The momentum graviton X

The momentum graviton X is the real part of the modular field Z = X + iY: a symmetric tensor field that sits beside the spacetime metric g = Im Z and describes how the split between position and momentum tilts from place to place. In this program it carries the geometry of momentum space. Its symmetries make it a second graviton-like field; a theorem excludes a second massless graviton that interacts with the first, and a symmetric tensor without its gauge symmetry has a ghost, so X must be massive. It acquires the mass when the vacuum selects a polarization, by a gravitational version of the Higgs mechanism, while the dark-energy field τ stays light. Ordinary matter feels X at gravitational strength but only over a short range, which the program gives as the reason it has not been seen. These are the program’s own structural derivations, resting on standard theorems; none of the chain is machine-checked.

Stands: shift symmetry, dual diffeomorphisms, the symplectic partner of the graviton, why X must be massive, the gravitational Higgs mechanism, τ is not eaten (at leading order), spectrum and parity, safety by range, Born’s answer · Superseded: no page of its own here; folded sections on the three-way classification of X, no second graviton under the original P2 and the v1 count of two standalone results, and the earlier forms kept on the spectrum, safety and Born’s-answer pages · Died: nothing yet · Open: τ–Stückelberg mixing (OP6), stability at equal Planck masses · Conjectures: two gravitational charges and the conjugate field equations (chat-only); popsci-only, folded: a “forty-year asymmetry”

Map#

Every page in this section is the program’s own material, not established physics: the results of what the program calls Debt 1, the missing field equation for the geometry of momentum space (the three debts). Most were derived in the calc notes notes/debt1a-ii-X-classification.md and notes/debt1a-iv-beta-coefficients.md and are restated in checkpoint #11 §7, which tags the chain [D + T] (§7.1) and the spectrum [D] (§7.5). The established spin-2 physics they rely on, from general relativity to Hassan–Rosen bimetric gravity, is in Spin-2 fields. The pages follow the chain of #11 §7.1 and then its consequences.

The chain comes first, four pages that fix what kind of field X is.

  • Shift symmetry: X is never auxiliary starts from the group of the rigid symplectic form, which contains the translations Z → Z + B. Constant shifts of X are therefore exact symmetries of any geometric action built from ω and H, so X can enter only through its derivatives: no potential, no mass term, no coupling without derivatives. The page explains why a shift symmetry forbids those terms, why it excludes the auxiliary outcome #5 had defined (and what that outcome would have given the dark-matter gate), and what it implies for the mass. It is the one step calc note #6 marks rigorous.
  • Dual diffeomorphisms: a second graviton makes the shifts local, δX_μν = ∂₍μλ_ν₎. After the background of linearized diffeomorphisms and the graviton [T], it shows why X’s symmetry type decides the matter: a symmetric field with this gauge structure is a second graviton-like field. It records what follows (both ways out closed, the symmetry broken by the vacuum) and that no document writes an action in which the symmetry can be checked.
  • The symplectic partner of the graviton is another graviton sets the program’s doubling beside double field theory’s. The fixed pairing decides the partner’s symmetry type, as a short standard computation shows: a symmetric pairing η admits an antisymmetric partner, the harmless B-field, and an antisymmetric pairing ω a symmetric one, a second graviton. The round-3 audit sampled both families with its own code. #11 §15 lists this as the one observation that stands regardless of the program’s fate; folded sections keep the v1 paper’s superseded count of two such results and a popular article’s “forty-year asymmetry”, a dating no technical document makes.
  • Why X must be massive assembles the chain. With its gauge symmetry, X is a second massless graviton, excluded by the multi-graviton no-go because X must couple to the metric and to dark matter; without it, the signature obstruction leaves a ghost. Mass is the exit, and the answer is “massive spin-2, conditionally healthy”, the condition being the tuning test. Folded sections record the superseded three-way question (healthy, auxiliary or ghost) that the chain answered, and the lab notebook’s v0.1 claim that there would be no second graviton at all.

The mass comes next.

  • The gravitational Higgs mechanism says where it comes from. After the Higgs mechanism and its Stückelberg form [T], it follows #6 step by step: no mass term by hand, the vacuum at X = 0, the shifts moving it, the would-be Goldstone modes absorbed, and M_X set by the polarization-breaking scale, with no value in any source. It records the consistency condition this leaves for the tuning test, the two unreconciled wordings of what X absorbs, and what follows from the mass.
  • τ is not eaten shows that the dark-energy field survives the mechanism as the light field, at leading order [D — leading order]. τ is a coordinate on the compact space of polarizations, while the absorbed modes belong to the gauge parameter of the shifts of X; the argument rests on that difference of origin. The page says what “leading order” leaves out and why τ’s lightness matters.
  • Open: τ–Stückelberg mixing (OP6) is what leading order leaves out: how strongly τ mixes with the absorbed modes through the potential. Calc note #8 logged it as expected small by compactness; no document computes it. It is item 6 of #11 §14 and not on the current work queue.

The consequences follow.

  • Spectrum, parameters and reciprocity parity counts what reciprocity leaves of the potential: the palindrome removes two of five coefficients and a flat aligned vacuum a third, leaving M_X and one cubic-shape parameter. It sorts the two spin-2 fields: the reciprocity-even combination is the exactly massless graviton of prediction row 6, and X is odd and massive, with equal Planck masses forced by the same symmetry. The parities assume reciprocity acts as the exchange of the two frames, and the page records #11 §6’s other action, Z → −Z⁻¹, under which the metric’s fluctuation changes sign too (flag 39 in review/wiki-build-flags.md). A folded section keeps calc note #7’s superseded reading of the graviton-mass bound as a mixing-angle check.
  • Safety by range: X and ordinary matter explains why ordinary matter shows no sign of X: it feels X at gravitational strength, but only within a Yukawa range of order 1/M_X. The page sets out the argument step by step, its scope (no experimental bound and no value of M_X cited), the open dual-stress enhancement beside it, and the three earlier forms the argument took across the Debt-1 notes.
  • Open: stability at equal Planck masses asks whether a heavy spin-2 field is stable in an expanding universe when reciprocity makes the two Planck masses equal, which closes the escape the cited stability results rely on. It came out of the review of the withdrawn soliton manuscript, records how its citations were checked, and is not among #11 §14’s items.

Two pages close the section.

  • Two gravitational charges and the conjugate field equations (chat-only) records two results that exist only in the original conversation (HANDOFF.md §4): dual stress as a second gravitational charge, and a comparison of the program’s field equations with general relativity’s, ending in massive X waves with a frequency floor. It separates the parts that have a counterpart in the documents from those that do not. Their status is conjecture.
  • Born’s answer: momentum space gravitates is the program’s summary of what the whole chain says about Born’s 1938 question, taken clause by clause with the tag of each part. It records what the answer leaves open, the stronger earlier forms it replaced, and the founding supposition it settles as “vindicated, transformed”.

Boundaries#