Convergence of interacting scale histories · Part 3

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packets and means. The underlying retained stationary and activity convergence is the history in-
put, whose extraction must use the common choices described in the construction. The lift retains
finite route labels, branch predicates and signed base shifts. Recovery preserves every finite smooth
expression and common bounded indicator, transporting derivative comparison with the original
retained-history dependencies.
For a fixed cutoffrow, let Jk be its retained finite path or route labels. Its coefficient-dependent
part has the reader
Tn(U) =
X
i∈Jk
cn,iSi(U).

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Realization retains Jk, the route words and their signed bases, giving T Z
n = Tn ◦π and section
recovery T Z
n σ = Tn. On the recorded local smooth domain, any tested derivative Dr obeys

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∥DrTn −DrT ′
n∥≤
X
i∈Jk
|cn,i −c′
n,i| ∥DrSi∥.

Finite labels and the retained slot bounds therefore recover the same derivative comparison. Common endpoint, cutoffand health predicates multiply the compared coefficient values with the same bounds; derivatives of the smooth expressions are recovered on their original domains before these hard predicates are evaluated. Signed reversal and carrier conjugation keep their recorded word order. Thus the finite physical-row comparison is transported through explicit readers with its actual cutoffand inherited-history inputs.

15.14 Mean, local sources and ordered chain comparisons

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Selected strain is free residual vision plus the physical row applied to saved log and hot fluctuation.
The draft fixes selection and hot sets; saved logs are the same all-argument local functions with
uniform derivative bounds, while dormant lens contributions vanish. Fine residuals compare base
stroke minus coarse vision. Noise base strokes are common and have derivative bounds on the
clean hull supplied by viability and selection; subtracting compared vision gives their comparison
on selected rows. The bill vector is transpose form applied to zero-fluctuation strains with orig-
inal scaling. The protected mean is minus the transpose window-column matrix applied to that
bill. Finite matrix operations compare both and the stationary defect −√β(plate μ + bill). The

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coupling-window lower endpoint makes square roots and reciprocals valid. Chain factors are ordered
negative plate–covariance defects followed by the terminal plate factor. Finite products preserve
order; quadratic fluctuation and linear defect contractions compare, as do smooth cone cutoffs in
squared norms. Blank chains use the viable empty store and identity field with the common plate
floor. Condensation is half the blank-minus-actual local determinant difference. The cutoffHaar
logarithm is smooth at all arguments: inside support its argument has strict margin within the
positive chart, outside it is locally zero, and the margin includes cutoffboundaries. This justifies
its use in products without extending an uncut logarithm beyond its domain. The actual packet
log is

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−β(chalky + crispy) +
X
visited
condensation +
X
plaza
HaarSource
−|pens| log(chartMass(β)).

Chalky is posted root energy minus selected quadratic costs; crispy is that cost at the protected mean. Costs sum squared strains over actual parent rows. Chart mass is positive and continuous

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at the positive coupling limit.
All indexing sets are draft-fixed and finite.
Thus strain, root-
energy and determinant comparisons recover the packet log with its original normalizer. The lift
retains transpose placements, terminal factors, chart margins and all finite index sets. Readout
preserves these expressions and derivative comparisons on the selected clean hull, including the
cutoffboundaries.
The protected-mean reader retains the window-column matrix V and physical bill b, so it is
exactly
μ = −V Tb,
δstat = −
p
β (Qμ + b),

with the physical plate and all transposes taken in its original coordinate metrics. Read these arrays componentwise before evaluating the formulas. Composition with readout preserves the displayed operations, and the section recovers both vectors and the sign of the defect. Likewise a finite chain reader records its ordered factor list () and evaluates in that order. Realization retains this list, so its contractions and terminal plate are recovered without commuting factors. The actual and blank lists remain separate records. Their local determinant readers consequently recover the original half-difference. Finally all finite sums in the packet logarithm, including the positive chart-mass reader and the pen count, commute with readout and section. This recovers the packet normalizer alongside its energy, Haar and chain sources on the original chart domains.

