Marked integration and coupling reset

Marked integration and coupling reset

What a marked integral records. Insertions must survive blocking as functions, including when a label occurs more than once. The marker algebra records occurrences separately; only the stated substitutions identify their final readings. This is why the numbered-tree estimates, repeated-occurrence factorials and distinct connected-set expansion all remain in the argument. The primitive integrand is the actual finite integrand with its chart Jacobian, Gaussian density, local interactions, insertion factors and masks. Its normalization is retained before any cluster regrouping.

The readout integrates this record and transports the prescribed insertion. Absolute weighted estimates justify the rearrangements and the passage from finite sums to a countable dictionary. Projection onto the coupling direction is followed by an exact reset identity; the unprojected part remains a residual activity, rather than disappearing. The loan cancellation restores the true kinetic base before the later bounds are used. At a group/logarithm interface, multiplication remains ordered: with exp, the neutral residual is exp, not the reversed product. Its off-chart value stays in the record. This local correction preserves the supplied primitive integration and all subsequent paid restoration estimates.

13.1 Insertion algebra, restoration and local prices

Fix finite markers and their allowed-mode census. A mode is an allowed subset . Multiplication joins disjoint subsets and discards products outside the census. Write [ and for coefficient and monomial; the empty mode is scalar. For positive

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tw(A) =
Y
v∈A
wv,
∥P∥w =
X
A
tw(A)|[P]A|.

Triangle inequality and) prove subadditivity and submultiplicativity by expanding disjoint coefficient pairs. On the zero-scalar nilpotent ideal, the finite exponential Exp has norm at most and obeys exponential addition for commuting arguments. An insertion representation consists of density exponent, primary functions Φ, forced profiles) and corrections), where) is support plus flagged inventory. With scalar profile and configuration flag set), restoration is exactly

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X
!
v
e{v}Φv(U) +
X
S
H(S, U)
X(U) = eE(U) Exp
!
×
X
Y
bq(U) +
X
v
e{v}ωv(q, U) + K(q, U)
.
P:Pattern(F(U),P)
q∈P

Pattern requires pairwise disjoint and inventory union). All sums are finite at fixed torus and census. Corrections have only modes of size at least two, are measurable, gauge invariant, supported on their regions and scoped to markers pinned there. Exponential corrections vanish off the healthy region and have zero scalar part. Flagged corrections requireand . Scalar multiplication preserves these permissions.

Correction prices use summable nonnegative envelopes with and

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X
i:x∈Si
aieecB(Si) ≤f,
sup
x

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where cB is the minimizing cover number. For e ≥0, if each nonzero term has a marker pin in its sup-
port, charge it to all such pins. At least one exists by scope and positive degree, and the exponential
weight is at least one. Summing the sitewise bound over pins proves P
i ∥Yi(U)∥w ≤|V |f. For the
unmarked background profiles the same argument charges to all sites, giving P
q ∥bq(U)∥w ≤|Λ|u,
where u is their sitewise activity budget. If wv|Φv(U)| ≤x and P
q wv|ωv(q, U)| ≤y, the exponen-
tial argument costs at most |V |(x+f) and all field factors Tq together cost at most |Λ|u+|V |(y+f).
Dropping pattern restrictions gives

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Y
q∈P
∥Tq∥w =
Y
q
(1 + ∥Tq∥w) ≤e
P
X
q ∥Tq∥w.
P

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Consequently
∥X(U)∥w ≤exp{E(U) + |V |(x + y + 2f) + |Λ|u}.

The lawful-density estimate supplies the uniform majorant. For a normalized mode integral use , integrate against the probability prior and divide by the positive density integral. Its lower bound and the history treasury give the final billΛ). This separates polynomial algebra from the vacuum density and denominator estimates. The lift retains the mode census, weights, pins, inventories and permission predicates. Coefficientwise readout is

an algebra map on that same census, commuting with finite exponential, pattern restoration and probability integration; the retained positive denominator then gives exactly the same normalized mode bill. Explicitly, for a mode-valued coefficient function define its lift by

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L P =
X
A
eA([P]A ◦π).

The allowed-mode census is fixed in this definition. Since the coefficient of a product is the same finite sum over disjoint allowed pairs before and after composition, one has

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L (PQ) = (L P)(L Q),
[L P]A = [P]A ◦π,
(L P) ◦σ = P.

