Marked integration and coupling reset · Part 2

13.7 Cluster logarithms with repeated occurrences

Let be symmetric incompatibility with) for every label, and nonempty supports such that incompatibility implies intersection. For real activities the hard-core gas is

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Z(w) =
X
Y
i∈A
wi,
A compatible

including the empty subset. The logarithm instead uses ordered occurrence tuples with repetitions. On their incompatibility graph, define

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φ(z) =
X
(−1)|E(H)|,
H connected
H⊆incompatibility graph
n
Y
L(w) =
X
X
1
n!
z∈In
φ(z)
j=1
wij.
n≥1

A singleton graph has coefficient one. Equal labels at different positions are incompatible, essential even though the gas uses distinct subsets. Breadth-first exploration partitions connected subgraphs into intervals with one fixed spanning tree as lower endpoint and an allowed edge set as upper endpoint. These intervals cancel by optional-edge toggling; each remaining interval is one tree. In each interval, if an optional edge exists, toggling that edge is a sign-reversing involution, so the alternating sum is zero. Thus is bounded by the number of compatible rooted spanning trees. Cancellation is performed inside the connected-graph sum before taking absolute values. Assume 1 and

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X
i:x∈Xi
|wi|e(a+δ)cB(Xi) ≤γ,
4(2B + 1)4γ
δ
≤q.
sup
x

The designated-root estimate bounds size- clusters with root-activity budget by, normalized by 11)!. For an unrooted cluster containing a label with a chosen property, sum possible marked root positions and divide by; its 1! normalization is bounded by that rooted expression. Tuple relabeling preserves graph coefficients and products, including repeated labels. If the union contains, root at an occurrence containing, with. A nonzero connected coefficient has overlap-connected union; meeting-support induction and0 give

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eacB(∪jXij ) ≤
Y
j
eacB(Xij ).

Summing+1 bounds the absolute logarithmic terms containing, with actual union-support price, by). Summing site root bounds proves absolute convergence at fixed torus with total at mostΛ). The lift retains occurrence tuples, incompatibility edges, root positions and factorials. Recovery commutes with the finite graph cancellation and preserves the union-support inequality. Nonnegative absolute majorants then justify the same infinite occurrence sum, without identifying tuples that share the same label set.

The interval cancellation is itself a recoverable finite operation. Order potential edges by their recorded endpoint orders. For each tree interval with an optional edge, choose its least optional edge and map to. This operation stays in the same interval, squares to the identity and reverses (. Readout retains endpoint orders, depths and the least-parent rule, hence recovers both and this involution before any absolute value is taken. For each finite occurrence

tuple the resulting is therefore identical on recovery. The section also recovers every tuple position, including positions carrying equal labels. The subsequent root-position sum keeps its ! normalization, while distinct-set relabelling keeps its separate! cancellation. Finally the nonnegative bound) controls the recovered size tail with its actual union support. This gives an absolute majorant for passage from finite tuples to the full lifted connected series.

13.8 The real logarithmic identity

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Positivity of the gas is a conclusion of the identity. First restrict to a finite alphabet of s labels
and put P(t) = Z(tw), L(t) = P
n≥1 lntn. Connected-component decomposition, designating one
occurrence and choosing its component and remaining graph, gives

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n
X
n[P]n =
k=1
klk[P]n−k.

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Position choices and component factorials give precisely this normalization. The cluster estimate
gives |ln| ≤Cqn−1. Choose u > 1 with qu < 1. The series and differentiated series converge
uniformly on compact subsets of (−u, u), using the summable derivative majorant Cn(qu)n−1. The
recurrence yields P ′ = L′P, so (Pe−L)′ = 0. Its value at zero is one, giving P(1) = eL(1). For
arbitrary alphabets direct finite subsets by inclusion. Every fixed gas subset and occurrence tuple
eventually appears unchanged. Dropping exclusions gives absolute gas majorant Q
i(1 + |wi|) ≤
e
P
i |wi|, finite by sitewise budgets at fixed torus.
The connected-series absolute majorant was
established above. Dominated convergence on both sides proves

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Z(w) = eL(w) > 0,
log Z(w) = L(w).

Removing all labels incompatible with a specified forced set only decreases absolute budgets, so the same identity applies. In the lift, coefficientwise recovery preserves the finite component recurrence. The retained absolute majorants justify passage through readout to the directed-alphabet limit; thus the recovered positive gas and its real logarithm obey this exact identity with the source’s occurrence normalization.

