The tuning test

The tuning test asked whether the potential that gives the momentum graviton X its mass is free of the Boulware–Deser ghost, the extra negative-energy mode that a generic mass term for a spin-2 field carries. Ghost-free mass terms form a narrow, known family: the Fierz–Pauli term at quadratic order, completed nonlinearly by the de Rham–Gabadadze–Tolley (dRGT) and Hassan–Rosen potentials, and “tuning” refers to the special coefficients that define it. Built from metric variables, the program’s potential misses that family twice, and both misses are checked by machine. Built from frame variables, the first-order quantities from which a metric is assembled, it lands in the family by construction, an argument that is structural and rests on the literature. The verdict, a conditional pass, leaves the program’s gravity as Hassan–Rosen bimetric gravity with the second frame confined to a compact space (constrained bimetric gravity). The passing potential was selected as the minimal consistent choice; deriving it from the program’s geometric action is open problem 1.

Stands: the misalignment potential, the metric tier fails twice (a negative result), the frame tier, the verdict: a conditional pass, constrained bimetric gravity · Superseded: the v0.1 action and its geometric term S_H; folded sections on a pass as the threshold for “theory” and safety through a dominant massless combination · Died: nothing yet · Open: deriving the wedge sector from S_H (OP1)

Map#

Every page here is the program’s own material. Most of it was derived in calc note #7 (notes/debt1a-iii-tuning-test.md) and is restated in checkpoint #11 §7.2, which tags the tuning test [D][V]. The [V] belongs to the metric-tier half: battery tests T3, T4 and T7 (the verification battery). The established theories the test is measured against are in Spin-2 fields. The pages run from the object under test, through its two halves, to the verdict and the picture that results, and end with the action the passing potential should come from.

  • The misalignment potential comes first because it is the object under test. Geometry gives X no potential, so its mass must come from the energy cost of a mismatch between the polarization J and the modular field Z, measured by the anticommutator {J₀, I[Z]} = [[S, A], [A, S]]. The page builds the ingredients, writes out the block multiplication, describes battery test T3, and gives calc note #7’s reading of the blocks: the mismatch vanishes at X = 0 with the metric pinned to a reference (see flag 3 in review/wiki-build-flags.md), so the potential involves two tensors.
  • The metric tier fails twice is the negative half. After the standard conditions a ghost-free mass term must meet, it sets out both failures: a quadratic mass term proportional to Tr(h²), with no Fierz–Pauli trace term (test T4, “66.58 vs 63.72 at seed 0”), and a nonlinear structure Tr(g²) outside the linear span of the elementary symmetric polynomials (test T7, rms residual 3.2853). It records what the tests cover and what rests on the derivation alone, including a second invariant with the same defect.
  • The frame tier: the symplectic vielbein is the positive half. It explains frames and wedge potentials [T], the symplectic vielbein E with H = EᵀE, the polarization frame E₀[J] and the first-order tier, and takes calc note #7’s argument step by step, marking the minimality step as structural. It also gives the wedge potential that calc note #8 then restricts.
  • The verdict: a conditional pass states the outcome: fail in second-order variables, pass in first-order ones. It explains why the pass is conditional, what was at stake (the failure branch would have left only a phenomenological shell), what a pass did not deliver, how the result is verified, and how the program reads it as forced convergence. A folded section records the superseded expectation that a pass would earn the word “theory”.
  • Constrained bimetric gravity is the picture that follows: Hassan–Rosen bimetric gravity whose second frame is confined to the ten compact moduli of Λ(d), with the graviton as its massless combination, X as its massive one and τ as its light direction. It shows how the palindrome and a flat vacuum cut five coefficients to two, and keeps calc note #7’s superseded safety argument in a folded section.
  • The geometric action S_H and the v0.1 action (superseded) records the lab notebook’s five-term action, its Einstein-like S_H written with ω and η frozen, and its argument that no ghost could arise. It explains why the form is superseded (strict P2 empties S_H, X is a second graviton, and the planned expansion of S_H was overtaken) and what survives of each term.
  • Open: deriving the wedge sector from S_H (OP1) closes the section with the derivation that is missing: the frame-tier class was “selected by minimality + consistency, not derived line-by-line”. The page sets out how the class was selected, what a derivation would start from, why it matters, and its history. It is item 1 of #11 §14, one of the three items on which the word “theory” waits, and is not on the current queue.

Boundaries#

Backward Born's answer: momentum space gravitates The misalignment potential Forward