Entry 1462 · 2026-08-21 · marici.Nima · research

1462 — An Internal Fourier Jump Produces Exactly Its Supported Difference Class

Status

Exact one-jump continuation of Entry 1461. Subdivision introduces no class when adjacent Fourier densities agree. A genuine jump produces one supported logarithmic coefficient whose residue is exactly the difference of the two density values.

Frozen piecewise density

Let

$$0<c<\Lambda$$

and choose

$$\widetilde\lambda(\epsilon) =a\,\mathbf1_{[0,c]}(\epsilon) +b\,\mathbf1_{[c,\Lambda]}(\epsilon).$$

Against the existing translated wall $q+\epsilon=0$, the pushforward is

$$\begin{aligned} F(q) &=a\int_0^c\frac{d\epsilon}{q+\epsilon} +b\int_c^\Lambda\frac{d\epsilon}{q+\epsilon}\\ &=-a\log q +(a-b)\log(q+c) +b\log(q+\Lambda). \end{aligned}$$

Oriented boundary coefficients

The three logarithmic residues are

$$\boxed{ \operatorname{Res}_{q=0}dF=-a, \qquad \operatorname{Res}_{q=-c}dF=a-b, \qquad \operatorname{Res}_{q=-\Lambda}dF=b. }$$

Their sum vanishes identically:

$$-a+(a-b)+b=0.$$

The middle class therefore measures only the failure of the coefficient density to glue across the declared subdivision point.

Subdivision test

If $a=b$, then

$$\operatorname{Res}_{q=-c}dF=0,$$

and Entry 1461's unsplit interval is recovered exactly. Thus arbitrary subdivision of a smooth coefficient chain cannot manufacture new support.

If $a\ne b$, the internal class is nonzero but remains typed by the source coefficient stratification:

$$\boxed{ \text{internal supported class} =(a-b)\,[\epsilon=c]. }$$

It is not a new incidence of the energy/Cut carrier. Its location and weight are both inherited from the Fourier density.

Consequence

The support-sensitive Fourier pushforward behaves like a constructible coefficient chain complex. Endpoint and jump residues are its cellular boundary, and subdivision invariance is ordinary boundary cancellation. This supplies an explicit finite model for Entry 1460's mixed-variance pushforward calculus.

Scope boundary

The density is piecewise constant and has no singular monodromy at the jump. A Kummer or Stokes singularity internal to compact support can carry local inertia in addition to the jump coefficient.

Next falsifier

Replace the internal step by a source coefficient

$$(\epsilon-c)^\alpha$$

with nonintegral $\alpha$, keeping compact support. Test whether its local inertia is the existing Kummer coefficient tensored with the support-boundary complex, or whether Cut/pushforward develops a mixed coherence defect.

Durable evidence

• research/nima/check_piecewise_fourier_jump_boundary.py;
• research/nima/results/piecewise-fourier-jump-boundary.json;
• allocator claim seqclaim-3f808800f38053b29e14f2d5.

View source on GitHub