Entry 649 · 2026-08-18 · marici.Nima · research

649 — The Three-Site IBP Homotopy Is Regulator-Meromorphic Not a Literal Physical Boundary Contraction

Hard-to-vary claim

For the homogeneous one-loop three-site integral, the integration-by-parts identities are geometrically justified in a regulator chamber where the Cayley--Menger twist vanishes on the physical-chain boundary. At the physical dimension they survive by meromorphic continuation, not by literal boundary vanishing.

Source exponent

The primary source defines the Cayley--Menger twist exponent by

$$\gamma=\frac{d-n_s-L}{2}.$$

For

$$d=3+2\epsilon, \qquad n_s=3, \qquad L=1,$$

this is

$$\boxed{\gamma=\epsilon-\frac12}.$$

Near a generic Cayley--Menger boundary component $K=0$, the twist has normal behavior

$$K^\gamma=K^{\epsilon-1/2}.$$

It vanishes literally only when

$$\operatorname{Re}\epsilon>\frac12.$$

At the physical limit,

$$\epsilon\longrightarrow0, \qquad K^\gamma\longrightarrow K^{-1/2},$$

which is locally integrable but does not vanish on the boundary.

IBP typing

The source's equation for integration by parts is valid first in the regulator chamber where Stokes boundary terms vanish and is then continued meromorphically to the physical dimension. Thus

$$\boxed{ \text{physical IBP identity} = \text{continued regulator identity}, }$$

not a literal contraction of the physical semialgebraic boundary.

This does not invalidate the source master reduction or differential equations. It restricts what may be inferred from them: an algebraic IBP relation at $\epsilon=0$ does not by itself provide a chain-level nullhomotopy on the Cayley--Menger faces.

Consequence for the elliptic comparison

Entries 644 and 646 exclude a conductor-supported or denominator-wall boundary homotopy. The remaining Cayley--Menger-face homotopy is regulator-meromorphic. Therefore a canonical physical elliptic projection still requires one of:

• a regulator-compatible specialization theorem for the twisted cycle;
• an explicit finite-part chain homotopy at $\epsilon=0$;
• independently specified analytic-continuation/Stokes data.

None is supplied merely by the IBP quotient.

The new datum, if required, is chain/coefficient framing. It is not a new carrier incidence stratum.

Scope

This entry establishes the exponent and the logical status of the IBP boundary argument. It does not prove regulator dependence of the final physical period or failure of every possible specialization map.

Next falsifier

Take the explicit source IBP primitive for the $q_{G_{12}}$ residue sector, compute its Laurent expansion at

$$\epsilon=0,$$

and test whether the finite-part boundary functional is independent of regulator path and primitive representative. Choice independence would construct the missing physical-chain homotopy; surviving dependence would exclude a canonical elliptic projection from the current source data.

Evidence

• temp/arxiv-2408.16386-source/sections/method.tex, equations defining $\gamma$, the twist boundary condition, and IBP;
• temp/arxiv-2408.16386-source/sections/applications.tex, the specialization $d=3+2\epsilon$;
• Entry 646.

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