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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. In the quantum instrument model with a supplied \(\hbar>0\), a preparation-independent test of the two motions obeys a disturbance bound. For every protocol of marks, made by any instruments at any times, a test that decides at error probability \(\epsilon\) between free motion and a constant force \(F\) over a duration \(\tau\), and must do so whatever the body’s initial state, the state under the force being allowed to differ by an unknown displacement, requires

\[\frac s8\sum_j\Delta(\hat D_j)+\frac J2\sum_j\Delta(\hat X_j)\ \ge\ \hbar\arcsin(1-2\epsilon), \qquad s=\frac{F\tau^2}{2m},\quad J=F\tau,\]

where \(\Delta(\hat D_j)\) and \(\Delta(\hat X_j)\) are the spreads of the momentum and position disturbances of mark \(j\) in the states the marks receive. Newton’s sagitta is paired with the impulse a record leaves undetermined, and his impulse with the displacement. A protocol that knows the body’s preparation escapes the floor, and pays instead in the body’s spread along the way. For non-adaptive marks with uncorrelated Gaussian probes, read by tests invariant under the unknown initial position and velocity, the floor takes the sharp form \(\tau\Delta E=F^2\tau^3/2m\ge24z_{1-\epsilon}^2\hbar\), about \(65\hbar\) at five per cent error, because a mark’s error and the impulse it delivers are canonically conjugate; a window of marks shorter than \(\tau_*=(48z_{1-\epsilon}^2m\hbar/F^2)^{1/3}\), used by itself, cannot tell the force from free motion. Correlated and grid-state probes lower that form, and the disturbance bound holds for them. The comparison also carries a quantity that no preparation of the body can move, a phase: Newton’s polygon inscribed in the parabola agrees with it classically at every vertex and differs quantum mechanically by \(F^2\sum_j\tau_j^3/(24m\hbar)\), the parabolic segments of the chords times \(F/2v\hbar\), so that each round of Archimedes’ exhaustion removes a phase. On the historical side, Newton’s Opticks contains a measured least length, the \(1/89000\)th part of an inch, corpuscles that carry an impulse, and a disposition whose period times the corpuscle’s momentum is invariant under refraction. The proposition that would join them to a floor is that a record leaves an undetermined impulse, and what Newton printed entails its negation. The modern disturbance bounds assume quantum states, instruments and a positive canonical commutator scale; they do not derive that scale from Newton’s axioms. On fixed collision-free intervals, the Kepler comparison recovers classical centroids from Coulomb evolution. The coherent-record argument requires coherence between launch points and a specified record topology; relativistic action thresholds require a selected point-source domain. These premises leave the independent necessity and universal calibration of the scale open.

Draft, 2026-09-17; revised 2026-09-23 and 2026-10-02. Synthesis of the mark-floor, derivation, probabilistic, mark-cost, recoil, additive-noise, disturbance, path-length and polygon-lift notes, which hold the proofs in full. An adversarial review in three batches on 2026-09-23, covering those notes and this paper, found no false result; its proof gaps, overclaims, historical corrections and missing prior art are repaired here and in the notes. The revision of 2026-09-22 restated two floors: the Gaussian floor needs uncorrelated Gaussian probes, and the aperture floor takes its area form only at the balanced aperture. The October 2 revision adds the Kepler result and selection-dependent endpoints in §10a, without treating them as a necessity proof. The historical submission obligations remain in §11. 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 (Corollary 3 of the first edition) makes the area between tangent and arc two thirds of the tangent triangle and the segment between chord and arc one third, “ex natura Parabolæ”, both cubic in the time; the area \(A\) is the first of these. It is also Democritus’s third: the sagitta grows as the square of the time as a cone’s section grows as the square of the height, so \(A=\frac13v\tau s\), one third of the rectangle, as the cone is one third of its cylinder (§9). 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 five parts. The geometry alone supplies no floor (§2). Recording supplies one: in a Gaussian model of marks it is set by a single number \(\kappa\) with the dimensions of action, sharply, in the combination \(F^2\tau^3/m\) (§3), and quantum kinematics fixes \(\kappa\ge\hbar/2\) because a mark’s error and its recoil are conjugate (§4). Within that quantum model, a preparation-independent test with probes and instruments of every kind obeys a bound on disturbance, paired with Newton’s sagitta and impulse (§5), and a phase that no preparation of the body can move carries the same quantity through Newton’s inscribed polygon (§6). Newton’s optics supplies both factors of \(\kappa\), a phase whose rate is proportional to momentum, and the premises from which the negation of the premise that joins them follows (§§7–8). Newton also drafted classical scholia 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 (§9).

