frac{F^2\tau^3}{24\,m\kappa},\qquad\text{hence deciding at error probability }\epsilon\text{ requires}\qquad \tau\Delta E=\frac{F^2\tau^3}{2m}\ \ge\ 48\,z_{1-\epsilon}^2\,\kappa ,\]

with \(z_{1-\epsilon}=\Phi^{-1}(1-\epsilon)\). With Theorem A this is \(\tau\Delta E\ge24z_{1-\epsilon}^2\hbar\), about \(65\hbar\) at five per cent error, for probes whose error and recoil are uncorrelated; Theorem C gives the correlated case. The proof is the derivation note’s proof unchanged: the same certificate, the same arithmetic–geometric step, the same integration by parts to a Dirichlet energy, with the separating hyperplane replaced by the optimal test direction. One step is sharper here: the invariance under the unknown velocity makes \(T\) vanish at both ends, so the Poincaré constant replaces the Cauchy–Schwarz bound and the constant becomes \(48\), which is sharp.

Newton’s premise M3 is Theorem A for his own probe: the interval of fits \(\Lambda\) is the corpuscle’s transverse position spread and \(p\) its transverse momentum spread, so \(\kappa=\Lambda p\) is the corpuscle’s own uncertainty product, and the heuristic value \(h/2\) of the mark-floor note differs from the theorem’s \(\hbar/2\) by a factor \(2\pi\). Section 6 sets out the single paper this supports. Exploratory; no ledger promotion.

1. What was actually wrong, stated exactly

The derivation note’s framework defines a mark’s resolution \(\delta\) and recoil width \(\Delta\) as worst-case interval widths: the datum confines the position to an interval of length \(\delta\) with certainty, and the records confine the delivered impulse to an interval of length \(\Delta\) with certainty. Three claims about that framework are now settled.

So the defect was the choice of currency, and it cost one constant and one proof. Standard deviations are the right currency because they are finite for quantum states, because they are what the uncertainty relation constrains, and because the derivation note’s proof never used containment beyond a quadratic bound, as Section 3 shows.

2. Theorem A: the mark’s error and its recoil are conjugate

Take the von Neumann model of a position mark. The body has \(\hat y,\hat p\); the probe has \(\hat Q_A,\hat P_A\) with \([\hat Q_A,\hat P_A]=i\hbar\); the mark is the impulsive unitary

\[U=\exp\left(-\frac{i}{\hbar}\lambda\,\hat y\,\hat P_A\right),\qquad\lambda>0 .\]

Theorem A. In the Heisenberg picture across the mark,

\[U^\dagger\hat Q_AU=\hat Q_A+\lambda\hat y,\qquad U^\dagger\hat pU=\hat p-\lambda\hat P_A,\qquad U^\dagger\hat yU=\hat y,\qquad U^\dagger\hat P_AU=\hat P_A .\]

Reading the pointer \(\hat Q_A\) and estimating the body’s position by \(\hat Q_A/\lambda\), the error operator \(\hat N=\hat Q_A/\lambda-\hat y\) and the delivered impulse \(\hat D=\hat p_{\rm after}-\hat p_{\rm before}=-\lambda\hat P_A\) satisfy \([\hat N,\hat D]=-i\hbar\), hence for every joint state

\[\delta\,\Delta:=\Delta\hat N\cdot\Delta\hat D\ \ge\ \frac\hbar2 .\]

Proof. With \(A=\frac{i}{\hbar}\lambda\hat y\hat P_A\), so that \(U^\dagger=e^{A}\), the commutators are \([A,\hat Q_A]=\frac{i\lambda\hat y}{\hbar}[\hat P_A,\hat Q_A]=\lambda\hat y\) and \([A,\hat p]=\frac{i\lambda\hat P_A}{\hbar}[\hat y,\hat p]=-\lambda\hat P_A\), both central, so the Baker–Campbell–Hausdorff series terminates and gives the four displayed relations. The error operator is then \(\hat N=(\hat Q_A+\lambda\hat y)/\lambda-\hat y=\hat Q_A/\lambda\), whence

\[[\hat N,\hat D]=\left[\frac{\hat Q_A}{\lambda},-\lambda\hat P_A\right] =-[\hat Q_A,\hat P_A]=-i\hbar,\]

and Robertson’s inequality gives \(\Delta\hat N\,\Delta\hat D\ge\frac12|\langle[\hat N,\hat D]\rangle|=\hbar/2\). \(\square\)