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The same finite calculation gives L (Exp P) = Exp(L P) on the zero-scalar ideal and ∥L P(z)∥w =
∥P(π(z))∥w. Record the flag set and all support–inventory barcodes so that the pattern predicate
is pulled back along with its coefficient. The restoration sum then is exactly L X, including zero
terms excluded by the predicate. Finally, for the realized probability prior νZ with π∗νZ = ν,
coefficientwise integration gives
Z
[L X]A dνZ =
Z
[X]A dν.

The displayed majorant ensures these integrals exist. Applying this also to the scalar density recovers the positive normalizing denominator, so the normalized mode integral and its stated bill are preserved.

13.2 Singleton restoration and response closure

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For represented density R, primary function Φ and summable forced family gi, define the singleton
density
coinR(Φ, g)(U) = R(U)Φ(U) + eER(U) X
i
nickelR,i(U; g).

The second term is the first activity-gas variation in direction. Absolutely convergent grouping by support-and-inventory barcode preserves that variation. Finite polynomial expansion shows the restored scalar mode is) and its coefficient is this coin: only one factor can supply a singleton, all others supply scalars; the exponential has scalar one, and higher corrections supply neither scalar nor singleton. The correction representation must preserve the actual singleton insertions after every step. Conversely suppose the polynomial is Exp() Field(), with zero scalar part of

original background scalar profiles in and prescribed forced singleton profiles. We show that if the singleton coefficients of the whole expression are coin), then the singleton coefficients of are precisely Φ. If its singleton densities are coin), subtraction cancels gas variations and givesΦ) = 0. Nonvanishing of) permits cancellation. Projection onto modes of size at least two splits into prescribed singleton plus higher exponential correction and each into background, prescribed forced singleton and higher flagged correction. The original density is preserved pointwise.

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For a misplaced forced singleton, take attached local responses ψv(S, U) supported on regions
containing pin ov and vanishing when that region is healthy. Put J = P
v,S e{v}ψv(S, U). Closure
joins overlapping background factors and responses into connected components, recording support
and flagged inventory. Component expansion proves exactly

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Field(Closure(K, ψ)) = Field(K) Exp J.

Replace by its closure and by to preserve the full polynomial. No-response terms reproduce; terms with a response have zero scalar part; terms with at least two responses have zero singleton part. A sole response is multiplied by the background attached to its support, giving exactly its hook in the singleton profile. Thus no unspecified singleton change is introduced. The converse then identifies primary functions from known singleton densities. Higher-mode projection yields the two higher correction families, preserves locality and permissions, and decreases weighted prices. For field budget, response budget and pin occupancy, the connected-component estimate bounds the higher closure budget by

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exp{W + M siteFactor(B)c} −1,

provided the support-exponent loss pays (2 siteFactor(. The forest-price argument below supplies this bound; marker-monomial reweighting transfers both higher families to the requested weights. These are the algebraic and numerical operations used in the one-step retained-insertion estimate. The source lift retains response attachments, barcode grouping and component partitions. Coefficientwise recovery commutes with finite projection and nilpotent exponential; absolute summability permits the same grouping. It therefore recovers both the exact closure identity and the nonvanishing cancellation establishing the prescribed singleton fields.

In the notation of the coefficientwise lift, let discard scalar and singleton modes. Its definition gives L = L

Preserve each response label and its attachment, even when coefficients coincide. The overlap graph is determined by the recorded supports, so readout and the section recover its vertices, edges, connected partitions and inventory unions. Each component coefficient is a sum of products of these same labelled factors. The algebra-map identity therefore gives

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L
Field(Closure(K, ψ))

 = L (Field(K)) Exp(L J).

Absolute summability permits the stated barcode grouping before or after composition. Multiplication by Exp(L)) then recovers the unchanged full polynomial. For the converse, the nonvanishing domain is exactly; there cancels in the lifted singleton identity. Section recovery returns the same Φ and the same two projected higher families, with their retained support permissions and weighted-price decrease.

