Physical labels and their detection · Part 3

= 0, equal planes containing zero and equal planes avoiding zero are the two possible types. For = 0, the planes share one other index, and the two placements of the derivative factor are the other types. Permutations fixing zero identify all coefficients of each type; antisymmetry fixes their signs. The three displayed tests kill three types. The cyclic identity leg + leg + leg paired with leg, kills the fourth. The legs span each radial jet space by the expansion

Searchable transcription

2
X
2
X
A = 1
i,j
legij(curl0(i, j)A),
B = 1
i,j,k
legk;ij(curl1(k, i, j)B).

Bilinearity therefore kills the entire mixed part. Reality recovers the complex coefficient from its real part. The three evaluations form an injective real map, proving the raw upper bound three. The two solution evaluations in (FC10) and the nonsolution pair (leg leg), on which = 1, prove independence of. Thus (FC11) is a basis and every raw vector has unique coordinates.

For later use, the divergence of the raw centered tensor with coefficients respectively diagonal mountain, diagonal total and off-diagonal shear, has these coordinates:

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Div(w) = 2(w0 −w2)V1 + (2w1 + 1
2w2)V2 + w2E.
(FC14)

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The product rule gives 2V1 for the first tensor and 2V2 for the second. For the shear use Bianchi and
interchange the ordered indices to obtain P
a,i Pai;abi = 1
2V2,b; deleting the a = b terms subtracts
2V1,b, while the other product derivative is Eb.
This proves (FC14).
Its coordinate matrix is
invertible, with inverse

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w2 = y2,
w0 = y0/2 + y2,
w1 = y1/2 −y2/4.

It defines the actual scalar contraction on raw vectors:

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ρ(Div(w)) = −4(w0 + 4w1) = −2y0 −8y1.
(FC15)

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If Q = 1
4
P
i,j⟨Fij, Fij⟩, the tensor trace is 4(w0 + 4w1)Q; hence ρ reads minus that trace in Q-units.
It kills E and the traceless conserved tensor w = (1, −1/4, 1). Since (FC10) proves that the raw
equation kernel is exactly RE, ρ annihilates that entire kernel. This is the scalar sensor used later,
with its sign and normalization fixed. The identification of the free quotient here uses the Maxwell
polynomial evaluation and the reflected Gram norm proved in (FM3)–(FM4) and in the detector’s
reflected polynomial calculation; it is not an additional assumption of nondegeneracy.