Physical labels and their detection · Part 2

then averaging error below4 by small, then projection error below4 by (IC4). One common mesh tail gives rank without extraction. Uniform trial and projection bounds give the required prelimit detector inequality. The lift retains placements, separation reserves, direction lists and exact refinement counts. Recovery preserves these correlations and finite differences, transporting (IC6) and all three errors in their specified order.

The realized detector probe is the same correlation reader:

Searchable transcription

c
Mi(f; x) =
Z
!
n
Y
◦π dμZ
βi = Mi(f; x).
f(x0)
ν=1
Fν(xν)

Store each prescribed physical position and its exact refinement count with the reader. The coefficient section recovers the tested insertion, and the field section recovers every placed slot. Hence (IC6) has the same separation and coefficient-ball bound for all admitted volumes. The detector’s finite probability average commutes with this equality. If bounds the displacement of its sampled placements, (IC6) bounds its averaging error by 4(, with0 for the fixed direction list. This recovers the ordered training, averaging and projection allowances on their common whole-mesh tail, including reflected columns and the exact position identity. No new placement or extracted subsequence is introduced by realization.

17.9 Complete corrected cylinders and common trial limits

For commuting square-zero markers set

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F(ε) =
X
A⊆{1,...,n}
Sfree(xA; qA)εA,
D = log F,
H(ε, U) = e−D(ε) Y
a
(1 + εaPa(U)).

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Empty moment is one; log and exponential are finite nilpotent polynomials.
Training bounds
every nonempty coefficient of H in replicated root seminorm by the common gapk →0. From
FH = Q
a(1 + εaPa),
Y
a∈A
Pa = FA +
X
B⊊A
FBHA\B.

Thus moment error is bounded by the finite absolute coefficient price of times gap. That price stays bounded near the fixed distinct placement, since all free submoments are continuous. This supplies training convergence in (IC6). Singleton correction is; the pair is

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(Pa −F{a})(Pb −F{b}) −(F{a,b} −F{a}F{b}).

Detector rows use at most two slots and retain this actual free cumulant. The pair correction is therefore its actual free cumulant at the rescaled placement; it cannot be replaced by a product of separately centered one-point terms. For each grade and training index finitely many lifted columns lie in a compact ball uniformly in both cutoffs. Countably many grade/training/column triples permit successive extraction and diagonal refinement of both cutofforders, preserving earlier hierarchy and projection limits. Replace placeholder limiting columns by these actual limits without changing finite-cutoffbounds or placements. The corrected-cylinder bound passed through the fixed-placement hierarchy gives convergence to the same free matrix. Stable rank and equation dimension make limiting projected columns span the physical quotient; retaining these trials under null-label removal gives the free quotient dimension at every grade. A single reference frame and separated point pair suffice. Frame images span the quotient and include a nonzero unit; extend it to a basis from those images, unit first. Real homogeneous frame and conjugation/bidegree-preserving

projection give the same properties to this quotient basis, without probe orthonormalization. The lift retains all corrected coefficients, common refinements and selected frame images. Recovery preserves finite marker convolution and the cumulant, and commutes with the retained compact coefficient limits and hierarchy comparisons.

Coefficientwise source recovery of the complete cylinder. Use the finite algebra with commuting square-zero generators,

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Nn = R[ε1, . . . , εn]/(ε2
1, . . . , ε2
n).

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Extend the configuration lift coefficientwise by
X
!
=
X
A
(fA ◦π)εA,
[εA]Lnf = ([εA]f) ◦π.
Ln
A
fAεA

This unital algebra map fixes the free scalar coefficients at the actual rescaled placement. Since the augmentation ideal is nilpotent, its logarithm and exponential are finite sums, so

Searchable transcription

LnH = e−D Y
a
(1 + εa(Pa ◦π)),
F LnH =
Y
a
(1 + εa(Pa ◦π)).

