The misalignment potential

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.

Status: Alive — [D][V] in checkpoint #11 §7.2; the block computation is [D — rigorous] in calc note #7 §2 — the blocks are machine-checked by battery test T3 at one sampled point (the verification battery); #7’s reading of their zero set, the compatible locus, sits under the same [D — rigorous] heading and has no test.

Why X needs a potential of this kind#

The geometry alone gives X no mass. Calc note #6 (the Debt-1 calc notes) found that the rigid symplectic group contains the shifts Z → Z + B, which move X and leave Y alone, and concluded: “Constant shifts of X are exact symmetries of any S_H built from (ω, H). Consequences: X has no potential and no non-derivative couplings from the geometric sector alone” (#6 §1). A mass term for X cannot be written in by hand.

The same note located where a mass can come from: “The vacuum ⟨Z⟩ = i·g sits at X = 0 and is moved by the shifts: the polarization selection (η-emergence) spontaneously breaks the local dual diffeomorphisms” (#6 §3). X then becomes a massive spin-2 field by a gravitational version of the Higgs mechanism (the gravitational Higgs mechanism), with “M_X = the polarization-breaking scale” (#11 §7.1). The potential that carries this breaking is the subject of this page. #6 also stated the condition it has to meet, using the standard result that a massive spin-2 field has one known consistent structure [T]:

X is a massive spin-2 field. The geometric sector is healthy iff the potential generated by the η-selection lands in the Hassan–Rosen class; a generic potential reintroduces the Boulware–Deser ghost. (#6 §3)

It then named the computation still to be done: “derive the X-potential from the η-selection (J–Z) sector and test it against the dRGT coefficient family” (#6 §5). dRGT is de Rham–Gabadadze–Tolley. The class and the ghost are described on dRGT massive gravity, Hassan–Rosen bimetric gravity and the Fierz–Pauli mass term and the Boulware–Deser ghost, and ghosts in general on ghosts and perturbative instabilities.

Calc note #7 opens with the object that computation needs (#7 §1):

The X mass must come from the η-selection (Higgs) sector: a misalignment potential penalizing incompatibility between the polarization J and the modular field Z. The natural incompatibility measure is the anticommutator {J, I[Z]} (zero exactly on the compatible locus). The test: does the resulting potential land in the ghost-free (Fierz–Pauli / dRGT / Hassan–Rosen) family, or in the generic Boulware–Deser-ghost family?

Both names appear in #6 §3, which speaks of “the potential generated by the η-selection” and of “the polarization selection (η-emergence)” that breaks the dual diffeomorphisms, and which calls the result a “Gravitational Higgs mechanism”.

The ingredients#

The modular field and its complex structure I[Z]#

Once only the symplectic form ω is rigid (P2′), the phase-space metric H ranges over the Siegel domain, and each H is labelled by a complex symmetric matrix Z = X + iY with Y > 0 (#11 §6; #5 §C):

H_Z = [[ Y + XY⁻¹X ,  XY⁻¹ ],
       [ Y⁻¹X      ,  Y⁻¹  ]]

The program reads Y = Im Z as the spacetime metric g and X = Re Z as the momentum graviton (#11 §6). From ω and H it builds the complex structure I = ω⁻¹H, with I² = −1 (#11 §5; the structures ω, η, H). Written for H_Z, this is the matrix #7 §2 uses, “with the verified Siegel parametrization”:

I[Z] = [[ −Y⁻¹X     ,  −Y⁻¹ ],
        [ Y + XY⁻¹X ,  XY⁻¹ ]]

#7 does not say how the parametrization was verified. Battery test T2 checks I[Z]² = −1, together with the symmetry, positivity and unit determinant of H_Z, at one sampled point (verify/battery.py T2, where I[Z] is computed as ω⁻¹H_Z).

