Local nonlinear geometry and Haar realization · Part 5

12.38 Compulsory packet factors and normalization expense

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For seed store A, visitd(A) consists of fine bonds with a coarse endpoint in its radius-d dome, and
pens is its nontrunk visit10B. Set mg(β) = chartScale(
√
2πβ−1/2)dim Color. The full exponent is

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steamLog = −β[quoteNear(A, u) + farBills(A)]
+
X
visit200B(A)
condensation(e) +
X
plaza(dome304B(A))
gentry(e, u) −| pens(A)| log mg(β).

FarBills is the nonnegative orientation-average of extracted mean half-squares on posted 200 rows. Multiply its exponential by the order profile, actual 300 tag tickets and sector profiles, and 304 tangent enforcement to obtain the nonnegative packet. Nonzero packet forces every individual enforcement cone there nonzero. Profiles are in [0 1]; ticket bound gives steam, paying its whole ticket cost once.

For nonnegative upper bounds on absolute condensation, absolute gentry and log, endpoint census gives visitshares() ballCost(Axis. The gentry plaza is contained in visit) and pens in visit), so dropping only farBills bounds ancillary expense by

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shares(A)[Q0 swept(200B + 2) + C0 swept(304B + 2) + K0 swept(10B + 2)],

where swept() = ballCost(Axis, in addition to quoteNear. Finally

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−log mg = −log chartScale −dim(Color) log
√
2π + 1
2 dim(Color) log β.

For 0 , monotonicity gives loglog(2 ). Thus the entire measure expense has an upper bound depending only on log(2 ) and fixed chart/color constants, and is eventually bounded above by Delay bill(), without a lower coupling-band assumption. The point-charge core bound and the packet annulus estimate show that the gain in quoteNear dominates these counted expenses. Core and annulus income pay these exact counts. The source lift retains frozen-measure normalization, all individual guards and the nonnegative farBills term, even when the bound drops that term by its justified sign.

12.39 The short correction family

Retain three local letter kinds with their fixed geometric sleeves. The signed energy letter outside the 200 seed neighborhood is

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−β orientation[enforce2a(u, booth(o)) light(u) + extracted(μ)],

and zero inside. Condensation is zero when either bond endpoint touches that dome. The Haar-log letter is retained only at hot links outside the 304 plaza.

On the original seeded sector, remote light equals the energy debit. Noise equality is definitional. For curvature turn fields, tags and store together: sector membership and positive likelihood supply Interior. If a queried retention threshold is below two, room puts that read within Interior and toll is at most2; if it is at least two, the universal chord bound supplies retention. The retained path field is therefore unchanged and exact curvature splitting identifies light with debit. This sector hypothesis is needed for that equality, not for the following all-field envelope.

The protected mean has path bound of rate1, so at most two parent half-squares bound extracted by. Let of rate2 bound light on a clean grafted hull with tangent size 4 fiery,

and let of rate2 bound remote condensation. If enforcement vanishes, its light product is zero. Otherwise its protection bounds every booth tangent by 4 fiery and distant grafting supplies the same clean hull; the profile is at most one, giving product bound. Seed separation supplies the required parent/hull margins. For 0 , orientation in [0 1], the energy letter is bounded by of rate has rate one, rate2, and rate2. Haar is bounded by fiery of that rate. Hence fiery + is an eventual nonnegative all-field envelope for all three kinds and admissible couplings. A bond touching the dome at an endpoint has its parent in the one-site enlargement; conversely, if it does not touch the dome its parent is outside it. These endpoint facts justify the condensation exclusion. The light, remote-condensation, grafted-signal, mean and Haar estimates supply the inputs to the common envelope. The lift preserves the three letter identities and the distinction between sector equality and all-field bounds, retaining actual enforcement rather than differentiating or dropping its hard support.

Provenance of the two bounds in the short correction family. The two bounds used by the short correction family are outputs of earlier constructions and not separate suppositions. The light bound is from the annulus estimates, with grafted tangent bound fiery(): the soft-square and mismatch terms have rate2 and the two quartic or squaredguide corrections have rate2, so after weakening the latter has rate2. The condensation bound is = dim gpulseHead(repair) from the local plate comparison: its repair envelope has rate2, its window census has rate zero and its fixed inverse-window constants do not depend on has rate2. In the realization these are the same scalar functions of the tagged level, read from the same annulus and plate records; the rate calculus (finite sums keep the larger rate, products add rates) is applied to identical numbers, so the lifted short family has the same two rates and no rate is introduced by the lift.

