Newton–Yang–Mills correspondence test

2026-10-01. An exact formula in each theory is not automatically an exact identification between the theories. The classifications below keep the objects and the map explicit. Sources retain their stated reading levels; this document introduces no new physical premise.

1. Exact common composition at the Gaussian/semigroup level

The Newton free kernel and a group heat kernel each obey Chapman–Kolmogorov: K_(t1+t2)(x,z)=∫K_t1(x,y)K_t2(y,z)dy. For gauge variables the integral is normalized Haar convolution, k_t1*k_t2=k_(t1+t2). In a linear Gaussian model, integrating an inserted variable gives a Schur complement; variances in series add and precisions in parallel add. These are exact common composition laws, with different state spaces and normalizations.

Sources: refinement composition, series/parallel, Gaussian blocking. Boundary: Newton’s deterministic trajectory insertion is not itself a heat semigroup or a measure on gauge connections. Compact-group series closure does not close the interacting transverse-plane integral in D>2.

2. Gaussian records versus parallel gauge faces: exact algebra, structural transfer

For a Gaussian quantum position instrument, forgetting y gives ρ(x,x′)↦ρ(x,x′)exp[−(x−x′)²/(8σ²)]. Its Wigner transform is exactly momentum convolution with v=ℏ²/(4σ²). Two simultaneous records satisfy σ_eff^−2=σ1−2+σ2^−2 and v_eff=v1+v2. Abelian Gaussian parallel-face weights multiply, so their inverse heat times add. Thus the precision algebra is exactly the same algebra, after naming the covariance/precision variables.

Source: Newton record move, §1–2, and the gauge Gaussian calculation in series/parallel. Boundary: a trace over a quantum pointer is a completely positive channel; a gauge mid-plane marginal is a Euclidean integral with nonabelian generated interactions. No constructed functor identifies these full objects. The Wigner representation may be signed, and ℏ is already supplied. Call the physical comparison a structural analogy, not a derivation of either goal.

3. Additive cut action versus generated interactions: structural only

For constant force, Kτ=F²τ³/(24m) and insertion at fraction s gives Kτ−Ksτ−K(1−s)τ=3s(1−s)Kτ. The general Dirichlet action decomposes into orthogonal Schauder-hat costs. This is an exact nonnegative additive measure on cuts of the specified classical path.

Eliminating gauge variables gives S_eff(U)=−log∫exp[−S(U,V)]dV, up to normalization. Connected cumulants generate terms supported on multiple plaquettes; they need not be nonnegative or a finitely additive cut measure. Sources: cut measure, 4D composition, small-field dressing, §36. The common question is the price of eliminating variables. There is no proved identification of the cut measure with all generated gauge couplings.

4. Universality versus dimensional transmutation: distinct mechanisms

Independent mass composition in the Brownian model yields mD=κ; compatible interacting rotation generators give a common action coefficient. Neither argument forces κ>0. Four-dimensional perturbative running instead has d(1/g²)/d log μ=2b0+… and supplies a matched scale of the form Λ≈a^−1 exp[−1/(2b0g²)], with further matching corrections.

Sources: stochastic route, rotation composition, parallel logarithm, UV/IR. The Newton coefficient has action units and may be zero. Λ has inverse-length units, and a physical positive gap needs a surviving positive coefficient and observable weight as well as a continuum construction. These are structurally useful scale questions, not the same selection theorem.

5. Zero action versus zero gap: structural diagnostic

The classical zero-action branch is an admissible law under several Newton consistency premises. A gauge zero gap is a statement about a reconstructed Hamiltonian’s spectral edge after volume/cutoff limits; a vanishing lattice gap can also be the correct scaling of a positive physical gap. Sources: fifth postulate, three continuum limits, lattice obligations. The common diagnostic is “which hypothesis excludes zero?” The zero objects and their limits differ. Goldstone protection in the pion comparison supplies an additional symmetry mechanism, not an action-normalization statement.

6. Valley lifting: a conditional mechanical implication

In the x²y² matrix-mechanical model a supplied transverse phase-space-area floor bounds excursions along the valley at fixed energy. The operator uncertainty argument supplies quantum confinement in the corresponding Schrödinger model. Sources: low-dimensional gap, action-floor model. This is a useful model-level conditional connection. Semiclassical phase-space counting does not prove a four-dimensional gauge gap. In D=4 a mass/energy gap and c alone do not reconstruct an action unit.

7. Tangent groupoid and “all zoomings”: exact ingredients, open synthesis

The tangent groupoid relates pair-groupoid arrows at nonzero scale to tangent vectors at zero. Time-graded trajectory refinement and the harmonic arbitrary-partition construction have their own exact composition laws. Quantum diffusive paths and Newton’s ballistic ultimate-velocity fibre have different scaling. Sources: groupoid trajectories, refinement §§3c–3d. A gauge double-groupoid organizing parallel moves is proposed, not proved. Use the language when it names state, composition and topology. The rhetorical claim that all these limits are one mechanism adds no theorem and should be omitted from the main narrative.

Consequence for the book

Keep exact insertion and state retention in the common opening part. Give Newton recording and gauge effective-action construction separate technical parts. Reunite them through explicitly scoped scale and spectral questions in the final part. This is the structure adopted in the book map.