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 met 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. The marks are fixed in advance, and tests are invariant under the two unknowns, which is minimax for this location family by the Hunt–Stein theorem (Lehmann and Romano, Testing Statistical Hypotheses); 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 non-adaptive protocol satisfying the mark trade-off with constant \(\kappa\), read by tests invariant under the unknown initial position and velocity,

\[d^2\ \le\ \frac{F^2\tau^3}{24\,m\kappa},\]

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

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

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, zero on \([\tau,\infty)\), and zero on \((-\infty,0]\) as well, since \(T(0)=\int_0^\tau N=\sum_iu_it_i=0\) by the invariance under \(v_0\); it has \(T'=-N\) 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 \(u^{\mathsf T}\Sigma u\ge2\kappa E/m\). Since \(T\) vanishes at both ends of \([0,\tau]\), the Poincaré inequality \((\int_0^\tau T)^2\le\frac{\tau^3}{12}\int_0^\tau T'^2\) applies, with equality for \(T\propto w(\tau-w)\) (the Euler–Lagrange equation is \(T''=\) const), and

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

Finally \(\Phi(-d/2)\le\epsilon\) requires \(d\ge2z_{1-\epsilon}\). For sharpness, take marks dense in \([0,\tau]\) with weights making \(N\) approximate \(2w-\tau\), so that \(T\) approximates \(w(\tau-w)\); the interior weights are negative where \(T\) is positive, so every \(u_jS_j\) has one sign and \(\sum_j|u_j||S_j|=E\); balance each mark, \(\delta_j|u_j|=\Delta_j|S_j|/m\), so the arithmetic–geometric step is an equality; and make the end marks sharp, which costs nothing since \(S=T=0\) there. Every inequality then becomes an equality in the limit. \(\square\)

Three features of the proof carry the claim that the bound holds for every protocol of the model. 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 the Poincaré constant of the interval, \(\tau^3/12\), which is where the combination in (1) originates. The sharp constant is approached, never attained, by dense protocols whose end marks are sharp, with a recoil that no invariant statistic sees.

The floor rests on invariance. The body’s initial operators enter every reading as \(\hat y_0+\hat p_0t_j/m\) and cancel from every invariant statistic, so the body’s state plays no part in Theorem 2. A protocol that knows the preparation escapes the floor even with Gaussian states: a contractive Gaussian state focused at time \(\tau\) and one sharp final mark decide at every force, which is Yuen’s protocol (PRL 51, 719, 1983). The boundary is knowledge of the preparation, and §10 prices the preparation route by the body’s spread along the way.

Corollary 3 (insertion). Marks inside a window of duration \(\tau'\), used by themselves, record the force only if \(F^2\tau'^3\ge96z_{1-\epsilon}^2m\kappa\). So a window shorter than

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

used by itself, is consistent with free motion at the stated confidence. Refinement continues below \(\tau_*\), and the sharp protocol of Theorem 2 needs marks far denser than that; what stops is local exhibition of the force. Newton’s polygon may be refined past \(\tau/\tau_*\) vertices, and its impulses remain in the geometry, though no record confined to one such window 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 \(36m\kappa\). There the initial velocity is bounded by the preparation’s recoil width instead of being free, so \(T\) need not vanish at \(0\) and the Cauchy–Schwarz step is the one available. 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 needs only the quadratic structure.

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 itself, whatever the strength of the coupling. 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 whose position and momentum are uncorrelated, a fresh probe for each mark, 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 for every invariant statistic; the body’s own state cancels from those and need not be Gaussian. With Theorem 4,

\[\tau\Delta E\ \ge\ 24\,z_{1-\epsilon}^2\,\hbar,\qquad \tau_*=\left(\frac{48\,z_{1-\epsilon}^2\,m\hbar}{F^2}\right)^{1/3}, \tag{2}\]

about \(65\hbar\) at \(\epsilon=0.05\) and \(24\hbar\) at \(\epsilon=0.16\).

Correlated marks. A probe whose position and momentum are correlated delivers an error and a recoil with correlation \(\rho=-\operatorname{Cov}(\hat Q_A,\hat P_A)/(\Delta\hat Q_A\Delta\hat P_A)\), and the Robertson–Schrödinger inequality gives \(\delta^2\Delta^2(1-\rho^2)\ge\hbar^2/4\). The cross terms enter the proof of Theorem 2 as \(2c_ju_jS_j/m\), and \(ax^2+2cxy+by^2\ge2(\sqrt{ab}-|c|)|x||y|\) replaces the arithmetic–geometric step, so every protocol whose marks have \(|\rho_j|\le\rho\) needs

\[\tau\Delta E\ \ge\ 24\,z_{1-\epsilon}^2\,\hbar\,\sqrt{\frac{1-\rho}{1+\rho}} , \tag{3}\]

sharp by the dense construction of Theorem 2 with every mark at correlation \(\rho\) of the helpful sign. Three marks at \(0,\tau/2,\tau\) with the test \(R_1-2R_2+R_3\) come within a factor \(4/3\), since the outer marks’ recoils are invisible to the test and the middle mark’s noise \(-2\xi_2+(\tau/2m)\iota_2\) is one quadrature of its probe: they decide once \(\tau\Delta E\ge32z^2\hbar\sqrt{(1-\rho)/(1+\rho)}\), at every force as \(\rho\to1\). This is the mechanism of Yuen’s contractive states (PRL 51, 719, 1983), which correlate the body’s position and momentum, applied to the meter, as in the back-action-evading measurements of Braginsky and Khalili. The proofs are in the mark-cost note, Theorem C and Proposition D.

