Phase decomposition and Wightman reconstruction · Part 7

terms of the primary lens are identically zero in this specialization. The candidate is bounded on the healthy compact chart region, uniformly in the finer terminal cutoffand period, and is multiplied by zero outside that region. Thus the bound for (AP1) is. Its read set is contained in the coarse window of radius equal to the stencil radius plus four: each inspected edge is within four sites of a stencil point, so the triangle inequality gives the inclusion. This proves the lens locality and uniform bound used for the real witnesses.

The finite coefficient inversion in the approximation is quantitative. In the coefficient norm suppose the recovered coarse coefficient vectors satisfy

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|ei −Jbi|1 ≤1
4

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for every coordinate basis vector ei. Let Av = P
i viJbi. Then

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|v −Av|1 ≤1
4|v|1,
|v|1 ≤2|Av|1.
(AP2)

The first inequality follows by expanding the difference and summing coordinate norms, and the second by the triangle inequality. Hence is injective and, in finite dimension with equal domain and codomain, surjective. For a terminal label, solve this linear system and use the injective coefficient map to recover

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L =
X
i
vi terminal(bi),
|v|1 ≤2ecolumnLogs(k)|L|1.
(AP3)

The last factor is the earlier backward coefficient budget. Denote its right side by stock Each basis lens, lifted back from scale to the fine lattice, approximates its recovered terminal observable in the root seminorm, with common error at most stakes. Their linear combination with (AP3) is a fine-lattice approximant to, with

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∥B∥∞≤stockk(L)Ak,
∥B −OL∥root ≤stockk(L)stakesJ,m
w
.
(AP4)

Back transport preserves the sup bound and sends the coarse window into the prescribed common fine chamber, by the atlas room condition. Subadditivity and homogeneity of the root seminorm prove the second bound after the exact terminal-label decomposition. Thus both locality and the approximation concern the same

The basis root bound comes from an actual many-copy insertion. Use the separated atlas copies of the observable difference, with the compulsory mark at each copy, and extract the coefficient containing every copy once. Exact RG transport preserves its Wilson integral. The insertion estimate bounds that coefficient by

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|M| wq ≤eqc,
c = (aJ/am)4 treasurym + 5,

where is the number of atlas cells and is the corresponding Wilson moment of the product of copied differences. The term ( follows from the exact cell census, and the remaining five units are the insertion and vault costs in this application. Taking the-th root gives the root estimate = stakes. The coefficient identity cancels the same positive Wilson partition function on both sides; it is not an estimate in a different normalization.

The coefficient construction requires

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w |am|4(1 + columnsk)
gainChart(k, m) ≤1
2ζd,m.
(AP5)

Fix the reporting data first. Choose at the upper value allowed by (AP5). Then the error in (AP4) is a fixed constant times the product of the stock and column factors divided by the gain chart. Those factors have subexponential growth, whereas the inverse gain chart contains the strictly decaying scale factor. Their product tends to zero; hence a sufficiently late, also satisfying the atlas room condition, makes (AP4) smaller than any prescribed positive tolerance. Fixing that fixes the sup bound as well, uniformly on all later mesh and period parameters. Taking the real part as above supplies the real bounded approximation needed for the reflection witnesses.

Transporting the approximation difference with its paid mark

The comparison used in (AP4) retains a single common source history. Starting from the primary coefficient at scale, the backward coefficient construction chooses a reference label and a residual history. Its identities at the upper comparison scale are

exp() = exp(primary) + exp(history( forced() = forced(history( (AP6)

These are exact identities for the evaluated exponential entry and the forced activity profile. They follow by linearity of the history map and cancellation of the reference label in the primary/reference decomposition. In particular the primary contributes no forced profile here. Its residual coefficient is, with norm at most4; the backward cone estimate gives this bound, while the terminal/reference coefficient is bounded by 2

The remaining residual cost at scale obeys

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

Transport the mark price from at by the common multiplicative schedule. The scalar reserve recurrence and (AP5) imply, throughout, that the transported price times the scaled residual cost is at most2. The same bound pays for the reference-label contribution at, using its coefficient bound 2. These are the two half-reserves in the construction.