15.15 Four vertex comparisons and marked primitives

Fix the common draft. After fixing the draft, the ordinary integrand has the same four vertex types as its absolute activity bound. Packet connectedness and containment/empty-flag tests are fixed; failed branches vanish in both histories. Otherwise the packet is its compared energy exponential times common bounded contracts, tickets and profiles and compared smooth cutoffon the enlarged plaza. Fine-input factors may vary with history but agree across the comparison; no ticket indicator limit is required. Their output-field dependence uses the already chosen smooth wire profile. Forced vertices use actual forced-coefficient comparison on the same fine input and the cutoffon their true footprint. Regular vertices compare logarithms, then exponentiate and subtract one. Inherited, Short, condensation, Haar and chain branches retain their signs and beta factors; posted-region, voyage and hot-exterior tests agree. The nonzero branches use the protected mean, covariance/window, local logarithm and row comparisons already needed for these expressions, with the actual signs and factors of retained. Warnings retain fixed mandatory/hot predicates, compared cutoffs and bounded common fine profiles or contract charges minus reference constants; exterior cones are smooth in compared fluctuations. Mandatory branches agree exactly. Finite products and density multiplication compare transformed integrands. Derivatives use the local smooth extended formulas, fixed fine input and compared steering/restart coordinates, identified with the actual differentiated integrand near the healthy base. These statements compare the true vertices rather than replacing hard indicators by smooth approximations.

A marked premium tests whether its fixed badge occurs. Absent badges give zero; present badges replace exactly one vertex by the paid ingredient and retain the erased-badge product. At fixed physical fluctuation the fine configuration is common. Mark-entry comparison on that possibly varying input, common enforcement cutoff, compared remaining product and Gaussian density give amplitude comparison on the eventually viable set. For derivatives the marked entry stays at the fixed reconstructed input; output derivatives act on smooth steered cutoffs and remaining vertices through the local reweighting identity. Steering, ordinary factors and Gaussian ratio/Jacobian comparisons therefore give the actual derivative integrand. The common Gaussian-polynomial majorant permits integration; finite forest and boundary averages compare the marked grove. Its

all-field bound also compares uncut amplitudes at arbitrary varying output fields, using the fixed-draft argument at each slice point. These supply regular one-mark stream and forced-reset inputs. This explains why convergence of mark entries under varying-configuration evaluation is sufficient here. The lift retains badge identities, all hard predicates and the exact local reconstruction. Readout preserves the branchwise products and derivative identities with the same viability scope and common integrable majorants.

For a fixed marked badge in molecule, the replacement reader has the explicit product form

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Ib,p = 1b∈p Mb(Ufine) Γb
Y
v∈p\{b}
Vv.

Here is the original paid ingredient, Γ its enforcement cutoffand the actual ordinary vertex readers. The density factor is then that of the same fixed local data. Realization retains the badge, erased list, common reconstructed fine configuration and all branch predicates, so composition with readout preserves this product and the section recovers it. The smooth local derivative identity holds with fixed: derivatives act on the recorded steering, cutoff, remaining vertices and reweighting factors. Thus recovery asks for the stated varying-input comparison of, with no additional derivative assumption on that evaluated mark entry. The common integral majorant and finite averages transport the same amplitude and derivative conclusions, including the absent-badge zero branch.

15.16 Uniform derivative comparison of a cluster stream

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A one-mark stream with r occurrences is its Ursell coefficient over r! times Pr
ν=1 lν
Q
μ̸=ν uμ. Its
support is the occurrence union when its fixed mark key speaks to the label inventory, and the
zero alternative otherwise. Supports agree across histories; silent streams vanish. Bounds aμ, bμ
through order j give ordered product bill (1 + r)jbν
Q
μ̸=ν aμ.
Inflate each bound by 2j, using
1+r ≤2r. Take ordinary bounds as the smaller primitive envelope and purchased budget, preserving
tails. A positive reserve d, selected by the one-mark budget argument, makes the inflated array
fμ = 2jaμ + d 2jbμ fit the connected-cluster reserve. Nonnegative expansion shows

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d
X
ν
2jbν
Y
μ̸=ν
2jaμ ≤
Y
μ
fμ.

