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The cost of a mark: Newton’s vanishing sagitta and a floor of order \(\hbar\)

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Abstract. Newton reads a force off a trajectory by letting the sagitta and the area between the inertial tangent and the curve go to zero, keeping their ratio to the time. We show that the same comparison, once it must be recorded, has a floor. For every protocol of marks, of any number and at any times, whose resolution and recoil obey \(\delta_j\Delta_j\ge\kappa\), deciding between free and forced motion over a duration \(\tau\) at error probability \(\epsilon\) requires

\[\tau\Delta E=\frac{F^2\tau^3}{2m}\ \ge\ 9\,z_{1-\epsilon}^2\,\kappa,\]

and for a mark realized by coupling the body’s position to a probe, the error operator and the impulse delivered are canonically conjugate, so \(\kappa\ge\hbar/2\) exactly. The floor is about \(12\hbar\) at five per cent error, and insertion of marks finer than \(\tau_*=(9z_{1-\epsilon}^2m\hbar/F^2)^{1/3}\) records free motion only. The historical claim is that Newton’s Opticks contains both factors of \(\kappa\), one of them measured as the \(1/89000\)th part of an inch, and that the single proposition joining them to a floor is an indeterminacy about the fits which Newton formulated the negation of. The distance between his system and a positive \(h\) is therefore one nameable proposition, and that proposition is Robertson’s inequality for his corpuscle.

Draft, 2026-09-17. Synthesis of the mark-floor, derivation, probabilistic and mark-cost notes, which hold the proofs in full. Two obligations remain before submission and are stated in §9. Exploratory; no ledger promotion.

1. The question

Galileo’s comparison is an inertial horizontal line against a falling parabola. With mass \(m\), transverse force \(F\), horizontal speed \(v\) and duration \(\tau\), the two paths are \(q_{\mathrm I}(t)=(vt,0)\) and \(q_{\mathrm F}(t)=(vt,Ft^2/2m)\), and the region between them has

\[A=\frac{vF\tau^3}{6m},\qquad s(\tau)=\frac{F\tau^2}{2m},\qquad \Delta E=\frac{F^2\tau^2}{2m},\qquad \frac{3F}{v}A=\tau\Delta E. \tag{1}\]

The sagitta \(s\), the area \(A\) and the product \(\tau\Delta E\) all scale as powers of \(\tau\) and vanish together. Newton’s Lemma X makes the first displacement quadratic in the time; Lemma XI, Corollaries 4 and 5, make the curved segment one third of the tangent triangle with both cubic in the time; Proposition VI reads the force off the limit \(2ms/\tau^2\); and Proposition I builds an orbit as a polygon struck by impulses at its vertices, the number of vertices “augmented in infinitum”. The construction is designed so that the quantities in (1) disappear and their ratios survive.

Our question is what stops that refinement when the trajectory has to be recorded rather than contemplated, and whether the stopping point can be reached from materials Newton possessed.

The answer has four parts. The geometry alone supplies no floor (§2). Recording supplies one, which is set by a single number \(\kappa\) with the dimensions of action, and every protocol of marks faces the same bound in the same combination \(F^2\tau^3/m\) (§3). Quantum kinematics fixes \(\kappa\ge\hbar/2\) by an exact and elementary argument (§4). Newton’s optics supplies both factors of \(\kappa\) and denies the inequality between them (§§5–6). Newton also attempted a preface founded on the ancients, and the ancient arguments that bear on his own method are the ones he passed over; stated as premises about records they generate a hierarchy whose first rung is Zeno’s arrow and whose second is the comparison above (§7).

2. The geometry has no floor

Proposition 1. For every partition \(0=t_0<\dots<t_n=\tau\) with steps \(\tau_j\) and mesh \(|\pi|\), the vertex impulse \(F\tau_j\), the step sagitta \(F\tau_j^2/2m\) and the step area \(vF\tau_j^3/6m\) tend to zero with the mesh; the areas sum to at most \(vF\tau|\pi|^2/6m\); and \(2m\,s(\tau_j)/\tau_j^2=F\) exactly at every step.

The proof is immediate from (1). Berkeley’s objection, that evanescent increments are “neither finite Quantities nor Quantities infinitely small, nor yet nothing”, is answered inside the geometry by the limit ratio, which is what Newton’s closing Scholium to Book I Section I provides. A floor must therefore come from the act of marking the curve, and the reading of “inserting a point” used here is physical: a point of the trajectory is inserted by making a mark on the body at a time.

