τ is not eaten

This result says that the dark-energy field τ is not one of the modes the momentum graviton X absorbs when it becomes massive. τ is an angle on the Lagrangian Grassmannian Λ(d), the compact space of all the ways of splitting phase space into position and momentum: it is the Maslov phase of the vacuum’s polarization. The modes X absorbs belong to a different sector, the gauge directions of the broken dual diffeomorphisms, which shift X; constant shifts of X do not generate the compact directions in which τ lies. So τ survives the gravitational Higgs mechanism as the light field the program’s dark-energy sector needs. Calc note #6 flagged the point as open; calc note #8 derived it at leading order [D — leading order]. How strongly τ still mixes with the absorbed modes through the potential has not been computed and is an open problem.

Status: Alive at leading order — [D — leading order] in calc note #8 (notes/debt1a-iv-beta-coefficients.md §3), carried by checkpoint #11 §7.5, a section tagged [D] — an argument from where each mode comes from, not a computation of the mixing; not machine-checked. The residual mixing is Open: τ–Stückelberg mixing (OP6), item 6 of #11 §14.

The statement#

Calc note #8 (the Debt-1 calc notes), §3:

τ is not eaten [D — leading order]: the Higgsing of the dual diffeomorphisms eats the vector π_μ and the longitudinal scalar of the λ-gradient sector. τ is a J-modulus on the compact Λ(d), a different sector with different origin; it survives as the light field. (Residual τ–Stückelberg mixing through the potential: logged, expected small by compactness.)

#11 §7.5, in its list of spectrum results [D]: “τ un-eaten (compact J-sector, not the Stückelberg sector).”

The two kinds of mode#

What X absorbs#

When the vacuum selects a polarization, it breaks the local shifts of X, δX_μν = ∂₍μλ_ν₎, and X absorbs the would-be Goldstone modes of that broken gauge symmetry, becoming a massive spin-2 field (the gravitational Higgs mechanism). These modes belong to the gauge parameter λ_μ. Calc note #6 calls them “the four would-be Goldstones π_μ of the broken λ-gauge” (§3); calc note #8 calls them “the vector π_μ and the longitudinal scalar of the λ-gradient sector” (§3), and no document reconciles the two wordings. Written as explicit fields, would-be Goldstone modes of this kind are Stückelberg fields [T] (Goldstone modes and the Higgs mechanism), and #11 calls them “the Stückelberg sector”.

What τ is#

Under the corrected postulate P2′ the vacuum chooses a real structure J, which fixes which directions of phase space count as position, and the possible choices form the Lagrangian Grassmannian: “the vacuum spontaneously selects a real structure J (an involution anticommuting with I), and η ≡ HJ. The space of H-compatible real structures on (ℝ^{2d}, I) is {J} ≅ U(d)/O(d) = Λ(d)” (#5 §B). Λ(d) is compact, of dimension “d(d+1)/2 = 10 for d=4” (#1 §2).

τ is one coordinate on that space, the angle around its Maslov circle (#1 §2):

τ ≡ f · arg det²(·)

#11 §6 places it on the boundary of the Siegel domain: “the dark-energy field τ is the Maslov phase of the boundary variable” (τ as the Maslov phase of the boundary). A coordinate on the space of J’s is what #8 calls “a J-modulus”, and the moduli of J together make up what #11 calls the “compact J-sector”.

The two kinds of mode therefore have different origins. The absorbed modes come from the gauge parameter of the shifts of X, which move X = Re Z through the non-compact Siegel domain. τ is a coordinate of the choice of J: changing it moves the polarization around the Maslov circle of the compact Λ(d).

The argument#

Calc note #6 stated the expectation and left it as a check: “(The Λ(d) moduli, τ included, are not the eaten modes — they carry the compact phase directions, which constant shifts do not generate; verifying τ survives un-eaten is flagged [O].)” (§3).

Calc note #8 settled it at leading order with the same distinction: the absorbed modes are those of “the λ-gradient sector”, while “τ is a J-modulus on the compact Λ(d), a different sector with different origin; it survives as the light field” (§3). The argument rests on where each mode comes from. The modes a gauge field absorbs are the would-be Goldstone modes of its own broken gauge symmetry, here the dual diffeomorphisms [T]; a mode from another sector can enter only through mixing, and that is what “leading order” sets aside.