12.40 Base stroke at a fixed multiple of the small-field scale

For integer set pcap(PlaqAxes. Axial flattening on radius + 1 and temp at most 2 give radius- chord bound pcap(. Here is fixed per finite query, not increased to

For a replica plaquette with clean radius-five hull use = 4. Its replica links are identity or gauged coarse links, bounded by pcap(4). Replacing exterior links by identity leaves its product unchanged; chord and curvature norm are at most 4 pcap(4). Every retained fine base-moment slot lands at such a plaquette within the original clean radius- hull: its actual path length is bounded by till(), and the room inequalities ensure this inclusion. Adjoint travel preserves curvature norm, so absolute path summation and the direct coordinate give

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∥Xf,R(replica(V ))∥≤4 pcap(4)(
q
directtop + pathstop)t =: C1t.

A common gauge for all slots is unnecessary because only each curvature norm is estimated.

For noise instead fix = predictionReach(). Flatten that fixed hull and extend its logarithm to with pcap(). Its linear copy obeys the same bound and exponential replica identity. Predictor locality with actual tags identifies the base noise with Test(). Its uniform Taylor bound gives size at most eventually, with and the chart radius. Gauge covariance returns the original bound. Thus bounds every clean-hull base stroke by a fixed multiple of; this is the crowns input to the global mean bound, obtained without spending an extra growing-radius factor. The lift retains which queries share

a gauge and which need only invariant norms; it does not combine distinct local gauges into an unsupported global preparation.

12.41 Global mean, stationary residual and Gaussian rearrangement

Keep the global estimate separate from the sharper remote path estimate. Use the selected rows, frozen cutofffield and inverse-window bounds. If base stroke is at most, guide at most oriented operator at most and saved field at most 2, selected-row cleanliness gives

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∥priced(0)∥≤scale(A0 + M + 2NγK)t =: b1t.

Unselected rows are zero. This uses the saved bound without a global gauge. The transpose price gives coordinate bill bound. Let be the inverse-column pad and . Its column price = payE then gives and color-field mean at most dim(Color). Symmetry of yields exactly

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Aμcount + b = −A(P −A−1)⊤b.

Plate price and pad error give residual bound distPay( Taking = 1 + max( dim(Color) 0) bounds all three global quantities. The residual has any prescribed rate, because the scraps estimate, hence distPay, has rate2 and has rate This supplies the mean/residual hypotheses of the correction budgets. The clean base-stroke and uniform maximum estimates supply

For the exact exponent retain, mean in count coordinates, bill, covariance 0 and. Expanding finite affine squares gives

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liftSq(μ + gz) = liftSq(μ) + g⟨z, b + D⊤Dμ⟩+ 1
2g2⟨z, D⊤Dz⟩.

This polarization expands each affine-row square and sums its finite color coordinates. Dormant mean is zero. With ghost = = 1 and, rearrangement yields

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−β liftSq(μ + gz) −ghost(z) = −β liftSq(μ) + ⟨z, bhβ⟩
+ 1
2⟨z, (C−1
ref −A)z⟩−1
2⟨z, C−1
ref z⟩,

Indeed before adding and subtracting the reference quadratic form the exponent is exactly

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−β liftSq(μ) −g−1⟨z, Aμ + b⟩−1
2⟨z, Az⟩.

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Here the linear coefficient is exactly bhβ = −g−1(Aμ + b), since Ssleepμ = 0. The last term is the
reference Gaussian, the preceding quadratic term its itinerary correction and the linear term its
first correction. Ghost supplies the dormant diagonal with normalized integral one. This pointwise
identity holds regardless of later profile zeros or sector events. The lift retains the same count
coordinates and every exponent term before any integration.

12.42 Pointwise Gaussian chain corrections

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For the completed plate and symmetric reference covariance set Serr = ACref −I and bhβ =
−g−1(A drift +b).
The first correction at a link is its finite color pairing with z, multiplied by
aperture: the cone profile at twice enforcement radius on hot links and one otherwise. An itinerary
of n + 2 link steps has half-bilinear quadratic correction with the ordered block product of n + 1

factors and terminal, multiplied by both endpoint apertures. At fixed starting link and , summing itineraries restores the intermediate link/color sums of matrix multiplication, hence exactly (. The terminal factor remains in its value and price.

Dormant plate/covariance rows and columns are identity, so vanishes if either endpoint is dormant. Mean and bill vanish there too. Induction along a nonzero chain shows both endpoints hot: the final cannot connect hot to a distinct dormant identity row. Let = gStep( , nonzero endpoint aperture gives atmos(, hence

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∥atmos(z)e∥≤g−1t(4a + K) = heat(a, K, t, β).