5. Every probe and every instrument: the record costs disturbance

The floors of §§3–4 assume Gaussian probes, and the assumption is needed. A grid state of the Gottesman–Kitaev–Preskill kind with a real wavefunction has no position–momentum correlation, yet as the middle probe of the three-mark protocol it turns the statistic into a comb of spacing \(\sqrt{2\pi\hbar\tau/m}\), and along the standard family of approximate grid states the teeth narrow, so the error tends to zero at any force whose signal is not a multiple of that spacing, at the price of a recoil that diverges. Grid states read both quadratures of a small displacement at once, the principle of the displacement sensor of Duivenvoorden, Terhal and Weigand (PRA 95, 012305, 2017). What holds for every probe state is a bound on the recoil, and for every instrument a bound on disturbance.

Theorem 5. For every protocol of momentum-transfer marks with their pointers read and probes in arbitrary states, deciding at error \(\epsilon\) whatever the body’s initial state under each hypothesis requires \(\arcsin(1-2\epsilon)\le\hbar^{-1}\min_{a,b}\sum_j\Delta_j|P(t_j)-a-bt_j|\), with \(\Delta_j\) the recoil spread of mark \(j\). For the constant force,

\[s\sum_j\Delta_j\ \ge\ 8\hbar\arcsin(1-2\epsilon),\qquad s=\frac{F\tau^2}{2m}, \tag{4}\]

which at certain decision reads \(s\sum_j\Delta_j\ge4\pi\hbar=2h\).

The proof moves the signal into the pointers: translating the body’s initial state along the best straight line \(a+bt\) turns the force into a translation of each pointer by \(\lambda_jc_j\), with \(c_j=P(t_j)-a-bt_j\) the line’s residual at the mark, and a pointer registers a translation only through the spread of its momentum, which is the impulse the body receives. Mandelstam–Tamm and the triangle inequality over the probes finish it, and running the same chain with the Bures angle in place of total variation gives the constant \(\arcsin(1-2\epsilon)\), the sharp single-shot one (path-length note, Theorem P). With sharp end marks the minimum over lines is the line through them, and the three-mark protocol with the best Gaussian middle probe, whose statistic then has spread exactly \(\hbar/\Delta_2\), comes within a factor \(z_{1-\epsilon}/\arcsin(1-2\epsilon)\) of that bound at every recoil, about \(1.47\) at five per cent error; uncorrelated probes come within \(\sqrt2\) times that. The constant-force form (4) charges the end marks as well, and their unbounded recoil makes its left side infinite. The proofs, the grid construction and the tightness computation are in the recoil note. It is the recoil clause of M3, and Theorem 6 completes it.

The recoil bound reaches well beyond von Neumann’s coupling. For any coupling and any pointer, the pointer commutes with the body’s momentum after the mark, which gives \([\hat N,\hat D]=-i\hbar-[\hat y,\hat D]-[\hat N,\hat p]\). So every mark whose error and impulse are independent of the body has conjugate error and impulse, which is Ozawa’s class of measurements with independent intervention (PRA 67, 042105, 2003; Ann. Phys. 311, 350, 2004); if it also leaves the position alone it is a von Neumann mark on some canonical pair of the apparatus, a dilation form of Davies’s translation-covariant instruments (1976); and Theorem 5 holds for all such marks, position-displacing ones included (additive-noise note). When the noise depends on the body, the offset between the hypotheses still has to be carried through each mark, and the cost of carrying it is fixed by what the mark does to the body.

Theorem 6 (every instrument). Let mark \(j\) be any unitary coupling of the body to an apparatus followed by reading a pointer, with momentum and position disturbances \(\hat D_j=U_j^\dagger\hat pU_j-\hat p\) and \(\hat X_j=U_j^\dagger\hat yU_j-\hat y\), operators on body and apparatus that may depend on both. Deciding at error \(\epsilon\) whatever the body’s initial state under each hypothesis requires, for every line \(a+bt\), \(\arcsin(1-2\epsilon)\le\hbar^{-1}\sum_j\sup\Delta(c_j\hat D_j-mc_j'\hat X_j)\) with \(c_j=P(t_j)-a-bt_j\), the supremum over the states that enter the mark in the interpolating processes of the proof; for the constant force,

\[\frac s8\sum_j\Delta(\hat D_j)+\frac J2\sum_j\Delta(\hat X_j)\ \ge\ \hbar\arcsin(1-2\epsilon),\qquad J=F\tau. \tag{5}\]

The constants \(1/8\) and \(1/2\) are the price of an initial state that may differ between the hypotheses: a test allowed to assume the same initial state under both faces the coefficients \(s\) and \(J\).

The proof is a hybrid argument that carries the offset \((c,mc')\) between the hypotheses through the marks one at a time. Passing mark \(j\) applies the unitary \(T^\dagger U_jTU_j^\dagger\), with \(T\) the translation by the offset; its generator, conjugated back to the state entering the mark, is \((c_j\hat D_j-mc_j'\hat X_j)/\hbar\), and Mandelstam–Tamm bounds its effect by that operator’s spread. The line of best uniform approximation gives \(|c_j|\le s/8\) and \(m|c_j'|\le J/2\) (disturbance note). Von Neumann marks give Theorem 5, and marks that read momentum pay in position jumps.

Equation (5) involves the disturbances alone, so the error–disturbance relations, Ozawa’s included (PRA 67, 042105, 2003), are unused. Newton’s two quantities appear with the partners they have in the aperture bound of §10: the sagitta with momentum, the impulse with position.

