Newton’s shrinking cell and a gauge lattice’s shrinking spacing both require compatible refinement laws. This tutorial follows their exact constructions, the resources needed to record them, and the still-open survival of a positive physical scale.
Use the arrow keys, Page Up and Page Down, or scroll. Every claim links to the note that holds its proof and states its status.
Galileo compares a body moving freely with the same body under a constant force F. Over a cell of duration τ the forced path separates from the inertial path by Fτ²/2m. The midpoint sagitta relative to an endpoint-matched chord is Fτ²/8m. The normalized limiting ratio recovers the force.
The matched-endpoint chord defect has action K; the inertial–parabola Galileo comparison has τΔE = 12K:
Newton takes τ → 0, and Kτ goes to zero with it.
Cut a cell at the fraction s of its duration. The cut spends exactly 3s(1−s)Kτ of the cell action, and the shares add over any sequence of cuts. Halving spends ¾; n halvings spend 1−4−n.
The stick of Zhuangzi 33, halved every day from one end and "not exhausted in ten thousand generations", spends only 6/7: the pieces it takes away are never cut again.
Cut-measure note, Theorems 1–2 (refereed).
In the independent Gaussian mark model, each position error δ and impulse spread Δ obey a supplied tradeoff:
For non-adaptive marks and tests invariant under unknown initial position and velocity, the supremum of squared distinguishability on a finite cut grid is the spent action divided by κ. Finite positive endpoint widths approach this value; they never attain it.
For arbitrary quantum instruments, the wider result prices momentum and position disturbance:
Here s = Fτ²/2m and J = Fτ. These coefficients apply when initial states under the two hypotheses are unknown. Each spread is bounded uniformly over the interpolating comparison states; a known common preparation requires a different bound. Prepared packets can evade a universal cell-action threshold; ℏ is supplied.
Cut measure, Prop. 7 · general disturbance · preparation dependence.
Within that non-adaptive independent Gaussian family, optimizing preparation-invariant tests gives the infimum error
For κ > 0 this optimized verdict tends continuously to ½ as F → 0. For κ = 0 arbitrarily precise admissible marks drive the error to zero for every nonzero force. Each fixed finite protocol can still be continuous; the singularity concerns optimization over the family.
"Lorsque la difference de deux cas peut estre diminuée au dessous de toute grandeur donnée in datis … il faut qu'elle se puisse trouver aussi diminuée … in quaesitis" — Leibniz, July 1687
Continuity note: the iff theorem holds for the stated mark family. Extending a uniform continuity principle to arbitrary adaptive apparatus, retained memory and complete records requires a physical premise; fresh repeated preparations are a different resource.
Book I of the Principia sets limits against indivisibles. The dispute about motion and its records runs through older and contemporary texts, each quoted in the notes from a named edition.
The arrow at rest in every now; the stick halved without end; halving that stops at a point. Paradoxes about division that set the problem.
A weight dropped from the thickness of a sheet of paper makes an impression "del tutto impercettibile": the beginning of fall argued from its record.
Leaps are pushed to ever smaller bodies unless atoms stop them; nature makes no leaps; a hundred thousand nothings cannot make something.
Parts can be distinguished by reason, "ex mathematica certum est"; whether nature can divide them "incertum est", and one experiment would decide.
Hard particles that never wear, so that natures stay the same in all ages: a stability premise that asks for a scale.
A floor on action fixes no length; it survives the similarity x → λx, t → λ²t, the scaling of quantum paths.
A lattice gauge theory refines by halving its spacing. Halving one direction inserts a new layer of links. Moves along a line compose exactly; the new mid-plane, coupled to its neighbours through the plaquettes, carries the whole defect.
The atlas of halving records what survives in each group and dimension: free fields, U(1), SU(2), SU(3), in 1+1, 1+2 and 1+3 dimensions.
For SU(2) in 1+2 dimensions, the target is a normalized comparison of the interacting mid-plane with a covariant Gaussian. On the controlled chart, response kernels decay uniformly in plane size. This input does not yet prove the full nonlinear group-integral comparison.
Small-field note: §§26–36 construct contact, connected-set and dressed-activity control in a weak-recoupling strip. §37 tests moving sources in an exact quadratic model; nonlinear centered control remains open.
Most links stay close to their classical values. A connected set H of links outside the small-field chart is rare jointly, with probability at most
Joint rarity uses the stated convexity and moment hypotheses. With the barrier undifferentiated, §35 now controls compatible connected bad components at every order in its weak strip, retaining rarity and tree decay; §36 constructs exact dressed activities. Physical recoupling and complete source bound (113) remain open.
The quadratic background one-loop coefficients enter through the Killing form. Under the order-t calculation’s hypotheses their group factor scales by N/2, hence 3/2 for SU(3). This coefficient transfer does not transfer the full nonlinear estimates or iteration.
In the stated twisted SU(3) finite-sector test, clock and shift matrices give eight nontrivial adjoint phase pairs and remove that zero mode. Centre-sensitive exact bridge cuts occur at thirds; ordinary halving does not access a nontrivial SU(3) centre element.
Order-t note, §7b; small-field note, §16.
Halving the lattice walks toward short distances, while confinement and the mass gap live at long distances. The steps connect through the running coupling: the conditional one-loop matching gives a shift governed by b0 = 11N/(48π²) in four dimensions, and after many steps the lattice spacing, measured in physical units, is set by a scale that no step contains. Dimensional transmutation identifies a scale on a matched trajectory. Depth-uniform generated-action and remainder control are still open; so are continuum construction, nontriviality and a positive spectral edge with physical observable weight.
Didactic note; in Newton's problem the analogous scale is an action, and ℏ stays explicit throughout.
| Result and source | Level / remaining obligation |
|---|---|
| Additive cut action; mark supremum | Exact/refereed; Gaussian invariant mark hypotheses; finite optimum not attained |
| General instrument disturbance | Refereed quantum bound, supplied ℏ and explicit resources; independent positivity open |
| Positive cost iff optimized verdict continuous | Proved in the declared Gaussian family; physical uniform complete-record premise open |
| Physical copies, nonlinear countertests, phase memory | Scoped results; packet stability uses supplied quantum free evolution. Physical history rule and nonzero scale selection open |
| Series/parallel factorization; compact-group cuts | Exact identities; full interacting comparison is separate |
| All-order connected bad sets and dressing, §§35–36 | Refereed weak-strip results; physical recoupling and full nonlinear comparison open |
| Moving-mean sources, §37 | Exact centered quadratic test for the subtracted barrier part of (113); nonlinear extension open |
| Four-dimensional generated action | Gaussian composition and finite one-step inputs; depth-uniform nonlinear matching open |
| SU(3) continuum existence and physical gap | Open; strong-coupling lattice regimes proved under their criteria, T2′ at all couplings is stronger than required |
Read the current dependency paths, complete archive and obstruction map. The theorem compendium is an entry point; latest notebook sections govern subsequent progress.