What survives refinement: the pion, the Yang–Mills gap and Newton’s action

Markdown source · PDF

Working conclusion, 2026-09-26. The common problem is to construct a limit while preserving a specified physical structure. In pure \(SU(3)\) Yang–Mills it is a positive energy threshold. In chiral QCD it is the broken global symmetry and observable spectral weight that require a zero threshold. In Newton’s comparison it is a positive action cost of physical records while the time partition becomes arbitrarily fine. All three require control independent of the regulator. The quantities controlled, and the reasons for their survival, differ.

User direction of this date makes this comparison the intended frame of the eventual paper. The two main proof goals remain the pure \(SU(3)\) mass gap and the necessity of a positive action scale in Newton’s comparison. QCD with massless and with nonzero quark masses is an orientation and a source of mechanisms, rather than a third full construction programme. This note supplies the definitions, two elementary limit criteria, and the plan. It proves neither main goal.

The Planck paper remains the developed Newton component; the Yang–Mills position and conditional theorem remain the technical map for pure gauge theory. Their results retain their stated hypotheses. The present note changes the destination of the synthesis.

1. Compare physical quantities, with the regulators separate

Use \(a\) for the spacetime lattice spacing, \(L\) for spatial box size, \(m_q\) for a renormalized quark mass, and \(\varepsilon=|\pi|\) for the largest step of a Newtonian time partition. Euclidean time extent is taken to infinity to select zero temperature. Write \(E_*\) for an energy threshold, \(M_*=E_*/c^2\) for a mass, and \(\mu_*=E_*/(\hbar c)\) for an inverse correlation length. This avoids the convention in older notes where \(m\) sometimes denotes an energy.

For an isotropic reflection-positive lattice theory, on a specified physical channel, normalize the transfer matrix by its vacuum eigenvalue and write

\[T_a=e^{-aH_a/(\hbar c)},\qquad \delta_a=\frac{aE_*(a)}{\hbar c}=a\mu_*(a).\]

A continuum theory with \(0<E_*<\infty\) therefore has \(\delta_a\to0\). A massless channel also has \(\delta_a\to0\). Their distinction is the limit of \(\delta_a/a\), together with convergence of the states and observables that detect it. A strong-coupling lattice gap at fixed bare coupling does not specify that limit.

The first three rows use a quantum theory with \(\hbar>0\) already in its definition. The last main goal asks how that positive action structure is justified. The pion supplies a useful check: positive \(\hbar\) and a dynamically generated strong-interaction scale can coexist with zero energy gap.

2. Pure Yang–Mills: retain coercivity through the limit

The target is a nontrivial four-dimensional \(SU(3)\) theory with the Wightman or Osterwalder–Schrader properties, and physical Hamiltonian \(H\) satisfying

\[H\Omega=0,\qquad E_{\rm YM}:=\inf\bigl(\operatorname{spec}H\setminus\{0\}\bigr) \in(0,\infty).\]

One chooses a vacuum sector and proves the corresponding vacuum and clustering properties. This target requires a gap above the vacuum; it does not prescribe every point of the spectrum above the threshold. The official statement explicitly couples existence, axioms and the gap (Jaffe–Witten, §§4–6, passage).

For the lattice route, fix a renormalized physical scale and tune \(g_0(a)\) toward zero. A sufficient spectral estimate has the form

\[H_{a,L}-E_{0,a,L}\ \ge\ E_-(1-P_{0,a,L}),\qquad E_->0,\]

uniformly along the trajectory in sufficiently small \(a\) and arbitrarily large \(L\), on the gauge-invariant Hilbert space. To pass this estimate to a continuum theory one still needs convergence of renormalized local observables, positivity, covariance, regularity and nontriviality. An upper estimate or a surviving finite-energy excitation prevents all non-vacuum states from escaping to infinite energy.

A corresponding correlation route is a bound \(C_O(t)\le K_Oe^{-E_-t/\hbar}\) for positive Euclidean time autocorrelations of a dense family of vacuum-subtracted local vectors. The constants must survive the cutoff limit for their chosen renormalizations. One rapidly decaying glueball correlator establishes only a channel statement. Reflection positivity and reconstruction are part of the route from correlations to the physical Hamiltonian; a Langevin relaxation rate by itself is a different object.

The existing H1/H2 route aims to deliver \(E_-=(\hbar c/a_*)\min(\gamma_*,\gamma')\). It remains conditional on the blocking and mixing estimates. Construction and nontriviality, called T4 in the obligations map, must remain explicit as well. We choose \(L\to\infty\) at fixed \(a\) followed by \(a\to0\) along the trajectory as the working order; interchanging these operations requires its own uniform estimates.