6. The part no preparation can move

The floors of §§3–5 are paid in spreads, of resolution, recoil or disturbance, and a protocol can move a spread from one quantity to another, as squeezing and grid states do. The comparison also carries a quantity with no spread at all. Newton’s polygon of Proposition I inscribed in the parabola, with each step’s impulse \(F\tau_j\) split equally between its ends, moves uniformly along each chord and has the parabola’s position and velocity at every vertex, so classically the two are the same state there.

Theorem 7. For every partition of \([0,\tau]\) into steps \(\tau_j\), the two evolutions satisfy \(W_{\rm poly}=e^{i\theta_N}W_F\) with

\[\theta_N=\frac{F^2}{24m\hbar}\sum_j\tau_j^3=\frac{F}{2v\hbar}\sum_jS_j,\]

\(S_j=vF\tau_j^3/12m\) being the parabolic segment cut off by the \(j\)-th chord. Inserting a vertex at fraction \(\lambda\) of a step lowers \(\theta_N\) by \(F^2\tau_j^3\lambda(1-\lambda)/(8m\hbar)\), the inscribed triangle over \(\hbar\), so midpoint halving removes it in the proportions of Archimedes’ exhaustion. The same increments composed in reverse order differ by the phase \(\Phi_N=F^2(\tau^3-\sum_j\tau_j^3)/(6m\hbar)\), and \(\Phi_N+4\theta_N=\tau\Delta E/3\hbar\) for every partition.

The proof uses the Weyl relations alone: an impulse \(J\) at time \(t\) is a phase-space displacement, two histories with equal endpoints differ by the enclosed phase-plane area over \(\hbar\) times the identity, and the per-step lobe is \(F^2\tau_j^3/24m\) (the polygon-lift note has it in full). Since \(W_{\rm poly}W_F^\dagger\) is a multiple of the identity, no preparation of the body, squeezed or mixed, changes \(\theta_N\). A control qubit that selects the history reads it at single-shot error \(\frac12(1-|\sin(\theta_N/2)|)\), so in one controlled pass a step is told from its chord at error \(\epsilon\) only if \(\tau_j\ge(48m\hbar\arcsin(1-2\epsilon)/F^2)^{1/3}\), for every state of the body; \(k\) passes accumulate \(k\theta_N\) and lower that threshold by \(k^{1/3}\), so the readout’s resource is the number of passes. The single chord lens and the closed loop of the earlier notes are instances. In numerical analysis the inscribed polygon is the kick–drift–kick (Strang) splitting of the propagator, exact for a linear potential up to this c-number; what is new is its reading in Lemma XI’s areas and Archimedes’ segments.

Archimedes’ exhaustion, read through \(\hbar\). Archimedes squares the parabola by inscribing, in the segment cut off by a chord, the triangle whose apex is “the vertex of the segment”, where the tangent is parallel to the chord; the two segments left over receive their own triangles, each round totalling a quarter of the one before, and his Proposition 23 sums the series, \(A+B+C+\dots+Z+\frac13Z=\frac43A\), so that the segment is four thirds of its first triangle (Proposition 24). On the Galileo parabola traced in time the vertex of a step’s segment is its midpoint in time, where the velocity equals the chord’s slope, so his construction is Newton’s refinement by halving, and the triangles he adds are those Theorem 7 removes. With \(T_j=\frac34S_j\) the first triangle,

\[\theta_{\rm step}=\frac{F}{2v\hbar}\cdot\frac43T_j =\frac{F}{2v\hbar}\Bigl(T_j+\frac{T_j}4+\frac{T_j}{16}+\cdots\Bigr),\]

and Proposition 23 becomes the statement that the phases removed by all the rounds add up to the phase between polygon and parabola. Exhaustion shows classically that the leftover segments, \(4^{-k}\) of the whole after \(k\) rounds, fall below any given area. With \(\hbar\) they are phases, and two scales appear: each segment’s phase falls below a radian once the steps are shorter than \(\tau_\hbar=(24m\hbar/F^2)^{1/3}\), and their total once \(4^k\) exceeds \((\tau/\tau_\hbar)^3\). The method began with Democritus’s cone, which Eudoxus proved to be a third of its cylinder (§9), and the Galileo area is the same third, \(A=\frac13v\tau s\). The two figures differ in what \(\hbar\) does to them: the cone is static, its sections can be marked without disturbing it, and Theorem 9(iii) finds no floor; the parabola is traced by a motion, and its exhaustion carries phases.

Theorem 8 (every force law). For a force history \(f\) on an interval of duration \(\tau\), let \(\mathcal K_\tau[f]=\int\frac12m(\dot y_f-\dot y_{\rm chord})^2dt =\frac1{2m}\iint G_\tau ff\) be the kinetic action of the motion relative to its chord, \(G_\tau(s,u)=\min(s,u)(\tau-\max(s,u))/\tau\). Newton’s polygon, with each step’s impulse split between its ends so as to match the step’s displacement, differs from the motion by the phase \(\sum_j\mathcal K_{\tau_j}[f]/\hbar\), which refinement never raises; and every protocol of uncorrelated Gaussian marks has \(d^2\le\mathcal K_\tau[f]/\kappa\), sharp for \(f\) of one sign. At \(\kappa=\hbar/2\) the best recorded deflection is twice the phase between the motion and its chord. The proofs are in §6 of the polygon-lift note. One functional of the force history, the part of the motion that Newton’s refinement removes, is converted by \(\hbar\) into a phase and by the mark cost into a statistical distance; for a constant force it is the chord’s segment times \(F/2v\).

