Physical labels and their detection · Part 2

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Choose a = [4(n + 2)]−1, B = Ha
k, ε = H−1
k . The original threshold obeys θk ≤H−1
k
eventually.
The terms on the right of (PC7d), including its outer B, have powers of Hk at most

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−1,
−1 + a(n + 2) = −3
4,
−1 + a(n + 1) < −3
4.
(PC7e)

Their fixed prefactors depend on, not on. Thus 21 eventually, proving (PC7) uniformly in those remaining indices. Also eventually, so the Wick polynomial fits its stated monitor price. These estimates use the original threshold and the actual rescaled covariance; no solved-column bound has been used to prove the inverse.

The unit is exact: its representative is the constant polynomial, contraction and truncation fix it, and both rescalings have weight zero on it. Thus. We retain (PC7) in its relative form; an absolute error on a few selected fields would not imply the inverse assertion that follows.

The triangle inequality gives

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|Av|1 ≤2|Tv|1,
|Tv|1 ≤3
2|Av|1 ≤2|Av|1.
(PC8)

Combining the first inequality with (PC4) proves

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|T −1z|1 ≤2Od,kad
k|z|1.
(PC9)

To justify existence of this inverse before using it, = 0 implies = 0 by (PC8), then by (PC4). An injective endomorphism of a finite-dimensional space is surjective. This proves both existence and the bound (PC9).

For a desired reference coefficient and a position magnification, let (. The exact solved calibration is (PC10)

The two diagonal operations use the distinct entries of the bidegree. The solution is unique. Real gives real, because conjugating the equation gives the same equation and uniqueness applies. If, the solution is by the exact unit identity. This constructs all the solved basis columns simultaneously for the one readout, rather than choosing a different readout for each column.

The finite cutoffbounds on give

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ξ−1 = θk
q
βN,p,k ≤2Hk,
(PC11)

where is the actual monitor at level are the finitely many chosen real representatives of a free quotient basis, (PC9)–(PC11) imply the uniform fixed-level bound

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s
X
∥uj∥≤2Od,kad
k
j′=1
|D2HkWρbj′|1.
(PC12)

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Indeed bound each component of Dξ−1 by the corresponding component of D2Hk, use ∥· ∥≤| · |1,
and enlarge the single basis-column bound to their finite sum. This is the compact ball used in
extracting the solved columns. It is allowed to grow with the reporting level.
For the subsequent scale-dependent prices write Hk = eBk, choose ρ = Hb
k, and put q = maxi fi.
Formula (PC10) bounds

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|Tu|1 ≤(2Hk)qρd|b|1,
Hk|Au|1 ≤2q+1|b|1Hbd+q+1
k
.
(PC13)

For fixed and, the finite prefactor is eventually paid by one further factor. The allowed polynomial coefficient price is therefore . This derives the degree-dependent price used for the calibrated polynomials. It also separates it from the fixed-level inverse bound (PC9) and from the inverse of the separating-test matrix used in (WS5).

The full free Maxwell polynomial hierarchy

The free comparison hierarchy is specified on all polynomial labels, with no restriction to its two-point function. Let be the real space of homogeneous curvature Taylor coefficients of order satisfying antisymmetry, Bianchi and the Maxwell divergence equation in (RJ3),(RJ5). Choose a real basis and write for its coordinate symbols. The free polynomial algebra is

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W = C[Xj,α],
weight(Xj,α) = j + 2.
(FM1)

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A real solution tower A = (Aj) evaluates these symbols in its real coordinates. Evaluation separates
polynomials by the finite-variable argument of (PF1).
The restriction map from radial labels
substitutes the radial potential reconstructed from Aj into each original jet coordinate. Its kernel
at horizon d is exactly the equation ideal: it consists of polynomials vanishing on every such solution
tower. Its image Winv
d
is the invariant sector of weight at most d. To see surjectivity onto this sector,
use the color-equivariant projection of radial curvature coefficients onto their Maxwell subspace and
then radial reconstruction to lift every linear solution coordinate. Polynomial substitution preserves
weight and equivariance; restriction of this lift is the identity. This is an algebraic operation on
coefficients, not a claim about interacting equations.
The basic covariance is fixed by

