Marked integration and the coupling reset · Part 3

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For a signed permutation with signs εi and permutation π, pullback of an affine jet changes its
slope to b′
ij = εiεjbπi,πj. Its constant term is

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a′
i = εi
aπi −1{axis i reversed}bπi,πi

.

The subtraction is necessary because a reversed positive link is read from the shifted base. Direct substitution of the lifted-coordinate increment proves this formula on the nonwrapping support. Its curl is the signed plane reframe of the old curl. Pullback of exponential link increments is a linear pullback of their direction fields; the Hessian chain rule at the critical identity therefore gives precisely the signed-plane covariance claimed for the projected form. No global identification of a nonperiodic affine polynomial on the torus is used.

Affine-Hessian coefficients and their bounds

Fix a unit color vector, which exists because the simple gauge algebra is nonzero. For each positive coordinate plane and color, let) be the affine wedge field with constant curl in that plane and zero curl in the other planes. It is linear in and commutes with adjoint color transformations. For the regular germ at the identity, define

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cab(f, o) = D2f(1)[Wa(ξ), Wb(ξ)].

For fixed, the bilinear form)] is invariant under simultaneous adjoint transformations, by gauge invariance of the germ. Differentiation makes it invariant under the infinitesimal adjoint action. The simple-algebra invariant-bilinear-form result therefore makes it a scalar multiple of the color inner product. Evaluating at fixes the scalar, giving

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D2f(1)[Wa(x), Wb(y)] = cab(f, o)⟨x, y⟩.

Since the primitive of a flux is the sum of its wedge fields, bilinearity gives

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D2f(1)[primitive(w), primitive(z)] =
X
a,b
cab(f, o)⟨wa, zb⟩.

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The invariant-bilinear-form argument and the reduction of affine Hessians to curls are the simple-
algebra and gauge-calculus inputs of this formula.
The constants in the projection estimate follow from actual normalized directions. The direction
tWa(ξ) is admissible in the paid second derivative bound; hence t2|caa| ≤cf. So is u = t
2(Wa(ξ) +
Wb(ξ)). At the critical identity, its repeated ordered left derivative equals its diagonal Hessian, and

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4D2f(1)[u, u] = t2(caa + cbb + 2cab).

Bounding the derivative on the left by and each diagonal term by yields. These estimates use only the paid directions at the identity, rather than a limiting periodic Fourier test.

The actual projected matrix is if the enlarged anchored support is nonwrapping, and zero otherwise. Its entry bounds hold in both cases. On a nonwrapping support it has exactly the flux-Hessian formula above. This cutoffis preserved by signed rotations and translations, as is the enlarged radius-two support. The pullback formula for flux Hessians then gives covariance of its quadratic form under the signed plane action, including the zero case.

The paid calculus used for these statements bounds all ordered left derivatives up toorder in the specified anchored direction class, on the closed healthy set. The underlying functions remain smooth on full neighborhoods of each such configuration. Thus it differentiates the unmasked germs,

not their discontinuous indicators. Constant multiplication multiplies the bound by the absolute constant; sums and differences add bounds. For an inner product the ordered Leibniz rule has at most 2 splittings, giving

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pairingCost(E, k) = 2k∥innerSLE∥.

Linearity of the derivatives and Hessian under subtraction follows on these full neighborhoods; enlarging the healthy support only restricts the configurations tested. These facts justify the residual and clover bills, including boundary points of the healthy sets.

Isotropy of the summed coupling projection

Let be the actual countable activity listing, with its regularity, paid bound and label permutations implementing every signed lattice motion. Define the matrix at anchor by

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To(a, b) =
X
{i: anchor(i)=o}
Πi(a, b).

These sums are absolutely convergent: the coefficient boundΠ is dominated by the summable activity bill. Finite sums over plane types can therefore be interchanged with this series. Reindexing by the label permutation for a signed motion, using its exact anchor, support and value transformations, gives

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QTaffine(s,b)o(w, z) = QTo(reframesw, reframesz).

In particular the nonwrapping cutoffin each projected entry has already been transported before the label sum is taken.