7. Both factors are in the Opticks

Newton’s optics supplies, on this paper’s reading, the two spreads whose product is \(\kappa\), in the form of two quantities of the probe; the identifications below are readings, and each says what it rests on.

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. We read it as \(\Delta\hat Q_A\). The reading is an analogy: the interval is a period along the ray, a longitudinal length, while \(\Delta\hat Q_A\) is the spread of a transverse mark’s pointer.

An impulse, inferred. Query 29 asks whether the rays of light are “very small Bodies emitted from shining Substances”. A body carries an impulse, and we read its scale as \(\Delta\hat P_A\); Query 29 itself speaks of sizes, and it is a query, so the impulse scale is our inference. Newton has no way to measure it.

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. On Newton-age evidence the product is therefore colour dependent, and it grows toward the red, since both the interval and the corpuscle’s size do; its constancy 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.

The interval times the momentum is a refraction invariant. Book II Part III Prop. XVII makes the intervals in two mediums “as the Sine of Incidence to the Sine of Refraction”, from Observation 10 of Part I, and Prop. X states its result on the hypothesis that light is “swifter in Bodies than in Vacuo, in the proportion of the Sines”, which Newton holds categorically in the Principia, Book I Section XIV. Together they make \(\Lambda v\), and so \(\Lambda p\), the same in every medium for each colour; that Newton’s interval varies inversely with the corpuscle’s speed, as de Broglie’s wavelength does, is an old observation (Sakkopoulos, Eur. J. Phys. 9, 123, 1988, metadata). A fit counted per interval along the path therefore counts the Maupertuis action \(\int p\cdot dq\) in units of \(\Lambda p\), and the inscribed polygon and the parabola of Theorem 7 differ by \((F/v)\sum_jA_j/(2\Lambda p)\) fits, with \(A_j\) Lemma XI’s tangent areas. Since the fit at arrival decides reflection or transmission (Prop. XII), the two are optically different corpuscles on Newton’s own terms once that count reaches half an interval: on this reading his determinate fit would tell them apart on a single corpuscle, where the quantum phase is relative between two histories and shows only in a superposition. The two are different in kind. Newton had no superposition of amplitudes: imaginary quantities enter his algebra only as “impossible” roots, and his mechanism for the fits, in Query 29, is that the rays “stir up Vibrations in what they act upon, which Vibrations being swifter than the Rays, overtake them successively, and agitate them so as by turns to increase and decrease their Velocities, and thereby put them into those Fits”, a wave that disposes the corpuscle, with no adding of amplitudes. Addition of amplitudes arrives with Young’s principle of interference (Phil. Trans. 92, 12, 1802). Counting \(2\pi\) per interval, a full return of the disposition, the fits phase is \(4\theta_N\) at \(\Lambda=\lambda/2\); the wave’s own phase, which advances \(\pi\) per interval, differs by \(2\theta_N\). Extending Props. X and XVII to a continuously varying speed is ours, and Prop. XV’s oblique rule is more complicated than a count along the path.

These passages, with Prop. XI below, are held verbatim with line anchors in the source companion, from the fourth edition of 1730.

8. The junction, and the proposition Newton’s commitments exclude

Theorems 4 to 6 say 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. The Gaussian floor also uses a clause M3 carries implicitly, that the undetermined impulse is unrelated to the mark’s error; eq. (3) prices that clause. Theorems 5 and 6 give M3 its universal form, in which the resolution drops out: a record of the sagitta leaves the body’s impulse undetermined, a record of the impulse leaves its displacement undetermined, and the exchange rate is \(\hbar\). 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 \(\Lambda p=h/2=\pi\hbar\), which differs from \(\hbar/2\) by a factor \(2\pi\).

What Newton printed entails the negation of M3, though he nowhere states M3 or its negation. 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”. A determinate periodic disposition is not by itself the negation, since a wave’s phase is determinate too. It becomes one when the disposition is carried by a corpuscle of Query 29 moving by ordinary deterministic mechanics: the ray’s fate at a surface is then fixed by its phase, what a prior record fails to supply is, on this reading, knowledge of a definite state, both position and momentum are definite, later records can recover the impulse a mark 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 follows from what he printed. 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.

Theorem 7 adds a second junction, one with no spread in it. The premise that Theorem 7 uses is the central extension, that an impulse \(J\) and a displacement \(d\) compose only up to the phase \(Jd/\hbar\) of their parallelogram, where Newton’s Corollary I composes them exactly. Newton’s fits already carry a phase whose rate is proportional to momentum, with the refraction invariant \(\Lambda p\) as its unit, and his determinism in Prop. XII does not touch it. A count of such a phase is order-sensitive in the parallelogram sense: a displacement \(d\) then a kick \(J\) counts \(p\,d\), the kick first counts \((p+J)\,d\), and the difference is the parallelogram’s area \(Jd\). The phase with a momentum-proportional rate is in the Opticks; the order-sensitivity is this paper’s construction on it. So a per-colour form of the quantum premise that squeezing cannot evade can be read into the Opticks; what it lacks is the universality of \(\Lambda p\) across colours.