3. Marks, and the floor they impose

Definitions. A mark at time \(t_j\) leaves a datum about the transverse coordinate. Its resolution \(\delta_j\) is the spread of the resulting position estimate’s error; its recoil \(\Delta_j\) is the spread of the transverse impulse it delivers to the body. A protocol is a finite set of marks; it decides the comparison at error probability \(\epsilon\) when some test on the full record does so with equal priors. A model satisfies the mark trade-off with constant \(\kappa\) when every available mark has \(\delta\Delta\ge\kappa\).

The statistical model. 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 \(\Delta_j\), independently, so that

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

with \(\theta=0\) under the inertial hypothesis and \(\theta=1\) under the falling one, and \(y_0,v_0\) unknown. Tests are invariant under the two unknowns, so the statistic is \(u^{\mathsf T}R\) with \(\sum_iu_i=0\) and \(\sum_iu_it_i=0\), and the optimal invariant test has error probability \(\Phi(-d/2)\) with \(d^2=\sup(u^{\mathsf T}P)^2/(u^{\mathsf T}\Sigma u)\) over such \(u\).

Theorem 2. For every protocol satisfying the mark trade-off with constant \(\kappa\),

\[d^2\ \le\ \frac{2F^2\tau^3}{9\,m\kappa},\]

independently of the number of marks and of their times. Hence deciding at error probability \(\epsilon\) requires

\[F^2\tau^3\ \ge\ 18\,z_{1-\epsilon}^2\,m\kappa,\qquad \tau\Delta E\ \ge\ 9\,z_{1-\epsilon}^2\,\kappa,\qquad A\ \ge\ \frac{6\,z_{1-\epsilon}^2\,v\kappa}{F},\]

with \(z_{1-\epsilon}=\Phi^{-1}(1-\epsilon)\).

Proof. The noise enters \(R\) in two ways: \(\xi_j\) affects \(R_j\) alone, and \(\iota_j\) affects \(R_i\) for \(i>j\) with coefficient \((t_i-t_j)/m\). Hence

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

and the arithmetic–geometric mean inequality applied termwise gives \(u^{\mathsf T}\Sigma u\ge\frac{2\kappa}{m}\sum_j|u_j||S_j|\).

Introduce \(N(w)=\sum_{i:t_i>w}u_i\) and \(T(w)=\int_w^\tau N\). Since \(\sum_iu_i=0\), \(N\) vanishes for \(w<0\) and for \(w\ge t_k\); \(T\) is continuous, piecewise linear, constant on \((-\infty,0]\), zero on \([\tau,\infty)\), with \(T'=-N\) jumping by \(u_j\) at \(t_j\) and \(T(t_j)=S_j\). Two identities follow. Fubini and one integration by parts, using \(P'(0)=0\) and \(T(\tau)=0\), give

\[u^{\mathsf T}P=\int_0^\tau P'N=-\int_0^\tau P'T'=\int_0^\tau P''T =\frac Fm\int_0^\tau T;\]

and since \(T'\) has bounded variation with jumps \(u_j\) and no other variation, while \(TT'\) vanishes at both ends,

\[\sum_ju_jT(t_j)=\int_{\mathbb R}T\,dT'=-\int_{\mathbb R}T'^2=-E, \qquad E=\int_0^\tau T'^2 .\]

Therefore \(\sum_j|u_j||S_j|\ge E\) and \(u^{\mathsf T}\Sigma u\ge2\kappa E/m\). Cauchy–Schwarz with \(T(\tau)=0\) gives \(|T(w)|\le\sqrt{\tau-w}\sqrt E\), hence \(\int_0^\tau T\le\frac23\tau^{3/2}\sqrt E\), and

\[\frac{(u^{\mathsf T}P)^2}{u^{\mathsf T}\Sigma u} \le\frac{(F/m)^2\frac49\tau^3E}{2\kappa E/m}=\frac{2F^2\tau^3}{9m\kappa}.\]

Finally \(\Phi(-d/2)\le\epsilon\) requires \(d\ge2z_{1-\epsilon}\). \(\square\)

Three features of the proof carry the paper’s claim to universality. The certificate \(u\) ranges over all invariant tests, so no monitoring scheme is privileged. The function \(T\) encodes the whole protocol in one object, and the bound depends on it only through the Dirichlet energy \(E\), which cancels. And the exponent \(3\) on \(\tau\) comes from a single Cauchy–Schwarz inequality against \(\sqrt{\tau-w}\), which is where the combination in (1) originates.