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An envelope U at least heat bounds each endpoint color coordinate, and aperture is at most one.
For coordinate first-correction size M, | qfirste | ≤dim(Color)UM, including dormant zeros.
Write Rc = dim(Color), dpay = distPay(η, r, Q, m, B) and Merr = Qdpay. Entry bounds are Merr
for discrepancy and Q for terminal plate. Summing its n + 1 intermediate color indices, with the
source’s extra convenient color factor, bounds each block-product entry by Rn+2
c
QMn+1
err . Endpoint
sums then give
| chain | ≤1
2U2Rn+4
c
QMn+1
err
= centreQ(U) etaSn,

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centreQ(U) = 1
2U2R4
cQ(Qdpay),
etaS = RcQdpay.

Nonzero itineraries are armed: every parent hop is at most 3, else the finite-range discrepancy/plate block vanishes. Define capQ by this geometric bound on armed itineraries and zero elsewhere. Splitting zero aperture, zero pairing and nonzero cases proves the pointwise bound for all. The lift retains ordered hops, full color census and endpoint apertures; these bounds use no extra Gaussian tail event and no deletion of the terminal plate.

12.43 Convergent ordered inverses and logarithms

Weighted absolute row/column prices multiply at any nonnegative width. Price 1 gives

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price qn; the finite identity (P
n<N En)(I −E) = I −EN and EN →0 prove P
n≥0 En = (I −E)−1

with absolute entry convergence. The two source orders remain distinct:

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AP −I = E =⇒A−1 =
X
n≥0
P(−E)n,
AC −I = E =⇒C−1 =
X
n≥0
(−E)nA,
C−1 −A =
X
n≥0
(−E)n+1A.

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They follow by multiplying the inverse of I+E on the appropriate side, without commuting matrices.
Absolute entry convergence permits each fixed multiplication and removal of the zeroth term.
For Gs = I + s(A −I) invertible on [0, 1], factor it near r as [I + (s −r)(A −I)G−1
r ]Gr. The
linear determinant coefficient is trace, giving ∂s log | det Gs| = tr((A −I)G−1
s ). Adjugate continuity
and compact nonvanishing allow integration to

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log | det A| =
Z 1
0
tr((A −I)G−1
s ) ds.

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Positive plates have positive determinant. Let Ps be the principal-column pad and Es = GsPs −I,
of uniform price ρ = scraps(B) < 1. Since GsPs = I + Es, one has G−1
s
= Ps(I + Es)−1; hence the
pad Ps occurs once in each term of the following expansion and only the residual −Es is raised to
the power n:
(A −I)G−1
s
=
X
n≥0
(A −I)Ps(−Es)n.

Each continuous diagonal is bounded by ( + 1) payE, uniformly in. The interval has length one, so the sum of the norms of the diagonal integrals is bounded by the same summable geometric majorant, independently of. Thus integral and series commute absolutely, before the finite coordinate sum. Writing its diagonal integrals as lact) and localBase =lact(0) gives

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log det A =
X
i
localBasei +
X
X
n≥0
lacti(n + 1).
i

The local diagnostic alone is the zeroth term; arms recover the full determinant. Applying the same interpolation to with discrepancy price below one gives

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Z 1
log | det(I + E)| =
X
X
0
[E(−sE)n]ii ds.
n≥0
i

Uniform bounds justify the exchange. For, positive determinants identify this with log det + log det, the actual covariance normalization defect. The lift retains ordered arms and their local prices, using uniform summability before finite coordinate sums; a global volume factor is not silently treated as a local bound.

12.44 Itinerary receipts, scalar stamps and address counts

An itinerary of hops retains an initial fine link and an ordered list of successors. A hop armed at radius has a pair of coarse endpoints within; since parents are within one site of either endpoint, successive parents are within + 2. The town of a link is the halo of its two coarse endpoints. It contains its parent and lies in its radius-( + 1) ball, hence in radius ( for the positive lattice scale. Write steady() = ballCost(+1). Its-cover uses at most this many tiles. The receipt includes the towns of all links, including initial and final links. If consecutive towns meet: the successor parent lies in both. The meeting-support covering inequality and induction give coverNumber(receipt) + 1) steady(

For tenure, points in the first town lie within + 1; points in the remaining receipt pay at most + 2 to its next parent and then its inductive bound. Thus, using + 1, the same induction puts every receipt point within ( + 1) of the initial parent. These are respectively the covering and tenure inputs to the address census.

For continuous matrix factors, define the stamp of a returning itinerary by integrating over 01 the color trace of its ordered block product; nonreturning itineraries contribute zero. Expansion of finite matrix multiplication gives exactly

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X
k-hop itineraries
stamp =
Z 1
0
Tr
Ps(0) · · · Ps(k −1)

 ds.

