Physical labels and their detection

Physical labels and their detection

From coefficient readings to physical labels. The free comparison hierarchy and its Taylor readers have supplied a finite coefficient language. The physical detector must now be built from actual local lattice cylinders. A coefficient vector and its chosen representative are different objects. The calibrated section selects a representative, while the quotient identifies precisely the equation directions that all allowed spectator readings annihilate.

The arguments below first bound coefficients before passing to limits, then prove equation cancellation using Haar integration by parts. The triangular and equivariant constructions preserve this cancellation through the chosen section, gauge averaging and corrected whole cylinders. Trials are placed in the same physical histories; their limits cannot be assembled from individually favorable but incompatible histories. The Kelvin and Wick tests supply the separation needed for the detector. What is recovered is the complete coefficient reading and its physical image on the stated finite degree spaces. The bare representative is recoverable only after a section is fixed; quotienting does not make it unique, and a finite reader is not an inverse on all unreduced histories.

17.1 A detector on physical coefficient space

At fixed degree let be coefficient space, its equation ideal and = dim(). The foundation identifies ker with an invertible image of, so rank-nullity gives dim ran uniformly in cutoffs; detector notation has = dim + 1. This dimension statement does not assert that every bare coefficient survives the equations. Choose trained vectors bounded by. Their limiting response matrix has full column rank. Compactness of its unit sphere gives a positive lower norm, stable under sufficiently small entry perturbations 0, with left-inverse bound Train at 4. Finite training and short-distance probe errors each cost. Replacing trials by adds a third error: the difference has zero projection, and equation cancellation makes its probe eventually below, simultaneously for finite. The projected response is within 3 and its columns are independent. The range dimension bound makes them spanning. With projection norm bound, their norms are at most; stable inversion yields

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∥πix∥≤ArvL ∥Liπix∥.

Choose training index first, then small positive probe scale, then a tail of the whole-mesh filter. Constants are independent of remaining cutoffs and volume. Equation cancellation uses banked insertion estimates (IC3)–(IC5) with flight exponent32; separated reflected Wick pairing below supplies the full-rank reference. For explicit stability let obey in sup norm and set. Column errors of at most imply

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∥(H −G)u∥≤rδ∥u∥,
∥u∥≤K∥Hu∥+ 1
2∥u∥.

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Thus L = 2K suffices. Reference injectivity follows from correlation functionals separating the
free quotient; their basis matrix is G. The lift retains equation ideal, projection, trained columns,
response functionals and ordered choices. Recovery preserves their finite rank and stability proof on
the same varying projected columns, with the banked and Wick dependencies retained explicitly.
For explicit coefficient-space recovery, let pE read coefficient data and jE be its section, with
pEjE = IE. Define
bπi = jEπipE,
bLi = LipE.

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Then pEbπi = πipE, and ran bπi = jE(ran πi).
The transported section norm makes this range
isometric to the physical range, preserving its dimension r and its detector inequality:

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∥bπiζ∥Z ≤ArvL ∥bLibπiζ∥.

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The lifted trial columns are jEπiej,i and have exactly the same response entries as the original
projected trials. Hence all three error allowances and the explicit stable inverse are recovered on
the same columns.
Equation-kernel recovery remains ker bπi = p−1
E (ker πi); section composition
returns the identified equation ideal. This transports rank, detector constants and the ordered
training/probe/mesh choices while distinguishing coefficient projection from the gauge-field read-
out.

17.2 Bounding physical coefficients before limits

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For difference coefficients xi let Xi = πixi, with π2
i = πi and detector inequality ∥Xi∥≤Z∥LiXi∥.
Bounded Wilson insertions and vanishing tested errors give ∥Lixi∥≤c, but the equation remainder
need not be coefficient small. Normalize instead by

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fi = (1 + ∥Xi∥)−1,
yi = fiXi,
zi = fi(xi −Xi).