Corollary 3 (insertion). Marks inside a window of duration \(\tau'\), used by themselves, record the force only if \(F^2\tau'^3\ge18z_{1-\epsilon}^2m\kappa\). So insertion stops at

\[\tau_*=\left(\frac{18\,z_{1-\epsilon}^2\,m\kappa}{F^2}\right)^{1/3},\]

and finer marks are consistent with free motion at the stated confidence. Newton’s polygon may be refined past \(\tau/\tau_*\) vertices, and its impulses remain in the geometry, though no record exhibits them.

A worst-case counterpart. If the marks instead confine position and impulse to intervals with certainty, the same three steps run with a separating hyperplane in place of the optimal test direction and give \(F^2\tau^3>9m\kappa\) for every protocol, with two marks sufficing above \(36m\kappa\). That version has no quantum instance, since certain position confinement forces full-support momentum by Paley–Wiener; it is recorded here because it shows that the argument depends on the quadratic structure rather than on any probabilistic assumption.

4. The cost of a mark is the probe’s uncertainty product

Theorem 4. Let a mark be realized by the impulsive coupling of the body’s position to a probe, \(U=\exp(-\tfrac{i}{\hbar}\lambda\hat y\hat P_A)\) with \([\hat Q_A,\hat P_A]=i\hbar\), the pointer \(\hat Q_A\) being read afterwards. Then

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

so the error operator \(\hat N=\hat Q_A/\lambda\) and the delivered impulse \(\hat D=-\lambda\hat P_A\) satisfy \([\hat N,\hat D]=-i\hbar\) and

\[\kappa=\delta\Delta=\Delta\hat N\cdot\Delta\hat D =\Delta\hat Q_A\cdot\Delta\hat P_A\ \ge\ \frac\hbar2\]

for every probe state, pure or mixed.

Proof. With \(A=\tfrac{i}{\hbar}\lambda\hat y\hat P_A\) the commutators \([A,\hat Q_A]=\lambda\hat y\) and \([A,\hat p]=-\lambda\hat P_A\) are central, so the Baker–Campbell–Hausdorff series terminates and gives the two displayed relations, while \(\hat y\) and \(\hat P_A\) are unchanged. Estimating the body’s position by \(\hat Q_A/\lambda\) leaves the error \(\hat N=\hat Q_A/\lambda\), and \([\hat N,\hat D]=-[\hat Q_A,\hat P_A]=-i\hbar\). Robertson’s inequality closes it. \(\square\)

The coupling strength cancels, so the cost is a property of the mark rather than of how hard it is made. The record and the recoil are themselves incompatible, since \([\hat Q_A+\lambda\hat y,-\lambda\hat P_A]=-i\hbar\lambda\), which is why no reading of the pointer determines the impulse. A deterministic probe, whose later position records fix its momentum, has \(\kappa=0\) and no floor; this is the exact point at which classical record models differ.

For Gaussian probe states and the linear dynamics of the comparison, the Wigner function is a probability density evolving classically, so the statistical model of §3 reproduces the quantum experiment exactly. With Theorem 4,

\[\tau\Delta E\ \ge\ \frac92\,z_{1-\epsilon}^2\,\hbar,\qquad \tau_*=\left(\frac{9\,z_{1-\epsilon}^2\,m\hbar}{F^2}\right)^{1/3}, \tag{2}\]

about \(12\hbar\) at \(\epsilon=0.05\) and \(4.5\hbar\) at \(\epsilon=0.32\).

5. Both factors are in the Opticks

Newton’s optics supplies the two spreads whose product is \(\kappa\), in the form of two least quantities of the probe.

A least length, measured. Book II Part III defines the interval of the fits as “the space it passes between every return and the next return” of the ray’s disposition to be reflected, and Proposition XVIII gives it: for the confine of yellow and orange passing perpendicularly into air, “the Intervals of their Fits of easy Reflexion are the \(1/89000\)th part of an Inch”, that is about \(0.285\,\mu\mathrm{m}\). A datum obtained through the fits repeats with that period, so it locates the corpuscle no better; Newton’s rings are a gauge of exactly this kind. This is \(\Delta\hat Q_A\).