The polarization J and the vacuum polarization J₀#

The polarization is the real structure J that the vacuum selects under P2′: “The neutral metric η is emergent — the order parameter, η = HJ, with the real structure J chosen spontaneously” (#11 §4). Calc note #5 describes J as “an involution anticommuting with I” (#5 §B): J² = 1, and a polarization compatible with the metric anticommutes with its complex structure. The anticommutator {J, I}, the sum of the products of J and I taken in the two orders, therefore vanishes for a compatible pair, and its size measures how far a given J and a given I are from compatibility.

The computation uses the vacuum polarization

J₀ = [[ 0, 1 ],
      [ 1, 0 ]]

the same block matrix that the lab notebook writes for η in the standard (x, p) basis (#1 §2). In verify/battery.py, test T3 forms the anticommutator with the script’s η matrix, which is this J₀.

The documents do not agree on the name of the real structure. #11 §4 writes η = HJ and #11 §5 defines J = η⁻¹H, while the toy-model note calls the real structure K, with η = ωK. This page follows #7 and writes J (the order parameter: J, η and K).

The computation [D — rigorous] [V]#

Calc note #7 §2 works “in the vacuum frame (g₀ = 1), Euclidean skeleton”, that is, in coordinates where the reference metric is the identity and with a positive-definite H. “Direct block multiplication gives”

{J₀, I[Z]} = [[ S, A ],
              [ A, S ]]

S = Y + XY⁻¹X − Y⁻¹,        A = [X, Y⁻¹]

#7 states only the result. The multiplication is short because J₀ exchanges blocks: multiplying I[Z] by J₀ from the left exchanges its two block rows, and from the right exchanges its two block columns. Written out as a reading aid:

J₀ I[Z] = [[ Y + XY⁻¹X ,  XY⁻¹ ],
           [ −Y⁻¹X     ,  −Y⁻¹ ]]

I[Z] J₀ = [[ −Y⁻¹      ,  −Y⁻¹X     ],
           [ XY⁻¹      ,  Y + XY⁻¹X ]]

Adding the two gives Y + XY⁻¹X − Y⁻¹ = S in both diagonal blocks and XY⁻¹ − Y⁻¹X = [X, Y⁻¹] = A in both off-diagonal blocks.

The machine check. Battery test T3 takes the random symmetric X and positive-definite Y of test T2 at d = 4 (seed 0), forms the anticommutator with J₀, and compares each of the four blocks with S and A numerically. The current gate run prints PASS T3 {J0,I[Z]} = [[S,A],[A,S]] (verify/battery.py T3). The algebra is exact; the test confirms it at one sampled point. #11 §7.2 records the blocks as “machine-verified”, and the handoff’s battery summary lists “T3 {J₀,I[Z]} = [[S,A],[A,S]] blocks” (HANDOFF.md §5).

What #7 reads off the blocks#

#7 §2 draws “Three consequences, each read off the blocks”. The first concerns where the misalignment vanishes:

  1. The compatible locus is X = 0 and Y = Y⁻¹ (i.e. g pinned to the J-frame reference metric): the Higgs sector is inevitably a two-tensor potential V(g, X; reference) — a massive-gravity-type potential, confirming the structural landing point of Chunk A-ii from the opposite direction.

At X = 0 the off-diagonal block A = [0, Y⁻¹] vanishes and the diagonal block reduces to S = Y − Y⁻¹, which is zero when Y = Y⁻¹. In the vacuum frame the reference metric is g₀ = 1. A potential of this kind compares the metric g with a fixed reference metric, which is how the potentials of massive gravity are built: from the dynamical metric and a second, reference metric [T] (dRGT massive gravity). “Chunk A-ii” is calc note #6, whose landing point was a massive spin-2 field in the Hassan–Rosen class. A suspected error in this reading of the zero set is recorded for a ruling in review/wiki-build-flags.md item 3; this page quotes #7 as written.

#7’s compatible locus is a different object from what #11 §6 calls “the η-compatible locus X = 0”, on which H takes the block form diag(g, g⁻¹) for any g (the compatibility theorem). #7’s condition involves the fixed vacuum polarization J₀, and in #7’s reading it also pins g.