Three features matter. The coupling strength \(\lambda\) cancels, so the bound is a property of the mark and not of how hard it is made. The result holds for every probe state, pure or mixed, Gaussian or not. And the two quantities are the probe’s own position and momentum spreads: \(\delta=\Delta\hat Q_A/\lambda\) and \(\Delta=\lambda\Delta\hat P_A\), so

\[\kappa=\delta\Delta=\Delta\hat Q_A\cdot\Delta\hat P_A ,\]

the mark’s cost is the probe’s uncertainty product, and the floor of the Galileo comparison is set by the probe rather than by the body. The contested error–disturbance relations play no part: Ozawa’s counterexamples concern instruments whose commutator \([\hat N,\hat D]\) differs from \(-i\hbar\), and the present statement is confined to the momentum-transfer class where it equals \(-i\hbar\).

The correlation of error and recoil. The same commutator bounds the joint covariance of the two quantities. With \(c=\operatorname{Cov}(\hat N,\hat D)\) the symmetrized covariance, the Robertson–Schrödinger inequality gives

\[\delta^2\Delta^2-c^2\ \ge\ \frac{\hbar^2}{4},\qquad c=-\operatorname{Cov}(\hat Q_A,\hat P_A),\]

with \(\lambda\) cancelling again. The correlation coefficient \(\rho=c/(\delta\Delta)\) is a property of the probe state. It vanishes for coherent states and for states squeezed along \(\hat Q_A\) or \(\hat P_A\), and it approaches \(\pm1\) for a state squeezed along a rotated quadrature \(\hat Q_A\cos\varphi+\hat P_A\sin\varphi\). Section 4 shows that \(\rho\) enters the floor.

Why the records cannot recover the recoil. The pointer record is \(\hat R=\hat Q_A+\lambda\hat y\) and the impulse is \(\hat D=-\lambda\hat P_A\), with \([\hat R,\hat D]=-i\hbar\lambda\). The record and the recoil are incompatible observables, so no reading of the pointer, however precise, determines the impulse. This is Proposition 5 of the derivation note seen from the other side: the far-screen construction recovers the impulse by measuring the probe’s momentum, and measuring the probe’s momentum is exactly what reading its position forbids.

3. Theorem B: the same proof, in the Gaussian statistical model

The model. Marks at \(0=t_0<t_1<\dots<t_k\le\tau\). Mark \(j\) returns \(R_j=y(t_j)+\xi_j\) with \(\xi_j\) centred Gaussian of standard deviation \(\delta_j\), and delivers an impulse \(\iota_j\), centred Gaussian of standard deviation \(\Delta_j\), all independent. The trajectory is

\[y(t)=y_0+v_0t+\theta P(t)+\frac1m\sum_{j}\iota_j(t-t_j)_+, \qquad P(t)=\frac{Ft^2}{2m},\]

with \(\theta=0\) under \(\mathrm I\) and \(\theta=1\) under \(\mathrm F\), and \(y_0,v_0\) unknown. A protocol satisfies the mark trade-off with constant \(\kappa\) when \(\delta_j\Delta_j\ge\kappa\) for every \(j\).

Invariance and the deflection. Tests are required to be invariant under the unknown \(y_0\) and \(v_0\), so the statistic is \(u^{\mathsf T}R\) with \(u\) orthogonal to the vectors \((1,\dots,1)\) and \((t_0,\dots,t_k)\). For Gaussian data the optimal invariant test has error probability \(\Phi(-d/2)\) with

\[d^2=\sup\left\{\frac{(u^{\mathsf T}P)^2}{u^{\mathsf T}\Sigma u}\ :\ \sum_iu_i=0,\ \sum_iu_it_i=0\right\},\qquad P_i=P(t_i),\]

\(\Sigma\) being the covariance of \(R\). This \(u\) is the statistical counterpart of the derivation note’s separating multipliers \(\mu\), and it obeys the same constraint \(\sum_iu_i=0\).