Continuity ensures integrability, and the fixed-order sum is finite. For the determinant arm the source chooses

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Ps(0) = A −I,
Ps(1) = pad(I + s(A −I)),
Ps(j) = −error(I + s(A −I))
(j ≥2).

An order- label has + 3 hops and restores the integrated grade-( + 1) intensity, with all discrepancy factors. For the source arm it chooses(0) = flawed() and flawed() for 1; the order- label has + 1 hops and restores the integrated grade- note. The color trace

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stays inside each label. If factor entries are bounded by lj, finite block multiplication and the final
color sum give
|stamp| ≤Rk+1
c
Y
j<k
lj.

An itinerary unarmed at the common factor range has a zero block and hence zero stamp. The interval length is one, so no additional integration factor enters the displayed entry-product price. These exact identities supply the pointwise arm prices.

Under source realization the factor at a hop from one link-color space to the next uses the section and readout of those endpoints. Intermediate readout–section compositions are identities. On a returning itinerary the resulting color block is conjugate by its starting-frame action to the physical ordered product; its finite color trace is unchanged. Recovery on the identity-frame section returns that same scalar stamp. The receipt retains the full ordered itinerary, even when links repeat, so this scalar invariance introduces no quotient of the address census.

An order- ledger with extra steps has hops. Its receipt is the prescribed union of enlarged neighborhoods visited along that itinerary. Set

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DL = L4|Axis|,
aL,d = ballCost(d + 2)DL.

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Here DL = density(L) and aL,d = abundance(L, d).
Every successor parent lies in the radius-
(d + 2) ball, and the fine-link caption census is DL. The parent census therefore bounds successors
per armed hop by aL,d, and thus k-hop itineraries from a specified initial link by ak
L,d. Assume
d + 2 ≤mB and B > 0, so the covering and tenure bounds just established apply. For a fixed
receipt site x, the tenure bound permits at most ballCost((n + u + 1)(mB + 1))DL starting links.
For nonnegative coefficients bounded by CQn on armed addresses and zero elsewhere, retain both
tarifffactors exp(2εn) and exp(Asupport coverNumberB). The inequalities n+u+1 ≤(u+1)2n and
the four-dimensional ball formula imply

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ballCost((n + u + 1)g) ≤ballCost((u + 1)g)16n.

The support tariffis at most exp( + 1) steady()). Multiply the starting-link count, armed-hop count and both tarifffactors. Consequently the order- total at is bounded by , where

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H = ballCost((u + 1)(mB + 1))DL au
L,d exp(Asupport(u + 1) steady(m)),
Rprice = 16aL,d exp(Asupport steady(m) + 2ε).

If2, the full sales bound is 2. To justify the countable ledger sum, bound any finite subset by its finitely many order fibers and then by this convergent geometric series; nonnegativity gives the full bound. This proves the asserted sales inequality and its summability, retaining every address even when different itineraries revisit the same link. In the lift, the realized tuple carries the entire ordered itinerary, receipt, factors, color trace and scalar stamp. Its literal readout restores the source address. Frame cancellation applies along the ordered product; it does not authorize commuting factors or merging repeated occurrences.

12.45 Pointwise arm prices and Gaussian-letter budgets

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Put Rc = dim(Color). Apply the stamp bound Rk+1
c
Q
j<k lj with k = n + 3 for the determinant
arm and k = n + 1 for the source arm. For a determinant arm the n + 3 ordered factors have
entry prices Q + 1, payE and n + 1 copies of scraps(B). Their common range is 3B: this includes

the plate’s2 stencil, the inverse-column window and the discrepancy’s enlarged window stencil. Interpolation 1] preserves the bounds and positive floors at every interpolation value. Thus the preceding trace-integral estimate gives, on armed addresses,

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| ArmAn,e,p | ≤Rn+4
c
(Q + 1) payE scraps(B)n+1 = centreA etaAn,

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centreA = R4
c(Q + 1) payE scraps(B),
etaA = Rc scraps(B).

For the source arm the + 1 factors each cost distPay(), and later interpolation multipliers have absolute value at most one. Their range is also 3, and

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| ArmSn,e,p | ≤Rn+2
c
(Q distPay(B))n+1 = centreS etaSn,

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centreS = R2
cQ distPay(B),
etaS = RcQ distPay(B).

Both arms vanish on unarmed addresses. These are scalar prices before exponential expansion; internal color indices are summed inside the actual arm and the integration interval has length one. For an armed-path envelope, halo4 and extra steps, the itinerary census gives a sales bound 2 expoHead( whenever

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16 abundance(L, 3B)eAsupport steady(i)+2εκ ≤1
2.