3. The pion: preserve the symmetry that requires a soft channel

Take \(SU(3)\) colour with two degenerate quarks, zero temperature, vacuum angle zero, and no electromagnetism. This fixes what is meant by the comparison with and without a quark mass. Additional flavours or electromagnetism would change the statement.

Massless case. Conditional on a local relativistic continuum theory and spontaneous breaking \(SU(2)_L\mathbin{\times}SU(2)_R\to SU(2)_V\), the Goldstone theorem requires three massless modes with nonzero axial-current matrix elements. These are the pion modes. The theorem’s symmetry-breaking premise is a dynamical obligation in QCD; the theorem does not establish it merely from the presence of fermions (Goldstone–Salam–Weinberg 1962, abstract; Leutwyler 1993, §§2–3, passage).

A possible order of limits for the per-flavour order parameter is

\[\Sigma_R=-\lim_{m_q\downarrow0}\lim_{L\to\infty}\lim_{a\to0} \langle\bar u u\rangle_{R,a,L,m_q}>0,\]

with a positive mass selecting the vacuum and the physical scale held fixed. The inner continuum limit is taken at fixed physical \(L,m_q\). An exactly symmetric finite-volume state has zero symmetry-breaking expectation at zero source; it cannot select the required vacuum by itself. Finite-volume zero modes and the infinite-volume order parameter must be distinguished (Damgaard–Fukaya 2009, introduction, passage). The displayed order is a proposed construction order, with existence at each stage an obligation.

Nonzero case. At fixed small positive \(m_q\), removing \(a\) retains explicit chiral breaking. In the conventional broken-phase low-energy description, put \(e_q=m_qc^2\) and let \(B_E>0\) have energy units. Then

\[E_\pi^2=2B_Ee_q[1+r(e_q)],\qquad r(e_q)\longrightarrow0 \quad(e_q\downarrow0).\]

This is the leading chiral mass relation, conditional on the broken phase and its controlled low-energy expansion; higher orders include chiral logarithms. In particular, \(a\to0\) at fixed \(e_q>0\) is different from \(e_q\to0\) (Leutwyler, Light Quark Masses, eq. (1), passage). We use this as a benchmark, without claiming a constructive proof of massive QCD or a lower bound on every channel from this formula.

There is a sharper identity underneath the expansion. Define \(\mu_q=m_qc/\hbar\) and \(\mu_\pi=E_\pi/(\hbar c)\), both inverse lengths. Assume the renormalized non-singlet Ward identity in compatible Euclidean conventions, away from contact insertions:

\[\partial_\alpha A^b_\alpha=2\mu_q P^b.\]

With pole amplitudes defined by \(\langle0|A^b_\alpha|\pi^d(k)\rangle=i\delta^{bd}{\cal F}_\pi k_\alpha\) and \(\langle0|P^b|\pi^d\rangle=\delta^{bd}{\cal G}_\pi\), its one-pion matrix element reads

\[\mu_\pi^2{\cal F}_\pi=2\mu_q{\cal G}_\pi.\]

The conventional continuation fixes the displayed phases. The equation is exact once the current identity and the pion pole exist; deriving the leading mass formula also uses the small-mass behaviour of the residues. A pole with nonzero residue must survive the limit. An unrenormalized divergent pseudoscalar susceptibility is insufficient: contact terms and ultraviolet divergences need separate control.

At finite \(a\) the regulator matters. A Ginsparg–Wilson Dirac operator has an exact lattice non-singlet chiral symmetry (Lüscher 1998, abstract). Wilson fermions instead require critical-mass tuning and a restored, renormalized Ward identity (Bali et al. 2019, §2, passage). Exact lattice chirality alone supplies neither spontaneous breaking nor a continuum construction.

For the final paper this comparison earns its place by testing the mechanism. A proposed pure-gauge estimate cannot be carried unchanged to chirally broken QCD if it gaps the pion channel. Colour gauge symmetry and global axial flavour symmetry have different roles. Supersymmetric cancellation of transverse zero-point energies, recorded in G07, is a separate mechanism from QCD’s Goldstone pions. Nor does a massless pion imply a massless coloured gluon.