A third junction is textual and of the same year as the Principia. In the mark model, M3 with \(\kappa>0\) is equivalent to the continuity at \(F=0\) of the best verdict on the comparison, \(\Phi(-\frac12\sqrt{K_\tau/\kappa})\) for the observer ignorant of the preparation (continuity note, Proposition L), and that continuity is Leibniz’s law of continuity, published in July 1687 as a test of laws of motion, read with outcomes measured by discernibility. Newton’s deterministic corpuscle gives \(\kappa=0\) and a best verdict that jumps from \(\frac12\) to \(0\) at \(F=0\), a jump of the kind Leibniz’s test condemns in Descartes’s rules of collision; read geometrically, as Leibniz applied it, the same test passes Newton’s laws. At a bounded preparation aperture the observer who knows the preparation reaches the same continuity through the probabilistic note’s Theorem 2, with \(\hbar\) in place of \(\kappa\). The records reading is that note’s step and remains a premise.

9. 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 founding study is McGuire and Rattansi, “Newton and the ‘Pipes of Pan’”, Notes Rec. R. Soc. 21 (1966), 108–143; the scholarship continues in 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; Schüller edited the scholia from Gregory’s papers. All are verified at metadata level here and still to be read.

So Newton did draft a set of scholia founded on the classics, with Gregory’s preface as their printed witness, 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.

Book I does carry two scholia that touch the ancient question, both from the geometers’ side, and the first edition’s Latin is held here (1687 witness; English renderings ours). The scholium after Lemma XI continues the series of contact angles without end: between any two, a new series “inseri potest”, can be inserted, “Neq; novit natura limitem”, and nature knows no limit. The scholium closing Section I opens by saying that the lemmas were put first “ut effugerem tædium deducendi perplexas demonstrationes, more veterum Geometrarum, ad absurdum”, to escape the tedium of the ancient geometers’ reductio: the method of exhaustion that Archimedes credits to Eudoxus, who used it to prove that the cone is a third of its cylinder, a result Archimedes says Democritus had asserted first without proof (preface to The Method). Newton prefers limits to the method of indivisibles, “quoniam durior est indivisibilium Hypothesis”, answers the objection that ultimate ratios would imply ultimate magnitudes, so that quantity would consist of indivisibles, “contra quam Euclides de incommensurabilibus, in libro decimo Elementorum, demonstravit”, and closes “cave intelligas quantitates magnitudine determinatas, sed cogita semper diminuendas sine limite”: do not take them as quantities of determinate magnitude, but think of them as always diminishing without limit. Euclid is cited against least magnitudes, the ancients’ exhaustion is replaced by limits, and the physical dispute about the cut, with the atomists who held least parts, is left aside. §6 reads that exhaustion through \(\hbar\): for a figure traced by a motion, each round of it removes a phase.

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 the theorems quantify. The aperture bound of §10 is protocol-universal 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 9. 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 \(F^2\tau^3/m\ge96z_{1-\epsilon}^2\kappa\), 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 arrow’s rung is attained by two marks: weights \((-1,1)\) at \(0\) and \(\tau\) make \(T\) linear, so the Cauchy–Schwarz step is an equality, with the first mark balanced, \(\delta_0=\Delta_0\tau/m\), and the last sharp. Newton’s rung needs the dense limit of Theorem 2.

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 \(\tau\Delta E\). The theorem covers \(n=1\) and \(n=2\); for higher \(n\) the form is dimensional analysis. 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. The Classical Scholia are reported to cite Plutarch’s De facie among the ancients said to have known the doctrine of gravitation (Casini 1984; McGuire and Rattansi 1966; metadata, not read here), which would put the sling in a text Newton was mining for authority while writing the propositions it models; whether they cite this passage is open.

Theorem 9 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 illustrated a conclusion already held. 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, cited from the GRETIL text and not yet held in a companion here) 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 9 says about records: within a window 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 approached through least parts, which the theorem replaces by a least action for records and no least magnitude, and one that what he printed about the fits excludes, in the single place where he had a measurement.

10. Unmarked preparations and prior art

Preparations of unbounded extent. Theorems 2, 5 and 6 concern 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

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

for an apparatus whose position and momentum spreads are bounded by \(L\) and \(P\). It is the integrated form of a kick bound, \(\int_0^\tau|f(t)|L(t)\,dt\ge\hbar\theta\) with \(L(t)\) the body’s position spread at time \(t\): a force is a succession of impulses, and an impulse is registered only through the position spread it acts on (path-length note, Theorem K). The floor equals \(\theta^2\hbar^2/(4LP)\) at the balanced aperture \(L=\tau P/2m\) and falls to zero as either side grows at fixed area, so a squeezed state of area \(\hbar/2\) evades it as a large one does. In Newton’s quantities the first inequality is \(F\tau\cdot L+\frac{F\tau^2}{2m}\cdot P\ge\hbar\theta\), the impulse of Proposition I against the position aperture plus the sagitta of Lemma X against the momentum aperture. It holds for every finite adaptive protocol of instruments, of any Kraus rank and with any apparatus memory, 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.