Section restriction recovers every coefficient simultaneously. For nonempty, the normalized Wilson reader therefore gives the same finite error sum

Searchable transcription



* Y
+
* Y
+
≤
−FA =
X
B⊊A
FB⟨HA\B⟩,
X
B⊊A
|FB|
gapk,
a∈A
Pa
a∈A
Pa
−FA

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with the source’s replicated-root control of each reader understood. The lifted integral is identical
by normalized measure pushforward. In particular, coefficient recovery retains the pair cumulant
at this placement.
Write c(λ)
g,k,i for the finite-dimensional coefficient columns on the common cutoffrefinement. The
lift of these columns is the fixed linear section jE, and recovery is pE, with pEjE = I. Consequently,
for every convergent column on the retained refinement,

Searchable transcription

lim
λ jEc(λ)
g,k,i = jEcg,k,i,
pE lim
λ jEc(λ)
g,k,i = cg,k,i.

Here the refinement is the source’s refinement of the ordered cutofflimits, retaining its earlier hierarchy limits. Enumerating all () gives one compatible retained family; it introduces no replacement for the prescribed cutofforder. The same section and recovery transport the selected real, homogeneous unit-first frame and its quotient dimension.

17.10 Separated Wick detector and Kelvin triangular responses

For, let be free two-point Wick pairing. The free quotient embeds in Maxwell solution-jet polynomials. Reflect every tensor index of the conjugate polynomial through the plane perpendicular to, giving Reflected positivity yields 0 for Hence )) is injective: basis expansion ofwould otherwise contradict positivity. The all-spectator null space therefore has no remainder beyond the equation ideal. Choose 1) for a generic pair. One row per basis partner, with one spectator each, gives precisely the finite detector.

To retain the joint-order positivity input, put and let be perpendicular reflection. A homogeneous order- real solution probe has response

Searchable transcription

reply(P)(y) = sjr(y)2R⊗2
y P(inv(y)),
sj = 4(j + 1)(j + 2)(−2)j normal .

Translate its inverted-argument polynomial byinv(), decompose into degrees and multiply each by, obtaining with diagonal. Their response sum is

Searchable transcription

r(y)2R⊗2
y sjP(inv(y) −inv(z)).

For, inversion difference is) with smooth and. Homogeneity gives factor, so jets below order vanish and the normalized order- jet is. The lift retains the quotient basis, reflected partner, Kelvin factors and triangular lower-order probes. Recovery preserves their finite polynomial transformations and Taylor jets, linking the detector to the same joint-order positivity construction continued below.

17.11 Joint-order reflected Wick positivity

Polarize the Kelvin line identity to recover symmetric derivative tensors. Reflection contributes derivative sign (, giving the transformed tower Gram form

Searchable transcription

n−1
X
j=0
cj(z)⟨Pj, Qj⟩tensor,
cj(z) = (−1)jsjr(z)j+2 > 0.

The probe change is triangular with identity diagonal and invertible; equivalently its positive Gram identity proves injectivity and finite dimension gives invertibility. Thus the original simultaneous reflected form is positive definite on the whole tower. Choose a real orthogonal tower basis with squared norms 0. Cross-insertion Wick contractions give

Searchable transcription

⟨eα, eβ⟩refl= 1α=β
Y
i
αi!dαi
i .

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Remove one left factor: only matching right indices pair, giving βidi, and induct from the empty
monomial. Within-insertion contractions are absent for these Wick products. Hence
*X
α
pαeα
+
α
pαeα,
X
refl
=
X
α
|pα|2 Y
i
αi!dαi
i
> 0

for a nonzero polynomial. This includes mixed engineering weights, field-count degrees and the constant. The lift retains tower order, triangular probe map and cross-insertion contraction rules. Recovery preserves polarization, the finite basis change and the exact factorial weights, establishing the same joint-order detector positivity.

