Open: deriving the wedge sector from S_H (OP1)

This open problem asks for the ghost-free potential of the momentum graviton X to be derived from the program’s geometric action instead of being chosen. The tuning test found that wedge polynomials of the symplectic vielbein E and the polarization frame E₀[J] form a ghost-free class (the frame tier), and argued that this is the minimal way to couple the polarization to the geometry. That argument selects the class; it does not obtain it line by line from an action. The only geometric action written out, the lab notebook’s S_H, predates the change of postulate that brought X into the theory. The problem is item 1 of the open problems in checkpoint #11 §14 [O], and one of the three items, with the partially-massless comparison and the Lorentzian dressing, on which #11 makes the word “theory” wait. Nothing has been computed toward it, and it is not on the current work queue.

Status: Open — [O] — item 1 of #11 §14; also item (i) of the open residue in calc note #8 §5 and the first half of open item 5 in HANDOFF.md §2; no derivation exists. Listed in open problems and the queue.

The open problem#

#11 §14 item 1, in full: “First-principles reduction of the frame-tier wedge sector from S_H (class selected, not derived) [O].”

The item was first stated by calc note #8 (the Debt-1 calc notes), in its close-out of Debt 1: “Debt 1: structurally paid. Open residue, none of it existential: (i) first-principles reduction of the symplectic-vielbein wedge sector from S_H (the class was selected by minimality + consistency, not derived line-by-line)” (#8 §5). The four other items of that residue became #11 §14 items 3, 6, 5 and 2.

“The frame-tier wedge sector” is the potential that gives X its mass, written as wedge polynomials of the two frames, (E, E₀[J]) (the frame tier). “S_H” is the program’s geometric action (the geometric action S_H). A first-principles reduction would start from the action and arrive at that potential, instead of selecting it.

How the class was selected#

Calc note #7 reached the frame tier in two steps of different kinds (#7 §3):

  • The literature step [T]. “In vielbein variables the ghost-free potentials are polynomial — wedge products ε(e ∧ … ∧ f ∧ …) of two frames (Hinterbichler–Rosen) [T]”. This is a result about which potentials are healthy (the vielbein formulation).
  • The program’s step, by minimality. “J and E live on the same geometric tier (first-order structures on the tangent, like the Born connection), while H = EᵀE is second-order. The geometrically minimal coupling of the polarization to the modular sector is therefore E-linear — wedge polynomials of (E, E₀[J])”. This is a reason to expect the program’s coupling to take that form, not a derivation that it does.

“Consistency” is the other half of #8’s phrase. The coupling had to lie in the one known consistent structure for a massive spin-2 field, “Fierz–Pauli/dRGT/Hassan–Rosen” (#11 §7.1, which drops the qualifier in #6 §3’s “exactly one known consistent structure”; see flag 7 in review/wiki-build-flags.md), dRGT being de Rham–Gabadadze–Tolley (the Fierz–Pauli mass term; dRGT massive gravity; Hassan–Rosen bimetric gravity), and the coupling built from metrics does not (the metric tier fails twice). Minimality picked the frame-level coupling, and consistency required a ghost-free one; the frame tier satisfies both. OP1 asks for a derivation from the action in place of that choice.

What a derivation would start from#

  • The only action written out is superseded. The lab notebook’s S_H = ½M_B² ∫_Σ √|H| R_B[H] was written “with ω, η frozen (P2)” (#1 §3). Under that postulate, “S_H is a functional of g alone: a pure metric theory. No momentum graviton” (#5 §B; #11 §5), so that form contains no X to give a potential to.
  • After P2′ the symbol stays, the action is not written. The notes use S_H for an action built from ω and H: “Constant shifts of X are exact symmetries of any S_H built from (ω, H)” (#6 §1), and “the field equation for momentum-space geometry is the Re Z sector of S_H” (#5 §C). No document writes this action out.
  • Where X’s potential must come from. The same paragraph of #6 §1 continues: “X has no potential and no non-derivative couplings from the geometric sector alone”. X’s mass comes from “the η-selection (Higgs) sector”, in which the polarization J enters (#7 §1; the misalignment potential), and the wedge sector involves J through E₀[J]. The documents do not say how the action from which the wedge sector is to be derived includes J.
  • The direct expansion was posed and not done. Calc note #5 defined the decisive computation as “expand the generalized curvature scalar of (ω, H_Z) to quadratic order in X about X = 0 on the section, with ω-generalized diffeomorphisms as the gauge symmetry” (#5 §D). It was “Not completed tonight”, and no later document reports it; calc notes #6 to #8 settled the character of X by symmetry and consistency arguments instead.
  • The connection under P2′. The v0.1 action is built on the curvature of the Born connection, which is unique for the full triple (ω, η, H). Whether ω and H alone fix a unique connection is open: “(5) connection uniqueness for the (ω, H)-only structure” (#5 §F). That item is not among #11’s numbered open problems.