One accounting. The kick bound pays for the force at the body and Theorem 6 pays for it at the marks, and any split of the force between the two gives a valid bound. For von Neumann marks and a force of one sign the best split is pointwise, \(\hbar\theta\le\int_0^\tau|f(u)|\min\bigl(L(u),R(u)\bigr)du\), where \(R(u)=\frac1m\sum_j\Delta_jG_\tau(t_j,u)\) is the recoil length, the marks’ recoil spreads weighted by the displacement each produces at time \(u\) relative to the chord and summed, with \(G_\tau\) the Dirichlet Green’s function of the interval: at every moment the force is paid in the smaller of the two lengths (path-length note, Theorem M).

Prior art. The combination \(F^2\tau^3\gtrsim m\hbar\) is the standard quantum limit for detecting a force on a free mass. It goes back to Braginsky and Vorontsov (Sov. Phys. Usp. 17, 644, 1975); the canonical statement is Caves, Thorne, Drever, Sandberg and Zimmermann, Rev. Mod. Phys. 52, 341 (1980); the standard accounts are Braginsky and Khalili, Quantum Measurement (1992), and Bocko and Onofrio, Rev. Mod. Phys. 68, 755 (1996). 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). The dense-mark limit of §3 is the continuous position measurement of Caves and Milburn (PRA 36, 5543, 1987), with the force limit worked out in Clerk, Devoret, Girvin, Marquardt and Schoelkopf (Rev. Mod. Phys. 82, 1155, 2010); what Theorem 2 adds to that model is the invariant-test criterion at a fixed error probability and the sharp constant. Hypothesis testing of a force waveform on a quantum probe is Tsang and Nair (PRA 86, 042115, 2012), the closest prior art to §10’s framing, and a quantum-mechanics-free subsystem removes the back-action floor altogether (Tsang and Caves, PRX 2, 031016, 2012). Theorem 4’s 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) set limits of this kind from field measurability and from conservation laws. That preparations beat the limit is Yuen’s point (PRL 51, 719, 1983); the two-packet preparation of this section is the delocalized force sensor of Childs, Preskill and Renes (J. Mod. Opt. 47, 155, 2000), and its sensitivity to displacements below the uncertainty area is that of Zurek’s sub-Planck structures (Nature 412, 712, 2001; Toscano, Dalvit, Davidovich and Zurek, PRA 73, 023803, 2006). The correlated probes of §4 apply the mechanism of Yuen’s contractive states to the meter, as Braginsky and Khalili’s back-action-evading measurements do, and the grid probes of §5 are the grid states of Gottesman, Kitaev and Preskill (PRA 64, 012310, 2001). The ingredients of §§5–6 and §10 are standard: Ozawa’s disturbance operators, Mandelstam–Tamm in the Bures angle (Uhlmann 1992; Anandan and Aharonov 1990), hybrid arguments (Bennett, Bernstein, Brassard and Vazirani 1997), Bures-angle speed limits for processes (Taddei et al. 2013), the position-spread form of force sensitivity (Tsang, Wiseman and Caves 2011), symmetry-based information–disturbance bounds of Wigner–Araki–Yanase type, and the Strang splitting behind Theorem 7. All these are cited at metadata level. This paper collects the quantified protocol and instrument bounds, their explicit constants, the split accounting of §10, the phase identities of §6 read in Newton’s and Archimedes’ areas, and the stated physical premises. Literature priority for that combination has not been established. The inequality \(F^2\tau^3\gtrsim m\hbar\) itself is standard.

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 the worst-case form of §3, 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.

10a. Kepler completion and the action-selection premise

Dated revision, 2026-10-02. The marked-comparison theorems assume a positive action scale. Here \(\hbar\) denotes the reduced phase constant, written \(h\) in the self-sufficiency and reachability notes; the full Planck constant in this paper remains \(h=2\pi\hbar\). To investigate whether motion itself requires a positive scale, the maintained self-sufficiency proof compares classical and quantum point-Coulomb dynamics. Classical radial inward or bound data collide in finite time and have no continuation under the original force field. The non-relativistic Coulomb Hamiltonian is self-adjoint on \(H^2(\mathbb R^3)\) for every \(\hbar>0\), and its evolution exists for all times. That comparison establishes an incomplete classical flow and one global completion. It does not exclude classical regularizations or other completions.

The same proof now establishes the classical comparison on a fixed collision-free interval. For a fixed orbit with minimum radius \(\rho_{\min}>0\), fix \(0<\delta<\rho_{\min}\) and a real \(V_\delta\in C_b^\infty\) equal to \(-k/r\) outside \(r<\delta\). For the normalized coherent preparations with width \(\ell_\hbar=\sqrt{\hbar L_*/P_*}\), fixed \(L_*,P_*>0\), centered weighted Sobolev estimates and Hardy’s inequality give \[\sup_{0\le t\le T}\|(V-V_\delta)U_\hbar^{(\delta)}(t)\varphi_\hbar\|_2 =O(\hbar^{3/2})=o(\hbar).\] Duhamel transfers the smooth-potential Gaussian approximation to the Coulomb evolution. Separate energy and position estimates give \(\langle x\rangle=q_t+O(\sqrt\hbar)\) and \(\langle p\rangle=p_t+O(\sqrt\hbar)\), uniformly on the fixed interval. The constants depend on \(T\), the preparation, derivatives of \(V_\delta\) and \((\rho_{\min}-\delta)/2\). No collision, long-time or shrinking-\(\delta\) limit follows. Hepp’s theorem supplies the smooth-potential limit [@Hepp1974; passage, §2]; the singular-multiplier and moment estimates are the October 2 written derivation in §5a of the maintained proof, which has not received independent verification.