13.3 Distinct-label component prices and exact closure bijection

Suppose supports and nonnegative obey

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X
i:x∈Xi
fie(a+w)cB(Xi) ≤γ,
a ≥0,
(2B + 1)4γ ≤w.
sup
x

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For finite labels D, let FD(Y ) sum Q
i∈A pi over A ⊆D whose every overlap component meets Y ,
including the empty set, where pi = fieacB(Xi). In each component choose a label meeting Y and
remove it. Every remaining component meets the removed support. Distinct original components
choose distinct labels. Chosen roots and their remainders reconstruct the subset, giving an injection.
Allowing incompatible remainders enlarges the nonnegative sum, so

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1 + piFD\{i}(Xi)

.
FD(Y ) ≤
Y
i∈D:Xi∩Y ̸=∅

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Strong induction gives FD(Y ) ≤ewcB(Y ): each factor is at most exp{fie(a+w)cB(Xi)}, whose expo-
nent sum is at most |Y |γ ≤(2B + 1)4cB(Y )γ ≤wcB(Y ). The same estimate also gives the sharper
FD(Y ) ≤e|Y |γ. A nonempty connected A whose union contains x is counted in FD({x}) −1. Or-
dering its supports so each meets the preceding union and applying the meeting-support inequality
proves cB(S
i∈A Xi) ≤P
i∈A cB(Xi). As a ≥0,

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X
eacB(∪i∈AXi) Y
i∈A
fi ≤eγ −1.
A̸=∅connected
x∈∪i∈AXi

For countable labels, each finite subsum lies in a finite and has the same bound. Nonnegativity proves summability without repeated-label or logarithmic expansion.

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Apply this to the disjoint union of background and response labels with γ
=
W +
M siteFactor(B)c.
Submultiplicativity prices each component by the product of label en-
velopes. Restricting to permitted components decreases the sum; grouping by output support and
inventory preserves it. This proves the closure price and support-exponent loss above. For the
exact identity, expand the field polynomial and Q
j(1 + uj) and join intersecting chosen labels.
A nonzero response is offits healthy region, so its support meets a flag, which belongs to a
chosen background inventory.
Its component therefore contains a background factor.
Original

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background supports are disjoint, and resulting nonempty inventories partition the original flagged
set. Conversely, disjoint permitted components recover the unique background and response label
sets by union, with unchanged coefficient products. This bijection proves the identity term by
j = 0, hence Q
j(1 + uj) = Exp(P
term. Singleton responses obey u2
j uj). The lift retains distinct
labels and component incidence rather than only unions of supports.
Recovery preserves the
decomposition bijection, polynomial products and finite-subset bound, and thus its countable
summability conclusion.

To make the recursion’s recovery map explicit, equip each finite with its recorded label order. In a component meeting , select the least label whose support meets . Record that root together with the exact remaining label subset of its component. The inverse takes their disjoint unions and restores each root. Thus the injection used for the product bound is defined by labelled incidence alone. Realization preserves the order and incidence, and readout recovers this same root–remainder record. In particular, equal supports never identify different labels. For a finite, every coefficient product and each cover number in ) is consequently recovered literally. Its two bounds

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FD(Y ) ≤ewcB(Y ),
FD(Y ) ≤e|Y |γ

have the same constants after readout. Taking the supremum over finite label subsets recovers the nonnegative connected-component sum and its1 bound. The closure bijection uses precisely these retained label subsets and inventory unions, so its algebraic identity and its price estimate are recovered by the same discrete construction.

13.4 Finite transport inventory and exact ticket expansion

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For a countable leaf family at one dictionary key with common support S, complex charges ri and
real-leaf bounds bi, assume P
i |ri|biescB(S) < ∞. Sum real and imaginary charge families separately.
Each sum is a real leaf bounded by P
i |ri|bi; locality, gauge invariance, measurability and eligibility
pass through absolute convergence. Sound and tuned transport identities pass the quote through
the sum. Charges 1, i therefore reproduce both original and transported profiles with total marker
price at most 2wv
P
i |ri|bi. An insertion tale has finitely many keys, so grouping within each key
preserves support weights and gives finite singleton leaves. A tale priced Av costs at most 2wvAv,

reproducing primary minus scalar displacement, original forced profiles, and coarse forced profiles minus response hooks.

