Phase decomposition and Wightman reconstruction · Part 8

Fix first and choose at this upper value. AP4’s error is a fixed constant times stock and column factors divided by the gain chart. Their growth is subexponential, while the inverse gain chart contains the strictly decaying scale factor, so the product tends to zero. A late enough also meets atlas room and any prescribed positive error tolerance. Once fixed, its sup bound is uniform on all later mesh and period parameters. Taking the real part then gives the real witnesses used by RS1.

The lifted approximation source retains the healthy mask, candidate coefficients, inversion columns, atlas, all-copy mark and common partition function. Its local reader is AP1, its coefficient reader is AP3, and its rooted reader is the normalized all-copy extraction. AP2 proves coefficient recovery and AP4 controls observable recovery with its explicit error. The finite coarse lens loses the residual observable only to that bounded error; retaining the transported discrepancy and paid mark restores the exact comparison, as below.

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Quantitative coarse-lens and coefficient recovery.
The framed local lens is the actual pull-
back
LZ
k,L = Lk,L ◦πk,
LZ
k,L ◦σk = Lk,L.

Its healthy predicate and candidate are read through the same map, so the bound and coarse physical read window are preserved. No unmasked polynomial value is used outside the healthy region. For the finite coefficient system in (AP2), set on its physical coordinate space. The coordinate-column hypothesis gives4, so its explicit recovery operator is

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A−1 −
r
X
n=0
En





1→1
≤(1/4)r+1
1 −1/4 .
∞
X
A−1 =
n=0
En,

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This preserves the source’s looser inverse bound 2 in (AP2)–(AP3). On the horizontal coefficient
carrier jdEd the system is AZ = jdApd with inverse jdA−1pd, so its solution recovers exactly the
same vi and terminal-label decomposition. The source fine approximant is then BZ = B ◦π, with
the same AP4 sup bound.
For every permitted finite atlas-copy list, configuration pullback and normalized measure recov-
ery give
* q
Y
+
* q
Y
+

Searchable transcription

i=1
(BZ
i −OZ
L,i)
Z
=
i=1
(Bi −OL,i)
= M.

The compulsory all-copy coefficient, exact RG identity and common positive Wilson denominator are retained in this reader before itsth root is taken. Hence the rooted error bound, cell-census factor, mark price (AP5) and stock bill are unchanged. The reporting data are fixed first; the same late auxiliary scale and atlas room condition then recover the prescribed tolerance for this very local, with its sup bound uniform on subsequent mesh and period tails.

19.44 The paid discrepancy history and backward cone

The AP4 comparison uses one common history. Starting with primary coefficient at, backward coefficient construction chooses a reference and residual history for, with

exp() = exp(primary) + exp(history(

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forced(Zk) = forced(history(−b)).
(AP6)

These are exact evaluated-entry and forced-profile identities by linearity and reference cancellation; the primary has no forced profile here. Its residual coefficient is, bounded by4 by the backward cone; the terminal/reference coefficient is bounded by 2. At intermediate scales,

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cost(Zi) + |Zcoeff
i
|1 ≤E/gainChart(k, i).

Transport the mark price at by the same multiplicative schedule. The scalar reserve recurrence and AP5 bound the transported price times scaled residual cost by2 throughout the interval. The reference contribution at is paid by the other half-reserve, using 2. At each atlas copy insert the negative terminal reference observable; transport its marked polynomial exactly from to and multiply by the coarse lens insertion there. AP6 identifies its first marked and forced slots with the residual slots. Paid transport carries them from to with the same migrating positions; the atlas supplies separation and occupancy at every scale. At the served-source

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support exponent and vault bounds 2, 1 follow from the same residual estimates. Neither history
nor mark schedule is reselected. For a finite census the new coarse factors are Q
i(1 + ziQi). A
multi-index coefficient is the subset sum over slots taken from these factors, all others supplied by
the transported negative fine observable. Finite coefficient extraction commutes with every RG
push by linearity and integrability. This sum is exactly the joint fine/coarse difference polynomial.
Integration of the push preserves its integral; restoration of the coupling gives the positive Wilson
partition function times the discrepancy moment. Thus the root extraction compares this actual
local difference in its original normalization.

Exact paid-discrepancy coefficient recovery. Retain the one reference label, residual histories, migrating copy positions and transported mark prices as one source record. At scale apply the coefficientwise configuration reader L to the evaluated entries in (AP6). It gives the exact identities

L exp( L exp(primaryL exp(history( L forced( L forced(history(

Here exp denotes the source’s evaluated entry; its stated linear identity is retained as such. The same coefficient section preserves, its4 bound and the reference 2 bill. Residual cost and mark prices are read from the retained history, so the two half-reserves and the scale- vault bills remain the original ones throughout the paid transport.

