The palindromic branch on The Broken-Born program

The palindrome βₙ = β₄₋ₙ

Mon, 01 Jan 0001 00:00:00 +0000

The palindrome is the constraint βₙ = β₄₋ₙ on the five coefficients β₀ to β₄ of the potential that gives the program’s second spin-2 field its mass: β₀ = β₄ and β₁ = β₃, with β₂ free, so the list of coefficients reads the same forwards and backwards. The potential couples two frames, one built from the dynamical geometry and one determined by the polarization that the vacuum selects, and exchanging the two frames reverses the order of its five terms. The program’s claim is that Born reciprocity, on the broken vacuum, acts as exactly this exchange, so a potential invariant under reciprocity must be a palindrome. The constraint places the program’s gravity sector in the interchange-symmetric branch of bimetric gravity and reads that branch’s two-element exchange symmetry as Born reciprocity. It is a derivation of the program [D] with no machine check, and the branch itself is known in the literature, where it contains the partially massless point.

The vacuum-energy identity V(1) = 2T

Mon, 01 Jan 0001 00:00:00 +0000

The vacuum-energy identity is a short algebraic result about the program’s spin-2 interaction potential once its coefficients obey the palindrome βₙ = β₄₋ₙ. At the aligned point, where the potential’s two frames coincide, the value of the potential, V(1), is exactly twice its tadpole T, the coefficient of the term linear in a small departure from that point. A flat aligned vacuum separately requires V(1) = T, and the two equations together force T = 0 and V(1) = 0, so the spin-2 interaction sector adds no vacuum energy there. Once the palindrome holds, that single flatness condition does the work of the two tunings that generic bimetric gravity needs. The integer arithmetic is machine-checked. The identity says nothing about the vacuum energy of matter loops, the “old” cosmological-constant problem, and whether it is new is the subject of an open literature comparison.

Open: the partially-massless comparison (OP2)

Mon, 01 Jan 0001 00:00:00 +0000

The partially-massless comparison is a pending literature check on the two results of this section, the palindrome and the vacuum-energy identity. A live literature search found that the palindromic branch contains the partially massless point of bimetric gravity, a parameter point with an extra gauge symmetry, and that Hossenfelder’s exchange-symmetric bimetric theory already cancels vacuum energy by a different mechanism. The comparison asks whether the identity is new, whether it holds up against that literature once conventions are aligned, and whether reciprocity is the origin of the partially massless symmetry. Until it is done, the identity is presented as adjacent to known results rather than as new. It is open problem 2 of the checkpoint and part of the first item of the current work queue, and no calculation for it exists yet.