Theorem B. For every such protocol, \(d^2\le\dfrac{F^2\tau^3}{24\,m\kappa}\), and the constant is sharp: it is approached, never attained, by dense protocols whose end marks are sharp, with a recoil that no invariant statistic sees. Consequently an invariant test with error probability at most \(\epsilon\) requires

\[F^2\tau^3\ \ge\ 96\,z_{1-\epsilon}^2\,m\kappa,\qquad \tau\Delta E\ \ge\ 48\,z_{1-\epsilon}^2\,\kappa,\qquad A_{\rm inertial,fall}\ \ge\ \frac{16\,z_{1-\epsilon}^2\,v\kappa}{F}.\]

Proof. The noise decomposes exactly as in the derivation note’s feasibility system: \(\xi_j\) enters \(R_j\) alone, and \(\iota_j\) enters \(R_i\) for \(i>j\) with coefficient \((t_i-t_j)/m\). Hence

\[u^{\mathsf T}\Sigma u=\sum_j\delta_j^2u_j^2+\sum_j\frac{\Delta_j^2}{m^2}S_j^2, \qquad S_j=\sum_{i>j}u_i(t_i-t_j),\]

the same \(S_j\) as there. By the arithmetic–geometric mean inequality applied termwise, exactly as in that note’s Lemma 2,

\[u^{\mathsf T}\Sigma u\ \ge\ \sum_j\frac{2\delta_j\Delta_j}{m}|u_j||S_j| \ \ge\ \frac{2\kappa}{m}\sum_j|u_j||S_j| .\]

Since \(\sum_iu_i=0\), Lemma 3 of that note applies verbatim with \(\mu\) replaced by \(u\): with \(T(w)=\int_w^\tau\sum_{i:t_i>s}u_i\,ds\) one has \(T(t_j)=S_j\),

\[u^{\mathsf T}P=\frac Fm\int_0^\tau T,\qquad \sum_ju_jT(t_j)=-E,\qquad E=\int_0^\tau T'^2 ,\]

so \(\sum_j|u_j||S_j|\ge E\) and \(u^{\mathsf T}\Sigma u\ge2\kappa E/m\). The constraint \(\sum_iu_it_i=0\) gives \(T(0)=\sum_iu_it_i=0\), so \(T\) vanishes at both ends of \([0,\tau]\) and the Poincaré inequality \((\int_0^\tau T)^2\le\frac{\tau^3}{12}E\) holds, with equality for \(T\propto w(\tau-w)\). Hence

\[\frac{(u^{\mathsf T}P)^2}{u^{\mathsf T}\Sigma u} \le\frac{(F/m)^2\frac{\tau^3}{12}E}{2\kappa E/m}=\frac{F^2\tau^3}{24m\kappa},\]

independently of \(u\), of the number of marks and of their times. The error probability \(\Phi(-d/2)\le\epsilon\) requires \(d\ge2z_{1-\epsilon}\), and \(F^2\tau^3\ge24m\kappa d^2\ge96z_{1-\epsilon}^2m\kappa\). Every step is an equality in the limit of dense marks whose weights make \(T\) parabolic, each balanced with \(\delta_j|u_j|=\Delta_j|S_j|/m\), with sharp end marks; the interior \(u_jS_j\) then share one sign. \(\square\)

The worst-case theorem and the statistical theorem therefore have one proof. The certificate changes meaning, from a hyperplane separating a feasible set to the direction of the optimal test, and every step after it is identical. This is the sense in which the derivation note’s content survived its formulation: the argument never needed interval containment, only a quadratic bound in the same two variables.

4. Gaussian quantum experiments, with and without correlation

For Gaussian probe states, a fresh probe for each mark, linear dynamics and pointer readouts, the Wigner function is a genuine probability density and evolves by the classical linear equations, so every distribution of an invariant statistic in the quantum experiment equals the corresponding distribution in the classical Gaussian model whose noise covariance is the probe’s Wigner covariance. The body’s own state enters every reading as \(\hat y_0+\hat p_0t_j/m\) and cancels from every invariant statistic, so it need not be Gaussian. That covariance gives mark \(j\) a jointly Gaussian pair \((\xi_j,\iota_j)\) with variances \(\delta_j^2,\Delta_j^2\) and covariance \(c_j=\rho_j\delta_j\Delta_j\), independent across marks. Section 3 is the case \(c_j=0\).