Write and for the gas and connected series restricted to a finite alphabet. For a parameterized physical activity family), its lifted family is)). The finite graph and occurrence identities established above give

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ZD(wZ) = ZD(w) ◦π,
LD(wZ) = LD(w) ◦π.

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At each admitted input the same absolute majorants control the directed limits, yielding these
identities for Z and L on the full alphabet. Since the real gas is positive there, scalar functional
composition gives
log Z(wZ) =
log Z(w)

 ◦π = L(w) ◦π,

and composition with recovers the original real logarithm. If a forced set is specified, keep it and its incompatibility incidences in the record. Restricting the alphabet then commutes with readout, so the same recovery identities hold for the forced-compatible gas with its unchanged, or decreased, absolute budgets. The logarithm is consequently recovered on exactly the positive domain supplied by the source identity.

13.9 Primitive marked integrals and independent tree averaging

A marked team is a finite collection of real local leaves with distinct keys, weight equal to the product of bounds and ingredient equal to the product of functions. At fixed coarse configuration and local tree choice, choose a permitted flag sector, local layout and finite warning and regularexponential selections. Together they form molecule address. The selected integrand is the paid team ingredient times local vertex factors, zero unless every team key occurs in. With local data , sector and selected integrand, its coefficient is

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Z
IX,k,p(h, z) dνX(h) dγX(z).
aX,k(p) = 1approved(X,k)

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Covariance and Gaussian density are those of this same tree choice; selectors and eligibility remain
inside the integral. Average over finite layout boundary choices, assigning zero to disallowed layouts
and retaining the original scalar overhead. The local vertex and integral estimates are developed
in the packet, non-packet and warning arguments below.
Expand warnings by Q
j(1 + dj) = P
Q
j∈D dj and regular exponentials by their absolutely
D
convergent subset expansion. Marked regular keys are dormant: their original coefficient is zero
and their unmarked increment e−0 −1 vanishes. Inserting and deleting these keys in selected lists

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is unique; the team ingredient supplies them. Forced marked keys must be present in the sector,
otherwise both expansions vanish.
Layout, sector and selection lists are recoverable from the
molecule address, giving an injective assignment on permitted data, with explicit inverse obtained
by reading these pieces from p. The activity majorant bounds integral norms by team weight times
primitive tariffs. Summability, Fubini and integration of summable-norm series yield
X

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p
overheadX aX(p) = P(team density)(U).

The Gaussian substitution is exact:, with that same density and tribute prefactor in the transformed integrand. Thus each nonzero molecule occurs with exactly the specified coefficient, and disallowed molecule addresses contribute zero.

A molecule depends only on tree choices in its enlarged ground. Project fluctuations to the finite coordinates used there. Data agreeing on that ground have equal projected covariance and hence equal Gaussian marginal measures; integrands agree on projection fibers. Their integrals coincide. Local measure-preserving gauge coordinate maps give the analogous gauge statement. Compatible grounds are disjoint, so finite independent tree averaging factors:

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Av
Y
p∈A
fp =
Y
p∈A
Av fp.

Factor the finite assignment sum over disjoint coordinate sets; unused coordinates contribute one. The overhead is tree invariant, preserving compatibility and inventory conditions after averaging, so averaging the team expansion yields the product of the averaged molecule coefficients. Every leaf key in a compatible molecule set belongs to exactly one molecule. Global distinct-key tickets therefore correspond bijectively to their local restrictions, with amplitudes factoring across molecules. Finite Mark-integral linearity gives exactly

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X
!
X
A compatible,permitted
overhead
Y
Y
i∈D
vi aD(p)
p∈A
D ticket
X
!
Y
= P
i∈D
vi teamDensityD
(U).
D

Primitive-integral and support-hit estimates supply the required summable norms and footprints. The lift retains local data, addresses, selected measures, ground projections and ticket partitions. Recovery commutes with these exact finite assignments and measure pushforwards; retained summability permits the same series integration. This reconstructs the literal Mark expansion used in the gas/log argument.

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The measure part of this recovery can be written for each fixed local data choice X. Let RX
read the physical gauge and fluctuation coordinates from the realized integration domain, with
realized measure bηX satisfying
(RX)∗bηX = νX ⊗γX.