Test this equality with fluxes supported in single planes and equal there to the fixed unit color vector. Translation and a permutation taking the reference plane (0 1) to any chosen plane show that every diagonal entry at every anchor equals ∆= 1)). For two distinct positive planes, there is an axis occurring in one and not the other. Reflecting just that axis changes the sign of their mixed pairing and fixes their plane labels. Its coefficient is consequently its own negative and is zero. Therefore

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To(a, b) = ∆1{a=b},
X
anchor(i)=o
QΠi(w, w) = ∆
X
a
∥wa∥2.

Applying this with the actual clover flux, and the same singleton healthy indicator on every term at that anchor, proves the projected-payout identity used in the reset exponent. It would not justify replacing different individual support indicators by that singleton indicator; the faults above are precisely what account for that change.

The diagonal coefficient estimate gives the sharper bound∆, where is the listing’s anchored paid budget. Indeed the triangle inequality bounds the anchor-zero diagonal sum by its local mass, and that mass is at most. The off-diagonal factor three is not spent here. Thus both the scalar and its bound are obtained at the original finite period from the original label orbits.

The local clover counterterm

The petal flux at an anchor is the based clover: the signed average of the four oriented plaquette curvatures in each coordinate plane, all based at. Every such oriented square touches, and all its visited sites lie within distance two of. Thus a healthy singleton anchor bounds each square chord

by, and the petal flux reads only the radius-two clover. Its gauge transform is adjoint transport by the gauge value at; signed lattice motions act by the corresponding signed action on plane indices.

For a word of length, curvature is the linear color projector applied to its matrix word minus the identity. The projector norm is at most the inverse color scale. The paid word derivative bound therefore gives the curvature bill

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t colorBill(G, l, P, k),
colorBill = G.colorScale−1feeWalk(G, l, P, k).

Adding the four clover terms with coefficients of absolute value 14 gives this same bill with for each petal. The ordered product derivative rule for their inner product gives

pairBill( pairBill = pairingCost(Colororder) colorBill(powerorder)

Here the pairing cost accounts for all ordered derivative splittings already used in the paid calculus. For a real matrix on plaquette types, the counterterm is

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2
X
Qq(U) = 1
a,b
qab⟨petals(U, o)a, petals(U, o)b⟩.

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Its bill is at most 1
2pairBill t2 P
a,b |qab|. It is smooth, gauge invariant, supported on the radius-two
clover, and linear in q.
Its normalization follows directly from affine tests. For the exponential link line U(t) = exp(tv),
differentiating a based word at zero gives the signed link sum along that word. For a square this is
the discrete curl
[vj(x + ei) −vj(x)] −[vi(x + ej) −vi(x)].

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For the nonperiodic affine field associated to a linear jet, on a nonwrapping neighborhood this equals
its constant curl. The signed clover average consequently has derivative equal to the jet’s positive-
plane flux w. The period n > 20 ensures the clover neighborhood has the required affine-coordinate
increments. Hence
Qq(U(t))/t2 −→1
a,b
qab⟨wa, wb⟩.
2
X

Gauge invariance for the simple group makes the identity a critical point of every regular germ. Taylor’s formula then identifies the left limit with one half its diagonal logarithmic Hessian. Therefore the counterterm Hessian on affine directions is exactly the quadratic form on their curls. Symmetry of the Hessian and polarization recover all mixed affine pairs from these diagonal tests. This establishes the counterterm normalization used in the residual decomposition.

Masked residual decomposition

For a regular term anchored at with support, enlarge the support to halo). Let be the counterterm built from its projected affine Hessian and the local petal flux. Define

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(
0,
D+ is healthy,
f(1) + 1{o} healthyqf(U) −1D healthyf(U),
otherwise.
rf = f −f(1) −qf,
df(U) =

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Then the exact identity is

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1D healthyf = f(1) + 1{o} healthyqf −df + 1D+ healthyrf.

On a healthy enlarged support all three smaller health tests hold and the equation reduces to the definition of. On its complement the enlarged-support term vanishes and the equation reduces to the definition of. This is why the constant(1) is retained even outside the original healthy support. If has paid supremum and derivative bound, its projected matrix entries obeyΠ . The quadratic counterterm bill is thus at most, where

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Mp = 1
2q2 3 pairBill(G, p).

Indeed its half-weighted double type sum has entries, and each petal pairing has bill pairBill(. It follows by the triangle inequality that both the rebate’s paid bound and the fault’s all-configuration absolute bound are at most (2 +. The value(1) costs at most The fault vanishes on a healthy enlarged support, and hence is a forced term with an actual failure witness.

