The palindromic branch

The palindromic branch is the family of Hassan–Rosen bimetric potentials whose five coefficients read the same forwards and backwards, βₙ = β₄₋ₙ. The program arrives there because Born reciprocity, on the broken vacuum, acts as an exchange of the potential’s two frames, and a potential invariant under that exchange must be palindromic. Inside the branch a four-line algebraic identity shows that a flat aligned vacuum gives the spin-2 interaction sector zero vacuum energy. Both results stand as program derivations. Whether the identity is new is open: the branch also contains the known partially massless point, and a different cancellation of vacuum energy by exchange symmetry is already in the literature.

Stands: the palindrome βₙ = β₄₋ₙ [D], the vacuum-energy identity V(1) = 2T [D][V] · Superseded: folded sections on the ad hoc branch given a reason to exist and the standalone-result claim · Died: nothing yet · Open: the partially massless comparison (OP2)

Map#

The palindrome βₙ = β₄₋ₙ comes first. It takes the frame-tier potential that passed the tuning test, a Hassan–Rosen potential of two frames with five coefficients, and shows what reciprocity does to it: on the broken vacuum reciprocity exchanges the two frames, the exchange reverses the order of the five terms, and an invariant potential must have βₙ = β₄₋ₙ. The page sets out the potential and the standard algebra of the exchange, calc note #8’s one-sentence argument that reciprocity is the exchange [D], and what follows: the interchange-symmetric branch, the parameter count, the parities and the equal Planck masses. The branch itself is known in the literature, where it contains the partially massless point. #8’s early claim that the literature had treated the branch only as “an ad hoc simplification” is kept there as a superseded section.

The vacuum-energy identity V(1) = 2T follows from the palindrome at the aligned point, where the two frames coincide. The page works through #8’s four lines: the values of the elementary symmetric polynomials at the unit matrix and their linear coefficients, the identity V(1) = 2T, the flatness condition V(1) = T, and the conclusion T = 0 and V(1) = 0. It is tagged [D][V]; battery tests T5 and T6 check the arithmetic (T5 in exact integers, T6 numerically), not the palindrome or the flatness condition. Its scope excludes the vacuum energy of matter loops (the cosmological-constant problem), and the page lists the result’s earlier names, from “Λ theorem” to “identity”.

Open: the partially-massless comparison (OP2) closes the section. It is the literature check that the identity’s novelty waits on: a convention-careful comparison with partially massless bimetric gravity [24S], whose parameter point lies inside the branch at α = 1; with Hossenfelder’s exchange-symmetric bimetric theory [25S], which cancels vacuum energy by a different mechanism; and with the Schmidt-May–von Strauss review’s no-protecting-symmetry statement [22S]. It also asks whether reciprocity is the origin of the partially massless symmetry. It is open problem 2 of the checkpoint and part of item 1 of the current queue, and the earlier “standalone result” claim is kept there as a superseded section.

All three are program pages, the last step of the program’s results on gravity and the momentum graviton. The established physics they use, Hassan–Rosen bimetric gravity, elementary symmetric polynomials, partially massless gravity and the cosmological-constant problem, is in Spin-2 fields.

Boundaries#