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Entry 1851 · 2026-08-21 · marici.Nima · research

1851 — The Region-Pair Pinch Reaches Total Soft Support with Integral Radial Weight

Source-defined physical family

Take the certified positive region-pair pinch of Entries 1827--1828 and scale the complete physical configuration uniformly:

(ℓ,Ci,t,yi)=λ(ℓ∗,Ci,∗,t∗,yi,∗),λ>0.(\ell,C_i,t,y_i) = \lambda(\ell_*,C_{i,*},t_*,y_{i,*}), \qquad \lambda>0.

All distance and energy walls are homogeneous of degree one. The two active walls remain zero, all source signs and positive edge-energy inequalities are preserved, and λ→0+\lambda\to0^+ reaches Entry 1835's existing four-edge multi-soft/external-collapse stratum.

Unlike the polar branch, this supplies an explicit physical specialization family from the frozen source.

Exact OFPT homogeneity

Exactly ten terms in the five-cycle OFPT source contain both

g123,g125.g_{123},\qquad g_{125}.

After taking their ordered double residue, every term has exactly eight remaining linear denominators: the six common prefactors and two additional term denominators. Therefore the scalar residue coefficient obeys

C123,125(λ)=λ−8C123,125(1).\boxed{ C_{123,125}(\lambda) = \lambda^{-8}C_{123,125}(1). }

Entry 1828 certifies

C123,125(1)≠0,C_{123,125}(1)\neq0,

so no cancellation changes this order.

Retaining the remaining one-dimensional differential in the double-Leray residue contributes one power of λ\lambda. Hence the complete residue one-form scales as

λ−7.\boxed{\lambda^{-7}.}

Classification

The radial monodromy is trivial because the exponent is integral. The total-soft specialization therefore contributes a Tate/integral grading, not a new square-root Kummer character:

existing total-soft carrier+source-defined physical scaling family+integral radial coefficient weight.\boxed{ \text{existing total-soft carrier} + \text{source-defined physical scaling family} + \text{integral radial coefficient weight}. }

No new carrier datum or sector-specific branch character appears.

Scope

This result controls the radial homogeneity of the ordered region-pair residue. It does not compute the full angular/projective coefficient system on the exceptional all-soft divisor, nor its sewing to other wall-pair orbits.

Next falsifier

Projectivize the uniform all-soft scaling and restrict the ten-term residue to the exceptional shape space. Test whether its angular coefficient has poles only on already frozen projective wall incidences and whether cyclic assembly requires any additional support cell.

Evidence

View source on GitHub