Thus times its ordinary absolute cluster weight gives a uniformly summable stream bill with small tails, retaining Ursell and factorial normalization. On healthy unions every occurrence is healthy; primitive comparisons and finite differentiation compare fixed streams through order Tail-ultrafilter comparison implies ordinary test-family convergence. If uniform order- bounds on allowed maps and healthy bases failed, select violating maps at bad indices; their sequence contradicts this comparison. Intersect the finitely many late cutoffs for 0. For arbitrary parameter directions use

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Lz = (k + 1)−1
k
X
ν=1
zνbvν,
bvν = vν/∥vν∥(vν ̸= 0),

with zero for zero directions. In maximum norm1 and. Pullback preserves allowed bounds; a test tolerance yields full multilinear operator bound Separate primitive bills supply smoothness at healthy bases; masks are not differentiated. At fixed period choose finite bounding 1 + dist(). Conversion to anchored directions costs at most , paid by shrinking tolerance first. Identity ensures the healthy test set is nonempty. Treat

real and imaginary streams separately, retain normalization and embed by zero in the common enumeration. The lift retains occurrence supports, allowed maps and normalizations. Recovery preserves this finite differentiation and contradiction argument, with its fixed-period conversion and summable tails. For an occurrence address), retain the exact stream reader

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r
X
ν=1
liν
Y
μ̸=ν
uiμ.
Sω = φ(ω)
r!

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The speaking predicate, ordered occurrences and marked position stay in the record. The algebra-
map identity gives SZ
ω = Sω ◦π and section recovery, including repeated labels and silent streams.
For any allowed physical parameter map a, its section lift σ ◦a satisfies SZ
ω ◦σ ◦a = Sω ◦a
on the same healthy domain. Thus all tested derivatives and the multilinear rescaling by L are
identical on recovery. The paid norm on section directions is transported from that original allowed
class. The retained nonnegative inflated-product bound consequently supplies the same summable
stream envelopes and tails. Applying the finite uniform-test argument and then the fixed-period
anchored conversion recovers the original paid derivative comparison, with its Ursell coefficient and
occurrence factorial unchanged.

15.17 Microscopic comparisons through projection

Regular comparison uses common supports, uniformly summable paid tails and arbitrarily small fixed-label derivative differences. Forced comparison uses uniform absolute tails and convergence at every varying configuration sequence. The latter implies uniform smallness by selecting violating configurations if necessary, without mask continuity. Finite-head/tail splitting proves dominatedseries comparison: finite label differences vanish and both uniform tails can be made arbitrarily small. This permits varying evaluations, barcode sums, fixed injective zero extensions and scalar combinations. Products compare by ); total absolute bounds times tail budgets control their tails. For the fixed weighted local section projection Π of price collected core is seed plus projected bulk sum, and

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|Π(f −g)|1 ≤Cread paidBound(f −g).

Linearity and summable projected tails give core comparison when seeds compare. Commonthreshold residual and rebate operations are linear; their fixed price controls enlarged-support regular residuals and all-field rebates of differences and preserves tails. Canonical zero extension keeps regular comparison. The new forced profile is old profile plus the threshold hook of collected rebates. Hooks sum support classes times compatible forced products. Common address data preserve discrete tests; compared numerical factors, finite products and summable product budgets preserve forced comparison after multiplying rebates and summing. The keeper barcode embedding preserves convergence and tails. Thus local projection and collection recover the formatted output comparison. The lift retains common support data, projection and keeper maps; readout commutes with their linear operations and absolute series, preserving the paid and all-field bounds without differentiating indicators.

The core collection reader has the explicit convergent expression

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cout = cseed +
X
i
Πfi.

The source object records the same weighted projection and its output coordinates. Define its realized projection by first recovering the regular germ; section recovery then gives exactly Π

The bound paidBound() is unchanged and summable, so finite partial sums followed by their limits recover. For two inputs and a finite head, its difference satisfies

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|∆cout|1 ≤|∆cseed|1 + Cread
X
i∈E
paidBound(∆fi) + Cread
X
i/∈E
(bi + b′
i).

