Radial jets, calibrated labels and the free Maxwell comparison hierarchy · Part 2

where is the original barrier threshold, not a restarted threshold. Define the linear readout by requiring that the polynomial obtained from (PC5) have coefficient vector This determines because every diagonal entry of is nonzero. It also explains the exact calibration identity: the retained polynomial, including its Gaussian contraction, is the polynomial with those coefficients. There is no assertion that the whole untruncated exponential insertion is polynomial.

The analytic comparison estimate for this construction is measured before the rescaling

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|Ad,k,N,pv −Td,k,N,pv|1 ≤1
2|Ad,k,N,pv|1.
(PC7)

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We derive this relative estimate before solving any calibrated columns. Let n bound the ordinary
polynomial degree, let z = θk/g ≥1, g = β−1/2
N,p,k, and multiply every variable of the unweighted
insertion polynomial by z; call that polynomial q. Its coefficient sum obeys coin(q) ≤B|Av|1. The
actual nonlinear transported-variable substitution has the form

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Xα 7−→fα + θ−1
k SθkAα,
(PC7a)

where is linear, multiplies every variable by, and every monomial of has ordinary degree at least two. Their coefficient sums are at most. For 0

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coin(t−1StAα) ≤tB,
coin(fα + t−1StAα) ≤2B.
(PC7b)

The first assertion follows term by term from for2; the second is the triangle inequality. Telescoping each monomial of degree at most, replacing one variable at a time, bounds the nonlinear substitution error by coin(. For a covariance with entries at most1, Wick’s finite pairing expansion bounds coin() by coin(. After the linear substitution this costs a further factor. These two errors give

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A(t, B, ε, n) = n(tB)(2B)n + ε[2(n + 2)]nBn.
(PC7c)

Engineering truncation can only decrease coefficient sum.

For the actual holonomy Taylor reading, the calibration before Wick contraction has error at most in polynomial coefficients. This is the Taylor remainder of the transported slots in (PF8)–(PF11), with their ordinary nonlinear part separated as in (PC7a). On the finite-dimensional image of the polynomial exposure map, fix a constant1 bounding its inverse from polynomial coefficient sum to frame coefficient sum. The total frame error is therefore

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|Av −Tv|1 ≤c{ε + BA(θk, B, ε, n)}|Av|1.
(PC7d)

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The covariance here is the rescaled covariance z−2Q, since scaling variables by z and Wick contrac-
tion commute with that change of covariance. The band estimate gHk ≤2θk implies |(z−2Q)αβ| ≤ε
when |Qαβ| ≤B and 4B ≤Hk.
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).

Source–readout formulation. Two coefficient coordinates of the same transported insertion are in play: the transported coordinate (a source-side record: the primary coefficient carried by the history) and the read coordinate (what the Gaussian-contracted polynomial shows in the coherent frame). Theorem 16.3 is a recovery theorem for the reader: it is a bijection of the finite label space at each (), with explicit two-sided bounds, so no label information is lost at fixed level, and (PC10) is its exact section for a prescribed target reading. What is not recovered is uniformity in is not bounded, and the lift does not improve it. The readout is defined on primary coefficients and has not yet been passed to the physical quotient; identifying those two is the business of the equation image below.

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Theorem 16.3 (Relative comparison and invertibility of the readout). Eventually in the level,
uniformly in N, p,
|Ad,k,N,pv −Td,k,N,pv|1 ≤1
2|Ad,k,N,pv|1,

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hence
|Av|1 ≤2|Tv|1,
|Tv|1 ≤3
2|Av|1,
T is invertible and
|T −1z|1 ≤2Od,kad
k|z|1,
Tvunit = e0.

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Proof. Let n bound the ordinary polynomial degree, z = θk/g ≥1, g = β−1/2
N,p,k, and let q be the
unweighted insertion polynomial with every variable multiplied by z; its coefficient sum satisfies
coin(q) ≤B|Av|1. The nonlinear transported-variable substitution is

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Xα 7−→fα + θ−1
k SθkAα,

linear, multiplication of every variable by, every monomial of of degree at least two, coefficient sums at most. For 0

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coin(t−1StAα) ≤tB,
coin(fα + t−1StAα) ≤2B,

by for2. Telescoping each monomial of degree at most one variable at a time bounds the nonlinear substitution error by coin(. For a covariance with entries at most1 Wick’s finite pairing expansion gives coin(coin(, and the linear substitution costs a further

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A(t, B, ε, n) = n(tB)(2B)n + ε[2(n + 2)]nBn.