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Finite tower and detector recovery.
For the finite solution-jet tower E = L
j<n Ej, use
coefficient section and recovery jE, pE with pEjE = I. Write Tz = I +Nz for the retained triangular
probe transformation. Its strictly order-lowering part satisfies Nn
z = 0, giving explicit source and
readout maps

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n−1
X
bTz = jETzpE,
T −1
z
=
a=0
(−Nz)a,
pE bTzjE = Tz.

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The identity for bTz is on the horizontal coefficient image; no inverse on redundant vertical coefficients
is asserted. In terms of homogeneous polynomial extraction [·]k, its blocks are exactly

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Tjk(z)P = sj
sk
[P( · −inv(z))]k,
0 ≤k ≤j.

Thus the source retains the translation of the polynomial argument together with its form-index reflection. The normalized jet reader is the physical jet reader after; it recovers both the zero jets below and the specified order- Kelvin jet, including the single opposite-point factor (

in the reflected pairing. For the simultaneous reflected form define its source form by

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GZ
z (ζ, η) = Gz(pEζ, pEη).

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It recovers the entire tower form on jEE. Its polynomial extension retains the Wick contraction
rule, so in the retained orthogonal coordinates the source and recovered polynomial norms are both
X

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α
|pα|2 Y
i
αi!dαi
i .

The empty multiindex contributes; mixed orders and degrees remain in this same finite sum. Finally the detector reader is

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DZ
x,y(ζ) =
Bx,y(pEζ, bi)

m
i=0,
DZ
x,yjE = Dx,y.

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Since the physical detector is injective by the retained reflected-partner argument, its source kernel
is precisely p−1
E (0) on this coefficient carrier. Horizontal restriction therefore preserves its rank,
including the unit column, at the specified generic pair (0, (1, 1, 1, 1)).

17.12 Normalized Taylor derivation of the Kelvin response

For symmetric order- tensor, use

Searchable transcription

DP f(x) =
X
∂ν1 · · · ∂νjf(x)
j!
P(eν1, . . . , eνj).
ν1,...,νj

Solution tensors have harmonic trace constraints. For = normal

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DSimG = (−2)j normal rj+1Sim(x, . . . , x).
(IC7)

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Induct from j = 0 with normalized recursion DP f = (j + 1)−1 P
a ∂aDP(ea,·)f.
Use ∂ark =
−2krk+1xa.
Tensor-polynomial derivatives sum to vanishing frozen traces; the other sum re-
solves P
a xaea = x.
The normalization cancels j + 1, leaving the next −2 factor.
Put Vi =
P
a xaSai(x, . . . , x). Bianchi and Euler give

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∂iVm −∂mVi = (j + 2)Sim,
X
a
∂aDSaiG = −2(j + 1)(−2)j normal rj+2Vi.

The first identity gets two copies from differentiating and from homogeneous derivatives; the second uses (IC7) and zero divergence. Contracting the two-curvature kernel with is twice the curl of this vector, ordered by the opposite-side derivative. Differentiation yields

normal

The bracket reflects both antisymmetric form indices by; the double radial contraction vanishes by antisymmetry. Degree- homogeneity changes(inv()), giving the stated Kelvin response. The lift retains normalized tensors, trace/Bianchi/divergence constraints and derivative ordering. Recovery preserves their finite contractions and induction, including every factorial and sign in (IC7) and the detector’s Kelvin coefficient.

The second-point order- sign ( differentiates and is included once. These Greenfunction identities hold offthe origin.

Differential recovery offthe diagonal. Regard the Green function and tensor responses as functions of the physical separation= 0, with their coefficient source independent of. For a smooth coefficient-valued response, finite dimensionality gives

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pE∂α(jEf) = ∂αf,
pEDP (jEf) = DP f.

Tensor contraction is read through before the physical contraction; thus trace, divergence and Bianchi constraints have exactly their stated physical kernels. In particular (IC7) is recovered as

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pEDSim(jEG) = (−2)j normal rj+1Sim(x, . . . , x).

On the two-point configuration domain, the same reader gives

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pE∂α
y jEf(x −y) = (−1)|α|(∂αf)(x −y).