The rebate is regular on, vanishes at the identity and has zero affine Hessian: the projected counterterm has exactly the affine Hessian of, while subtraction of(1) changes no derivative. This identification uses the nonwrapping support and period hypotheses of the affine-curl construction. Gauge invariance and measurability pass to the counterterm, rebate and fault. The fault uses the enlarged support’s field data and its final-threshold flags, precisely as permitted for forced terms. Signed rotations and translations send the petal flux and the projected matrix covariantly, and send healthy sets by their actual affine images, so they commute with this decomposition. Projection and the counterterm are linear in; consequently differences of rebates and faults are the rebate and fault of the difference. These facts justify applying the decomposition to the common parameterized source series in the reset.

Low-path atoms and the loan cancellation

The low indices are a plaquette type together with either one direct symbol or an ordered pair of path-row labels (). For a direct index let its mark bedirect, its length zero, and its plain value the plaquette price. For a pair let

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m = cicj,
l = length(si) + length(sj),
P(U) = 1
2⟨si(U), sj(U)⟩.

Its analogue on a flux is2 in the direct case, and2 in the pair case. The proxy) is this analogue evaluated on the local petal flux at the anchor. The home is the anchor’s radius- halo together with both true path stencils in the pair case; in the direct case it is just the halo. Let and be the healthy indicators on this home and on the anchor singleton. The live value) is the clipped plaquette price for a direct index, and 1) for a pair. The freshness and loan functions are exactly

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(
0,
hH = 1,
(α + ∆)J −∆mhoQ,
hH = 0.
F∆= ∆m(Q −P),
Lα,∆=

The masked plain value is. On a healthy home the selector is kept and the direct plaquette is below threshold, so. Also = 1. Substitution gives

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(α + ∆)J + hHF∆= αhHmP + ∆mhoQ + Lα,∆.

On an unhealthy home the same equality is the defining formula for the loan, with both homemasked terms zero. This proves the low loan cancellation in all cases, including a healthy anchor inside an unhealthy home.

These functions have the required locality despite the selector’s apparent infinite test. Outside the radius- aperture its exponential threshold already exceeds the maximum chord two, by the decent-budget hypothesis. Inside it the plaquette corners lie in the aperture unless the test is automatically true. Thus the kept predicate is determined locally. Slot values read their true route and plaquette stencils, and proxy values read only the anchor’s radius-two clover, contained in the home. Gauge conjugation preserves chords, prices, kept predicates and the inner products of the transported slots. All these functions are measurable; a loan is a finite combination of continuous functions and measurable healthy/kept indicators. Its possible extra flag read is exactly the final-threshold read allowed in the definition of the forced dictionary.

Writethreshold andrelaxation. A kept slot of length has norm at most Using the chord bound two and the clip kink inequality for the direct term, both cases give the all-configuration estimate

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On a healthy anchor the proxy has the derivative bill 1
2pairBill(G, p)t2. Consequently

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|Lα,∆(U)| ≤|m|el/ρ 
2t|α + ∆| + 1
2|∆| pairBill(G, p)t2
.

A nonzero loan therefore has an actual unhealthy home, and its bound retains the path length and relaxation factor needed for its support charge. It is linear in ( ∆), as is the freshness in ∆, so their difference estimates use the same formulas with parameter differences.

Finally the freshness has zero affine Hessian. Along an affine exponential link field, a slot divided by its scalar parameter tends to the corresponding constant plaquette curl. This follows by differentiating the finite link product; the two nonconstant affine edge terms cancel to the constant curl. On a nonwrapping home, both the plain value divided by the parameter squared and the proxy divided by that square tend to the same flux analogue. Their difference therefore has zero quadratic limit. The freshness is smooth in a healthy neighborhood, vanishes at the identity, and the affine Hessian limit identifies its quadratic coefficient with this limit; polarization gives zero mixed affine Hessians. The nonwrapping and period hypotheses are retained here, as in the reset projection argument.