A least impulse, posited. Query 29 asks whether the rays of light are “very small Bodies emitted from shining Substances”, which gives each a transverse momentum scale. Newton has no way to measure it. This is \(\Delta\hat P_A\).

Colour dependence, measured. Observation 13 of Book II Part I reports the ratio of the red to the violet interval as “greater than as 3 to 2, and less than as 13 to 8”, and “By the most of my Observations it was as 14 to 9”. Query 29 makes the violet corpuscles the least and the red the biggest. The product \(\Delta\hat Q_A\Delta\hat P_A\) is therefore colour dependent on Newton-age evidence, and its universality across the spectrum is what Planck’s constant later supplies.

Inflexion. Query 1 asks whether bodies “act upon Light at a distance, and by their action bend its Rays”, most strongly at the least distance, which is the effect that limits a mark’s resolution when the beam is narrowed to improve it.

All five passages are held verbatim with line anchors in the source companion, from the fourth edition of 1730.

6. The junction, and the proposition Newton denied

Theorem 4 says what the missing premise is. In the mark-floor note it was stated as M3, that a mark fixing the position within \(\delta\) leaves the delivered impulse undetermined within \(\kappa/\delta\). Theorem 4 identifies \(\kappa\) as the probe’s own uncertainty product, so

\[\text{M3 for Newton's corpuscle}\quad\Longleftrightarrow\quad \Delta\hat Q_A\cdot\Delta\hat P_A\ \ge\ \text{a positive constant},\]

which is Robertson’s inequality stated in Newton’s two quantities. With M3, §§3–4 give the floor and the insertion mesh from Newton’s own materials, with \(\kappa=\Lambda p\) in place of \(\hbar/2\); the identification \(\Lambda=\lambda/2\), \(p=h/\lambda\) makes \(\Lambda p=h/2\), which differs from \(\hbar/2\) by \(\pi\).

Newton denied M3, and the denial is explicit rather than inferred. Proposition XII of Book II Part III states that every ray is “put into a certain transient Constitution or State, which in the progress of the Ray returns at equal Intervals, and disposes the Ray at every return to be easily transmitted”. The disposition is a determinate periodic property carried by the ray, so its fate at a surface is fixed by its phase, and what a prior record fails to supply is knowledge rather than determination. On that reading a probe’s position and momentum are both definite, later records can recover the impulse it delivered, \(\kappa=0\), and the comparison has no floor. The countermodel is explicit: let the corpuscle travel freely after the mark to a screen at distance \(D\), and two position records fix its transverse momentum to within \(p(\delta+\delta_s)/D\), which vanishes as \(D\) grows.

The historical claim is therefore narrow and checkable. Newton’s optics contains both factors of the product that bounds the recorded comparison; his mathematics contains the comparison and takes it to zero; and the one proposition that would join them is an indeterminacy about the fits whose negation he asserted in print. 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. No counterfactual about what Newton might have discovered is needed, and none is offered.

7. Newton and the classics

What Newton attempted

In the early 1690s Newton drafted a set of Classical Scholia to Propositions IV to IX of Book III, arguing that the oldest philosophers had known universal gravitation and its inverse-square law, so that the Principia recovered a lost wisdom rather than announcing a novelty. He never printed them. He did pass material to David Gregory, whose Astronomiae physicae et geometricae elementa of 1702 opens with the programmatic sentence that lest the physics delivered here seem something new and unheard of in astronomy, eandem vetustissimis Philosophis notam … ostendam, I shall show it was known to the most ancient philosophers, naming Anaxagoras, Archelaus, Euripides and Pythagoras. That preface is the printed witness closest to the unpublished scholia, and the source companion holds its programmatic paragraph. The scholarship is Casini, “Newton: The Classical Scholia”, History of Science 22 (1984), 1–46, and Mosley, “The Origins and Sources of Newton’s Classical Scholia”, Erudition and the Republic of Letters 9 (2024), 171–217, both verified at metadata level here and both still to be read.

So Newton did attempt a preface founded on the classics, and the attempt was doxographic. He sought ancient authority for a law he had already proved, and the authority he sought was for the conclusions of Book III.