The abundance and fixed-step prefactor have rate zero. A negative-rate therefore satisfies this condition eventually; a rate- envelope gives a rate- sales bound, uniformly in geometry. The itinerary address count and halo-cover estimate supply the quantities expoHead and steady used here. Apply this with = 3 and () = (centreA etaA), and with = 1 and ) = (centreS etaS). All four quantities have every prescribed negative rate, so both budgets have any prescribed rate. The blank geometry receives its own determinant-arm bound. One scalar remains attached to each actual arm; colors are not separately relabeled.

For the Gaussian chain retain separately drift +) and shell access. 2 betaF() put

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burst(x) =
q
2 betaF(x),
gStep(β)−1 =
p
β ≤burst(x).

Burst has rate 12. The common cutoff-field envelope

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U(x) = burst(x) fiery(x) muRate(C, H)

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has rate zero, is nonnegative and independent of β and the Gaussian variable. Together with the
mean bound it supplies the pointwise chain hypotheses. If E has rate k −1/2, then the linear
residual definition gives |bhβ,i| ≤burst(x)|E(x)| =: b(x). Thus b has rate k, and the first correction
envelope J(x) = dim(Color)U(x)b(x) has rate k. The caption is the parent of the fine bond. Its
town lies in radius (m + 1)B, giving sales price

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ballCost((m + 1)B)L4|Axis|J(x)eAsupport ballCost(m+1),

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also of rate k, since its geometric caption and covering factors have rate zero at fixed parameters.
Higher quadratic-chain envelopes are the source’s capQ evaluated at U. Their center is proportional
to 1
2U2 flagP4 Q2 payE scraps(B), and their ratio is flagP Q payE scraps(B). The ratio has every
prescribed negative rate, and so does the center because U has rate zero. Their pointwise bounds
retain one scalar per actual arm, rather than splitting a trace into independently labeled colors.
The itinerary estimate yields the rate-k chain sales bound for arbitrary prescribed k, retaining

series-index support charge and halo4, together with coupling and residual factors. More generally a finite caption family of density, radius- supports and constant envelope costs at most ballCost(

Indeed at most ballCost( supports contain a fixed site, and each has-covering number at most ballCost(). This is the fixed-envelope count for the first correction and local Short letters. In the source lift all envelopes, captions, receipts and rate hypotheses remain components of the realized record. Literal recovery therefore preserves each pointwise inequality and its eventual threshold; the geometric-series argument uses precisely the retained address census.

12.46 True walk supports, bundle budgets and connected tiling

The source’s native support native() is the finite set of all visited sites, including start and endpoint. Induction on steps puts each visit within length() of the start. A walk of length at most fits in one radius- tile. Otherwise split off steps; the two supports meet at the split endpoint. Induction and the meeting-support covering inequality give, for

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M coverNumberM(native(w)) ≤M + length(w).

Length counts every repetition and winding even if the visited set is smaller. A curvature slot’s support is exactly the native support of its transport walk concatenated with its four-step plaquette: concatenate visit lists and wrap cover coordinates. Its slot length is that concatenated length. A -slot bundle rooted at is the union of slot supports with inserted. These supports all meet at coverNumber(bundle) + totalLength

Every bundle site lies within totalLength of, since each slot length is at most their nonnegative sum. Forhalo(parent(bundle)), the parent-distance estimate bounds its extent from parent( + 1)(1 + totalLength). Coarse footprint conversion at gives

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baseFee = preCost(l) + conversion(L, l)(1 + d),
legFee = conversion(L, l)/B,
originFee = lB + 1,
coverNumberB(Y ) ≤baseFee + legFee totalLength .

Apply the anchored-to-hit principle with degree equal to true totalLength, extent originFee and these covering coefficients. At a fixed fine root the input is exactly the original slot-family length budget. The root delta and that budget form a product budget; only the parent population

supplies the root count, with no additional volume factor. This recovers the source’s bundle_hit bound and its explicitly paid tail for totalLength at least. Atom footprints are precisely their slot-leg bundles; an ordered path pair is the two-slot bundle with the sum of both actual lengths.

For recovery of this support construction, retain the ordered visit list as part of the source path data, alongside its finite native set. Realization wraps each visit in the original period, and readout leaves that site address unchanged. Taking the image of the list, concatenating two lists, and then forming their finite union therefore commute with readout. Consequently the realized bundle has exactly the original native support, root and total length; repeated visits may collapse in the set while remaining in the length and label budget. The identity-frame section recovers each visit and