Local nonlinear geometry and Haar realization · Part 4

and two Bool coordinates imply workL workD , of rate2: widening preserves the rate, and both products have rate2. The layered-extension and interior-gap estimates supply these bounds. All terms share actual

The light estimate is decomposed into four already specified energy types: the soft curvature/noise remainder, the correction, the coarse correction and the mismatch corrections. Sum their four envelopes into nonnegative of rate2, weakening the rate-2 bounds if needed. A selected fine curvature row extracts its two real-path half-squares, while a selected noise row extracts one, so the soft bounds give absolute remainder estimates there. At unselected curvature/noise rows extraction is zero and flaw plus half-squares are nonnegative; only a lower bound is asserted. Such a row therefore satisfies the lower bound without an absolute-value claim. The coarse row has no extracted parent contribution and uses its envelopes; the row uses its envelope.

These claims require a clean radius- coarse hull and grafted rate-2 tangent bound. Before a signed turn restrict to2+1, yielding2 afterwards and paying the reversed-link shift; read the maximum condition in that same background. On the packet annulus inside 200 seeds but outside , Wide bounds through 300, tangent control through 304 and 2 width) give grafting and cleanliness. Actual toll bounds make retained selectors one, hence original debit equals light and debit is at least. This uses no derivative of a hard selector and no globally small gauge. The soft-remainder, outer-correction and grafted-signal estimates of this chapter provide these four envelopes and the required shell access. The lift retains each selection status and this enforcement domain; it neither differentiates a hard selector nor converts the lower bound for unselected rows into an absolute bound.

12.28 Physical operator allowances and path-series bounds

Searchable transcription

For width η ≥0 and cover price P
z eη|z|∥Kαβ(z)∥≤C, each routed summand has norm at most
its absolute coefficient: adjoint travel is isometric even when winding. Endpoint congruence selects
exactly one input/output site for a fixed displacement. Absolute summation before periodization
gives row price |β|C and column price |α|C, retaining all coincident-endpoint displacements. For
the fine/coarse row the per-type price is C/L2 with input type comprising the fine residue and
two-layer plaquette type, and L4 residues; the corresponding row and column allowances are a/L2

and. Noise and tight rows have common allowance. The source takes the finite type-census maxima with one independently of the later choices.

A nonzero cut term has endpoint distance at most route length, bounded by The parent map foreshortens adjacent fine steps. The route from the lifted fine residue to ends at the embedded coarse site parent()+wrap(); this remains true for wrapping routes by additivity of wrapping and the cover route endpoint identity. This is incidence support, not a claim about every datum read by the operation.

For full path labels () retain the nonnegative matrix

Searchable transcription

Mpu =
X
i
|ci|1{d length(si)≤R, landing(si,p)=u}.

Searchable transcription

For each source plaquette type, the weighted budget P
i |ci|eσ length(si) ≤P gives row sums P. Each
fixed label/type fixes the source site for a landing, hence column sums qP. Absolute convergence jus-
tifies exchanges. Curvature chord and transport bounds give ∥shorten(U, p)∥≤P
u Mpu toll(U, u).
At a kept source the selector bounds toll at distance k by tek/ρ; landing distance at most slot length
and σρ ≥1 yield shortening size Pt.

Searchable transcription

√
Thus Astage = (q + 1)P +
direct pays rows, columns and direct clamp. Both thinned branches
have size at most Astaget: shortening uses the kept estimate and the direct branch uses min(1, t/r),
with zero curvature at r = 0. Reach margins hold eventually: for t = temp(x + a), tek/ρ < 2 and
t ≥(x + a)−1/2 imply k/ρ < log x once 2√x + a ≤x. Since B ≥(log x)2, for sufficiently large log x
this gives k + 1 ≤B/50, and the cut length plus one also fits; parent distances add at most one.
These observations supply both the shortening stencil reach and the discarded-witness reach used
above.