Recovering the actual kinetic base from the path atoms

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The low pair sum is the square of the original path row, with all ordered labels retained.
If
fi = cisi(U) and P
i ∥fi∥< ∞, then

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X
i
fi



2
.
X
1
2⟨fi, fj⟩= 1
2
i,j

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Absolute convergence follows by domination by 1
2(P
i ∥fi∥)2; continuity of the inner product then
permits the two sum exchanges. Applying this to each row proves that the total live low terms are
exactly the clipped direct term plus the kept shortening half-square. The pair selector is common
to its whole row, so the assertion holds on both selector branches.
For flux analogues the same square identity gives half the squared norm of the actual zero-
momentum symbol row. If Sab = R.symbol0(a, b), the path normalization is

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c.direct δbi +
X
a
SabSai = δbi.

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Expanding the symbol-row norm squares and applying this matrix identity proves
X
2
X

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low d
md analogued(w) = 1
a
∥wa∥2.

All series converge absolutely by the weighted path coefficient bounds. Multiplying by the common healthy singleton indicator and evaluating on the petal flux is therefore legitimate.

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The layer operation is the sum over translated signed views divided by the number of signed
orientations; translations are summed, not probability averaged. For each view-dependent term,
translation covariance changes its anchor-zero value to its value at the translated anchor. Reindex-
ing the finite sites by the signed permutation then proves that the low live layer plus the cubic and
quartic masked layers is exactly the stipulated kinetic energy. It also proves that the low proxy layer
is P
o headsp(o, U): signed reframe preserves the sum of squared plane norms, and the orientation
count cancels its inverse. Thus both identities used in the reset’s path cancellation come from the
original symbol normalization and actual path rows.

On an everywhere healthy configuration every masked plain low atom is unmasked. Their layer is then the original unclipped second term, namely the orientation average ofdirect price+ This specialization does not remove masks in a configuration with any flags.

The high freshness terms retain the actual cubic and quartic atoms on their own footprints and have coefficient + ∆). An atom of arity = 3 or 4 has paid bill its startup coefficient times ; since 01, both are bounded using. Along any exponential link line every slot is first order, so the atom divided by the line parameter to the power has a finite limit. Dividing instead by its square leaves a factor tending to zero. Taylor’s formula at the critical identity then gives zero diagonal Hessian for each high freshness term, and polarization gives zero mixed Hessian in arbitrary directions. This proves the required vanishing without changing the cubic/quartic masks or attaching an extra boundary term.

The unmarked local scalar

The normalization in the reset uses the same local pool 0 as the density expansion. Its trained connected-log series has sum log ) and is absolutely convergent by the paid log estimate. The incoming-to-current label equivalence preserves the actual union support. For each of these fixed labels use minus its stream as the regular coefficient on that support. The support-mask identity for the trained stream says that this masked coefficient is minus the corresponding trained log term. Reindexing its absolutely convergent series therefore gives the regular sumlog

Adjoin the literal path-score terms with their original supports. Their masked sum is

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−γR(V ) −β
billcubic(V ) + billquartic(V )

 −billloop(V ).

Thus the combined regular dictionary has additive sum equal to this expression minus log No anchor is selected from an evaluated field: the anchor for each stream support is chosen once from that nonempty support, and its address is retained along the label equivalence.

All these complete sums are invariant under a signed translated view. For the path bills, reindex the finite signed-seat average. For the pool logarithm, reindex the trained connected labels by the exact incoming-label motion; its support and eligibility data move with it. The regular dictionary now uses a separate full-seat average: each pulled coefficient is multiplied by the reciprocal of the number of seats, including translations. The healthy mask is pulled with its actual support. Each view has the same complete sum, so summing these averaged coefficients preserves that sum exactly. This full-seat normalization differs from the orientation-normalized layer operation on kinetic paths.

The field-independent scalar token is the original sector prefactor times the incoming vacuum exponential times the fixed deducted chart factor. It is positive. SetSite log(token). The actual scalar overhead is then

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(
)
τ|Sitec| −β averageorientations
X
i
baseTerm(V, i)
O(V ) = exp
.

This follows by separating the token from the exact clipped-base exponential in the overhead; the threshold in that base is the same one in the dictionary.

Insert the combined regular sum into the frozen exponent, whose channel decomposition contains exactly that base, the cubic and quartic bills at price, and the loop bill at its stated price. The latter three cancel the path-score terms. The receipt correction cancels the receipt in the regular sum. The remaining exponent is log ) + log ). Therefore

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exp{Efrozen(V ) −γR(V )} = O(V )C(V ).