The ancient arguments he passed over

The ancient material that bears on the method of Book I is the debate over division and the cut, and it is absent from the scholia. Six positions, each held here in a primary witness:

Lemmas X and XI perform exactly the operation these positions dispute. The scholia cite the ancients for what Book III concludes and leave them silent on how Book I proceeds.

The last four entries differ in kind from the rest, and they matter most here. The Greek and Chinese items are paradoxes about division, which set a problem. The Indian entries answer it, and they answer it three different ways: the arrow’s flight is many motions carried by an impression (Vaiśeṣika), or it is no motion at all because each thing perishes where it arose (Vasubandhu), or it is possible only because a medium stands ready to permit it (Umāsvāti). Each also carries a stopping rule for division: the Nyāya paramāṇu than which nothing is smaller, the directional-parts reductio by which a touched atom would have parts, and the Jain atom that has no space-points at all. The Vaiśeṣika sūtras in particular state a positive theory of propagation: motion is a succession of numerically distinct events, each carried to the next by a quantity the previous one deposits, and transmission through a medium is a chain in which what arrives is a later member than what was sent. That is the structure Theorem 2 quantifies. Its protocol-universality holds because the signal is a phase-space path whose total variation is unchanged by subdivision, which is the modern form of the claim that impulsion supplies only the first motion while the impression carries the rest. An ancient author had therefore already framed propagation as a bounded succession rather than a continuous traversal, and Newton’s classical preface reached for none of it.

The hierarchy the ancient premises generate

Read as premises about records rather than about geometry, the cone and the arrow are the first members of a sequence whose next member is Newton’s comparison. The apparatus of §3 covers all of them; what changes is the order of the signal against the nuisance it must beat.

Theorem 5. In the model of §3, with every mark obeying \(\delta_j\Delta_j\ge\kappa\):

  1. The arrow. Distinguishing rest from uniform motion at speed \(v\), with the initial position unknown, requires

\[m\,v^2\tau\ \ge\ 8\,z_{1-\epsilon}^2\,\kappa, \qquad\text{that is}\qquad \tau\,E_{\rm kin}\ \ge\ 4\,z_{1-\epsilon}^2\,\kappa .\]

  1. Newton. Distinguishing uniform motion from constant force, with the initial position and velocity unknown, requires \(F^2\tau^3/m\ge18z_{1-\epsilon}^2\kappa\), which is Theorem 2.

  2. The cone. Measuring a static taper requires nothing. With no recoil to propagate, the deflection grows without bound as marks accumulate, so a static solid has no floor.

Proof. For (i) the nuisance is the constant, so \(u\) ranges over vectors with \(\sum_iu_i=0\) alone. With \(P(t)=vt\) the identity of §3 reads \(u^{\mathsf T}P=v\sum_iu_it_i=v\,T(0)\), and \(T(\tau)=0\) with Cauchy–Schwarz gives \(|T(0)|=|\int_0^\tau T'|\le\sqrt\tau\sqrt E\). The noise bound \(u^{\mathsf T}\Sigma u\ge2\kappa E/m\) needs only \(\sum_iu_i=0\) and so is unchanged. Hence \(d^2\le v^2\tau E\,m/(2\kappa E)=m v^2\tau/(2\kappa)\), and \(d\ge2z_{1-\epsilon}\) gives the claim. For (ii) the nuisance is the linear space, \(u^{\mathsf T}P=\frac Fm\int_0^\tau T\), and the proof of Theorem 2 applies. For (iii) the marks leave the object unchanged, so \(\Sigma=\operatorname{diag}(\delta_j^2)\); with \(N\) marks of resolution \(\delta\) at each of two heights separated by \(\Delta h\), the taper \(\vartheta\) gives \(d^2=N\vartheta^2\Delta h^2/(2\delta^2)\), unbounded in \(N\). \(\square\)

The two dynamical cases share one invariant. If the signal is the \(n\)-th order departure \(P\) with \(P(0)=\dots=P^{(n-1)}(0)=0\), the quantity the theorem bounds below is

\[m\,\bigl(P^{(n)}\bigr)^2\,\tau^{2n-1}\ \gtrsim\ \kappa ,\]

which has the dimensions of action for every \(n\): at \(n=1\) it is \(mv^2\tau\), twice the kinetic energy times the duration, and at \(n=2\) it is \(F^2\tau^3/m\), twice \(\tau\Delta E\). Newton’s comparison is the second rung of a ladder whose first rung is Zeno’s.