Why it matters#

Selection against derivation. #11 §2 warns of “the selection-versus-derivation bias”, acknowledged “explicitly where it operates (§7.4)”. §7.4 applies it to the novelty of the vacuum-energy identity, which it downgrades pending a literature comparison. The parenthesis of §14 item 1, “(class selected, not derived)”, records the same distinction for the frame-tier class (claim tiers and selection bias).

What rests on the selected class. The positive half of the tuning test, the frame tier, and with it the verdict: a conditional pass. The potential that reciprocity then restricts, the palindromic branch, including the palindrome βₙ = β₄₋ₙ and the vacuum-energy identity. The picture of constrained bimetric gravity, and the program’s statement of Born’s answer. All of these take the selected class as their starting point.

The word “theory”. #11 §15 ends: “That is a live research program; the word ’theory’ still waits on items 1–3 above.” Items 2 and 3 are the partially-massless comparison (OP2) and the Lorentzian dressing of the Euclidean skeleton (OP3) (the assessment).

How the notes ranked it. Calc note #8 called the whole residue “none of it existential” (#8 §5), and #11 opens its list of open problems by saying that none of them is “any longer, about whether the structure can exist” (#11 §14). The item is ranked as unfinished work on a structure the notes consider settled, and at the same time as one of the three conditions for calling the program a theory.

History of the item#

  • Calc note #8 §5 lists it first in the open residue of Debt 1, and the lab notebook’s v0.6 addendum repeats it: “Residual open items (none existential): first-principles wedge reduction from S_H; Lorentzian dressing; τ–Stückelberg mixing; dual-stress enhancement parametrics; literature check on the Λ = 0 observation” (the lab notebook).
  • The v1 program paper, superseded by #11, makes it open problem 1 in nearly the same words as #8 (#9 §10 item 1), and makes “theory” wait on it and the Lorentzian dressing: “The word ’theory’ awaits items 1–2 of §10” (#9 §11; the checkpoint).
  • The handoff pairs it with the Lorentzian dressing as open item 5: “First-principles wedge reduction from S_H; Lorentzian dressing of the Euclidean skeleton” (HANDOFF.md §2; the handoff).
  • The standalone manuscript #14, since withdrawn after its round-3 review for reasons unrelated to this item (Obituary: superheavy conjugate solitons), put the caveat to its referees: “That the symmetry-breaking sector lands on the ghost-free potential is argued from a first-order (vielbein) formulation and is a structural claim, not a line-by-line derivation” (#14 §1).
  • Checkpoint #11 makes it §14 item 1, and §15 adds the partially-massless comparison to the items on which “theory” waits.

No document reports work toward a derivation.

The queue#

The handoff’s ranked queue of next actions did not include the item (HANDOFF.md §8). The current queue, in CLAUDE.md (‘Since the handoff’), is the handoff’s §8 items 5 to 7, namely the partially-massless comparison, the † literature battery and the phase-space-diameter note, followed by an optional round-5 review of the toy-model note “but only before any external use”. This item is not on it (open problems and the queue).

Sources#

  • papers/broken-born-checkpoint-v2.md §14 item 1 — the open problem [O], and the opening line of §14; §15 — “theory” waits on items 1–3; §2, §7.4 — the selection-versus-derivation bias; §7.1 — the one consistent structure for massive spin-2; §5 — S_H a functional of g alone under strict P2.
  • notes/debt1a-iv-beta-coefficients.md §5 — the open residue of Debt 1, “none of it existential”.
  • notes/debt1a-iii-tuning-test.md §1, §3 — the η-selection sector; the frame-tier argument by minimality.
  • notes/debt1a-ii-X-classification.md §1, §3 — S_H built from (ω, H) gives X no potential; the health criterion.
  • notes/debt1a-geometric-core.md §B, §C, §D, §F — S_H under strict P2; the Re Z sector of S_H; the expansion posed and not completed; connection uniqueness for (ω, H).
  • notes/broken-born-theory.md §3 — S_H written with ω and η frozen; ‘Addendum (v0.6)’ — the residual open items.
  • notes/broken-born-program-v1.md §10, §11 — the v1 listing and “theory” awaiting items 1–2.
  • notes/knot-dark-matter-standalone.md §1 — the caveat as stated in the withdrawn manuscript.
  • HANDOFF.md §2 — open item 5; §8 — the ranked queue at the handoff.
  • CLAUDE.md ‘Since the handoff’ — the current queue.
  • review/wiki-build-flags.md item 7 — “the unique consistent structure” without the “known” of #6 §3; awaiting Paul’s ruling.