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This is the same selected tree’s measure, including its covariance. Lift the entire integrand, including
selectors, by bIX,k,p = IX,k,p ◦RX. Then

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Z
bIX,k,p dbηX = aX,k(p).
1approved(X,k)

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The section recovers its coordinate formula. For a ground-coordinate projection QG, composition
gives (QGRX)∗bηX = (QG)∗(νX ⊗γX). Thus equal local marginals and fiberwise integrands give the
same locality identity after realization. All address and tree-choice labels are retained, so the finite
independent average and the global-to-local ticket bijection commute with these integral equalities.
The summable integral-norm majorant then permits the same molecule series to be integrated on
either side. No tree covariance is replaced during this recovery.

13.10 Forced-dictionary fusion and scalar normalization

Old forced labels have support, nonempty inventory and coefficient Attached logarithms have nonempty supports and are nonzero only when meets the flag set. Assume summable uniform envelopes for both. Use the disjoint atom alphabet with factors and1. A new label is a finite overlap-connected atom set containing an old forced label, whose old forced supports are mutually disjoint. Its support and inventory are the respective unions, its coefficient the product restricted to exact inventory and original eligibility.

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These labels are countable and include every singleton old label; enumeration by N only names
them. Expand the old gas times Q
j(1 + (elj −1)), justified by absolute summability. In a nonzero
term each attachment meets a flag supplied by an old inventory and hence an old support. Every
component therefore contains an old forced label. Components give disjoint new supports with
inventories covering all flags. Conversely a permitted new collection recovers the unique old label
and attachment sets by union; nonempty atom supports prevent shared atoms. Exact inventory
and disjointness recover local old eligibility. Products agree, proving

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Gasnew(U) = Gasold(U) exp
X
j
lj(U).

Regular labels and coefficients retain their regular role. Finite products and eligibility indicators preserve gauge invariance, measurability and union-support locality. Prior widening is an injective old-label placement in a universal dictionary with zero unused coefficients, preserving gas and logarithms by absolute reindexing. The regular dictionary and its original coefficients remain the regular part of the fused representation. Thus decomposition and union are inverse on all nonzero terms, with identical products. This choice only names the labels and does not change their functions or bounds. The dictionary fusion used twice in the coarse construction has an explicit formula.

For switched forced budget and attached-log budget at exponent + 1, inflate old envelopes to0, and assign attachments 2, where1 in the admitted logarithm range. Use there. The combined budget is. Every new component contains an inflated forced atom, so undoing inflation pays at least; other undone factors are at most one. Eligibility is switch independent. If1, the connected-component bound spending one exponent unit gives output price1) at exponent. The selected payment inequalities fit this within output allowance. Unmarked values follow the same fusion formula. Exterior-switch silence passes factorwise: omission of the required switch parent from the union means every factor and eligibility indicator is unchanged when erasing it. These supply value, price and scope after both coarse dictionary changes. The fusion estimate for switched coefficients follows from the distinct-component forest bound already proved.

For intermediate frozen exponent, reset receipt with coefficient, and consumed attached-log sums, retain the exact normalization equations

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Efinal + L2 = E0 −αR,
exp{E0 + L1 −αR} = overhead Z0,
eEfinaleL1eL2 = overhead Z0.

The last follows by addition of exponents. The scalar normalization needed after these fusions is an equality of exponents, not a comparison of partition functions. The lift retains atom types, exact inventories, switch scopes, both component decompositions and the reset receipt. Readout preserves their inverse union construction and these scalar equalities, with the paid-envelope bound transported through the same nonnegative connected-component sum.

Explicit fusion recovery uses the atom-set label itself, rather than its enumeration index. For a permitted connected atom set, write for its support union, for its old-inventory union and 1 for its exact eligibility indicator. Then

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KC = 1elig,C
Y
a∈C
ha,
KZ
C = 1elig,C ◦π
Y
a∈C
(ha ◦π) = KC ◦π.

The atom tags and incidence record recover and the inverse component union; composition with recovers each coefficient. Absolute summability then transports the gas identity, including its attached exponential, with the same paid-envelope majorant. Apply this to both fusions, retaining their distinct attached sums. The recovered scalar multiplier is exactly

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eEfinaleL1eL2
 ◦π = (overhead Z0) ◦π.

Thus the switched expression uses the same recovered as its unmarked normalization; subtracting log cancels that factor on its positive domain. No separate normalizing constant enters the lift.

