The tuning test on The Broken-Born program

The misalignment potential

Mon, 01 Jan 0001 00:00:00 +0000

The misalignment potential is the energy cost that the program assigns to a mismatch between the polarization J, the split of phase space into position and momentum directions that the vacuum has chosen, and the local phase-space geometry, which the modular field Z carries. The momentum graviton X, the real part of Z, gets no potential from the geometry alone, because constant shifts of X are a symmetry (the shift symmetry), so its mass has to come from this term. The calc notes call it the η-selection, or Higgs, sector. The natural measure of the mismatch is the anticommutator {J, I[Z]} of the polarization with the complex structure built from Z, which is zero when the two are compatible; its block form is derived exactly and checked by machine. Calc note #7 reads off the blocks that the mismatch vanishes only at X = 0 with the metric pinned to a reference metric (see flag 3 in review/wiki-build-flags.md), so that the potential involves two tensors, as the potentials of massive gravity do. Whether such a potential avoids the Boulware–Deser ghost is the question of the tuning test.

The metric tier fails twice

Mon, 01 Jan 0001 00:00:00 +0000

The metric tier is the naive way, in calc note #7’s word, of writing the mass term of the momentum graviton X: as a function of metric-level, second-order quantities, the blocks of the misalignment potential. It fails the ghost-freedom test in two independent ways. At quadratic order the natural potential gives the metric fluctuation h a mass term proportional to Tr(h²) without the −(Tr h)² term that the Fierz–Pauli form requires. Beyond quadratic order it contains Tr(g²), which is not a linear combination of the elementary symmetric polynomials from which every ghost-free Hassan–Rosen potential is built. Either failure lets the Boulware–Deser ghost, an extra mode with negative kinetic energy, propagate [T]. Calc note #7 calls the result a “theorem-grade negative”, and tests T4 and T7 of the verification battery check both failures by machine. The way out was a change of variables, to the frame tier.

The frame tier: the symplectic vielbein

Mon, 01 Jan 0001 00:00:00 +0000

The frame tier is the reformulation that rescued the tuning test: the mass term of the momentum graviton X is built from frames, or vielbeins, instead of from metrics. A frame is a set of local reference axes from which a metric is assembled quadratically, as H = EᵀE, so the program calls frames first-order variables and metrics second-order ones. In frame variables the ghost-free potentials for two spin-2 fields are plain polynomials, wedge products of the two frames in the form found by Hinterbichler and Rosen (the vielbein formulation), with no matrix square roots and no ratio of coefficients to get wrong. The program has a frame of this kind, the symplectic vielbein E, which sits on the same first-order footing as the polarization J. Calc note #7 argues that the minimal coupling of the polarization to the geometry is therefore a wedge polynomial of E and a polarization frame E₀[J], and such a polynomial lies in the ghost-free class by construction, with the Fierz–Pauli structure automatic. The argument is structural and rests on the literature. The class was selected by minimality and consistency rather than derived from the program’s geometric action, and no machine check covers it.

The verdict: a conditional pass

Mon, 01 Jan 0001 00:00:00 +0000

The verdict of the tuning test is a conditional pass. The potential that gives the momentum graviton X its mass carries the Boulware–Deser ghost when it is written in metric, or second-order, variables, and it is ghost-free when it is written in frame, or first-order, variables. The condition is the choice of variables, and the calc notes argue that the first-order formulation is the natural one anyway, because the polarization J and the symplectic vielbein E are both first-order objects. Before the test, calc note #6 had spelled out what a failure would leave: only a phenomenological shell of gated self-interacting dark matter and a dark-energy field on the Maslov circle, with the geometric story demoted to heuristics. Checkpoint #11 tags the tuning test [D][V]. The machine checks cover the metric-tier failures; the frame-tier pass is a structural argument with no test, and its potential was selected rather than derived.

Constrained bimetric gravity

Mon, 01 Jan 0001 00:00:00 +0000

Constrained bimetric gravity is the picture of the program’s gravity that the tuning test leaves. Hassan–Rosen bimetric gravity is a ghost-free theory of two interacting metrics, each with its own dynamics. In the program’s version the second frame is not a free field: it is the polarization frame, built from the polarization J, and J ranges only over the compact space Λ(d) of splittings of phase space into position and momentum, which is ten-dimensional at d = 4. Calc note #7 calls the result a predictive truncation of bimetric theory, with fewer free functions. In it the massless combination of the two spin-2 fields is the ordinary graviton, the massive one is the momentum graviton X, and the light direction of the compact space is the dark-energy field τ. Reciprocity then restricts the potential to its palindromic branch, and with the condition for a flat vacuum two free parameters remain. The picture is a structural derivation; the standard bimetric facts it uses come from the literature, and nothing in it is machine-checked.

The geometric action S_H and the v0.1 action (superseded)

Mon, 01 Jan 0001 00:00:00 +0000

The v0.1 action is the complete action written in the program’s lab notebook in September 2026: a sum of five terms, for the phase-space geometry, the dark-energy angle τ, the Standard Model, conjugate matter, and an unspecified mixing term. Its geometric term S_H is an Einstein-like action for the phase-space metric H, built from the curvature of the Born connection and written with both the symplectic form ω and the neutral metric η frozen, as the original P2 required. From that rigidity the notebook argued that no second independent metric, and so no Boulware–Deser ghost, could arise. Both points were later overturned: postulate P2′ made η emergent, the momentum graviton X was found to be a second graviton, and ghost freedom now comes from the frame tier. This form of the action is therefore superseded, and checkpoint #11 writes no full action. The name S_H survives as the action from which open problem 1 asks for the ghost-free potential to be derived.

Open: deriving the wedge sector from S_H (OP1)

Mon, 01 Jan 0001 00:00:00 +0000

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.