Searchable transcription

The shift a = G. slopeConstant log L ≥0 follows from the minus-Killing sum of squares. At fixed
shift eventually temp(x) ≤2temp(x+a) using √x + a ≤2√x and monotonicity of the nonnegative
logarithmic temperature factor. With A = Atop + Abelow and operator allowance C, the common
d = 2A(1 + C) > 0 gives

Searchable transcription

∥vision ∥≤dt,
∥Mv∥≤dL−2t,
∥Mw∥≤dt.

Indeed coarse vision is bounded by 2, and the fine/noise row allowances give these three bounds. Both stage lane counts use this same fixed before, with late cutoffchosen afterwards. These prove the operator and maximum hypotheses used throughout the preceding energy estimates. Readout preserves the full label census, selector branch and dimensional factors, rather than collapsing repeated paths or replacing incidence support by total-read locality.

12.29 Fine income and discarded-row accounting

Searchable transcription

Let a, b, p, t > 0 be the direct quadratic/linear coefficients, noise penalty and fine threshold. For
plaquette chords ri, let K be the listed large plaquettes ri > t and Z = P
i∈K ri. Actual plot
conditions put listed nonlarge plaquettes in [t/4, t] and list every large plaquette in the required
five-radius dome. The direct and second components are

Searchable transcription

Di = clipt(ri) + 1
2∥v0
i ∥2 −⟨v0
i , shooti⟩,
Ei = 1kept(i)
1
2∥ui∥2 + 1
2∥v1
i ∥2 −⟨v1
i , ui⟩.

Searchable transcription

Use ∥shooti ∥≤√ari, clipping ar2
i /2 below threshold and btri above, and response bounds ∥vν
i ∥≤
εt. Square completion gives Di ≥(√ari −∥v0
i ∥)2/2 for ri ≤t. Thus ε ≤√a/8 pays at2/128 on
each listed small plaquette, while ε√a ≤b/4 pays 3btri/4 on large ones. Unlisted summation rows
have ri ≤t and hence nonnegative direct part, hence
X

Searchable transcription

i
Di ≥(3b/4)tZ + (a/128)t2|costs \ K|.

A discarded row has a witness with. Such a row lies within radius log( of, including all plaquette types, for fixed relaxation length 0. Its witness ball contains at most) rows,PlaqAxes log. Raise the last inequality to the fourth power for1. Local witness reach puts in the five-radius dome, so makes it a member of. Thus the discarded set obeys on the original fine lattice, without an factor.

Searchable transcription

For shortening majorant Mij ≥0 with row/column prices C, support reach ensures all con-
tributing j /∈K have rj ≤t. Therefore ∥ui∥≤Ct + P
j∈K Mijrj for i ∈X, and summing uses
Ct|X| ≤CHρZ plus the majorant column bound to give P
i∈X ∥ui∥≤C(Hρ + 1)Z. Kept rows
have Ei = ∥ui −v1
i ∥2/2 ≥0; discarded rows have Ei ≥−εt∥ui∥. Choosing also εC(Hρ + 1) ≤b/4
proves
X

Searchable transcription

i
(Di + Ei) ≥(b/2)tZ + (a/128)t2|costs \ K|.

Searchable transcription

This argument does not assume a good halo at a discarded row. The lift retains witnesses, listed
sets and shortening multiplicities on the same fine lattice, so no unearned positivity or extra block
census enters the sum. For noise rows ni = wirei, the exact energy is at least (√p ni −∥vi∥)2/2.
Listed aisles have ni ≥wt, and ∥vi∥≤Ct with C ≤√p w/2 pays pn2
i /8 each; other noise rows
are nonnegative. Choose the response constant C before L, then ε = C/L2. First enlarge L for
all three fine-income restrictions, next choose w ≥2C/√p, and only then take the late scale. The
resulting fine coefficient remains exactly a/128 > 0.

12.30 Macro multiplicity and spending census

Searchable transcription

Put R = j + 8.
If twice macro radius (j + 7)L is below the fine period, coincident periodic
displacements there are identical, so M(i, z) ≤1. Nonzero multiplicity implies dist(i, parent z) ≤R,
by multiplying the coarse distance by L, bounding the embedded-base displacement by (j + 7)L,
adding the at-most-L/2 parent offset, and applying the triangle inequality and integer rounding.
Therefore P
i M(i, z) ≤(2R + 1)4. Grouping the finite displacement/plaquette sums proves the
port identity with actual multiplicities; the period condition is needed for the column bound, not
that identity. The macro census used here has a period qualification, already included in the step
hypotheses.