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Now insert, at each separated atlas copy, the negative terminal reference observable, transport
its marked polynomial exactly from N to k, and multiply there by the corresponding coarse lens
insertion. Equation (AP6) identifies the resulting first marked slot and forced slot with the residual-
history slots. The paid insertion transport theorem then carries those slots from k down to m, along
the same migrating copy positions. The separation and occupancy bounds are obtained from the
chosen atlas at each intermediate scale. The resulting insertion at m has the served-source bound
with support exponent of that scale and the two vault bounds 2, 1 used in the root extraction. The
residual-history estimates pay both bounds; no new history or mark schedule is chosen during this
transport.

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The coefficient identity relating this insertion to the actual fine-lattice discrepancy is algebraic.
For a finite census, the new coarse insertions multiply as Q
i(1 + ziQi). Extracting a prescribed
multi-index coefficient sums over the subset of slots taken from this product; every other slot is
the coefficient transported from the negative fine observable. Coefficient extraction commutes with
each finite RG push by linearity and integrability. The resulting subset sum is exactly the joint
fine/coarse difference polynomial. Integrating the push does not change its integral, and restoring
the fine/coarse coupling gives the positive Wilson partition function times the discrepancy moment.
This is the identity used before taking the all-copy coefficient in (AP4).
Thus the comparison
bound, its normalization, and its locality all refer to the same transported difference.

The backward-cone estimate used here follows from the one-step coefficient and remainder bounds, as follows. Use the coefficient norm, including its unit coordinate. At a step of block

size, put and is the finer coefficient, let

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yα = Ld−deg(α)vα,
Y = |y|1.

All exponents are nonnegative, so . The step supplies an invertible linear map, commuting with deletion of the unit coordinate, with1. Its coefficient error and remainder costs0 satisfy

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

Applying the inverse bound, the triangle inequality, and core , gives

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Y ≤K
|u|1 + e(Y + t)

,
r ≤KY + et.
(AP7)

Let1 be the column allowance at scale, and put. The room and bank choices give (AP8)

For precision, the bank condition is

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q ≥100000 M6(1 + Ci+1)2,

where is the neighboring price bound. It implies Use and to get the last three inequalities in (AP8). The room condition , with the stipulated monotonicity of stage prices, gives the first.

Suppose. By (AP7) and

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4K|u|1 ≤Y + et.

Substitute this bound on back into (AP7). The first smallness inequality in (AP8) absorbs the resulting -term and gives. Consequently

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

The same argument with = 0 also holds directly. Thus the cone propagates from the coarser endpoint to the finer endpoint, while the remainder cost gains the factor in the opposite direction.

To obtain the history to which this estimate applies, start with the given terminal formatted insertion and its remainder bound. Descend the finite interval one scale at a time, using the same one-step marked transport and its linear coefficient decomposition. At each step update only the newly reached component of the history. Previously obtained components and their step identities are retained. This finite backward induction constructs the entire matched history; 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 at that endpoint zero, while preserving every density and step identity. Its nonnegative remainder cost therefore satisfies the cone there. This explains why the cone is available for the residual history, rather than being an assumption on an arbitrary primary history. Let be its remainder cost on the interval [], and its residual coefficient. Iterating (AP9) gives

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

.
(AP10)

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8

The identity used in this iteration is gainChart() = gainChart( . The terminal pairing bound is

Since the cone implies 2, it follows that. Using yields the claimed residual estimate in its explicit form:

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gainChart(k, i)(ci + |vi|1) ≤4(1 + Ck)w.
(AP11)

The constant used in the approximation argument is this terminal bound with its fixed coefficient and stock factors.

The numerical choices required for (AP8) can be made before selecting a history or a volume. At each stage choose its column allowance above the room threshold, and choose its block lower bound so that exceeds the displayed bank threshold. This is possible since; enlarging either the prescribed block lower bounds or the later delays preserves the choice. Equations (AP6)– (AP11), followed by the paid marker transport and root extraction, therefore establish the bounded local approximants on the original common family.