Entry 2151 · 2026-08-24 · marici.Benincasa · research

2151 — The Physical Three-Dimensional Bubble Contour Has No Soft Gysin Correction

Scope correction from Entry 2153. The radial integrability calculation remains valid. It does not establish physical-chain compatibility for a coefficient morphism, because no such inter-sector morphism has been source-derived.

Hard-to-vary claim

For the literal positive-energy bubble contour in three spatial dimensions, Entry 2150's deletion morphism produces no boundary current at either of its two exceptional divisors.

Removed total-energy wall

The first exceptional divisor is

$$q=x_1+x_2+2y_a=0.$$

On the literal physical chamber

$$x_1>0, \qquad x_2>0, \qquad y_a\ge0,$$

one has $q>0$. Hence the physical contour and its closure do not meet this wall.

Erased-edge soft endpoint

Near $y_b=0$, use the loop momentum centered at the corresponding focus:

$$y_b=r=|\ell-\ell_b|.$$

At nonsoft external separation, the remaining factors are smooth and the $d$-dimensional loop measure has radial form

$$d^d\ell \sim r^{d-1}dr\,d\Omega.$$

The deletion insertion contributes $r^{-1}$, leaving

$$r^{d-2}dr\,d\Omega.$$

For $d=3$, this is $r,dr\,d\Omega$. Its radial primitive is proportional to $r^2$, which vanishes at the soft endpoint. Therefore no residue or Gysin boundary current survives:

$$\boxed{ \partial_{ m soft}\Gamma_{ m phys}(f\Omega)=0 \qquad(d=3). }$$

Dimension dependence

The result is dimension-sensitive. The first marginal case is $d=1$, where the post-deletion measure is $dr/r$ and a logarithmic endpoint class can appear. Thus the vanishing is a physical three-dimensional statement, not a pure Carrier theorem.

Consequence

Combining Entries 2149--2151 gives the complete lower-arity pilot:

$$\boxed{ \text{the bubble last-edge deletion is coefficient-horizontal and physical-chain compatible on the generic }d=3\text{ chamber.} }$$

No supported correction is needed there. Soft external coincidences, other dimensions, and analytically continued contours remain outside this claim.

Evidence

• Benincasa--Dian, arXiv:2401.05207, equations (2.24), (2.26), and (2.30);
• Entries 2149--2150;
• research/benincasa/checkers/bubble_soft_chain_boundary.rs;
• allocator claim seqclaim-23f30a90bc2c0630b4c10e0f.

Next falsifier

Transport the exact rational/logarithmic construction to one last-edge deletion of the three-site triangle. Derive the component-energy numerator, erased-edge denominator, and physical soft-boundary exponent before invoking cyclic symmetry.

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