Then1, and closure under bounded scaling and subtraction of bounded families keeps admissible. Idempotence gives = 0. Literal Wilson equation cancellation makes its tested observables vanish, eventually2. Consequently

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Z∥Li(Xi −xi)∥≤1
2(1 + ∥Xi∥),
∥Xi∥≤Zc + 1
2(1 + ∥Xi∥),
∥Xi∥≤2Zc + 1

Thus projected primary costs are fixed before limits. The unprojected remainder has zero projection, hence vanishing and fixed cost by cancellation. Subtracting it and the original approximation error gives fixed cost for the difference insertion, using the actual detector copy count. For slot correlations, a position difference in slot changes only its factor. Compact-space continuity gives integrability, so finite subtraction commutes with Wilson integration. Iteration expresses mixed lattice differences exactly by shifted/differenced slot insertions. The detector and cancellation estimates therefore give their stated fixed-cost bounds. The lift retains idempotent projection,

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admissible normalized families and slot shifts. Recovery preserves this normalization and finite
subtraction, transporting the absorption argument and its literal Wilson cancellation dependency
without replacing it by a coefficient-norm hypothesis.
Using the coefficient section jE, realize the normalized families as bXi = jEXi, byi = jEyi and
bzi = jEzi. Their transported norms give

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fi = (1 + ∥bXi∥Z)−1,
bπibzi = 0,
bLibzi = Lizi.

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Thus the identical cancellation input and detector bound produce ∥bXi∥Z ≤2Zc + 1, recovered
by pE and the section. For a lattice shift h of slot j, retain both slot addresses and define its
difference reader by finite subtraction. Coefficientwise lifting is linear, so the difference of lifted slot
readers equals the lift of that difference. Exact Wilson measure pushforward then preserves the
equality between the correlation difference and the integral of the differenced slot product. Iterating
preserves every prescribed mixed difference and its actual detector copy count. This recovers the
fixed-cost conclusion with the same admissibility and equation-cancellation dependencies.

17.3 Equation cancellation in the physical detector

At grade, the marking identity for gives zero Wilson expectation of its marked insertion against bounded measurable spectators outside a finite neighborhood on sufficiently large lattices. A finite basis of makes radius and size thresholds uniform by taking maxima; linearity permits arbitrary complex coefficients. Fix reporting level. Fine chambers eventually contain this neighborhood strictly. Place translated copies in the prescribed chambers, distinguishing chamber zero; all other copies form a bounded measurable exterior spectator. Therefore

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Eβ
Y
chambers a
fa = 0.

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The distinguished copy keeps its exact scale normalization, so its replicated absolute-root seminorm
is zero on the whole mesh and every volume, with k fixed first. For terminal chart Zi and cali-
brated invertible change Pi, a kernel family has yi = P −1
i
Z−1
i
xi and ZiPiyi = xi. The source also
identifies Ziyi with the marked equation carrier. The calibrated inverse bound preserves admissible
subexponential coefficient growth, with size multiplier at most two. The trained chart gives

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upper(ZiPiyi) −hallmarks(yi) = Oroot(flight−1/32(i)).
(IC3)

Indeed expand in the fixed finite signed-orbit frame tally. Each basis comparison costs flight scalar bounds sum subexponentially and are eventually at most flight/flight Finite

recombination is exact, and signed motions/conjugations commute by covariance. The hallmark seminorm is zero by chambers, hence

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upper(xi) = Oroot(flight−1/32(i))
(xi ∈ker πi).
(IC4)

Apply to by idempotence. Projected and unprojected trials have identical limiting responses. The equation carrier is an invertible image of, and the free observable kernel is exactly , giving free quotient rank by rank-nullity, without assuming independence of all bare coefficients. Reference rows are linear free correlation functionals, variable quotient field in the first slot and fixed basis spectators elsewhere. Their separating family has zero common kernel, so its quotient-basis matrix is injective. Stability and (IC4) therefore act on the same projected trial columns. The lift retains marking neighborhoods, chamber copies, trained charts and finite orbit tally. Re-

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covery preserves the local spectator identity and exact chart recombination, transporting zero root
seminorm and its flight error with the original growth and limit-order hypotheses.
The chamber cancellation has an exact source integral reader. Let F = Q
a fa be the original
finite chamber product, with its distinguished copy and exterior spectator recorded. Exact Wilson
pushforward gives
Z
Z
F ◦π dμZ
β =
F dμβ = 0.

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The section recovers each translated copy and its scale normalization; its support addresses retain
the exterior-spectator condition. Taking the same absolute-value root therefore gives the identical
zero seminorm.
For the chart step, retain the inverses in their stated order: yi = P −1
i
Z−1
i
xi
and ZiPiyi = xi. Coefficient readout and section recover these finite linear operations and the
signed-orbit tally. Each flight majorant is the same scalar reader, so the subexponential coefficient
payment transports (IC3) and then (IC4) with exponent −1/32. The same recovered projected trial
columns consequently have the original cancellation and detector responses on the fixed-reporting-
level whole-mesh tail.