The packet limit and the complete record of a coherent swarm test different preparations and observables. The reachability proof, Theorem 5, shows exactly for quadratic forces that an incoherent mixture of shrinking coherent packets recovers Newton’s branch-weighted coordinate density at each fixed time and regular record value. Its coherent-stream contrast instead requires coherence across launch points and a record that resolves the interference. The same classical swarm does not choose between those attachments. A physical rule that selects and preserves the required coherence remains an additional obligation.

At finite \(c\), Proposition 2 of the self-sufficiency proof distinguishes the selection premises. For Dirac–Coulomb, essential self-adjointness gives \(\hbar\in[2k/(\sqrt3c),\infty)\); requiring finite separate potential energy gives \((k/c,\infty)\); a critical Schur-domain or the specified cutoff limit extends the latter to \([k/c,\infty)\) [@EstebanLewinSere2019; passage, Theorems 1.1 and 1.11, arXiv v7]. The critical domain is defined by the original Esteban–Loss closed Schur form [@EstebanLoss2007; passage, Definition 3 and Theorem 4]. The maintained proof records an angular-channel counterexample to the literal compact domain formula printed in arXiv v7, equation (1.32); that rewrite is not a proof premise. If extra boundary conditions are allowed, self-adjoint Dirac realizations exist at every \(\hbar>0\). For Klein–Gordon, strict coercivity in the usual energy topology and the cited charge-space construction give \((2k/c,\infty)\) [@LangerNajmanTretter2008; passage, Example 8.3, Theorems 8.5–8.6]. Its principal radial Friedrichs operator includes \(\hbar=2k/c\) with a no-log branch [@DerezinskiRichard2017; passage, §2.3 and Theorem 5.5]. That radial selection is not a proof of full critical Klein–Gordon dynamics: the critical combined form does not control the singular multiplier needed in that construction.

Probability conservation justifies Dirac self-adjointness; it does not choose an extension. Finite separate energies, a particular point-source limit, Friedrichs closure and an energy topology are further premises. Before a selection-dependent threshold can establish action necessity, those premises must be justified independently of the desired gap. There is consequently no single unqualified admissible interval \([\hbar_{\min},\infty)\) here, and no continuum Yang–Mills mass-gap result follows from this mechanical comparison. The action threshold concerns a family of selected theories; a spectral gap concerns a fixed operator.

11. Obligations before submission

The historical obligations and the scope of the new mathematical material are stated separately 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 §8’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 §9 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 Koyré and Cohen’s variorum for the Latin, the first edition of which is now held for Book I Section I, and from Cohen and Whitman for the English, in place of the Motte text and this paper’s own renderings. The firebrand passages of the Abhidharmakośabhāṣya are to be added to that companion, and whether the Classical Scholia cite De facie 923 itself is to be checked in McGuire and Rattansi and in Casini.
  3. October 2 mathematical additions. The localization, moment estimates and replacement selection proposition are written derivations supported by the cited source passages. The September 23 and October 1 reviews predate them and supply no independent verification of these additions. Their stated domains and fixed-data limits must be retained in submission. Any stronger action-necessity claim requires an independent physical justification of the selection premise; full critical Klein–Gordon dynamics remains open.

Three mathematical items remain open and are not obligations of the paper: the worst-case interval \(9\le F^2\tau_*^3/(m\kappa)\le36\) of §3, which concerns a framework without quantum instances; whether the minimum of the one accounting of §10 over splits and lines is attained by some protocol, which would make it the exact floor; and whether the constants \(1/8\) and \(1/2\) of (5) are attained, the line of best uniform approximation being optimal for the momentum term alone.

12. Consequence for STATE

This synthesis collects the conditional mark bounds, disturbance estimates and polygon phases together with Newton’s source materials. The quantum results supply a positive canonical scale; their record costs do not derive that scale from Newton’s axioms. The foundations content is §§3–6: the sharp Gaussian floor, the mark cost, the disturbance floor for every instrument and the phase that no preparation of the body moves. The history content is §§7–9, resting on the source companions. The junction is §8: M3 is Robertson’s inequality for Newton’s corpuscle, its universal form is that a record costs disturbance, what Newton printed entails its negation, and Newton’s fits carry a phase whose rate is proportional to momentum and whose unit \(\Lambda p\) is a refraction invariant. Section 11 records this draft’s historical obligations and the proof status of the mathematical additions. The October 2 §10a separates the closed fixed-interval Kepler localization and first-moment problem from the open physical selection, coherence and action-necessity obligations. STATE maintains the research queue, including the continuum Yang–Mills obligations.