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G0(z) =
1
4π2|z|2 ,
Cab,kl(z) = δbk∂a∂lG0 + δal∂b∂kG0 −δbl∂a∂kG0 −δak∂b∂lG0,
z ̸= 0. (FM2)

Colors contract by their invariant Euclidean inner product. This is the curl at the first endpoint and the opposite curl at the second endpoint of the potential covariance; the opposite endpoint derivative accounts for its signs. Its two legs are closed and divergence free away from zero, by commutation of derivatives and harmonicity of

The contractions of higher solution coordinates use normalized Taylor coefficients, so the factorial convention must be retained. If , first extend it linearly to the ambient two-form Taylor tensors. Apply it to the order- normalized Taylor tensor of the first leg of (FM2), obtaining a two-form in the second indices and color. Pair its order- normalized Taylor tensor with an extension of , and multiply by (. Denote the resulting scalar by). Both extensions are immaterial: at each point away from zero the differentiated leg is in the corresponding solution subspace, so their differences annihilate it. This defines the contractions on precisely the constrained variables in (FM1).

For monomials at distinct points, regard every power of every symbol as that many distinguishable occurrences. Sum over perfect matchings of all these occurrences, allowing a pair only if its two occurrences belong to different insertion points. A pair () contributes ). The product over its pairs, summed over the permitted matchings, is the monomial moment. The empty matching contributes one and an odd total number contributes zero. Extend by complex multilinearity in the polynomial inputs:

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Y
!
X
Mn(x; P1, . . . , Pn) =
X
Y
i
ci
pairs
Kp,q(xi −xk).
(FM3)
monomial choices

perfect matchings no pair within one insertion

The restriction on pairs is the Wick ordering of each composite insertion. There are finitely many terms for every tuple of polynomials. Pulling (FM3) back by the restriction map defines the free table on the original coherent label frames.

The elementary contraction is symmetric under exchange of endpoints and legs. Reindexing matchings therefore proves permutation symmetry. It depends only on differences, proving translation invariance. The chain rule applied to (FM2), orthogonality of the color action, and orthogonal covariance of the two curls give the full orthogonal tensor law. All entries are real on real coefficients, giving the reality law. Inserting the constant polynomial one introduces no occurrences, so it removes that slot. These prove the algebraic hierarchy laws for every arity.

Each derivative of a contraction of orders is bounded by a constant times a power of the inverse separation; at order zero the power is 4 +. This follows by differentiating the rational homogeneous function (FM2), with the finite normalization factors included in its constant. Taking products and finite sums in (FM3) proves analyticity on the separated configuration space and a power bound there. For the product bound add the exponents and multiply constants; for a sum take their sum and enlarge to a common exponent using1 when 01. These are the positional bounds needed for its smeared hierarchy.

Positivity and strictness of the reflected free pairing

We derive the reflected one-particle positivity needed for polynomial detection. Fix an axis, let reflect that coordinate, and let act on all tensor indices of a probe by this reflection. Take finitely many real probes of possibly different jet orders, at points with ( 0. Coincident points among these sources are allowed. Float all sources by0, and put

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H(t) =
X
i,j
K(pi, Rapj; xi + tea, θa(xj + tea)),

where the last two arguments indicate the endpoints of the difference kernel. Let be the sum of their curvature pulses. The derivative in the first endpoint and the opposite motion of the reflected endpoint give

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H′(t) = −2L(t),
L(t) =
X
i,j
−K(∂api, Rapj; xi + tea, θa(xj + tea)).
(GP1)

Here is the raised probe, with the normalized derivative convention of (FM2). The factor two is the sum of the two endpoint motions.