13.11 The exact coupling-reset exponent

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Both fusion-normalization sides use the same unmarked gas Z0 as the switched formula; subtraction
of its logarithm cancels that normalizer. Retain the stationary path and vacuum choices. For source
v on listing X, the kinetic path family R and fixed threshold, support scale and Taylor order p, use
the actual healthy masks. Set ∆= δ(X. read(v)) and retain the committed loop coefficient α. The
reset is
β′ = β + α + ∆,
τ ′ = τ −|Site|−1 X
i
v. logReadi(1).

The receipt) averages the masked plain low terms along the original paths. It is not obtained by changing paths to match a later choice. Regular-source decomposition into identity value, projected payout, fault and masked rebate consists of absolutely summable series, bounded by the original paid bill and fixed multiples. The calibrated projection, using the simple-group hypothesis and period 20, sums payouts at their anchors and gives

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(X. read(v)). additivep(U) =
X
i
v. logReadi(1) + ∆
X
o
headsp(o, U)
−
X
i
dusti(U) +
X
i
untaggedi(U).

For averaged low and high correctionsand loan sum, termwise identities give

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(α + ∆)KR + F∆+ Hα,∆= αR + ∆
X
o
headsp(o, ·) + Lα,∆.

The kinetic base splits into live low terms and high summits; the high correction cancels ( times each summit. Each low term obeys

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(α + ∆) live + fade∆= α maskedPlain +∆sketch + loanα,∆.

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Sketch sums equal head sums. Path bounds make live, masked plain, proxy, fade and summit sums
absolute; the identity also proves loan summability. Finite seat averaging preserves this, justifying
all cancellations.
The reset additive sum is untagged plus F∆+ Hα,∆; its attached log sum is
dust plus Lα,∆. Substitution into τ ′|Site|−inChannels(β′)−additive, using channel decomposition,
yields
Ereset(U) +
X
i
logi(U) = Efrozen(U) −αR(U).

Exponentiation gives the unnormalized-evaluator identity with its exact constant and signs. It holds when the activity gas vanishes: that gas multiplies both sides and is never divided out. Fusion absorbs the attached log exponential. The lower projection, live-base and loan formulas retain the same threshold and paths. In the lift those paths, masks, projection hypothesis and paid series are realized data; readout recovers them before summation, so absolute convergence transports the exact reset and its vacuum scalar without a new path choice.

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An explicit reset reader retains the pair of scalar outputs as well as the transformed dictionary.
Put
X
c0 =
v. logReadi(1),
r(β, τ, v) = (β + α + ∆, τ −c0/|Site|) .
i

The listing, projection, paths, threshold masks and finite seat average are included in the same realized input tuple. Their scalar readers give exactly and ∆, so componentwise readout commutes with r, and the section recovers its original two outputs. Define the reset residual

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C(U) = Ereset(U) +
X
i
logi(U) −Efrozen(U) + αR(U).

The termwise low/high and projected-payout identities give = 0 with the original absolute sums. Composing each finite partial sum with and taking the same convergent limits yields 0. Exponentiation consequently preserves the unnormalized evaluator, and multiplication by the recovered activity gas preserves the identity at every input, including zeros of that gas. This recovers the vacuum shift and coupling shift together with the dictionary reset, rather than treating the shifts as independent parameters.

13.12 Gauge invariance and invariant color forms at the flat germ

Let(increment(1)) be the regular germ. Constant gauge transformations act linearly asAd. The restriction of(0) to each edge is an invariant linear form. Differentiation along exp() shows it kills all brackets. Its kernel is an ideal, hence by simplicity is zero or the whole algebra. The zero case would be injective and force brackets to vanish, contradicting nonabelianness. Thus(0) = 0 after summing edge directions. Twice differentiating constantgauge invariance gives adjoint invariance of(0). For site field, set ( Gauge link exponentials by exp()) and take local logarithms, giving a smooth chart with

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C(0, 0) = 0,
DC(0, 0)[u, t] = u + t∇φ.

The three link factors contribute). The local logarithm recovers the gauged configuration near the identity, and gauge invariance gives Its mixed second derivative has no right-side term, and the term vanishes since Hence ] = 0, and symmetry gives the reversed pairing. Locality transfers Hessians between fields agreeing on support, by comparison of diagonal exponential lines and polarization. These are smooth unmasked germs.

