Local nonlinear geometry and estimates · Part 4

tangent change and the carrier adjoint preserve the norm of each color vector. Thus the straightened and oriented rows have the samebound as the cutoffflat row. At = 1, every carrier is 1 and the straightened row is exactly the cutoffflat row.

Gauge covariance is exact. The carrier transforms as, because its endpoint is y. Combining its adjoint with the adjoint on u cancels (), leaving the common factor )) at the row root. The flat row commutes with this common color action. Signed tangent transformation commutes with the gauge change after pulling the gauge to the signed sites; hence the oriented row transforms by the adjoint at its signed root as well.

The locality and comparison bounds can be proved without controlling the background outside the queried hull. A carrier to y depends only on the radius-d hull about x if 4 separation( link chords there are at most0, replace all links outside the hull by 1. The carrier is unchanged and its chord is at most length by the walk estimate. Similarly, if two tangent inputs differ by at most M on the radius R+predictionReach+4 input stencil, truncate their difference to zero outside this stencil. The resulting global field has norm at most M, and linearity and locality give a row difference at most

It follows that, when 4( + predictionReach +4) and the input is bounded by M on the needed stencil,

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∥straightenedR(V, u) −barFieldR(u)∥
≤Kst · 2 colorScale ·dhM.

Indeed each carrier changes the input by at most 2 colorScaledh M; apply the preceding row comparison. The signed case has one additional error: at a reversed stored link the tangent transformation at V differs from that at 1 by at most 2 colorScale h M. A signed link root moves by at most one site, so radius d+1 bounds in the original orientation supply the required radius-d bounds after turning. Adding the carrier and reversal errors gives

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∥orientedRowR(V, u) −orientedRowR(1, u)∥
≤Kst · 2 colorScale ·(d + 1)hM.

All fields in this inequality refer to the same signed row and input; no independent branch outcome is selected. For a single-link input of color a, the signed tangent transformation and the carrier spin still produce a single-link input with color norm. Its parent distance from the row root differs from the original signed distance by at most one. The flat impulse estimate above therefore yields

Searchable transcription

exp(widthtop · distAt(link, signedRoot))
∥orientedRowR(V, singlelinka)∥
≤2Kst exp(widthtop(predictionReach +7))∥a∥.

This bound requires no smallness of V. Locality at = till() follows from the same carrier reach and the room inequality, using a half-sized hull before the signed change to absorb its one-site root shift.

Affine accuracy on the protected stencil

The local hypothesis here records both actual field agreement and a tangent bound. Say that U is grafted to V by u on a radius-d hull if it agrees there with exp() replica( ) and on the same hull. This condition is preserved by a coarse gauge transformation with u changed by its adjoint action. Under a signed coordinate change it is preserved on radius d1, because the stored

positive branch of a reversed link may shift the root by one site. Restriction to a smaller hull and enlargement of H also preserve it. Fix a row p, its root i,4 and = till(). Suppose V agrees with exp b on that hull, , and U is grafted by u there. Replace u outside this hull by zero to obtain v with and set. The logarithm-product estimate supplies w with

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exp w = exp v exp a,
∥w −v −a∥≤JH2, ∥w∥≤(1 + J)H

for eventually small H. The equality is linkwise and uses the exact replica exponential identity. Locality of the entry row now identifies its actual value C with entry(exp w,exp b), and its background value Y with entry(exp a,exp b). Let bound the error between these two logarithmic entries and their corresponding full linear costs; let bound the full linear cost for w. These are the uniform entry-near estimates at the enlarged envelope (1+J)H. Then. To estimate the affine remainder write = linearCost( = linearCost(). Linearity gives the exact decomposition

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C −Y −straightened(V, u)
= (C −z) + uncut(w −v −a) + (z0 −Y )
+(uncut(v) −truncated(v))
+(truncated(v) −straightened(V, u)).

The five terms have bounds respectively, where, for the common stencil constant K and exponential-chart constant A,

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q1 = KJH2,
q2 = KH exp(−width · R),
q3 = K · 2 colorScale ·B · AH2.

The first uses the full stencil bound on wva. The second uses its exponential tail bound. For the third, V’s links in the hull have chords; the adjoint transport comparison bounds the background-to-flat row difference by the displayed radius-times-quadratic error. Locality first replaces u by v, so the bound never uses u outside the hull. The room inequality + predictionReach +4) ensures that every needed input lies inside it.

Consequently the common affine error can be chosen as

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Eaff= 2E0 + q1 + q2 + q3.

It also boundslinearCost(. If H has rate1/2, the entry-near estimate gives rate 1 and rate1/2; the quadratic, cutoff, and radius-weighted quadratic terms all have rate Thus has rate1/2 and has rate1. The predictor entry-near estimate and the uniform full and tail stencil bounds of this chapter supply the constants used here.

The protected mean and its stationary comparison

The protected mean is the stationary comparison variable retained when vertical fluctuations are integrated. Its affine accuracy is tested on the prescribed stencil. The proof first estimates the discrepancy of the actual nonlinear rows from their flat counterparts, then inserts those bounds in the stationary identity. The cancellation concerns these actual rows; it does not assert that every nonlinear minimizer is equal to the flat one.

Here the scale, geometry, selected rows, and gauge forest are held fixed throughout. Write A for the counting-coordinate plate, D for the selected linear form, b for its linear bill, and S for the diagonal dormant-coordinate operator. For every fine tangent field v the pricing identity gives

Searchable transcription

Av + b = DTpriced(v) + Sv.