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On the reflecting plane the reflected pulse is Ft,bc 7→εbεcFt,bc, where εa = −1 and all other
εb = 1. Substitute these signs into the bilinear Maxwell stress. Its normal-normal component is
P
−1
b,c ∥Ft,bc∥2. The mixed terms with exactly one normal index change sign; skew symmetry
2
gives Ft,aa = 0, and combining the remaining terms yields exactly this negative half-sum. Integrate
its divergence over the positive half-box, removing small balls about its finitely many upper sources.
The plane term has the negative outward normal; the small-sphere terms are the translation case
of the pulse-current identity (FS2). The reflected sources are in the negative half-space and cause
no extra sphere term. The other box faces disappear as their size tends to infinity. Indeed each
pulse of order d is O(|y|−4−d); its bilinear stress is O(|y|−8−di−dj), while the three-dimensional face
measure is O(R3). Their fluxes are therefore O(R−5−di−dj). Summing the finitely many terms
proves

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Z
X
L(t) = lim
R→∞
1
2
b,c
∥Ft,bc(ya = 0)∥2 d3y ≥0.
(GP2)
[−R,R]3

The same argument bounds each finite-box integral by), by monotonicity of the nonnegative integrand. Since every source-to-reflected-source separation tends to infinity, the pole bounds give 0. Equations (GP1)–(GP2) make decreasing, so0. This proves the real positive

semidefiniteness of every finite reflected one-particle matrix. Symmetry of that matrix follows from endpoint exchange and reflection covariance. Its complex extension is Hermitian positive semidefinite by splitting a coefficient vector into real and imaginary parts.

The equality case is stronger when all source points equal one point with 0. Suppose (0) = 0. Monotonicity and nonnegativity imply) = 0 for all0, hence) = 0 for Equation (GP2), together with continuity, forces every component of to vanish on the entire reflecting plane: a nonzero value would have a neighborhood with positive squared integral inside some finite box. Translation covariance then makes the unshifted summed pulse vanish on for every 0. Each scalar color component is real analytic on, which is connected. The analytic identity theorem extends this half-space vanishing to every

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Apply now the radial version of (FS2) to this zero summed pulse and any polynomial Maxwell
solution f. It yields
X

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i
(di + 2) pi(jdif(x)) = 0,
(GP3)

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where jk is the normalized solution jet of order k. For any homogeneous solution tensor Ak, the
field f(y) = asWave(Ak)(y −x) has normalized jet Ak at order k at x, and zero jets there at all
other orders. Thus (GP3) gives P
i:di=k pi(Ak) = 0, since k + 2 > 0. Summing over the finitely
many jet orders proves that a zero reflected norm annihilates every solution tower. In particular,
for any finite alphabet of independent solution-coordinate probes, its one-point reflected matrix is
positive definite: evaluate the last identity on a tower with a single basis coordinate equal to one
to force each coefficient to be zero. This establishes strictness on the constrained solution spaces
themselves, including mixtures of jet orders.
We pass from legs to polynomial words algebraically. On finitely many complex variables define

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Y
!
⟨P, Q⟩F =
X
i
νi!
PνQν.
(GP4)
ν

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Every weight is positive; hence ⟨P, P⟩F > 0 for P ̸= 0, including the scalar polynomial. Factor a
finite reflected leg matrix as L∗L, which is ordinary finite-dimensional positive matrix factorization.
Associate to a leg i the creation polynomial li(Z) = P
k LkiZk. To represent a word, process its
occurrences successively: append its creation polynomial and, for each possible contraction to a
previous occurrence on the same side, add that covariance times the polynomial for the remaining
word. This finite recursion is Wick’s pairing recurrence. Pairing the two resulting polynomials by
(GP4) contracts their remaining occurrences across the reflection plane, since ⟨li, lj⟩F = (L∗L)ij.
The factorial Q νi! counts the bijections between equal creation occurrences. Induction on word

length therefore identifies the reflected Wick moment with this polynomial inner product, including all permitted contractions internal to either half. A finite linear combination of words has reflected norm equal to the nonnegative squared norm of its associated polynomial. Expanding polynomial insertions into monomials proves reflection positivity at arbitrary arity, and smearing preserves it by integration.

At a single source point there are no contractions within either half because each insertion is Wick ordered. The recursion is then simply the substitution). The one-point leg matrix is positive definite by (GP3), so is invertible; substituting gives the inverse polynomial substitution. Thus a nonzero polynomial remains nonzero, and (GP4) makes its reflected pairing strictly positive. This is the strict pairing used in the hosting argument and the detector construction. It proves that the free all-spectator null space has precisely the equation ideal already removed in forming the solution coordinates.