Why the arrow and not the sling

One feature of the list needs saying. Almost every ancient entry is rectilinear or static, while the case Newton uses to define centripetal force is the sling: Definition V has the stone whirled about in a sling endeavouring to recede from the hand, and names the force that “retains it in its orbit” after the planets “perpetually drawn aside from the rectilinear motions, which otherwise they would pursue”.

The exception is Plutarch, and it is a pointed one. De facie 923C has the moon kept from falling “just as missiles placed in slings are kept from falling by being whirled around in a circle”, which is Newton’s apparatus applied to Newton’s case a millennium and a half earlier. The inference runs the other way: for Plutarch the whirling prevents the fall, so speed sustains; for Newton the cord pulls the stone inward, and what it diverts is a straight line that would need no cause. The same object supports opposite accounts, and what separates them is which motion is taken to be free. Plutarch’s next clause, that “each thing is governed by its natural motion unless it be diverted by something else”, is as near as the passage comes, with natural motion still doing the work inertia later does. Newton had read him: the Classical Scholia name Plutarch among the ancients said to have known the doctrine of gravitation, so the sling was available in a text he was mining for authority while writing the propositions it models.

Theorem 5 explains why the exception stayed isolated, and Aristotle supplies the other half of the reason. De caelo I.2 makes uniform circular motion the natural motion of the aether, needing no cause, which is the exact inverse of Definition V. For the heavens, then, curved motion was the default and straight motion the anomaly, so the sling illustrates a conclusion already held rather than posing a problem. The claim that the ancients treat rest as natural holds below the moon and is incomplete above it; the reference is at metadata level here. The ancient question is the first rung, whether the thing moves at all, with position as the only nuisance. The sling is the second rung, whether a force acts, with uniform motion as a further nuisance, and that rung cannot be stated until straight-line motion is held to need no account. Ancient physics largely does not hold that, so the sling has no work to do in it. The Vaiśeṣika comes closest by charging only the first motion to impulsion, and stops short of making a straight continuation free and a curved one costly.

Indian sources do supply one whirled object, and use it for a third purpose again. The firebrand circle, alātacakra, appears in Vasubandhu (Pradhan 189.23–24) to argue that because contact with the parts is successive, the awareness of a whole is really of the parts, and at Pradhan 33.9 with āśuvṛttyā, by rapid action. That is an argument about sampling, and it is the ancient form of what Theorem 5 says about records: below the mesh a succession is indistinguishable from the continuous thing it mimics. The companion note sets this out.

What each ancient premise buys

The section’s claim about Newton is therefore double. He looked to the ancients for authority and found the wrong ancients, taking the doxography of gravitation while passing over the dispute about division that his own Lemmas turn on. And the premise that would have converted that dispute into a theorem, an indeterminacy in the least parts of light, is one the atomists supply in the form of a least part and one he denied in the single place where he had a measurement.

8. What the theorem does not give

Preparations of unbounded extent. Theorem 2 concerns protocols that mark the trajectory. A protocol that prepares once, waits and measures once faces a different bound. In that case the two hypotheses differ by the phase-space displacement \((-F t^2/2m,\ Ft)\), whose components have product exactly \(\tau\Delta E\), and Mandelstam–Tamm in the geometric form of Anandan and Aharonov (PRL 65, 1697, 1990), with Helstrom’s discrimination bound (Quantum Detection and Estimation Theory, 1976), gives

\[\frac{F\tau L}{\hbar}+\frac{F\tau^2P}{2m\hbar}\ \ge\ 1-2\epsilon, \qquad\tau\Delta E\ \ge\ \frac{(1-2\epsilon)^2\hbar^2}{4A},\]

for an apparatus whose position and momentum spreads are bounded by \(L\) and \(P\) with \(A=LP\ge\hbar/2\); and this holds for every finite adaptive protocol of instruments, because the displacement is a phase-space path whose total variation does not grow when it is subdivided. The bound is tight: two narrow packets separated by \(\pi\hbar/(F\tau)\) decide the comparison with error probability of order \(\tau\Delta E/\hbar\), for arbitrarily small \(\tau\Delta E\), at the price of an apparatus whose extent diverges. So the floor of order \(\hbar\) requires either that the trajectory be marked or that the laboratory be bounded, and a non-Gaussian preparation of unbounded extent evades it.