4. Newton: retain a record cost while the polygon error vanishes

Use \(M\) for the body’s mass, constant force \(F\), horizontal speed \(v\), and comparison duration \(T\). The specified Galileo comparison is

\[q_I(t)=(vt,0),\qquad q_F(t)=(vt,Ft^2/2M),\] \[A(T)=\frac{vFT^3}{6M},\qquad {\cal A}(T):=\frac{3F}{v}A(T)=\frac{F^2T^3}{2M}=T\Delta E.\]

For a partition \(\pi\) of \([0,T]\), the action difference associated with its inscribed chord segments is

\[K_\pi=\frac{F^2}{24M}\sum_j(\Delta t_j)^3, \qquad 0\le K_\pi\le\frac{F^2T}{24M}\varepsilon^2\to0.\]

This follows by \((\Delta t_j)^3\le\varepsilon^2\Delta t_j\) and \(\sum_j\Delta t_j=T\). In the supplied quantum lift the phase difference is \(K_\pi/\hbar\), which also tends to zero at fixed \(\hbar>0\). The polygon note separately gives an ordering phase \(\Phi_\pi\) with

\[\hbar\Phi_\pi+4K_\pi=\frac{{\cal A}(T)}3.\]

Thus one geometric action defect vanishes while the ordering comparison retains a finite action at fixed \(T\). Reading that phase requires a relative-phase experiment. Its derivation already assumes the Weyl relations and positive \(\hbar\).

The proposed \(h\) gap concerns what a physical record must cost. For a fixed class \(\mathfrak M_\varepsilon\) of calibrated marks, define

\[\kappa_\varepsilon =\inf_{M_j\in\mathfrak M_\varepsilon}\delta_j\Delta_j,\]

where \(\delta_j\) is reading error and \(\Delta_j\) impulse uncertainty. The class, available preparations, apparatus resources and meaning of the uncertainty must be specified, with feasible informative records of finite cost retained in the limit. A restricted theorem might establish \(\inf_{0<\varepsilon<\varepsilon_0}\kappa_\varepsilon\ge\kappa_*>0\). The general-instrument formulation instead bounds the disturbance functional defined in the disturbance note. These are distinct formulations of record cost; a single-mark product does not by itself establish a universal protocol bound.

The current paper proves such conditional bounds using quantum kinematics or an assumed positive mark trade-off. The main necessity goal is to supply independently justified physical premises that force a positive scale and its universality, without placing that scale in the premise. Identification with \(h=2\pi\hbar\) is a further calibration obligation. Newton’s optical fits provide a candidate product \(\Lambda p\); the paper’s §8 keeps its extra indeterminacy and cross-colour universality premises visible.

Two refinement operations must stay separate: making more marks within fixed \(T\), and asking each shrinking cell to decide the force on its own. The Gaussian result bounds the latter decision window; it permits marks much denser than that window. A positive action cost is compatible with continuous time and with convergence of the polygon. It is also compatible with zero energy gap, as the quantum free particle illustrates.

Gap thesis (user, 2026-10-01). The Newton goal is now stated as a gap in the space of theories: Newton’s mechanics exists only with a positive scale and no zero branch is continuously reachable. The reachability note locates it. The Gaussian comparison above is immune: with Gaussian preparations and marks the record family is the classical Gaussian process plus kicks of variance \(\hbar^2/(4\sigma_j^2)\) for every \(\hbar\) (its Theorem 1). The gap lives in complete records of determinate swarms that fold: there, wherever streams cross, the record has an \(\hbar\to0\) limit only if signed cross amplitudes cancel, while smeared records converge to the classical branch sum (Theorems 2–3), and in the anomaly record Hooke keeps the zero branch for all time while a Kepler swarm loses it once its windings overlap, at the lapping time of order \(T^2/\Delta T\) for a narrow swarm spanning more than half a turn (Theorem 4). The object discontinuous at zero, in the place the mass holds in §2, is the leading fringe visibility \(2\sqrt{\rho_1\rho_2}/(\rho_1+\rho_2)\) of crossing streams, fixed by the classical branch densities, the same along the family \(\hbar>0\) and zero at \(\hbar=0\); the dilation \(\hbar\mapsto\mu\hbar\) makes it a contrast, not a spectral gap, and it generates no scale, where the Yang–Mills gap is a scale (refereed by GPT-6.1 Sol (Codex), 2026-10-01, corrections applied). The order of limits is the same as in §2: the record limit, resolution to zero, is taken before the constant’s, as the continuum limit is taken before the coupling’s. For the paper this moves Newton’s gap argument from the Gaussian parabola to the Kepler swarm and makes its two premises, complete records and determinate preparations, the hypotheses to state beside coercivity in §2 and the soft channel in §3.