A store has at most four base sites per share, because a fine plaquette has at most four corner parents, a noise bond at most two pins, and a flag one site. Hence faces in its radius- dome number at most) shares,, and for1. Apply this separately to original and spending stores. Each flag touches a coarse face above temperature, and each face has at most four corners, bounding flags by four times those faces.

Every coarse face with toll above fixed 0 has a spending witness within eventually. Otherwise all macro-port fine tolls are at most spending threshold, using listed witnesses and unlisted; all four wires are at most using 2. The local predictor estimate would bound its toll by box((

at late scale with fixed, a contradiction. The distance bounds put every relevant input in the plot dome, as required for this local predictor estimate. Explicitly 0, while ) with subpower. For each fixed constant, eventual bounds give . Also(logpermits. The pre- constant

Searchable transcription

N = V (R) + V (2) + R + (2R + 1)4

pays macro columns/reach,-large face counts per spending share, and outer/affected counts

per respective share. Readout preserves the original/spending-store distinction and actual multiplicities rather than assigning an independent copy to repeated periodic data.

12.31 Affected departure: macro excess and operator transfer

Fix the same plot. For its affected face set, use local allowance constants0 and define

Searchable transcription

high. direct + d,
v =
√
B = L−2
X
swell(t)
toll U +
X
aisles
wire(Q, V ),
u =
p
low. direct + d,
g = √p.

HerePlaqAxes is the number of plaquette types. Retain nonnegative fine excess precisely on the swell at and noise excess on listed aisles. For macro multiplicity), row sum is at

most box(( and column sum at most on the physical torus. The exact port identity retains coincident-image multiplicities. Define

Searchable transcription

H(p) =
X
e∈∂p
F(e) + DL−2 X
z
M(p.1, z)E(z).

Searchable transcription

The four boundary terms count repetitions. Each fixed boundary-position map from face base to
bond is injective, so their total costs at most 4q P F. Macro columns give
X
p
H(p) ≤4q
X
F + DqNL−2 X
E ≤(4q + DqN)B =: b.

On each usable plaquette the predictor estimate reads tollfuel( port(), with the local predictor-curvature comparison constant. Thus macroLarge( and = macroLarge( The five-radius plot supplies toll and wire unlisted inputs use their upper thresholds, while listed inputs retain their full size in the excess. Every three-radius face is usable because its required radius-one coordinate box lies inside the listing dome; their boundary and macro stencils remain in the five-radius dome by reach and The predictor-curvature comparison therefore gives toll, where

Searchable transcription

macroSmall(j, D, L, W) = 8W +DL−2Sj,L. The four wires contribute 8Wt and the macro row sum
contributes DL−2Sj,Lt. This row census needs no injectivity between distinct cover displacements.
For the 2m tested Boolean slots the actual operator row/column allowances give
X
∥Φ∗Xf∥≤2maL2ut + 2auB,

Searchable transcription

X
∥(Φ′)∗Z∥≤2mag(2Wt) + agB,
X
∥Xn∥≤2mvrt + 2vb.

Searchable transcription

Separate uniform small input from its nonnegative excess.
Fine small input uses column price
aL2, while large input uses row price a/L2 and excess at most 2u P E, retaining the inverse-square
saving. Noise uses narrow prices a and coarse input its direct excess bound. Stencil reach sends
each tested two-radius dome into the permitted input dome.
The lift retains this local excess
split and both matrix directions; swapping a row and column price would lose the stated block
factor. For departure L, vision has norm at most dt and pairs with Φ∗Xf + (Φ′)∗Z −Xn. Cauchy–
Schwarz bounds its sum below by −dt times the preceding three norm sums. For departure D
the residual size is at most (d + ad + ad + a2d)t: the wide aL2 cancels the dt/L2 response, with
narrow and tight terms adt, a2dt. Pairing and the two half-weighted slots give workD(p) ≥−d0t2,
d0 = d(d + ad + ad + a2d).