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For a higher coefficient h in mode A, split real and imaginary parts into two leaves with charges
1, i. If |h| ≤bA, their combined price is 2tw(A)bA. Coefficientwise scope, locality and permissions
remain.
Include only modes nonzero somewhere.
A correction budget f becomes at most 2f.
Dormant regular slots of zero original coefficient exist for every nonempty support, and forced
slots for every admissible nonempty inventory with exact eligibility. Place exponential and flagged
corrections in these respective slots. Nonzero higher corrections have at least two scoped markers
and hence nonempty support. Together both families cost at most 4f, using this explicit history
property: the existence of these dormant dictionary slots is part of the history construction. Their
primary sum is P
v e{v}(Φv −νv)+P
S HS; the forced sum is P
v e{v}ωv(q)+Kq. Higher leaves have
no singleton quote. Occupancy converts singleton root budgets to M siteFactor(B)d; with 4f this
is the numerical budget T.

At a regular key its finite tagged monomials have nonempty modes and = 0. Thus

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X
!
=
Y
i
(1 + ui) = 1 +
X
i
ui +
X
Y
Exp
i
ui
i∈J
ui.
J:|J|≥2
disjoint allowed modes

Overlapping modes and products outside the census vanish. Each remaining product is a real product leaf at the same key, with product charge, union mode and product price. Composite leaves have degree at least two, leave singleton quotes unchanged and occur only at regular keys, so forced-slot contributions are untouched. This exactly replaces the exponential by one plus the enlarged key sum. The activity expansion now selects an old scalar or one marked leaf at each key; a ticket uses no key twice. Every term is the unselected old activity product times selected marked ingredients. Only finitely many keys are perturbed; lawful absolute summability permits

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barcode regrouping and interchange with the finite ticket sum. The result is the exact fine restored
density, factoring out Exp(P
v e{v}νv). Expanding the represented activity gas now gives the finite
transport expression. The exponential at a regular dictionary key must still be represented exactly.
For key k, let pk sum its unweighted leaf prices. A marker pin in the common support implies
0 ≤pk ≤T. Composite prices are bounded by
Y

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i at k
(1 + fi) −1 ≤epk −1 ≤epkpk ≤eT pk.

The middle inequality integrates ]. Composites share the support of their factors, and distinct keys use disjoint original labels. Weighted sitewise summation costs at most giving total inventory price. Finally a normalized monomial has fee and value ). For 0, multiplication recovers = 0, positive marker weights imply = 0; both sides vanish under the zero-inverse convention. The weighted norm is at most one, and its square is zero since. The remaining input is the Mark-step reserve on this fully priced inventory. The lift retains key labels, charges, zero-fee convention and ticket exclusions. Coefficientwise recovery commutes with each finite expansion; absolute convergence justifies its countable leaf grouping. Thus the inventory reduction preserves the insertion itself and its transported quotes, not merely their bounds.

For the countable reduction, let denote the real leaf functions at one fixed key, with If exhausts its labels, then

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≤
X
i/∈IN
|ri|bi −→0.
X
sup
U
i/∈IN
rifi(U)

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Composition with π satisfies the same tail bound. Hence the regrouped coefficient obeys the literal
identity
 X
!

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◦π =
X
i
ri(fi ◦π),
i
rifi

and the section recovers the original sum. At every transported profile the source’s sound and tuned transport identities give the corresponding quote identity retained above. Finite key and mode labels are kept through the real/imaginary split and through each compatible composite subset. Thus every ticket has exactly the original distinct-key exclusion. The algebra lift carries its positive-fee normalization, including the zero-fee branch, term by term. Together with the lawful absolute summability of the remaining scalar activity series, this proves recovery of the exact ticket expansion as well as its inventory price.

13.5 Repeated-occurrence rooted-tree estimate

Put 0 and ) = max. For nonnegative vertex weights with sitewise budget at exponent

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K(Y, i) = 1Xi∩Y ̸=∅wie(a+δ)cB(Xi)
tF(Y )
.