Accordingly the free physical field space is canonically

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Ed/Nd ≃Winv
d .
(FM4)

Under this identification the horizon inclusion is literal polynomial inclusion. A derivative in direction is the polynomial derivation determined on a linear symbol by

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Dv symbolj(p) = symbolj+1(p ◦lowerv).
(FM5)

Here lower is the order- Taylor coefficient of the derivative of the homogeneous polynomial represented by; its coefficient is + 1 times contraction with. Thus (FM5) has the actual Taylor normalization. The Leibniz rule differentiates one occurrence at a time. Differentiating a matching product in (FM3) selects its incident covariance edge, while replacing an occurrence by (FM5) differentiates precisely that same edge. Summing proves the insertion-derivative identity. It raises the horizon by one and stays in the invariant sector. Conjugation and signed transformations obey (QD6). Hence the free differential system is the same type of all-spectator system as the interacting one. For(4) the free field action is the polynomial substitution induced by the tensor action on the solution tower. It preserves weight, equations and color invariance, with inverse given by ; conjugating it through (FM4) gives the full field action. Infinitesimal skew rotations likewise act by the derivation obtained by differentiating that linear tensor substitution. Both descriptions are on the same field space and preserve the moment law (FM3).

The free symmetric traceless stress is the actual weight-four invariant polynomial

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T free
ab
=
X
c,i
F c
aiF c
bi −δab
X
c,i,k
F c
ikF c
ik.
(FM6)
4

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Products here mean the Wick polynomial product used in (FM3). Adjoint invariance follows from
preservation of the color inner product, symmetry by exchanging its two factors, and reality from
real coefficients. Its trace is zero by summing the four diagonal entries. Its divergence is zero as a
polynomial on solution jets. Indeed the derivative of the first term is P
c,i F c
ai∂aF c
bi, since ∂aF c
ai = 0.
Bianchi and antisymmetry give

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X
a,i
F c
ai∂aF c
bi = 1
4∂b
X
a,i
F c
aiF c
ai;
(FM7)

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multiply the Bianchi equation by F c
ai, sum, and interchange a, i in its second derivative term to
obtain the factor two on the left and the half derivative on the right. This cancels the divergence
of the subtracted term in (FM6). Evaluation on all real solution towers then proves the polynomial
identity. Through (FM4) it gives a real, symmetric, conserved free field at horizon four, with the
exact normalization used in the later free comparison.

The free low-degree stress and vector spaces

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The low-degree classification uses simplicity of the actual color Lie algebra, without assuming that
the matrix group is connected. Differentiate its adjoint action along the actual exponential curves.
This gives
d
dt Adexp(tx) y

0
= [x, y].
(FC1)

The span of brackets is a nonzero ideal, hence is the whole simple nonabelian color algebra. An invariant linear functional annihilates each bracket by (FC1), so it is zero. An invariant real bilinear

form satisfies]). Its alternating part is zero: three cyclic applications to ), with bracket antisymmetry, identify this value with its negative, and brackets span. For the symmetric part writewith self-adjoint. Invariance says Every eigenspace of is therefore an ideal. A real self-adjoint operator has an eigenvalue; its nonzero eigenspace is the whole algebra, so is scalar. Applying this argument to real and imaginary parts proves that every invariant complex-valued real bilinear form is a complex multiple of the fixed color inner product.

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A monomial of weight at most five has at most two fields, since each variable has weight at least
two. With two fields its jet orders are either (0, 0) or (0, 1); the latter has weight five. Decompose
a polynomial into its constant, linear and these quadratic pieces by ordinary field number and
engineering weight. Invariance of each piece follows from the linearity of the color action. The
linear piece vanishes by (FC1). Polarizing the (0, 0) piece, and separating the two jet orders in
the (0, 1) piece, gives invariant bilinear forms. Resolve their spatial tensor coordinates; (FC1) then
makes every color contraction a multiple of the color inner product. Consequently all weight-four
labels are spanned by the unit and ⟨Fab, Fcd⟩; through weight five one adds ⟨Fab, ∂eFcd⟩. This is a
spanning assertion about the actual invariant labels before further free equation relations.
Put Qab,cd = P
c F c
abF c
cd, with products interpreted as in (FM3), and define