All charged third-range faces lie in and have toll sum at most. Coarse temperature at most 2 therefore incorporates their linear charge. With= low linear, the exact choices give

Searchable transcription

ltemp
X
tides
toll V −talll(s) ≤CpretB + Cseatt2m,

Searchable transcription

Cpre = d(2au + ag + 2v macroLarge(N, D)) + 2lmacroLarge(N, D),
Cseat = 2d(aL2u + 2agW + vr) + d0 + 2lr.

To see the coefficient collection explicitly, the pairing contributes at most

Searchable transcription

d(2au + ag + 2v macroLarge(N, D))tB + 2d(aL2u + 2agW + vr)t2m

to the negative departure. The pairing adds, while the payments relation, whose charged third-range faces are all in, adds at most 2). Substituting = macroLarge(

yields exactly the two displayed constants. The first is fixed before; the second may depend on both. Since shares(spend), the seat charge is at most ( shares(spend), with a positive coefficient. Fine excess remains cut at; only the spending census uses its larger threshold. The lift keeps this distinction and the same affected set, so coarse charges are paid by the actual fine/noise income rather than a differently truncated substitute.

12.32 Three coarse clipping ranges

Let = toll( = temp( = low direct 0 and = low linear. Set otherwise = min(1). Shortening terms cancel exactly: kept rows contribute their matching half-square and discarded rows have both tested terms zero. The direct remainder is

Searchable transcription

workC(i) = a(α −α2/2)∥curvature(V, i)∥2 −clips(r).

Choose group-only 0 giving curvature square at least and 34 for. In the small range = 1 gives workC at least2. For, one has 0 and2, giving

Searchable transcription

workC(i) ≥as
2r∥curvature ∥2 −bsr.

Thus on, the curvature bound and clip hypothesis8 give explicitly

Searchable transcription

workC(i) ≥(3a/8 −b)sr ≥(a/4)sr.

On, dropping only the nonnegative curvature term yields workC at least Finally the supplied temp(1 makes when The common late scale therefore supplies all three ranges with small debit coefficient2 and fixed group-only middle cutoff. The lift preserves the exact branch values, including the zero-toll convention, without differentiating selectors or replacing the outer lower bound by an absolute estimate.

12.33 Threshold ratios and stitch absorption

Fix first. Set for its fixed coupling shift,(log = temp( and. At these late scales temp( exp((log log). For every fixed real shift , the logarithm of temp()/temp(

Searchable transcription

(log log(x + k))2 −(log log x)2 −1
2[log(x + k) −log x].

The bounded derivative of (log ), applied at the logarithmic arguments, bounds the first difference by twice the log difference eventually. Hence the ratio tends to one. For every fixed 0 because eventually2(log, every fixed logarithmic power is), and the fixed shift leaves this comparison unchanged. The monitor boundstemp(

Searchable transcription

for any p > 0 eventually imply
Bn temp(x) →0,
Bnt/T →0,
BnT →0,
BnT 2/t →0.

The first ratio uses200, leaving the decaying power; the last uses the lower bound on and exponent, not, that is small relative to Quadratic logarithmic growth of makes eventually for each fixed 0, using + 100000 and log2 log. Thus0 and0. For stitch size and, expanding nonnegative terms and using gives

Searchable transcription

Bnh →0,
Bnh2/t →0,
Bne →0,
Bnhe/t →0.

The first two stitch limits use the preceding bounds with exponents + 10 and + 20. Expand for the other two and use; each nonnegative term is controlled by a preceding limit, and the mixed tail also uses0. These limits pay dome-census stitch losses without a period or share-count factor. The lift preserves the fixed- order and original exponents, rather than treating these as simultaneous limits in all construction parameters.