The retained uniform tails and fixed-label comparisons recover each term of this bound. Residual, rebate and hook readers retain their original support predicates; the canonical keeper map is recorded with its inverse on its image and zero extension elsewhere. Their linear identities and summable product majorants consequently give section recovery of the full formatted output, including its forced barcode profiles.

15.18 Common shot inputs and all-configuration comparison

For fixed capture and frame index, the shot merges primary insertion, regular remainder labels and forced barcode labels from the same input book. Common orientation fixes threshold, addresses and profiles; keeper stowing gives a common forced ingredient. Regular input prices are mass() and forced prices mass(, with 0. Capture wealth makes these summable. Primary tariffs use the larger heading and heading-plus-core price; the same array bounds output cores and remainders. Actual regular ingredients are real/imaginary fixed masked functions at their logarithmic keys, with weight at most. Forced ingredients are common stowed functions at gas keys, with weight at most; zero inputs give zero outputs. Nonconstant primary terms use the fixed masked section; constant frame terms give identical scaled units and zero forced profile, independent of keeper presentation. These are genuine fixed ingredients across the two histories, although their auxiliary preparation data may vary.

Fix scale, block, degree and dictionary. Their couplings approach each other, their regular and forced activity coefficients have the weighted comparison already proved, and their stationary kernels and path responses have the common limits. The established activity, coupling, kernel and path comparisons supply Gaussian, mean, saved-field and vertex comparisons. Common-chart conversion preserves each ingredient, wing and weight exactly, even if preparation data vary. Each fragrance address has the same support, containing zero. Mixed marked-integral comparison gives arbitrarily small fixed-address paid derivative differences for both real and imaginary bulk terms, with common threshold/order and uniform marked-cluster tails. Its hypotheses are now the identical ingredient just identified, the compared Gaussian data and vertices, and their common summable derivative envelopes. We have therefore proved the bulk comparison, including both regularity on the common support and the summable tails needed for projection. Multiplication by , recombination and zero extension preserve this bulk comparison. Forced output first restores missing ordinary logs, then marked slots, then resets on common shelves. Ordinary reset comparisons and marked integral comparisons, finite subsets, exponentials and summable product bounds preserve tails. The same path, threshold and rebate operations give

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Fn,α(Un) −F ′
n,α(Un) −→0
for every configuration sequence Un.
(HL1)

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If uniform convergence failed, choose violating configurations on a bad subsequence and arbitrary
ones elsewhere, contradicting (HL1). Uniform tails then give convergence of summed profiles by
finite-head/tail splitting. The lift retains fixed input ingredients, wing/address supports and com-
mon shelves. Recovery preserves exact ingredient equality and the paid majorant, transporting
(HL1) and its uniform consequence without mask continuity.
For each fixed ingredient label j, write Mj for its common real or imaginary input function. Its
realized reader is MZ
j = Mj ◦πf, so the two histories have identical input readers even when their

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auxiliary preparations differ. Section recovery gives MZ
j σf = Mj. Retain the same logarithmic
or gas key, wing, tariffand fragrance support with this function. The recovered summable tariff
therefore bounds both its output core and remainder with the original forced scale κ1. For the
forced output let ∆Fn,α = Fn,α −F ′
n,α. Surjective coarse readout, supplied by its section, gives

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sup
z |∆Fn,α(πc(z))| = sup
U
|∆Fn,α(U)| −→0.

Thus (HL1), its uniform consequence and its uniform-tail summation are recovered on precisely the original all-configuration domain. Real and imaginary coefficients retain their separate readers and fixed normalization before recombination. The constant-frame branch likewise recovers its identical unit and zero forced profile through either keeper.

15.19 Summing local insertion comparisons

Real and imaginary forced comparisons recombine exactly as

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Ld(Fn + iGn) −Ld(F ′
n + iG′
n) = Ld[(Fn −F ′
n) + i(Gn −G′
n)].