If an earlier dictionary fusion has already been performed, its attached-log sum is added to the exponent on the left, giving exactly the scalar normalization formula stated above. Positivity of ), supplied by the convergent real log identity, justifies exp(log

The forced gas in this same frozen representation is the canopy-pool sum. Its convergent density identity is ) gas( ) = push( ). Multiplying the scalar equality by this gas proves the early evaluator identity for every field, including fields where the gas vanishes. This identifies the unnormalized push without dividing by its gas or its value at a selected field.

Pool averaging and the literal score series

The regular pool uses integrated averages of the single-forest activities. If every forest value has absolute value at most a common tariff, the probability average has the same bound, by the triangle inequality and cancellation of the finite averaging cardinality. Thus the original thin and thick tariffs also bound the averaged entries and deposits. The renewed reserve gives the connected-log convergence hypotheses for these averaged entries, with the same supports and compatibility relation. Applying the already proved real gas/log identity gives

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log C(V ) =
X
connected labels q
Gas.term(entriesPool(V ), q),

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X
!
C(V ) = exp
q
Gas.term(entriesPool(V ), q)
> 0.

It is this averaged gas, not an average of logarithms of separate forest gases, that occurs in the scalar proof.

For the full density, split each nonzero molecule family into components with nonempty flags and components with empty flags, retaining the requirement of nonempty ground. The two subtype maps have an injective union map and recover every nonzero family. Thus the integrated full gas reindexes as a sum over thick families and thin families. With a thick family fixed, excluding the thin supports it meets amounts to setting those thin entries to zero. Their absolute bounds can only decrease, so the same log theorem applies. The omitted-cluster restoration identity factors this excluded gas as

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!
−
X
q
missingA(q, V )
C(V ) exp
.

Multiplying by the banked thick-family factor and summing gives the pool density identity used above. Throughout this step the forest averaging is performed at the level justified by disjointground factorization, rather than assuming arbitrary products commute with a common forest average.

The literal path score can also be read directly term by term. A low score label consists of a signed translated seat and a low atom. Its coefficient isSigned times the atom’s marked plain value in that view; its support is the view of that atom’s full home. Pulling its healthy mask through the view gives exactly the corresponding summand of. A cubic, quartic or loop score label consists of a seat and an atom label, with its actual transformed footprint. Its coefficient is respectivelySignedSigned orSigned times the unmasked atom. Applying the footprint mask produces its placed atom.

The absolute atom budgets justify summing these labels. Reindexing the seat’s translation by its signed action identifies the seat sum with the orientation average of the sum over all anchors. This proves the exact score total used above, with no change of masks. Finally the channel decomposition is simply the expansion of its definition:

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inChannels(β) = β average
X
baseTerm + β(billcubic + billquartic) + billloop.

These are convergent equalities before any smallness estimate or cancellation of cubic coefficients is imposed.

Why a nonzero detector forces a warning

The quiet-watch condition at o says every fluctuation coordinate with anchor within B of o has t/g. Since the lifted field is+gz, each color coordinate of its difference from is at most t. If the color dimension is d and, the elementary Euclidean bound by the sum of coordinate absolute values gives

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∥(μ + gz)e −μe∥≤dξt,
∥(μ + gz)e∥≤(M + dξ)t

on those links. A watch on radius 2B gives watches of radius B at nearby sites, by the distance triangle inequality. Choose a positive error margin and the fine/noise plateau bounds

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2/√cdirect(A/L2 + τ) < 1/4,
2/√cpenalty(A + τ) < W.

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These choices are possible in the required order. First take τ = √cdirect/32; next choose L large
enough that A/L2 ≤τ; finally choose W larger than 2(A + τ)/√cpenalty. The stationary maximum
A is chosen before this block-length choice.

The physical local path estimate has three contributions to its discrepancy bill: the effect of the fluctuation away from, the protected mean bill, and the nonlinear path-approximation remainder. Choose 0 with1 and with the first contribution at most3. The source’s coefficient bound makes this possible by taking no larger than divided by one plus three times its displayed linear-response coefficient. The mean and approximation errors have reference-coupling order and are eventually each at most3: multiplying either by sqrt(reference coupling) tends to zero, while t is at least the inverse square-root coupling. Their sum is therefore at most

Away from the mandatory bad-seed region, the clean-patch and grafted-field estimates apply to these same links. They place the likelihood site and pin tags on their plateau and identify