12.34 Finite-store budget inequalities

Searchable transcription

Let a = lineartop, p = penalty, t = fiery and u = largeT ≥t. For listed fine sets Ft, Fu above these
thresholds and R = P
Ft toll, one has t|Ft| ≤R, u|Fu| ≤R. The supplied tally (a/2)tR+dt2|costs\
Ft| therefore pays, when m ≤a/8, m ≤d, cF ≤a/4,

Searchable transcription

tallyF ≥mt2|costs| + mtu|Fu| + cF tR.

Searchable transcription

Indeed 2m + cF ≤a/2 and the cost partition pays each term. Original shares, spending subset and
mismatch fee remain separate.
Listed noise has wire at least Wt ≥t.
With RI = P
listed wire2, t2|aisles| ≤RI and
tu|spend_aisles| ≤RI. If C ≤pW/16, then Ct wire ≤(p/16) wire2. Using m ≤p/32, the supplied
(p/8)RI gives
tallyI ≥mt2|aisles| + mtu|spend_aisles| + Ct
X
listed
wire .

The coefficient budget is 28, since32: the first two terms each cost at most, and the fee costs at most (. The lower wire bound is the actual endpoint Plot condition at a bond endpoint belonging to the store dome.

Split coarse outs into toll at most = temp, between and, and above. With, the three-range estimates give respectively, at least 4 fordirect andlinear toll. Sum these disjoint ranges and enlarge the nonnegative small-range error count to all outs. Hence

Searchable transcription

tallC + linearbelow s
X
toll>δ
toll +Bquartics4|outs| ≥4mt2#{s < toll ≤δ}.

Payments bounds flags by four times-large tides and-large tides by shares(spend). Subtracting the latter subset givesflags toll shares(spend)]. These are finite algebraic consequences of the producer estimates and Payments, with no added period compatibility. The lift retains gains at separately from original seed shares at and carries every store set on the same configuration.

12.35 Point-charge gain on the common store

Searchable transcription

Keep the same store, input, predictor and background. Set t = fiery(x), s = temp(x), u = largeT(x)
and B = radiusB(x). The fine producer gives (lineartop /2)t times its large-toll sum plus dt2 per
other listed fine toll, d > 0 after the block/width choices. Noise gives (penalty /8) P wire2. Choose
m > 0 at most lineartop /8, d, penalty /32, directbelow /64. The hit mismatch estimate is

Searchable transcription

tallL,D(hit) + cF t
X
large fine
toll +Ct
X
listed
wire +Zt2B4 shares(spend)
≥linearbelow s
X
δ-large coarse
toll,
cF = C/L2.

Choose to makelinear4, then1 withpenalty16. Listed-wire size makes its quadratic income absorb the linear loss; fine income absorbs . The coarse budget pays its large-toll charge. The flag census costs four times the middle-range count plus 4 shares(spend), and output count is at most shares. At fixed preceding choices take the common late scale so and

Searchable transcription

(4mK + ZB4)t2 ≤mtu,
Cquartics4KB4 ≤(m/4)t2.

Indeed the spending remainder is explicitly

Searchable transcription

[mtu −(4mK + ZB4)t2] shares(spend) ≥0.

The first pays spend flags and mismatch remainder from spending income; the second pays the small coarse error. The unaffected mismatch costs at most another ( shares, and tally is nonnegative. These scale choices follow from0: divide the first inequality by, and use for the second. They are uniform in live period.

Split hit/unaffected rows and add the finite-store fine, noise, coarse and flag inequalities. Positive remainders pay per original share before the two quarter losses, giving

Searchable transcription

assessed ≥(m/2)t2 shares .

The source lift retains the same store and all its contributions, with block length paying the fine loss, paying noise, then positive fixed and the final delay paying remaining errors. These positive remainders come from the exact fine, noise and coarse budget rearrangements above. No later retuning of or incompatible witness is used.