17.4 Haar integration by parts and triangular local realization

Vary a stored positive link byexp(, with derivative. Product Haar is invariant and

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Here Ee is the sum of the incident plaquette curvatures, with a backward incidence transported by
the inverse preceding link. For plane i < j, the axis-i incidence is Fij(x)−AdU−1
x−ej,j Fij(x−ej); axis

gives the negative analogous difference, with other types zero. Ordered four-factor differentiation supplies inverse-link signs. Curvature projection and skew-adjointness give normalization

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−Re tr(colorMat(A) mat(g)) = repDim⟨A, curvature(g)⟩.

For continuous and derivative, and bounded measurable spectator unchanged by this variation, invariance yields (IC5)

The group law extends the derivative to every Compactness bounds the continuous Wilson-weighted derivative; multiplying by the spectator bound and the fundamental theorem gives a uniform integrable difference-quotient majorant. Differentiate the constant Haar integral and divide by its positive partition function. Spectators need neither continuity nor gauge invariance. For = 1, nonzero gives exact equation expectation zero. Marking retains multiplier derivatives on

varied links. The local identity holds outside a common radius, above a common torus size, for all nonzero real beta. Finite sums, scalars and conjugation preserve it by taking maximal radii/sizes and using the real Wilson law. One must also arrange the desired continuum label while enforcing these exact identities: the finite graded construction supplies a linear pair), with gauge-invariant local polynomial of leading jet. Strictly higher lexicographic errors make triangular and injective by its first nonzero graded piece; finite dimension makes it invertible. The same argument preserves descending weight filtrations under. Thus has leading jet, retained power filtration and exact equation marking. Paired construction imposes conjugation, signed symmetry and fixed unit on Enclose finitely many frame-basis polynomial stencils in one centered signed-invariant box containing zero; rewrite in its coordinates and extend linearly. Basis equality recovers on all coefficients. The lift retains varied-link incidence, marking radii, graded ordering and this common stencil. Readout commutes with Haar integration and finite triangular inversion, preserving (IC5) and the exact leading-jet realization.

The varied-link operation has an explicit realization on frame data. Write for the physical color direction and) for the stored link. Keep fixed and replace its link coordinate by

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Ae(t) = rx exp(ta)r−1
x Ae.

Its readout is = exp(, with every other physical link unchanged. Conditional Haar invariance therefore recovers the original variation. Differentiating a smooth reader composed with gives the physical reader exactly; the exterior spectator is unchanged under this realized variation too. Exact Wilson pushforward then transports (IC5), with its original bounded measurable spectator and positive partition function. Section recovery gives the same incidence signs and color normalization. For the finite graded realization, retain the linear matrices

and the polynomial stencil together. Reading the leading jet before and after returns, while the exact equation marking is recovered under the same finite linear combinations. These are local operations chosen before subsequent scheduling, so recovery preserves that order of choices and the fixed stencil’s derivative credits.

17.5 Graded and equivariant assembly of the fixed section

With, the fixed section obeys(coords(, mapping the equation ideal to the exact marking class. Compact polynomial bounds supply derivative credits. Choose section and once at grade, before block, period, coupling, refinement or derivative-budget scheduling. This explains why the local equation cancellation survives all subsequent scheduling choices. For each weight, let be homogeneous labels and. Use the ordinary linear polynomial lift on. On a finite basis of, positioned equation lifts give pairs () with higher-weight , prescribed form, power bound, gauge invariance and marking identity. Extend linearly and choose algebraic projection. Then

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Fm(p) = F eq
m (Lmp) + F ord
m (p −Lmp).

On the ordinary term vanishes; in general both preserve grade and leading label. Since the unit maps to (1 1). Finite homogeneous summation gives; homogeneity of the ideal preserves marking on its entire image. For signed label action andpull), average

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Fav = |S|−1 X
s∈S
φsFθs−1.

Each summand preserves leading label, higher weights and marking. Weights sum to one, retaining the unit; group reindexing proves equivariance. For semilinear real structure c), set

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FR(p) = 1
2(Fav(p) + cFav(p)).