Theorem C. In this model, if every mark has \(\delta_j\Delta_j(1-|\rho_j|)\ge\kappa_{\rm eff}\), then \(d^2\le F^2\tau^3/(24m\kappa_{\rm eff})\) and deciding at error probability \(\epsilon\) requires \(\tau\Delta E\ge48z_{1-\epsilon}^2\kappa_{\rm eff}\). For momentum-transfer marks with \(|\rho_j|\le\rho\),

\[\tau\Delta E\ \ge\ 24\,z_{1-\epsilon}^2\,\hbar\,\sqrt{\frac{1-\rho}{1+\rho}} .\]

Proof. The statistic’s noise is \(\sum_j\bigl(u_j\xi_j+S_j\iota_j/m\bigr)\), so the variance acquires the cross terms \(2c_ju_jS_j/m\):

\[u^{\mathsf T}\Sigma u=\sum_j\Bigl(\delta_j^2u_j^2+\frac{2c_j}{m}u_jS_j +\frac{\Delta_j^2}{m^2}S_j^2\Bigr).\]

For \(a,b\ge0\) and \(|c|\le\sqrt{ab}\), the arithmetic–geometric mean inequality and \(2cxy\ge-2|c||x||y|\) give \(ax^2+2cxy+by^2\ge2(\sqrt{ab}-|c|)|x||y|\), so each term is at least \(2\delta_j\Delta_j(1-|\rho_j|)|u_j||S_j|/m\ge2\kappa_{\rm eff}|u_j||S_j|/m\). From here the proof of Theorem B runs unchanged with \(\kappa_{\rm eff}\) in place of \(\kappa\). For the quantum value, the section-2 inequality \(\delta^2\Delta^2(1-\rho^2)\ge\hbar^2/4\) gives \(\delta\Delta(1-|\rho|)\ge\frac\hbar2(1-|\rho|)/\sqrt{1-\rho^2} =\frac\hbar2\sqrt{(1-|\rho|)/(1+|\rho|)}\), with equality for pure Gaussian probes. \(\square\)

At \(\rho=0\) this is \(\tau\Delta E\ge24z^2\hbar\): \(65\,\hbar\) at \(\epsilon=0.05\), where \(z=1.645\), and \(24\,\hbar\) at \(\epsilon=0.16\), where \(z=1\). The dense construction of Theorem B, with every mark at correlation \(\rho\) of the sign opposite to \(u_jS_j\), attains the bound; three marks already come within a factor \(4/3\).

Proposition D. Marks at \(0,\tau/2,\tau\) and the test \(R_1-2R_2+R_3\) decide the comparison when \(\tau\Delta E\ge32z_{1-\epsilon}^2\hbar\sqrt{(1-\rho)/(1+\rho)}\), where \(\rho\) is the middle probe’s correlation.

Proof. The weights \(u=(1,-2,1)\) annihilate the unknown position and velocity, and \(u^{\mathsf T}P=F\tau^2/4m\). The recoils of the outer marks have \(S_1=S_3=0\): the first is absorbed into the unknown initial velocity and the last acts after every reading. So the outer marks may be made as sharp as desired at no cost, and the noise is that of the middle mark, \(-2\xi_2+(\tau/2m)\iota_2\). With the coupling chosen so that \(2\delta_2=\tau\Delta_2/2m\) and a pure Gaussian probe of correlation \(\rho>0\), its variance is \((2\tau/m)\delta_2\Delta_2(1-\rho)=(\tau\hbar/m)\sqrt{(1-\rho)/(1+\rho)}\). Hence \(d^2=\frac{\tau\Delta E}{8\hbar}\sqrt{(1+\rho)/(1-\rho)}\), and \(d\ge2z\) is the stated condition. \(\square\)