5. Two elementary criteria for carrying spectral information to a limit

These statements are standard consequences of the spectral theorem and weak convergence of positive measures, included to state exactly what the comparison needs. They are written derivations, with no numerical test and no claim of novelty.

Proposition 1 (a surviving lower bound). Suppose renormalized, vacuum-subtracted local vectors \(O_n\Omega_n\) have finite positive energy spectral measures \(\nu_n\) converging weakly to \(\nu\) in a reconstructed limiting theory. If every \(\nu_n\) is supported in \([E_-,\infty)\) for one \(E_->0\), then so is \(\nu\).

Proof. Every nonnegative continuous test function compactly supported in \([0,E_-)\) integrates to zero against every \(\nu_n\), hence against \(\nu\). Such test functions exhaust the interval. If the statement holds on a dense family of local vectors orthogonal to the vacuum, the spectral projection of \(H\) onto \((0,E_-)\) is zero. A nonzero limiting measure supplies surviving finite-energy spectral content. \(\square\)

For positive autocorrelations \(C_n(t)=\int e^{-Et/\hbar}\,d\nu_n(E)\), a uniform physical decay rate is another way to obtain the support condition. The normalization of the observables and their nonzero limit are part of the hypotheses.

Proposition 2 (a surviving zero threshold). Suppose instead that \(\nu_n\Rightarrow\nu\), that \(\nu(\{0\})=0\), and that for every \(\eta>0\) there is \(w_\eta>0\) such that, for all sufficiently large \(n\),

\[\nu_n([0,\eta/2])\ge w_\eta.\]

Then \(\nu((0,\eta))>0\) for every \(\eta>0\), and the channel’s excited energy threshold is zero.

Proof. The closed-set inequality for weak convergence gives

\[\nu([0,\eta/2])\ge\limsup_n\nu_n([0,\eta/2])\ge w_\eta.\]

Removing the zero atom leaves positive weight in \((0,\eta)\). \(\square\)

The no-atom hypothesis distinguishes arbitrarily soft excitations from an extra vacuum contribution. It belongs to a selected pure vacuum and a connected observable. The chiral comparison suggests obtaining the required surviving weight from an axial Ward identity and a nonzero order parameter, with the Goldstone theorem supplying the continuum implication.

Why convergence of eigenvalues alone is too weak. For \(E_0>0\) let \(\mathrm d_x\) denote unit point mass at energy \(x\), and set

\[\nu_n=(1-e^{-n})\mathrm d_{E_0}+e^{-n}\mathrm d_{E_0/n}.\]

Every \(\nu_n\) has lowest energy \(E_0/n\to0\), yet \(\nu_n\Rightarrow\mathrm d_{E_0}\). The soft state’s observable weight disappears. Conversely, a positive gap at each \(n\) can collapse with nonvanishing weight. These examples identify the estimates a continuum argument must actually carry. Compact \(U(1)\) gauge theory in three dimensions realizes the collapse as a theorem: every lattice has a positive Debye mass, and at fixed coupling the continuum limit is the massless free field, so the uniform \(E_-\) of Proposition 1 fails; the series/parallel note, §4 traces the failure to the error density of one refinement step in physical units.

For Newton the analogous surviving object is a calibrated record and its information about the comparison. No spectral measure is supplied by classical mechanics for this purpose. The useful transfer is the proof discipline: compatible observable limits, nonvanishing response, and a bound uniform over admitted refinements. Any stronger equivalence requires a constructed mathematical map.

6. The mechanism worth developing across the two main goals

The action-floor note G08 supplies one concrete bridge. A transverse oscillator of frequency \(\Omega(x)\) obeys

\[\frac{p_y^2}{2M}+\frac12M\Omega(x)^2y^2 \ \ge\ \frac12\hbar\Omega(x).\]

When \(\Omega(x)\) grows along a classically escaping valley, this all-state operator bound can confine that valley. Its force comes from the action scale together with the non-abelian commutator potential. The solved matrix models establish the mechanism in a specified Hamiltonian; extension to four-dimensional gauge theory requires uniform control of the additional modes and renormalization.

The pion comparison asks which flat directions an interaction is allowed to lift. A broken exact global symmetry protects Goldstone directions. Pure Yang–Mills has no corresponding axial flavour symmetry. The absence of that protection makes lifting possible; the positive bound still has to be proved. Finite quark mass then supplies a controlled way to lift the protected direction, which is why both pion cases are useful as orientation.