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Na =
X
i
Qai,ai,
Ra =
X
i,k: i̸=a, k̸=a
Qik,ik,
Pab =
X
i
Qai,bi.
(FC2)

The sums in are over ordered pairs. Consider centered real symmetric tensor fields transforming under the signed coordinate group as rank-two tensors. Reflection in one axis multiplies by the product of its two axis signs. Thus a diagonal component permits only products in which every axis occurs evenly; for two antisymmetric pairs this forces the same unoriented plane twice. The permutations fixing axis have just two orbits: planes containing and planes disjoint from They give. A constant could also occur, but centeredness removes it: the Wick quadratic one-point functions are zero and the unit expectation is one.

For the off-diagonal component, the parity character is the product of the signs of The two planes must therefore share exactly one other axis, and be, up to antisymmetry and exchange of the two factors. Permutations fixing identify the possible remaining axes, giving precisely. The characters of the other products differ in at least one reflection, so averaging against the required character kills them. Permuting axes then determines every tensor component from 00 and 01. Reality makes the coefficients real. The whole centered signed tensor space is therefore spanned by

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Aab = δabNa,
Bab = δabRa,
Cab = 1{a̸=b}Pab.
(FC3)

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For a real polar vector of weight at most five, fix component zero and average under its stabilizing
permutations and reflection character. Constants and FF terms have the wrong parity. In an
Fab∂eFcd term, parity requires that the symmetric difference of the two plane-index sets be {0, e}.
If e = 0, the planes coincide, giving the first two sums below. Otherwise they share a remaining
index; the location of the derivative leg gives the last two sums. Thus its zeroth component is
spanned by
n =
X
i̸=0
⟨F0i, ∂0F0i⟩,
c =
X
⟨Fij, ∂0Fij⟩,

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i,j̸=0
i̸=j
a =
X
⟨F0i, ∂jFji⟩,
h =
X
⟨Fji, ∂jF0i⟩.
(FC4)
i,j̸=0
i̸=j
i,j̸=0
i̸=j

The free divergence equation gives = 0. Bianchi gives Summing over ordered distinct nonzero gives 2. Hence only are needed. A signed vector is determined by its zeroth component, since exchanging axis zero with any axis transports that component. The polar vector space has real dimension at most two. These relations hold on all solution towers, so they are zero in the actual free quotient by (FM4).

The lower rank bounds are obtained by explicit separated correlations, not by declaring the formal expressions independent. Let) be (FM2) and set

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Qab,ik;ce,fm(z) = Hab,ce(z)Hik,fm(z) + Hab,fm(z)Hik,ce(z).
(FC5)

The correlation of with at 0 is (dim g), because the two cross matchings force the same color. At, writing, the Hessian and third derivative of the Green function are

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ν−1∂r∂j∂iG0(et) = 8δijδtr −48δtiδtjδtr + 8(δriδtj + δrjδti).
(FC6)

These follow by differentiating. Substituting them into the four signed Kronecker terms in (FM2), and applying the two-factor product rule to (FC5), gives every sample below. The common nonzero factor removed is = (dim g Take the divergence component zero of (FC3), correlate it at with and at zero, divide by, and take real parts. The two resulting linear functionals have columns

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A
B
C
Q01,01
−256
0
128
Q12,12
0
−512
−128
.
(FC7)

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For example the first A entry is P3
k=0 ∂0Q0k,0k;01,01(e0)/ν2 = −256; the corresponding B entry is
the same sum over ordered i, k ̸= 0, which is zero. For C sum ∂rQrk,0k;01,01 over r ̸= 0, k. The
other row uses 12, 12. These finite sums and (FC6) give respectively the three displayed columns. A
third, undifferentiated sample correlates tensor component 01 at e0 with Q02,12 at zero; its columns
are (0, 0, −16), directly from (FC5). It separates the last tensor direction.
Thus A, B, C are independent, so the tensor space has dimension exactly three.
The two
divergence samples in (FC7) are independent; combined with the upper bound two from (FC4),
they identify the entire polar vector space and prove surjectivity of divergence. Rank-nullity gives
a one-dimensional conserved kernel. Solving (FC7) yields its generator