Prior art. The combination \(F^2\tau^3\gtrsim m\hbar\) is the standard quantum limit for detecting a force on a free mass (Braginsky and Khalili, Quantum Measurement, 1992; the canonical statement is Caves, Thorne, Drever, Sandberg and Zimmermann, Rev. Mod. Phys. 52, 341, 1980), and its status has been disputed since Yuen’s objection: Caves defended it (PRL 54, 2465, 1985) and Ozawa exhibited a measurement breaking it for free-mass position (PRL 60, 385, 1988). That non-Gaussian preparations beat the limit, which is §7’s counterexample, is stated as a principle by Giovannetti, Lloyd and Maccone (Science 306, 1330, 2004). The contribution here is universality over protocols with an explicit constant, the exact mark cost of Theorem 4, and the identification of the premise. The inequality itself is not new.

The classical theorem belongs to a literature it does not cite. Worst-case recovery of a linear functional from noisy linear data is optimal recovery in the sense of Micchelli and Rivlin (1977), and the apparatus series this programme built, together with Theorem 1’s worst-case form, are results of that kind; information-based complexity (Traub, Wasilkowski and Woźniakowski, 1988) is the same setting. The invariant-test step of Theorem 2 is standard (Lehmann and Romano). A referee from either community will ask why these go unmentioned, and the honest answer is that the theorems were derived without them.

Instruments outside the momentum-transfer class. Theorem 4 holds where \([\hat N,\hat D]=-i\hbar\). Its coupling is von Neumann’s (1932), its inequality Robertson’s (1929), and its ancestor the Heisenberg microscope (1927); Bohr and Rosenfeld (1933) and Araki and Yanase (1960) are where limits of this kind from field measurability and from conservation laws are set. Ozawa’s counterexamples to the naive Heisenberg product concern instruments where that commutator differs, and a general class would need the proved calibration relation of Busch, Lahti and Werner (PRL 111, 160405, 2013) or an argument of another kind. No error–disturbance relation is used above.

9. Obligations before submission

Two, both on the history side, and both stated so that a reader can see what the present draft rests on.

  1. Historiography, to be read rather than cited. On the Opticks side, Shapiro’s Fits, Passions, and Paroxysms (1993) on the theory of fits and how determinate Newton meant it, which is where §6’s central claim must be tested, and Sabra’s Theories of Light from Descartes to Newton (1981) for the context of Query 29. On the Principia side, Guicciardini’s Reading the Principia (1999) and his Isaac Newton on Mathematical Certainty and Method (2009) for what the limit arguments of §2 were for, De Gandt’s Force and Geometry (1995) directly on the sagitta, and Bertoloni Meli’s Thinking with Objects (2006) on the sling and the pendulum as objects to think with. On the scholia, McGuire and Rattansi, “Newton and the ‘Pipes of Pan’” (1966), beside Casini and Mosley. On the ancient comparison, Sorabji’s Time, Creation and the Continuum (1983), which treats Zeno, Diodorus Cronus, Epicurean minima and kalām atoms together; von Rospatt (1995) on momentariness; Dhanani (1994) on kalām atomism and the Indian influence question; and Lloyd and Sivin’s The Way and the Word
    1. as the methodological standard that protects §7 from the charge of naive parallelism. Clagett (1959) for impetus, where Philoponus’ rhopē is the Western counterpart of saṃskāra.
  2. Editions. The Opticks passages come from a transcription of the 1730 fourth edition and should be cited from that printing; the Principia passages should be cited from Cohen and Whitman in place of the Motte text used here.

Three mathematical items remain open and are not obligations of the paper: extending Theorem 4 beyond the momentum-transfer class; the general force law, for which the proof of Theorem 2 already pairs \(\int_0^\tau P''T\) against the Dirichlet energy and so bounds a norm of \(P''\); and closing the constants between the worst-case form, the statistical form of Theorem 2 and the single-shot form of §7.

10. Consequence for STATE

This is the synthesis the goal asked for: the Planck gap established with Newton-age arguments and their modern equivalents, in one document whose historical and technical halves are load-bearing for each other. The foundations content is Theorems 2 and 4 with §7’s honest positioning; the history content is §§5–7, resting on the source companions; the junction is the identification of M3 with Robertson’s inequality for the corpuscle. STATE’s queue reduces to §9’s obligations and the three open mathematical items.