For the all-copy coefficient on a finite set, writez) for the transported negative-fine insertion polynomial. The exact finite coefficient reader is

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!
N(z)
Y
i
(1 + ziQi)
=
X
B⊆A
[zA\B]N(z)
Y
i∈B
Qi.
[zA]

This is the square-free all-copy instance of the source’s general multi-index rule. Finite coefficient extraction and configuration pullback commute term by term; every RG push commutes with it by the same linear integral identity. Restoring the coupling thus recovers the same positive Wilson partition function times the actual fine/coarse discrepancy moment. Section restriction recovers the finite polynomial itself, including both primary and forced slots. The support, occupancy, mark schedule and normalization in the rooted estimate therefore belong to this one transported discrepancy.

Here is the quantitative backward cone. In the coefficient norm including the unit coordinate, a step of size has and

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yα = Ld−deg αvα,
Y = |y|1,
|v|1 ≤Y.

The step’s invertible commutes with deletion of the unit coordinate, with 1. For costs

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|u −By|1 ≤e(| core v|1 + t),
r ≤K| core y|1 + et.

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Consequently
Y ≤K
|u|1 + e(Y + t)

,
r ≤KY + et.
(AP7)

For column allowance1 put. Room and bank choices imply

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ai ≥4K2,
4Ke ≤1,
4Kai+1e ≤1,
(4K2 + 1)qe ≤1.
(AP8)

Explicitly the bank requires, with neighboring price bound implies1; use and for the last three bounds. Room and monotonicity of stage prices give the first. If, AP7 gives 4 Substitution in AP7 and absorption using AP8 give. Thus

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ai+1|v|1 ≤ai+1Y ≤t,
qr ≤q(KY + et) ≤t.
(AP9)

The = 0 case follows directly as well. The coefficient/remainder cone propagates coarse to fine, with the opposite-direction remainder gain. The first smallness inequality in (AP8) absorbs the resulting -term and gives

Construct the matched history by finite backward induction from the terminal formatted insertion and remainder bound. At each step use the same marked transport and linear coefficient decomposition, updating only the newly reached component while retaining all earlier components and identities. Its terminal density is exactly the prescribed insertion plus the reference history. Subtract the reference coefficient determined at the coarsest endpoint: linearity makes the residual coefficient there zero and preserves all density/step identities. Its nonnegative cost therefore starts in the cone. This is why the cone applies to the residual history, not to an arbitrary primary one. For costs and residual coefficients], iterate AP9:

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ai|vi|1 ≤ci,
gainChart(k, i)ci ≤ck,
gainChart(k, i) = exp
(scale(k) −scale(i))/8
.
(AP10)

The recursion is gainChart() = gainChart( . The terminal pairing bound and cone inequality 2 give. Since, this proves gainChart( (AP11)

The approximation constant is this terminal bound with its fixed coefficient and stock factors. Choose column allowances above room thresholds, and block lower bounds so exceeds the bank threshold, before selecting any history or volume. Larger block bounds or later delays preserve the choice. AP6–AP11, paid transport and root extraction therefore construct the bounded local approximants on the original common family.

The lifted discrepancy source retains each component and step identity of this matched history, together with its paid price schedule and finite subset-coefficient polynomial. Coefficient extraction is its observable reader; commutation with the exact push proves recovery of the original discrepancy moment. AP9–AP11 give a quantitative reader bound without discarding the residual or its unit coordinate. Thus the local approximation, the real reflection witness and its common Wilson normalization are recovered from one source history.

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Backward-cone and matched-history recovery.
Use coefficient section and recovery ji, pi at
each scale, retaining all coordinates including the unit. Put bv = ji+1v, by = ji+1y, bu = jiu, and use
section norms. The diagonal scaling reader and step map are

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yα = Ld−deg αvα,
BZ = jiBpi+1,
piBZji+1 = B.

Core deletion is transported with these same sections. Thus its commutation with, the inverse bound, and both one-step coefficient/remainder inequalities recover (AP7) exactly. For clarity the source’s absorption in the cone can be read as

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Y ≤1
4(Y + et) + KeY + Ket ≤1
2Y + (K + 1
4)et,
Y ≤4Ket,

using 41 and1. The remaining (AP8) bills then recover both conclusions (AP9) with the same direction and factor

A source history is the full finite tuple of coefficient components, remainders and step identities. Its readout applies componentwise and leaves the recorded nonnegative costs and bank prices fixed. At each backward-induction step only the newly reached component is changed, so this reader commutes with the finite induction. Subtraction of the same reference history recovers the zero coarse residual coefficient before initiating the cone estimate. Consequently its iterated source bounds are precisely

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ai∥bvi∥1 ≤ci,
gainChart(k, i)(ci + ∥bvi∥1) ≤4(1 + Ck)w.

The gain-chart product telescopes by the same scale identity (AP10). Thus the original terminal pairing bound, unit coordinate, room and bank choices, and history-before-volume quantifiers all recover through one matched tuple; no arbitrary primary history is substituted for the residual history in this cone argument.