12.36 Unaffected mismatch and exact core assignment

The point-charge producer combines E01–E09 with this additional unaffected-row estimate derived from the mismatch bounds. The stitching bound, its two displayed scale limits, and the census and maximum estimates of this chapter supply its inputs. For an unaffected coarse plaquette row outside the hit subset of the store outputs, put = fiery. Vision is bounded by, and stitching gives one gauge with fine/coarse chords at most = stitchCost on radius 25, retaining forest and tags. Choose the delay to contain 3 till +4 + predictionReach and meet the chart radius. With = widen(till), local gaps give

workL + workDloss loss = 2

Two slots give the first factor two; the two half-weighted terms give the last. There are at most shares such rows. The stitching-scale limits are0 and0. Dividing their total bound by gives 20. Thus every prescribed 0 eventually bounds total unaffected loss by shares, with unchanged actual input and predictor.

The resulting point-charge order is block lower demand, sufficiently large, then positive, then a common late. It is pointwise and uniform over coarse sides at least 100(1 + funded data, forests and actual Plot inputs, giving assessed at least times shares. Passage from the supplied grant intersects its delay with positivity of and specializes its universal query; the fine, noise, coarse-clipping and flag budgets provide the energy combination.

Row locations are fine plaquettes, noise bonds, coarse plaquettes bundling rows. Their principal values are workF, workI, workC+workL+workD and workN, all at the same actual input, coarse background and predictor. The coarse anchor is the parent of a fine-plaquette root,

and the coarse root for the other types. Each fine plaquette parents two moment rows, noise one, others none, with every moment row parented once. A signed row is selected when its signed coarse anchor lies outside the ambient 6 seed dome; its surcharge is its real affine path half-square when selected, zero otherwise. Extracted sums parent surcharges, debit is principal minus extracted, and liftSq is the orientation average of extracted rows. Exact finite recombination gives

Searchable transcription

wholeGap +
X
noise
monetary = averages
X
rows
debits + liftSq .

The finite subtraction of extracted rows cancels exactly against liftSq, with still bundled at their common coarse plaquette. Signed noise covariance carries actual transformed tags and preserves the finite bond sum. For a store contained in the ambient seed place, posted 2 core rows are inside 6, hence unselected with zero extraction. Their principal/debit sum is exactly assessed, retaining the bundle. Wide through implies Plot on its restriction and is preserved by simultaneous rotation of fields, tags, forest and store. Apply point-charge in each rotated slice and average, with shares unchanged. The exact core assignment and point-charge inequality proved in this chapter supply this step. The lift retains this entire recombination; extracted energy is accounted for, not discarded when assigning the core.

12.37 Packet annulus and tariff

For a store in the ambient seed place, assessed is its posted 2 debit sum and assigned is the corresponding 200 sum. Their difference is the finite annulus sum. Each row is at least and the posted-row census per orientation is at most shares() ballCost(200 +1) spots(). Hence assessed minus) times shares is at most assigned, where spots() counts the row types per coarse site and ballCost(200 + 1) spots() retains rate2, since the radius factor has rate zero. Averaging the same orientations and using point-charge gives

Searchable transcription

quoteNear(Z, u) ≥[μt2 −E(x)] shares(Z).

The domain retains Wide through 300, tangent control through 304 and shell access. Connectedness of is not needed.

Ancillary expenses use the all-field condensation bound, eventual Haar-log bound one, sweptneighborhood/ticket costs of order bill and frozen-measure negative log of order bill. A fixed 0 therefore gives steamLog plus tollFee times shares at most quoteNear + bill shares.

For an admitted nonzero packet, seed consistency supplies ambient containment and actual flags. Its nonzero ticket/sector profiles supply the same 300 Wide domain, and enforcement gives hot bound 2 through 304. The width condition width pays frozen tangents. Thus the quote applies to that actual field, forest and tags. Since and has rate2, eventually bill. Substitution of the quote bound into the full expense estimate gives the exponent

Searchable transcription

[−βμt2 + βE + Kup bill5] shares(Z) ≤[−βμt2 + (1 + Kup) bill5] shares(Z).

Searchable transcription

Exponentiation proves

Pack(exp) bill(] shares(

Zero packets satisfy it immediately. This is cap one, expense (1 +) bill and gain per share, pointwise before frozen-Haar/Gaussian integration. The lift retains the whole packet once, including its tag and guard costs, and applies the income only on the nonzero-enforcement domain that earns it.