Truncation only decreases coefficient sums. The calibration before contraction has error at most (Taylor remainder of the slots, (PF8)–(PF11)). With1 bounding the inverse of the polynomial exposure map on its finite-dimensional image,

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|Av −Tv|1 ≤c{ε + BA(θk, B, ε, n)}|Av|1,

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the covariance being the rescaled z−2Q, with entries at most ε when |Qαβ| ≤B, 4B ≤Hk, by the
band estimate gHk ≤2θk. Choose a = [4(n + 2)]−1, B = Ha
k, ε = H−1
k ; eventually θk ≤H−1
k , and
the terms of (PC7d) have powers of Hk at most

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

with prefactors depending on only. Hence 21 eventually, which is (PC7); the triangle inequality gives (PC8). If = 0 then = 0 and = 0 by (PC4), so the endomorphism is injective, hence bijective, and (PC4) with (PC8) gives (PC9). The unit is exact: its representative is the constant polynomial, fixed by contraction and truncation, and both rescalings have weight zero on it.

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The order of choices in the theorem is: degree d and level k first, then the eventual threshold
in the level, uniformly in the cutoffN and period p. The factor Od,kad
k is the price of inverting the
reader at level k and is retained in every later use.

16.4 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
pairs
Kp,q(xi −xk).
(FM3)
i
ci
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.

Source–readout formulation. The free hierarchy is an imported comparison realization: a second realized sector over the same label algebra, built from the explicit kernel (FM2). It is not the interacting theory and nothing in its construction refers to the Wilson history. Its role in the supplied construction is that of a calibrated yardstick: detection, charge and Ward-kernel statements proved for it are transferred to the interacting hierarchy only through the comparison estimates that follow. The label algebra, with its restriction map, is the interface between the two sectors in the sense of the method: both hierarchies are read on the same labels, and discrepancies between them are retained as interface residuals.

The realized sector of the free hierarchy is explicit: its source record is the kernel on the fixed real symbol space, its readout is the family of moments of (FM3), and the only operations are multilinear extension, differentiation in the insertion points and restriction to smaller label sets. Each of these commutes with the moment formula because the sum over matchings is finite and the kernel is real analytic away from coincident points. No statement about the interacting measure is contained in this subsection.

16.5 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

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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
X
4∂b
X
a,i
F c
ai∂aF c
bi = 1
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.

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Source–readout formulation.
The free readout (all separated correlations with all spectators)
has a kernel on labels, and (FM4) computes it exactly: the information the free sector cannot see is
the equation ideal and nothing else. The quotient map from radial labels to Winv
d
is the readout; the
color-equivariant projection followed by radial reconstruction (Theorem 16.1) is its section. This
is a quotient equivalence in the method’s classification, proved by a strict positivity argument and
not assumed as nondegeneracy.

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Theorem 16.4 (Strictness at one point). If all sources sit at one point x with xa > 0 and H(0) = 0,
then
X

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

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for every polynomial Maxwell solution f, and therefore P
i:di=k pi = 0 on Sk for every k.
In
particular the one-point reflected matrix of any finite family of independent solution-coordinate
probes, of mixed jet orders, is positive definite.

Proof. Monotonicity and nonnegativity give) = 0 for 0 and by (GP2) and continuity vanishes on the reflecting plane. By translation the unshifted summed pulse vanishes on for all 0, hence on a half-space, and by real analyticity on the connected set it vanishes identically. The radial case of (FS2) applied to this zero pulse gives (GP3). For a homogeneous solution tensor the field) = asWave() has jet at order and zero jets at other orders at, and

Strictness at one point is the form of the statement that is used: it says that the readout “reflected pairing against all polynomial Maxwell solutions” has no kernel on solution-coordinate probes beyond the equation ideal, for probes of mixed jet orders at the same point. The weights + 2 are the engineering weights of the probes and are retained; they are what separates the homogeneous components in the proof.

16.6 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