As \(\rho\to1\) the three-mark protocol decides at every force. The middle probe’s kick is correlated with its reading error so that the kick’s effect on the third reading cancels the error in the second: the mechanism of Yuen’s contractive states (PRL 51, 719, 1983, metadata), which correlate the body’s position and momentum, applied here to the meter, as in Braginsky and Khalili’s back-action-evading measurements. In the convention of Section 2 the favourable sign is \(\rho>0\), that is \(\operatorname{Cov}(\hat Q_A,\hat P_A)<0\). The floor of order \(\hbar\) is therefore a statement about marks whose error–recoil correlation is bounded, and Theorem C gives its value for every bound. The insertion statement follows as before: marks inserted into a window of duration \(\tau'\), used by themselves, record the force only if \(F^2\tau'^3\ge96z^2m\kappa\), so a window shorter than the mesh cannot exhibit the force by itself

\[\tau_*=\left(\frac{96\,z_{1-\epsilon}^2\,m\kappa}{F^2}\right)^{1/3} =\left(\frac{48\,z_{1-\epsilon}^2\,m\hbar}{F^2}\right)^{1/3},\]

and marks confined to such a window are consistent with free motion at the stated confidence. Refinement below the mesh continues, and the sharp protocol of Theorem B needs it: what stops is local exhibition of the force.

This coexists with the counterexample of the probabilistic note, which distinguishes the hypotheses at arbitrarily small \(\tau\Delta E\) using a two-packet preparation of separation \(\pi\hbar/(F\tau)\). The boundary between the two results is knowledge of the preparation. A test that must work whatever the body’s initial position and velocity faces the floor of Theorems B and C; a protocol that knows the preparation escapes it, even with Gaussian states: a contractive Gaussian state focused at time \(\tau\), whose position spread there is as small as desired, and one sharp final mark decide at every force, which is Yuen’s own protocol. What prices the preparation route is the body’s spread along the way: the two-packet preparation, and the contractive state, pay in the aperture bound of the probabilistic note, \(2m\arcsin^2(1-2\epsilon)\hbar^2\tau/(2mL+\tau P)^2\) at aperture \((L,P)\). Newton’s refinement, which learns the trajectory from the marks alone, is the first case, which is why the floor is the relevant statement for the insertion question.

Gaussian probes are needed, 2026-09-22. Theorems B and C concern Gaussian probes, where the Wigner function is a density. A grid probe with a real wavefunction has \(\rho=0\) and still decides the comparison at any force, and for probes in arbitrary states the statement that holds is the recoil bound \(s\sum_j\Delta_j\ge8\hbar\arcsin(1-2\epsilon)\) of the recoil note.

Both escapes use one resource. A correlated probe and an unbalanced aperture each stretch an uncertainty ellipse of fixed area along a direction the test does not see: the rotated quadrature of the middle probe in Proposition D, the long side of the aperture in the probabilistic note. Robertson’s inequality fixes the area; the floor needs the shape bounded as well, and then \(\hbar\) fixes its value.

5. Newton’s premise is Robertson’s inequality for his corpuscle

The mark-floor note posited M3, that a mark fixing the position within \(\delta\) leaves the delivered impulse undetermined within \(\Lambda p/\delta\), and observed that Newton’s optics supplies \(\Lambda\) and \(p\) while his determinism denies the premise. Theorem A says what M3 is. The corpuscle is the probe; \(\hat Q_A\) is its transverse position and \(\hat P_A\) its transverse momentum. On this note’s reading the interval of fits \(\Lambda\) is the scale of the first and the corpuscular impulse \(p\) the scale of the second: the interval is a period along the ray, a longitudinal length, and a gauge built on the fits resolves no better than one interval, which is the analogy used; and Query 29 speaks of the corpuscles’ sizes, so the impulse scale is an inference. The premise asserts that the product of the two cannot be reduced. So

\[\kappa=\Lambda p\quad\longleftrightarrow\quad \kappa=\Delta\hat Q_A\cdot\Delta\hat P_A\ \ge\ \frac\hbar2 ,\]