The paired semilinear composition is complex linear. Real structure commutes with signed action and preserves grades, forms and marking, so the midpoint is real and signed-equivariant with triangular leading map and fixed unit. Perform these averages before common-stencil construction or cutoffchoice. The lift retains homogeneous projections and finite symmetry averages. Recovery commutes with this linear assembly and triangular map, preserving the section’s exact marking property and scheduling independence.

Recover the pair-valued graded lift by leaving its label coordinate unchanged and lifting its polynomial coordinate: (). Record the signed pullbacks and their induced source action so that their readouts are the original signed pullbacks. Finite linearity then gives

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bFm(p) = bF eq
m (Lmp) + bF ord
m (p −Lmp),
bFav = |S|−1 X
s
bφs bFθs−1.

Section recovery returns the original pairs in both formulas. Complex conjugation of coefficient readers commutes with composition with, so the same midpoint recovers. Leading labels, the unit and higher-weight differences are therefore unchanged, and the equation subspace still has its exact marking reader. The common finite stencil and triangular inverse can consequently be retained with the same grade and power filtrations, chosen before all later scheduling parameters.

17.6 Finite polynomial gauge averaging and equation-ideal generation

For a powered local polynomial, the constant-color orbit(gauge) spans a finite-dimensional space: fixed representation and finite polynomial degree give finitely many coefficient tensors. Gauge change of variables preserves its marking identity and spectator radius and bound. Paired orbit vectors (1) are finite dimensional; continuous first-coordinate and evaluation functionals separate them and hence span the dual. Coordinate Haar integration yields an element of their span with first coordinate one and second

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 ̄f(U) =
Z
G
f(gaugeg U) dg.

It is a finite linear combination of orbit pairs. Leading label, higher weights, power and marking therefore remain those of an actual finite polynomial. General gauge action on transported polynomials uses its root value; Haar right invariance proves full gauge invariance. Equivariance and uniqueness identify the color-invariant leading form in label space.

On degree- jets choose projection onto the injective free gauge-jet range. Variable substitution defines algebra map, and equation labels are exactly = 0, since polynomial evaluations separate. Split each linear functional as (. Telescoping products against projected products expresses every kernel polynomial as a finite sum with a literal equation-linear factor. Projection preserves jet degrees, so homogeneous terms keep their bidegrees; homogeneous components remain in the ideal and no nonzero field-count-zero constant belongs to it. To lift, express the equation functional as a finite combination of linearized transported lattice forces. For each force choose other-factor stencils avoiding its varied edge; their product is unchanged under that variation, so (IC5) gives exact local silence. Combine only after obtaining each identity. Avoiding ordinary factors agree with their linear functionals modulo field-count at least two; finite product errors have count at least + 1, and after the force at least + 2, above target + 1. Retained higher linear force jet weights also remain strictly above target. Truncation and finite combination thus give the target leading term with higher-weight remainder and exact silent lattice function. This constructs positioned raw lifts of homogeneous ideal labels with the required triangular form. Every resulting term contains one literal factor vanishing on free solution jets. The lift retains finite

orbit tensors, jet projections and individual avoiding stencils. Recovery commutes with their finite algebra and normalized Haar averaging, preserving both exact silence and the full higher-weight bookkeeping. The equation-factor decomposition has the explicit finite identity

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n
Y
n
Y
n
X
k=1
(lk −lkPj)
Y
r<k
(lrPj)
Y
r>k
lr.
k=1
lk −
k=1
(lkPj) =

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Its coefficientwise realization preserves every product, jet degree and literal equation factor. Section
recovery returns the original polynomial identity; finite monomial summation therefore recovers the
full ideal construction and its higher-weight errors. For gauge averaging, each realized orbit reader
is fg ◦π. At a fixed realized input,
Z

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G
fg(π(z)) dg =  ̄f(π(z)).

The retained finite-dimensional orbit span represents this average by the same finite polynomial coefficients. Its paired first coordinate remains one, preserving the leading label. Each force summand retains its own edge-avoiding stencil and exterior-spectator identity before the finite sum is formed. Recovery thus preserves the actual polynomial realization, exact marking and triangular jet bookkeeping together.