The other two consequences test the potential against the ghost-free family, and both are failures. Expanded to quadratic order, the natural invariant Tr({J, I}²) gives the metric fluctuation a mass without the Fierz–Pauli trace term; on the slice X = 0 it contains Tr(g²), which lies outside the Hassan–Rosen family. Both are machine-checked, by tests T4 and T7, and set out on the metric tier fails twice.

How the documents state it#

  • #11 §7.2 [D][V], the current statement: “The Higgs sector is a misalignment potential in {J, I[Z]} = [[S,A],[A,S]], S = Y + XY⁻¹X − Y⁻¹, A = [X, Y⁻¹] (blocks machine-verified).” #11 goes on to the failures and does not restate #7’s compatible locus.
  • The v1 program paper (#9 §5 item 4, superseded by #11; the checkpoint): the same blocks, “(computed exactly)”, under the tag “[D — both branches computed]”.
  • The lab notebook, v0.5 addendum: “Explicit block computation of the misalignment structure {J, I[Z]} shows” the two metric-tier failures (the lab notebook).

Scope and caveats#

  • Euclidean, and in the vacuum frame. The computation assumes a positive-definite H, the “Euclidean skeleton” in #7’s words. Extending the program’s Euclidean results to the Lorentzian signature of spacetime is open problem 3 (Open: Lorentzian dressing (OP3)).
  • What T3 confirms. T3 checks the matrix identity at one random point. It does not check #7’s reading of the zero set, nor that a potential built from these blocks is the one the theory contains. A verification script confirms arithmetic, not the model it encodes (verification confirms arithmetic, not the model).
  • Which function of the blocks. #7 does not derive one potential. It tests “Any potential of the class Tr({J,I}²)-with-positive-norm” and “A second natural invariant”, and reports that “Every metric-tier (second-order, H-coupled) misalignment invariant tested lands off the tuning” (#7 §2). The potential that passes is written in frame variables instead (the frame tier), and it was selected rather than derived (Open: deriving the wedge sector from S_H (OP1)).

Where it leads#

The blocks are the input to the rest of the test: the metric tier fails twice, the frame tier passes, and the combined outcome is the verdict: a conditional pass. The mass the potential produces is set by its coefficients βₙ, which the pass did not deliver (#7 §3). After reciprocity and the existence of a flat vacuum restrict them (the palindromic branch), #11 §7.5 counts two free parameters, one setting M_X and one a cubic shape (spectrum, parameters and reciprocity parity).

Sources#

  • papers/broken-born-checkpoint-v2.md §7.2 — the current statement, [D][V], with the blocks “machine-verified”; §7.1 — M_X as the polarization-breaking scale; §7.5 — two parameters after the palindrome.
  • papers/broken-born-checkpoint-v2.md §4, §5, §6 — η = HJ in P2′; I = ω⁻¹H and I² = −1; Z = X + iY, Im Z = g, Re Z = X and the η-compatible locus X = 0.
  • notes/debt1a-iii-tuning-test.md §1 — the object to test; §2 — the block computation [D — rigorous], the compatible locus, and the tested invariants; §3 — the βₙ not delivered by the pass.
  • verify/battery.py T3 — the machine check of the four blocks; T2 — I[Z]² = −1 and the properties of H_Z.
  • notes/debt1a-ii-X-classification.md §1, §3, §5 — the shift symmetry; the Higgs mechanism and the health criterion; the computation still to be done.
  • notes/debt1a-geometric-core.md §B, §C — J as an involution anticommuting with I; the Siegel parametrization H_Z.
  • notes/broken-born-theory.md §2 — η in the standard basis; ‘Addendum (v0.5)’ — the notebook’s account.
  • notes/broken-born-program-v1.md §5 — the v1 statement, “(computed exactly)”.
  • HANDOFF.md §5 — the battery summary for T3.
  • review/wiki-build-flags.md item 3 — the suspected error in #7’s zero set, awaiting a ruling.
Backward The tuning test The metric tier fails twice Forward