and M3 is Robertson’s inequality for the corpuscle, stated in the two quantities Newton had. The floor uses M3 with a second clause that the mark-floor note left implicit: the undetermined impulse is unrelated to the mark’s error. Theorem C prices that clause, since a correlation \(\rho\) multiplies the floor by \(\sqrt{(1-\rho)/(1+\rho)}\). Newton measured the first as \(1/89000\) inch and had no access to the second. His Prop. XII states the fits as a transient constitution that returns at equal intervals and disposes the ray at every return, a determinate periodic property. That alone is compatible with the inequality, since a wave’s phase is determinate too; what excludes it is the disposition carried by a corpuscle of Query 29 moving by ordinary deterministic mechanics, whose later records recover the impulse it delivered. So what Newton printed entails the negation of the inequality, which he nowhere states; the source companion holds that passage and the other four with their line anchors. The modern value \(\hbar/2\) against the identification \(\Lambda p=h/2=\pi\hbar\) differ by a factor \(2\pi\).

The historical claim this licenses is narrow and checkable. Newton’s optics contains both factors of the product that bounds the recorded Galileo comparison; his mathematics contains the comparison and takes it to zero; and the single proposition that would join them is a claim of indeterminacy about the fits, whose negation follows from his determinate periodic disposition carried by a corpuscle. The gap between the Principia’s vanishing sagitta and the Opticks’ finite interval of fits follows from that single commitment, and the separation of the two books is incidental to it.

6. The single paper, valid for both audiences

The user’s requirement is one paper that a foundations-of-physics referee and a history-and-philosophy-of-science referee both accept. That is attainable, because the historical analysis now does argumentative work. The structure is:

  1. The comparison and its limit. Newton’s Lemmas X and XI and the projectile Scholium, with the sagitta, the inertial–parabola area and the identity \((3F/v)A=\tau\Delta E\). Berkeley’s objection and its resolution inside the geometry, with Guicciardini on what the limit arguments were for.
  2. The two theorems. Theorem A, that a momentum-transfer mark’s error and recoil are conjugate, and Theorem B, that every protocol of such marks needs \(\tau\Delta E\ge48z^2\kappa\), sharp, with Theorem C for correlated error and recoil. Proof by certificate, arithmetic–geometric mean and Dirichlet energy. This is the foundations content and it stands alone.
  3. The two factors in the Opticks. The interval of fits as a measured length, the corpuscular impulse as a posited one, and Shapiro’s account of what the theory of fits was and how determinate Newton meant it. This is the history content and it stands alone.
  4. The junction. M3 is Robertson for the corpuscle. The counterfactual is replaced by an exact statement: the premise that the theorem needs is one Newton formulated the negation of, so the distance between his system and a positive \(h\) is a single proposition, nameable in his own vocabulary.
  5. What the theorem does not give. The aperture counterexample, the standard-quantum-limit literature and Ozawa’s class, and the honest prior-art label: the combination \(F^2\tau^3\gtrsim m\hbar\) is the standard quantum limit, and the contribution is universality over protocols, the exact mark cost, and the identification of the premise.

The venues that take this shape are Foundations of Physics and Studies in History and Philosophy of Modern Physics, both of which publish papers whose historical and technical halves are load-bearing for each other. Two obligations remain before submission: reading Shapiro, Guicciardini and Sabra rather than citing them, and replacing the Project Gutenberg Opticks and the Wikisource Principia with the fourth edition and the Cohen–Whitman translation.

7. Consequence for STATE

The derivation note’s Proposition 6 is reinstated in substance with a correct proof and the constant \(\hbar/2\) in place of \(2\hbar\), and its worst-case formulation is replaced by standard deviations. The Planck gap for marked trajectories is \(\tau\Delta E\ge24z_{1-\epsilon}^2\hbar\sqrt{(1-\rho)/(1+\rho)}\), protocol-universal for marks whose error–recoil correlation is at most \(\rho\), with the insertion mesh \(\tau_*=(48z^2m\hbar/F^2)^{1/3}\) at \(\rho=0\); the constant is sharp, and Proposition D comes within \(4/3\) with three marks. A floor with no shape bound needs a quantity that squeezing cannot move, which is the task of the next note. The remaining programme is item 6’s two obligations plus: extending Theorem A beyond the momentum-transfer class, where Ozawa’s instruments live; the general force law, which Theorem B’s proof already reduces to a norm of \(P''\); and closing the constants between the worst-case, statistical and single-shot forms.