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The primary seed already compares; linear projection, residual/rebate bounds and common keeper
extension give core convergence and arbitrarily small collected remainder on the original input
book. Local insertion differences keep common coefficient supports and orientation; append the
negative second labelled remainder to the first to form a legitimate difference. If each pair has
summable common bound fi, its difference costs at most 2fi. For target r > 0, choose finite E
with P
i/∈E 2fi ≤r/2 and eventually use r/[2(|E| + 1)] on each index in E. Total price is at most
r, so the paid merge gives that remainder bound for the full difference. Absolute convergence and

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common enumeration identify it with the difference of merged insertions. Core sums converge by
the same finite-head/tail split. Finite frame and prescribed motion-chip sums preserve comparison
of shot inputs captured on the same coordinate axes. Site marks therefore have vanishing core
difference and arbitrarily small common remainder. Each actual coefficient readout is its site’s
core; recorded capture is capture under the common output keeper. The keeper difference identity
makes the difference of captures the capture of the formatted difference, whose norm is bounded
by its arbitrarily small remainder price. Thus both coefficient-isomorphism outputs and recorded
captures compare to zero, supplying the fixed-input transfer comparisons. This proves the required
forced comparison for the raw complex preparation. The lift retains labelled negative remainders,
merge enumeration, frame chips and keeper identities.
Readout preserves absolute merges and
finite sums, recovering both output comparisons with the same quantitative remainder budget.

For the merged difference retain two disjoint labelled copies, one for each history, and put the minus sign on the second coefficient reader. Finite partial merges then satisfy the exact identity

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X
!
=
X
i∈E
(L Mn,i −L M′
n,i).
i∈E
(Mn,i −M′
n,i)
L

The summable 2 majorant and the retained merge bounds pass this identity to the full countable merge and recover it through the section. The chosen finite-head prices and tail prices sum to at most on both sides. Applying the same keeper difference reader gives

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∥capture(Mn) −capture(M′
n)∥≤remainderPrice(Mn −M′
n) ≤r

eventually. The recovered finite frame and motion sums preserve this inequality. Core readers recover the coefficient outputs separately, so both components of the transfer comparison have exactly the original vanishing bounds and common support format.

15.20 Transfer comparison metric

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In the fixed finite coefficient frame let |x|1 = P
ν |xν|. Gram data contain complex-linear captures gP
with ∥gP x∥≤a|x|1. Define P ∼ε Q by ε ≥0 and ∥gP x−gQx∥≤εa|x|1. This may be nonseparating;
it is symmetric, triangular and universally bounded by two. Coefficient-map comparison is |(S −
T)x|1 ≤ε|x|1. Finite coordinate-axis bounds imply both uniform comparisons. Adjacent actual
integrated transfers have flow Φn and isomorphism Sn, with

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|x|1 ≤b|Sn(P)x|1,
|(Sn(P) −Sn(Q))x|1 ≤cε|x|1,
Φn(P) ∼ε/2 Φn(Q).

At fixed input, flows and maps are Cauchy. Outside eventual admission the zero capture and identity coefficient map complete total sequences. For normalization write with map differences bounded by, and. For

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gXi −gZi = (Tk −Uk) + gZ(Wk −i).

The terms cost and. Thus, using only tested coefficient space. Escaping-subsequence comparisons on finitely many axes imply uniform map bounds, then flow bounds by this identity. Failure of a double-tail estimate would select bad contradicting those subsequence comparisons. For Cauchy inputs, insert a fixed late between output flows: outer errors contract by half and the middle is small by fixed-input convergence. For coefficient maps the outer errors cost times input error. Input tolerance + 1)] and middle tolerance2 give the uniform comparison. Finally, if an eventual pairwise tail bound

failed, one could choose bad indices; both escape to infinity and contradict the assumed subsequence convergence, which turns the subsequence statement into an actual double-tail Cauchy estimate. The lift retains capture maps, coefficient isomorphisms and this possibly nonseparating relation. Recovery preserves the tested-space inverse normalization and finite-axis bounds, transporting contraction and the actual double-tail Cauchy property.