17.7 Force-span and edge-avoiding probes

Linearized root-transported forces along source-edge-avoiding paths have finite expression

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d−2
X
X
a<b
incidenceν,a,b(F (l)
ab (x) −F (l)
ab (x −ei)).
l=0

Carrier derivatives drop out because identity curvature is zero; incidence signs are those of (IC5). A transverse difference of face averages is the three-unit-interval integral of a transverse derivative by the fundamental theorem. Rescaling base and mesh by, its order-homogeneous expression, with the extra transverse factor, scales by. If every force probe vanishes, multiply orders by

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and test integer addresses whose physical rescalings approach arbitrary x. At points t indext(x),
which tend to any prescribed x ∈R4, the resulting continuous expression vanishes.
As t ↓0,
lower orders vanish and interval averages shrink, leaving P
i ∂iF (j)
iν (x) = 0. Bianchi already holds
for radial-potential curvature jets, so the top jet is free.
A top-jet functional annihilating the
free range annihilates the common force kernel. Finite-dimensional annihilator duality expresses
it as an actual finite combination of force probes. Ordinary probes avoiding one forbidden edge
span the entire finite jet dual.
Outside finitely many sampling addresses, both approach path

and plaquette boundary can avoid the edge. Vanishing safe samples imply curvature polynomials vanish at all integer points outside that set. Scaling to nonzero eventually avoids the exception; top homogeneous curvature vanishes, then continuity includes zero. Descending induction and injective radial reconstruction give zero jet array. Thus safe probes have zero common kernel and finite-dimensional duality gives finite representations of all ordinary functionals. Finally

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⟨X, Adg Ee⟩= ⟨Adg−1 X, Ee⟩.

An edge-avoiding carrier makes this color test invariant under varying. Expand in a fixed real color basis; each scalar coordinate, avoiding multiplier and remote spectator is unchanged. Apply

(IC5) with = 1 termwise and sum finitely. The locality radius includes the force edge, and a large enough side prevents periodic collision with remote edges. This completes exact positioned marking. The lift retains safe routes, incidence signs, finite probe combinations and locality radii. Readout preserves the finite-dimensional kernel argument and termwise Haar identity, recovering the same equation lift and jet filtration.

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The finite probe representation is recoverable as a linear certificate. Choose a finite spanning
subfamily of the force probes λ1, . . . , λm for the relevant annihilator, and record the coefficients of
leq = Pm
q=1 aqλq. For each forbidden edge record likewise the finite safe-probe representation of
each ordinary factor. With coefficient readout pE, composition gives

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m
X
leq ◦pE =
q=1
aq(λq ◦pE).

The coefficient section recovers this exact certificate and its homogeneous jet ordering. Every lattice probe reader keeps its own carrier and sampling stencil, so edge avoidance survives readout. Its transported color test, avoiding multiplier and remote spectator are consequently unchanged under the realized link variation already constructed. Termwise recovery of (IC5), followed by the recorded finite color sum, preserves the exact marking identity with the same locality radius and torus-size restriction. Thus the finite dual certificate and its local lattice realization are recovered together.

17.8 Uniform detector-trial placement

For fixed trained spectators define the actual lattice probe

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n
Y
Mi(f; x) = Eβi[f(x0)
ν=1
Fν(xν)],
i = (N, p).

Compact-space continuity makes it complex linear in the first insertion. Linearity follows by expanding the first factor under the finite Wilson probability integral; all local polynomial insertions are continuous on the compact configuration space. Forward differences there equal differences in its first position; iteration and the detector’s finite averaging preserve linearity on coefficients. Trained generic placements lie inside one physical box. Choose closed generic neighborhoods with 1; pair-coordinate gaps have a positive minimum. Physical periods at least four preserve this reserve for every volume, including the smallest admitted torus, since differences below two cannot be closer to a nonzero period translate. Unextracted insertion estimates give uniform local continuity on bounded coefficient balls. Replace the 4( + 1) coordinates of nearby placements one at a time, staying inside the neighborhood. Every intermediate replacement stays within their original physical distance and within the chosen neighborhood. Each costs at most, yielding

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|Mi(f; x) −Mi(f; y)| ≤4(n + 1)An+1R/r.
(IC6)

Reflected labels retain uniform bounds by their finite real-linear action. A scale- detector average displaces points by at most 2 per tested direction, so (IC6) makes its error arbitrarily small uniformly on the whole-mesh tail and all volumes. Placement is exact because count( Unaveraged trained columns approach free reference responses by finite-level training, whose error tends to zero by the trained insertion estimates, and smooth free evaluation at approximate generic positions, whose error tends to zero by smoothness of the free hierarchy on the distinct-position configuration space. Intersect the finitely many row/column tails. Choose training error below