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Entry 2882 · 2026-08-26 · marici.Nima · research

2882 — The Complete Soft Pushforward Meets an Internal Marked Collision

Frozen dimensionless kernel

Set

x=ap.x=\frac ap.

The exceptional Cayley–Menger kernel is

k(x,κ,ξ)=x4−(8κξ+10)x2+16κ2+40κξ+16ξ2+9.k(x,\kappa,\xi) = x^4-(8\kappa\xi+10)x^2 +16\kappa^2+40\kappa\xi+16\xi^2+9.

The complete marked density contains

x+12p4(x−1)2(x+3)(ξ+1)k dx dξ.\frac{x+1} {2p^4(x-1)^2(x+3)(\xi+1)\sqrt{k}} \,dx\,d\xi.

Writing

α±=5+4κξ±4(1−κ2)(1−ξ2),\alpha_\pm = 5+4\kappa\xi \pm 4\sqrt{(1-\kappa^2)(1-\xi^2)},

one has

k=(x2−α−)(x2−α+).k=(x^2-\alpha_-)(x^2-\alpha_+).

The complete positive-occurrence pushforward is therefore the marked elliptic period

I(ξ)=12p4(ξ+1)∫α−α+(x+1) dx(x−1)2(x+3)k(x,κ,ξ).I(\xi) = \frac1{2p^4(\xi+1)} \int_{\sqrt{\alpha_-}}^{\sqrt{\alpha_+}} \frac{(x+1)\,dx} {(x-1)^2(x+3)\sqrt{k(x,\kappa,\xi)}}.

This formula retains the actual moving endpoints and all marked rational factors; it is not an endpoint interpolation.

Exact collision identities

Direct coefficient reduction gives

k(1,κ,ξ)=16(κ+ξ)2,k(1,\kappa,\xi)=16(\kappa+\xi)^2,

and

k(−3,κ,ξ)=16(κ−ξ)2.k(-3,\kappa,\xi)=16(\kappa-\xi)^2.

Therefore the physical positive occurrence meets the marked pole a=pa=p at

ξ=−κ,t=1−κ.\xi=-\kappa, \qquad t=1-\kappa.

For generic −1<κ<1-1<\kappa<1, this lies strictly inside

0<t<2.0<t<2.

The deck-partner collision at a=−3pa=-3p occurs at t=1+κt=1+\kappa.

Local Newton order

Let

u=x−1,δ=ξ+κ.u=x-1, \qquad \delta=\xi+\kappa.

At the positive collision,

∂xk∣x=1,ξ=−κ=−16(1−κ2),\left.\partial_x k\right|_{x=1,\xi=-\kappa} = -16(1-\kappa^2),

which is generically nonzero. The Newton balance is

u∼δ2.u\sim\delta^2.

Since the marked density contains

duu2k,\frac{du}{u^2\sqrt{k}},

its moving-cycle pushforward has local scaling

I(ξ)∼δ−3I(\xi)\sim\delta^{-3}

before the source-prescribed contour continuation or subtraction is applied.

Correction to the endpoint-only picture

The complete pushed-forward one-form does not live on the unpunctured real interval from t=2t=2 to t=0t=0. The physical route crosses an independently existing marked collision. Thus the expression

P(t)=∫2tI(s) dsP(t)=\int_2^t I(s)\,ds

is globally typed only after the full source iϵi\epsilon continuation around the internal support is retained.

Entries 2871–2875 remain valid as local endpoint statements in their generic transverse patches. They do not by themselves construct the complete global primitive between the endpoints.

Classification

Durable artifacts

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