It follows that the normalized factorial, curvature curl ordering and second-point sign are preserved by one differential readout. These maps act on the smooth off-diagonal domain used in this passage; they do not supply a new contact extension at zero separation.

17.13 Selecting trials on the physical history

Fix a finite detector panel, real reference representatives and positive comparison width. Arbitrarily late comparison indices supply actual metering, restarted representations, admissible paths and one-/two-insertion and marked estimates. For completeness, the passage from arbitrarily late indices to a sequence makes all the geometric limits simultaneous. For all sufficiently fine they hold for every volume; representations/readouts may depend on, while and its bounds do not. Only insertion arities at most two are needed, since each panel row has at most one additional insertion. One finite linear label transport serves every row, retaining filtration, reality and unit. Its inverse gives real lifts with

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sup
N,p,ν
∥xN,p,ν∥≤Dk < ∞.
(IC8)

Complete exceptional early refinements by unit tuples and enlarge; no estimate uses them. Comparisons propagate from interval midpoint to starting width through the actual intermediate representations. Reference coupling stays between half and all its value at, so metering remains valid. Conditioned insertion and path propagation give the starting-width bill with common shipping error0. This is a statement about the transported physical insertions: it requires neither an assertion of a stronger density comparison nor a newly chosen scale history. Intersect finite panel thresholds on this same history. There are finitely many panel rows and coefficient requests, so all their eventual thresholds can be met together. To make this statement include the finitely many refinements preceding the availability threshold, put every member of the tuple equal

to the unit there, and enlarge to include its norm. Writing starting width) and positive testing scale), admissible indices have0. Calendar depth proves the first: a bounded width would contradict eventual calendar increase beyond its plateau depth. Intersect the arbitrarily late supply with, banking, grade and finite fit conditions. Recursively select increasing) and decreasing), all cofinal or tending to zero as required. Choose (IC8) lifts at each selected index. Prescribed lattice marks fit the quarter box with separation and stencil admissibility and approach fixed physical positions. Conditioned estimates for all sufficiently fine and every, followed by stencil realization, identify trial entries with corrected errors) uniformly in volume. This supplies the actual trial sequence with common readout, unit and conjugation properties. The lift retains the selected physical history, finite transport and quantifier order. Recovery preserves its bounded lifts and cofinal selection, including the uniform volume scope of the detector trials.

Trial carrier and uniform-volume recovery. Retain the selected calendar index, finite label transport, admissible path, and physical marks as part of the source trial record. If is that transport and its fixed reference representatives, then on available refinements its coefficient source and recovery are

Searchable transcription

xk,N,p,ν = L−1
k,N,pbν,
bxk,N,p,ν = jExk,N,p,ν,
pE bxk,N,p,ν = xk,N,p,ν.

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With the section norm, (IC8) reads ∥bxk,N,p,ν∥= ∥xk,N,p,ν∥≤Dk. The exceptional early refinements
use the prescribed unit tuple on both sides. Their completion has no role in the eventual comparison
estimate. For each selected kn there is one threshold N0(kn) for the finite panel, with quantifier
order
∀n
∃N0(kn)
∀N ≥max{kn, N0(kn)}
∀p.

All rows use the same at these indices. Realization and normalized measure recovery give exactly the same trial matrix:

Searchable transcription

MZ
n,N,p = Mn,N,p,
MZ
n,N,p;rν −Mref
n,N,p;rν
 ≤ε(kn).

Here the reference entry and corrected comparison are those at the prescribed lattice marks; their physical configuration limit is retained. This equality preserves the source’s uniform volume bound while allowing the representations and readouts to depend on

For an explicit cofinal selection, at theth recursive step intersect the same arbitrarily late supply with max, and min, as well as and the stated banking, grade and fit conditions. The source’s eventual limits make this intersection nonempty. This records one sequence with all required limits, on the already retained physical history. Section restriction recovers its unit, real structure, marks and every finite trial entry before compact extraction.