Every finite row sum is at most one: its numerator sum is bounded by ). A forest on numbered internal occurrences has parent map to with external root 0 supported on and ranks strictly decreasing toward each internal parent. Child and parent supports must meet. Distinct occurrences may carry the same activity label. For a fixed admissible parent map, the sum over label assignments of its kernel product is at most one. Sum a maximal-rank vertex first: it has no remaining children, so its label occurs only in its row factor. Remove it after applying the row bound and repeat. Nonnegative finite restrictions prove the assertion for arbitrary label sets. If counts children of vertex, including the root, then!. For use one exponential-series term; below one use!1. Multiplication over parents cancels each internal extra support weight in, leaving only the root factor. Thus the original exponent- overlap weight is at most

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


m
Y
j=0
kj!

m
Y
i=1
K(Sp(i), i).
eδcB(Y )tm

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Label summation and division by the occurrence factorial m! give eδcB(Y )tm Q
j kj!/m!. For all maps
of m numbered objects into r bins, let Am(r) sum the products of bin-size factorials. Appending
an object multiplies the total by P
j(kj + 1) = m + r, so A0(r) = 1 and Am+1(r) = (m + r)Am(r).
At r = m + 1,
X
m
Y
 
2m
m
!
m! ≤4mm!.

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j=0
kj! =
p

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This counts all maps and hence bounds admissible maps. The forest price is at most eδcB(Y )(4t)m ≤
eδcB(Y )qm. If an internal designated root has weights ui ≥0 with P
i uie(a+δ)cB(Xi) ≤b, multiply
the external-root bound at Xi by uieacB(Xi) and sum. This gives bqm with factorial m! for the other
occurrences. For q < 1, the size tail is bqm/(1 −q). The argument includes m = 0, empty external
root and arbitrary branching with unchanged normalization. All child factorials are paid together
by an exact parent-map count. The lift retains numbered occurrences, activity assignments, parent

ranks and the occurrence factorial separately. Recovery thus preserves repeated labels and the kernel elimination order; the finite nonnegative bounds pass through readout before the countable summation. The occurrence reader is explicit if) denotes the activity label assigned to numbered vertex . For a fixed admissible parent map, set and for 0. Its kernel product is

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m
Y
Kh(λ) =
r=1
K(Sh(r), λ(r)).

The realization keeps the pair ()), so readout never replaces an occurrence assignment by its label range. Every finite assignment sum of this product is unchanged, including assignments with repeated labels. A maximal-rank occurrence can therefore be eliminated by exactly the same substochastic row bound. The section restores the assignment and parent map, hence also! and the bin-count payment. The source’s final geometric bound uses a chosen with 4; its summed tail additionally uses These conditions are retained with, and the designated-root sum is recovered before passing to its nonnegative size tail. Thus the occurrence multiplicities and the convergence parameter are both preserved explicitly.

13.6 Graph intervals and distinct connected sets

Order occurrence vertices injectively and root at 0. For a connected graph, let) be shortestpath depth and choose each nonroot parent as its least ordered neighbor at depth1. These edges form a ranked spanning tree, whose parent map uniquely determines its depths. Let consist of original graph edges changing depth by at most one and, for an edge from to precedinglevel, satisfying order(order()). Then graphs with breadth-first parent are exactly

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stem(p) ⊆H ⊆Up.

The forward implication follows from shortest depths and least-parent selection. Conversely the stem realizes assigned depths, the upper edge constraint precludes shorter paths, and the order constraint recovers the same least parent. These intervals partition connected subgraphs. If an optional edge lies instem(), toggling it is a parity-reversing pairing and its alternating interval sum is zero. Otherwise the interval is a single tree with absolute sum one. Summing proves the rooted spanning-tree bound on the absolute connected coefficient.

For distinct labels of size, a parent map on and a bijection produce a numbered forest. This assignment is injective: its label range recovers, its assignment recovers the bijection and transport recovers the original parent map. The! bijections have equal product weights divided by!, so their sum is exactly the unnumbered weight. Restriction to this image proves the distinct-parent estimate and geometric size tail. A finite connected support union meeting can be rooted at, joining it to supports containing and taking a spanning tree. The parent-map sum includes this tree and bounds its product. For the stronger sitewise prefactor choose a label containing as internal root and charge it to the site’s budget. Apply the internalroot estimate to other labels and sum possible roots, only enlarging the bound. Numbering distinct nonroot labels cancels their factorial; permitting the root label among nonroots in the majorant is harmless. Under readiness, the connected-set sum with actual union-support weight is therefore at most). This is the distinct-set bound for positive-multiplicity restoration. The lift retains occurrence ordering, parent map, optional edges and distinct-label bijections. Readout recovers the sign-reversing pairing before absolute values, and preserves the exact factorial cancellation in passage to sets.