Mass parameters move into the geometry: deformed Amplituhedron fixes the double-box span and the two-loop walking dimension

The geometric description of four-point Coulomb-branch amplitudes extends from a two-parameter mass configuration to general external kinematics carried by a six-parameter embedding-space Gram matrix. The earlier deformed Amplituhedron covered one particular two-parameter mass configuration, with inequalities deformed at one and two loops to give that vacuum's loop integrands. At one loop the canonical logarithmic form of the resulting positive geometry is the box integrand; at two loops, under explicit copositivity conditions, the allowed logarithmic form is restricted to the span of the two planar double-box integrands.

What shapes the geometry is copositivity of the inverse Gram matrix, meaning the matrix takes non-negative values on non-negative vectors. In the massless case positivity of the momentum-twistor matrix already fixes the geometry at all loop orders; the deformed case needs extra conditions at two loops. Several algebraic intersections satisfy the one-loop positivity constraints yet violate two-loop mutual positivity, becoming residual strata whose union forces the canonical numerator to vanish on them.

In the three-parameter mass subfamily the box and double-box integrals are obtained analytically directly in four dimensions by finite differential equations. This subfamily admits a physical realization with all external particles of equal mass and massless propagators joining distinct loop lines, and the integrals are infrared finite in four dimensions. The one-loop result agrees with the Feynman-parameter representation to 45 digits at several Euclidean kinematic points; the double-box result agrees with explicit Feynman-parameter numerical integration to 13 digits and is given in Goncharov polylogarithms.

In the walking high-energy limit the logarithm of the proposed two-loop amplitude cancels quartic and cubic logarithmic divergences, leaving a double-logarithmic behavior set by the walking anomalous dimension. The one-loop contribution alone carries none of these higher scalings, so the cancellation requires an exact match between the two-loop amplitude and the square of the one-loop amplitude. The two-loop computation reproduces the earlier one-loop interpolation and fixes the two undetermined constants of the all-loop ansatz to be equal and to half the cusp anomalous dimension, supporting at two loops that the walking dimension is fully determined by the cusp and octagon dimensions.

The conclusions hold for four-point amplitudes in planar super Yang–Mills whose external kinematics satisfy the embedding-space orthogonality condition; explicit analytic integration is done only in the three-parameter mass subfamily, while the six-parameter geometry is treated at the level of constraining the integrand numerator. The two-loop double-box decomposition is assumed by extension from known configurations, with its normalization checked by infrared exponentiation.

The two-loop double-box decomposition and its relative normalization to the amplitude are assumed, with infrared exponentiation as support rather than independent proof; that the walking dimension is fully determined by the cusp and octagon dimensions is a conjecture raised by the equality of the two-loop constants. The polylogarithmic function space is treated as special to four-point massive kinematics, and whether it persists for generic six-parameter configurations is left open.