#8 states what is set aside in the same passage. Through the potential the two sectors can still mix, and #8 records that mixing only as “logged, expected small by compactness”. No calculation backs the expectation: the τ–Stückelberg mixing matrix is open item 6 of #11 §14 (Open: τ–Stückelberg mixing (OP6)).

Why it matters#

τ is the program’s dark-energy field: “τ is hilltop quintessence on the Maslov circle” (#11 §10; τ: hilltop quintessence on the Maslov circle). That role needs τ to be light. The lab notebook assumed it from the start: “that τ is the lightest modulus, with the other nine directions heavy, is assumed [P]” (#1 §2). The tuning test’s picture builds the same split into its geometry, Hassan–Rosen bimetric gravity with the second frame confined to Λ(d), “with the light direction of the compact space being τ (the dark-energy field) and the massive spin-2 being X” (#7 §4; constrained bimetric gravity).

If τ had been among the modes X absorbs, it would have become part of the massive spin-2 field instead of a light field of its own [T]. The result shows, at leading order, that the mechanism giving X its mass leaves τ alone. It does not explain why τ is light: the lightness itself remains the postulate above, and the origin of τ’s potential is open (Open: the origin of the ρ-preserving potential).

What remains open#

The residual mixing. The τ–Stückelberg mixing matrix, open item 6 of #11 §14 (OP6).
The other nine moduli. At d = 4, Λ(d) has ten directions: “Λ(d) σ-modes (10: τ plus nine, of which inverse-Higgs constraints likely eliminate most [O])” (#5 §D; inverse-Higgs constraints). #6 places all the Λ(d) moduli outside the absorbed modes; what happens to the nine is “(4) inverse-Higgs accounting of the nine heavy moduli” in calc note #5’s open list (§F), a question no later document takes up (Open: field content of the broken vacuum).
Lorentzian signature. The mass mechanism belongs to what calc note #6 calls “the Euclidean-rigorous skeleton” (§3); the extension to spacetime signature is open item 3 (Open: Lorentzian dressing of the Euclidean skeleton (OP3)).

The calc notes write Λ(d), counting its dimension at d = 4; #11 writes Λ(d) in §6 and Λ(4) in §7. At d = 4 these are the same space.

History#

Flagged. Calc note #6 §3 marked the point [O], and its closing list repeats it: “Also now open: verify τ is not eaten (§3)” (§5).
Ranked as a check. Calc note #7 lists “(ii) τ-not-eaten verification” among the remaining Debt-1 items and adds “Items (ii)–(iv) are checks; item (i) is the substance” (§5), item (i) being the coefficient derivation. The lab notebook’s v0.5 addendum lists three logged checks: τ not eaten, the Lorentzian dressing, and the graviton-mass and mixing parametrics against gravitational-wave bounds.
Derived at leading order. Calc note #8 §3, quoted above. The v0.6 addendum: “τ confirmed un-eaten at leading order (it lives in the compact J-sector, not the Stückelberg sector)” (the lab notebook).
The papers. #9 §5 item 6, inside “Spectrum and parameters [D]”: “τ is not eaten (it lives in the compact J-sector, not the Stückelberg sector)”; #9 §10 item 5 lists the mixing as open, “expected small by compactness [O]”. #11 §7.5 is the current statement, and #11 §14 item 6 the open mixing matrix.

Sources#

notes/debt1a-iv-beta-coefficients.md §3 — the leading-order derivation [D — leading order] and the logged residual mixing; §5 — the mixing matrix in Debt 1’s open residue.
notes/debt1a-ii-X-classification.md §3 — the point as first flagged [O], the eaten modes in #6’s wording, the Euclidean-rigorous skeleton; §5 — “verify τ is not eaten”.
papers/broken-born-checkpoint-v2.md §7.5 — the current statement [D]; §6, §7 — τ the Maslov phase, Λ(d) and Λ(4); §4 — P2′; §10 — τ as quintessence; §14 item 6 — the open mixing matrix.
notes/debt1a-iii-tuning-test.md §4 — τ as the light direction of the compact space; §5 — the check ranked.
notes/debt1a-geometric-core.md §B — the space of real structures J is Λ(d); §D — the ten Λ(d) moduli; §F — the “inverse-Higgs accounting” left open.
notes/broken-born-theory.md §2 — τ ≡ f·arg det², the dimension of Λ(d), τ the lightest modulus [P]; ‘Addendum (v0.5)’, ‘Addendum (v0.6)’ — the check logged and confirmed.
notes/broken-born-program-v1.md §5, §10 — the v1 statement and the open mixing.