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1
2A −1
4B + C = T free.
(FC8)

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For its diagonal entry use P
i,k Qik,ik = 2Na + Ra; its off-diagonal entry is Pab. This is exactly
(FM6), already shown conserved. The last sample is nonzero on it, so the generator is nonzero.
Every conserved centered real signed tensor in this weight-four space is its real scalar multiple.
Any later nonzero scalar normalization, including either sign, preserves the rank and the conserved-
line statements; the normalization of shell fluxes is a separate argument.
There is a raw degree-five vector direction which must be kept until the equation quotient is
taken. In polynomial notation its components are

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Eb =
X
i,a
pairDeriv(b, i, a, a, i),
(FC9)

that is, the contraction of with the equation, in the normalized radial-jet coordinates used above. The two other raw vector directions are the polynomials called momentum1 and

momentum2 in the preceding classification. Every real signed-equivariant raw vector has a unique expression. To see the independence after imposing the Maxwell equations, take a unit color vector and the two solution jets with respectively

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(F (0), F (1)) = (leg12, leg0;12),
(leg01, leg0;01 −leg2;21).
(FC10)

Here the legs are radial potential jets, with the normalization made explicit below. Both satisfy the linear divergence equation and Bianchi identity. The first reads3 in component zero; the second reads = 1. An equation polynomial reads zero on both. It follows first that then that = 0. Conversely vanishes on every solution jet, directly by the divergence equation. A non-solution jet with nonzero divergence shows that itself is a nonzero raw polynomial. Thus the raw vector space has dimension three, and the kernel of its map to the free vector quotient is exactly the one-dimensional space

This statement is independent of coefficient coordinates. For a coherent degree-five frame, apply its inverse coordinate map to each component of a real signed-equivariant coefficient vector. This is a real linear isomorphism to the raw polynomial vector space, because the coordinate map intertwines signed permutations and conjugation. Composing with the free quotient projection gives a surjection whose kernel is the inverse-coordinate image of. A section is obtained by applying the corrected polynomial lift to each quotient component. That lift commutes with both symmetries, and its quotient projection is the identity. These facts justify the vector kernel subtraction used later: it removes precisely the equation direction, and does not discard either of the two nonzero free vector classes.

To make the raw coordinates completely explicit, write

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Pab;kcd = ⟨Fab, ∂kFcd⟩,
(V1)b =
X
i
Pbi;bbi,
(V2)b =
X
i,j
Pij;bij,
Eb =
X
i,a
Pbi;aai.
(FC11)

The pairings are the actual invariant polynomial labels, evaluated on radial potential jets; imposing the Maxwell equation is a later quotient operation. The basic legs in (FC10) are the homogeneous potentials

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(legab e)i(z) = 1
2(δibza −δiazb)e,
(legk;ab e)i(z) = 1
3(δibza −δiazb)zke.
(FC12)

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Taking their exterior derivatives defines the curvature legs, including the factor 1/3 in the first
curvature derivative. This factor explains the 4/3 sample in (FC10). A constant curvature is auto-
matically divergence free. For a linear curvature the Maxwell condition is exactly P
i curl1(i, i, b) = 0
for every b; direct differentiation of (FC12) checks it for the two combinations in (FC10). Higher
homogeneous jets can be set to zero.

Here is also the raw upper-dimension argument, before using those solution samples. A weight-five label decomposes as a constant, an invariant linear term, an term and an term. Simplicity kills the linear term as above. Total axis inversion fixes the two’s but negates a polar vector, so both its constant and terms vanish. Resolve its remaining invariant bilinear color form into the color inner product. To recover its spatial coefficients it suffices to use the mixed evaluations m (FC13)

where occupies jet order zero and order one. Take, in component zero, the three pairs

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(leg01 e, leg0;01 e),
(leg01 e, leg2;21 e),
(leg21 e, leg2;01 e).

If all three readings vanish, every mixed coefficient vanishes. Indeed reflections require the symmetric difference of its two plane-index sets to equal, where is the derivative index. For