The candidate unifying question is therefore: which structures enforce an irreducible cost, which protect a soft direction, and which survive the removal of the regulator? Quantum kinematics supplies an action unit to both QCD theories. Their different spectra depend on dynamics and symmetry. A successful Newton necessity argument must explain the action structure at the preceding level.

7. Work toward the final paper

The intended paper can be called What survives refinement: action, symmetry and the continuum. Its architecture is: Galileo’s comparison and the three regulators; the pion benchmark with and without explicit breaking; physical spectral limits and reconstruction; the established Newton record bounds; the valley-lifting bridge; and the two remaining proof obligations. Historical claims stay tied to the held source companions and the sibling newtonlean work. The present Planck paper becomes a component of this synthesis. Since 2026-10-01 the Newton component is organized around the gap thesis of §4: the Kepler swarm as the carrier, the Gaussian parabola as the immune case in which the Planck paper’s record bounds live, and the two premises, complete records and an \(h\)-independent coherence length between the launch points of crossing streams, stated beside coercivity and the soft channel. This working architecture does not designate either open goal as solved.

The research order is deliberately bounded.

1. Next constructive step: the \(SU(3)\) small-volume bridge. Develop the H3 estimate for the zero-mode Hamiltonian plus the even torus valley potential in the Feshbach note. Seek an explicit lower bound \[E_{\rm small}(L)\ge (d_3-Cg(L)^{2/3})g(L)^{2/3}\frac{\hbar c}{L},\qquad d_3>0,\] with an explicit admissible coupling interval. State the operator, gauge sector and cutoff dependence. A confining potential or a positive ground energy alone does not establish the excitation gap; the first excited energy and the ground energy need comparison. Stop at one proved estimate, or the precise uncontrolled term in that estimate. Fixed-cutoff H3 would be a result, with its boundary retained; it would not close H1/H2 or T4 of the full mass-gap map.
2. Newton necessity: construct the missing physical premise. Use one explicit model of records and their composition, with calibrated lengths, impulses and time. Seek a refinement-independent positive cost from dynamical or optical premises that have their own justification. The deliverable is one implication with its full admissible class, or an explicit identification of the additional premise needed. A result starting from the canonical commutator or M3 belongs to the existing conditional branch. For the optical route, observable relative phase and a common action unit across probes must be established separately. Attainment of the existing disturbance bounds remains useful, but secondary to this task.
3. Use the pion as a bounded test of a proposed mechanism. Carry a non-singlet Ward identity and its nonzero residue through the chosen regulator, or state these as explicit inputs. Recover the zero-mass case and its small-mass lifting. This is one comparison section; a full fermionic-QCD construction is outside the active queue.
4. Return the successful bridge to the field theory. Determine whether the small-volume estimate survives eliminating nonzero modes with constants uniform in \(a\), and then whether it supplies the order-one-coupling blocking estimate missing from H1. Preserve the chosen vacuum, physical scale and observable normalization. Reconstruction and nontriviality stay separate obligations; their order may interact with the estimates rather than follow a single irreversible chain.
5. Integrate proved results into the final synthesis. Replace the corresponding conditional statements only when their hypotheses are discharged. Complete the historical reading obligations in the Planck paper before submission. Keep the pion’s role explanatory unless it actually changes one of the two proofs.

Status on 2026-09-29 (pointers only). The active gauge route since 2026-09-27 is the ultraviolet halving construction organized by the halving atlas, rather than item 1’s small-volume bridge, which did not move this week. Its cell 2 now has one normalized small-field step for \(SU(2)\) in \(1+2\), uniform in plane size (small-field note, §§13–14), the covariant link-kernel estimate conditional on two local jet bounds (§20.3), a reduction of the third response and an all-order polymer criterion (§21), and a transcription to \(SU(3)\) (§16; order-\(t\) coefficients in the order-\(t\) note, §7b). For item 2, the missing premise has an equivalent Newton-age form, Leibniz’s law of continuity read on records (continuity note), and the cut measure coincides with the record measure at every finite stage (cut measure, Proposition 7); the premise itself remains open.

8. Consequence for STATE

The destination becomes a joint paper on survival under refinement, with the Newton action necessity and pure \(SU(3)\) continuum mass gap as the two main goals. Pion physics, at zero and nonzero quark mass, is the symmetry and limit-order benchmark requested by the user. The next bounded constructive task is the H3 small-volume bridge; the Newton necessity task retains the independent-premise requirement. The three review batches of the existing Planck paper are complete; attainment and its submission obligations remain open. This note is the live plan, and adds no claim of a four-dimensional construction or an independent derivation of \(h>0\).