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The comparison relation itself has an exact recovered model. Let J be the section realization of
the insertion-data seminormed space, with its transported seminorm, and define bgP = JgP . Then

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∥bgP x −bgQx∥Z = ∥gP x −gQx∥,

so is equivalent to the lifted comparison for the same. The coefficient frame and its

norm are retained unchanged; readout recovers and each finite-axis test. Consequently the inverse-normalization identity has exactly the original two bounds and threshold (1 + after recovery. The relation continues to be possibly nonseparating. Its triangle and contraction inequalities, fixed late input, and bad-index subsequence construction are recovered with the same errors, establishing the same double-tail Cauchy statements on the actual integrated transfer data.

15.21 Fixed-level convergence of transported insertions

At fixed physical period integrated Gram data satisfy eventually). Flows halve comparison error and preserve the two-part Cauchy property. All admitted inputs compare within two. Replacing the input levels up by fixed admissible data therefore costs at most . For fixed finite transfers give a Cauchy approximation. Choose this error below a quarter tolerance and use triangle inequality to prove fixed-level Gram Cauchy comparison, using only finitely many eventual identities. For coefficient transport suppose eventually, Cauchy, uniformly Cauchy on bounded inputs and. Bounded imply bounded, and

Both terms tend uniformly to zero on tails, and the inverse bound proves Cauchy. Finite-dimensional norm equivalence gives ordinary convergence. Apply adjacent actual relations normalized by the reference inverse. The lowest coefficient is eventually fixed terminal; Gram comparison and each paid inverse bound inductively give Cauchy coefficients at every fixed level, without an infinite future product bound. For captures,

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gnxn −gmxm = gn(xn −xm) + (gn −gm)xm.

Uniform capture bounds control the first term and Gram comparison on bounded coefficients the second, giving captured-data Cauchy sequences. Escaping horizons eventually include every fixed level with original common choices and prescribed stationary paths. History conditions retain lawful, scalar, activity and path budgets. Bounded coefficients times fixed normalization give uniform insertion payment. Exponentials and forced profiles are measurable sums of masked regular terms, with continuous pullback and the prescribed conjugation or identity. The lift retains reference inverses, actual adjacent maps and capture identities. Recovery preserves finite-depth approximation and these Cauchy estimates, together with fixed-level payment and measurability for integration.

The recovered finite-level inverse argument can be written quantitatively. If and , the actual adjacent identity gives

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|yn −ym|coeff≤b
|xn −xm|coeff+ εn,mMy

.

Readout retains and the coefficient norm, so the same inequality holds for the realized coefficient readers and is recovered by the section. Similarly the transported capture seminorm gives

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∥gnxn −gmxm∥≤a|xn −xm|1 + ε′
n,ma|xm|1

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when the Gram comparison error is ε′
n,m. Both right sides vanish on double tails at each fixed level.
The realized finite-depth approximation retains its 2 2−d error, and every proof at a chosen depth
uses the same finite set of eventual history identities. Thus coefficient and capture convergence
are recovered on the original escaping horizons, with the stated fixed-level payment bounds and
measurable readout functions available for the ensuing integral limit.

15.22 Uniform fixed-level insertion integrands

Use eventual uniform boundedness and uniform Cauchy comparison over the entire configuration set. Eventual equality, fixed pullbacks, finite sums and products preserve this property; continuous maps preserve it on bounded finite-dimensional ranges by uniform continuity on compact balls. For captured data, its masked regular sum and barcode forced profile satisfy

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|Lx(U)| ≤∥x∥,
|Fx,B(U)| ≤k∥x∥.

Paid health bounds and masks bounded by one prove the first; restriction of the forced budget proves the second. Absolute convergence makes both linear in, including differences under fixed masks. Capture Cauchy therefore gives uniform remainder and profile Cauchy bounds. Each placed core axis is continuous on compact finite-lattice configuration space and bounded. Its mask cannot increase that bound. Finite-axis expansion gives a uniform coefficient-Lipschitz bound, also for the constant axis. Core Cauchy thus implies uniform masked-core comparison. Format and capture identities identify actual exponential entries with core plus regular remainder, and actual forced profiles with captured profiles. Fixed motions, required conjugation and normalization preserve these comparisons at every fixed anchor. No new normalization is chosen. Eventually