For adjoint-invariant color-bilinear, differentiation gives]). Its antisymmetric part obeys). Cycling and bracket antisymmetry identify this value with its negative, so it vanishes. For fixed) kills brackets and is zero by the preceding kernel argument. Thus is symmetric. Writewith real self-adjoint . Invariance and adjoint skew-adjointness yield]. The finite-dimensional spectral theorem supplies a nonzero eigenspace, which this identity makes an ideal. Simplicity makes it the whole color space, so and. Skew-adjointness follows from the real matrix-trace

inner product and cyclicity of commutators. The proof uses no complex or infinite-dimensional color-representation assumption. In the lift these are physical gauge directions in the recovered link space. Frame-fiber directions remain the separate kernel of the frame readout. The recovered smooth chart and its differential transport the physical-gradient null pairing, while the section pulls back the Hessian as a form; neither operation identifies these two notions of verticality.

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The Hessian recovery here includes the nonlinear readout chain rule. Choose compatible local
charts at the identity and let Π be the readout germ, with Π(0) = 0, DΠ(0) = p and section
differential j, so pj = I. For bF = F ◦Π, the full second derivative is

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D2 bF(0)[ζ, η] = D2F(0)[pζ, pη] + DF(0)[D2Π(0)[ζ, η]].

The last term vanishes by the recovered critical-germ identity just proved. Consequently

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D2 bF(0)[ju, jv] = D2F(0)[u, v],
D2 bF(0)[j∇φ, η] = 0.

A frame-fiber vectorker also gives a zero pairing, by the same pullback formula. These identities distinguish the lifted physical gradient from the frame-fiber kernel while recovering both null statements with their correct domains. Constant adjoint transformations and the color inner product are retained, so recovery of the invariant color forms preserves their scalar coefficient as well.

13.13 Affine tests reduce to constant curl

Use centered lifts) = lift( and the linear jet

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Ai(x) = ai +
X
j
zj(x)bij,
Fij = bji −bij.

On tested support with 10 dist(4, centered coordinates increment ordinarily under either sign of a unit step: the old lift plus the step and the new lift wrap to the same point and remain in the centered range. Thus the discrete curl is there. If = 0, symmetry of makes

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φ(x) =
X
i
zi(x)(ai −1
2bii) + 1
2
X
i,j
zi(x)zj(x)bij

a primitive:). The diagonal subtraction removes the extra half-diagonal term, and symmetry combines cross terms. Hence curl-free affine fields agree locally with gauge gradients. Locality and the gradient-null Hessian identity show that equal-curl affine fields have the same Hessian pairings; the form depends only on both curls. For positive plane the wedge with sole component equal to has curl in that plane and zero in all other positive planes. Summing therefore gives a primitive with exactly the prescribed flux. A thresholdscaled unit-color wedge is paid-admissible sincedist() and allowed growth dominates 1 + dist(); averages of two are admissible by triangle inequality.

For signs and permutation, affine pullback has

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b′
ij = εiεjbπi,πj,
a′
i = εi

aπi −1{i reversed}bπi,πi

.

The subtraction records the shifted base of a reversed positive link. Substitution of centered-coordinate increments proves the formula on the nonwrapping support, with curl equal to the signed-plane reframe. Pullback of exponential increments is linear on direction fields; the Hessian chain rule at the critical identity gives projected-form covariance. No global torus affine polynomial is identified. The source lift retains the center, tested support, lifted coordinates and reversal shift. Recovery therefore preserves the local primitive and signed pullback, transporting the curl-only Hessian without extending its nonwrapping domain.

This reduction defines a recoverable bilinear form on local flux data. Let be the affine directions tested on the stated nonwrapping support and let collect their constant positive-plane curls. The wedge construction gives a linear right inverse of curl. Define

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Bf(F, G) = D2f(1)[W(F), W(G)].

For any affine with curls, the differences) and) have zero curl. The local primitive formula and the recovered gradient-null identity annihilate every Hessian term containing either difference. Hence

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D2f(1)[A, A′] = Bf(F, G).

In particular is independent of the chosen affine primitives on this domain. Lifting its directions through and using the preceding Hessian recovery gives the same on readout. The recorded signed pullback, including the reversed-link base shift, recovers the same curl reframe and thus the same covariance of this flux form. Both the quotient construction and its section recovery retain the original local support restriction.

13.14 Affine-Hessian coefficients and paid bounds

Choose a unit color in the nonzero simple algebra. The affine wedge) has curl in positive plane, is linear in and commutes with color adjoints. Define

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cab(f, o) = D2f(1)[Wa(ξ), Wb(ξ)].