Coordinates are expressed in the fixed orthonormal color basis. All norms of fields in this argument are supremum norms; a color-vector norm is bounded by the sum of the absolute values of its color coordinates. Let Y bound both the row sums of D and the row sums of. Its matrix entries vanish beyond distance B/4.

We first produce a comparison field in the same gauge as the actual background. A clean background on a ball of radius d+1, with, admits an axial gauge in which each link in the radius-d ball has chord at most

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u(x) = 16(2(32B(x) + 1) + 1)5|PlaqAxes| · 2 fiery(G, x, L).

The axial-patch estimate supplies this bound using the clean plaquette toll and the room temperature comparison. The local logarithm extension then gives a globally defined bounded logarithmic field a agreeing with that gauged background on the smaller ball, with). Thus the bound u has rate1/2. The axial-gauge and bounded logarithmic-extension argument of this chapter proves this assertion.

For clarity, the logarithm-product estimate used by the subsequent comparisons is a genuine group-chart estimate. Set) = log(exp exp). Differentiating at (0,0), using the derivative of the matrix exponential and the inverse-chart identity, gives. Analytic Taylor expansion on a sufficiently small ball therefore gives, with fixed positive K,r,

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exp h(p, q) = exp p exp q,
∥h(p, q) −p −q∥≤K max(∥p∥, ∥q∥)2,
∥h(p, q)∥≤K max(∥p∥, ∥q∥).

Applying this link by link proves the same uniform bounds for the logarithm of two successive increments. No identification of a group-valued density with a tangent vector is made here. Let spring() be the stationary response with zero values on the chosen forest trunks. It is linear in a. The stationary response estimate bounds each color coordinate for

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C0 = Response.total(j, η, C) + exp η.

Consequently its field norm is at most, where

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Cs = max(1, max(0, C0) dim(Color)).

Indeed this is the coordinate bound summed over colors for unit input; rescaling a by

proves the general case, and linearity handles = 0. Signed coordinate changes preserve the field bound and commute with the linear copy. The latter assertion follows either from the copy formula or by differentiating the exact identity for its exponentiated replicas.

If z is the axial gauge transformation, put) spring(). Then v vanishes on trunks and w. Gauge covariance of the selected path, away from the seed region, identifies its value at v with

(gauge((gauge( spring(

The source uses one common clean hull for all rows and columns subsequently read. On this hull the uncut stationary identity is exactly

Searchable transcription

linearCost(linearCopy(a), a) −linearGuide(a)
+uncutField(spring(a)) = 0.

To see this, the stationary shifted field is the sum of the linear copy and spring(). Linearity of the uncut field and the defining stationary equation identify its image with linearGuide(). This proves the displayed cancellation before any estimate is applied.

Subtract this identity from the gauged real path. There are four errors: the base-stroke linear approximation, the guide linear approximation, replacement of the background-oriented row by the flat truncated row, and replacement of the truncated row by the uncut row. Denote the first two envelopes by and; the guide and local affine comparison estimates give each rate bounds chord(exp a) byand bounds the stationary stencils, the last two errors are bounded respectively by

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U1x = Kst · 2 colorScale ·(B(x) + 1) · Cchw(x) · Csw(x),
U2x = KstCsw(x) exp(−width · till(x)).

The first has rate1 because it is quadratic in w with only the radius factor; the second has rate1 by the cutoffdecay. Hence

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A0 = A1 + A2 + U1 + U2

has rate1 and bounds every path of v whose root lies outside the 20B seed neighborhood and whose radius-2B background hull agrees with exp a. These bounds use the previously specified guide, affine-row, stencil, and stationary identities on that same hull. Now let P be the nonsymmetric inverse-column pad,, and. Symmetry of A gives , whence the exact identity

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μ −v = ETv −P T(Av + b).

For a coordinate i, only the column window of P at i is read by the second term. If on that window and, the inverse-window estimates already proved imply

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|μi −vi| ≤payE · M + scraps(B) · W.

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This uses only the elementary inequality | P
j Mij tj| ≤K P
j |Mij| when |tj| ≤K on the
nonzero entries of the row. It requires no bound on t outside that row support.
For coordinates outside the 11B seed neighborhood, S v = 0: on a trunk v = 0, and any other
such coordinate is active. The pricing identity and the transpose row bound therefore show that
path bounds A0 within distance B imply

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|(Av + b)i| ≤Y · scale · A0.

For a selected path p, locality and the row bound of D also imply

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∥realPathp(u) −realPathp(v)∥
≤dim(Color)scale−1Y U

whenever on the radius-B coordinate neighborhood of its root.

Take a path root outside the 30B seed neighborhood. Its clean hull has radius 8B+1. Flatten on radius 8B and use the resulting single comparison field v for every coordinate needed below. Its paths within distance 6B satisfy the bound. Each column window read at distance B from the root lies within distance 3B, so the residual bound above applies throughout that window. All these rows remain outside the required 11B seed neighborhood. Thus, with

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W = Csw, M = Y scaleA0,
U = payE · M + scraps(B) · W,
Emean = A0 + dim(Color)scale−1Y U,

we obtainrealPath. All envelopes are eventually nonnegative. The scraps bound at any prescribed negative power gives rate1 for U and hence for. The root is selected by the same seed-distance condition. This proves the protected-mean estimate used in the watch argument, with one common choice of gauge and comparison field for its entire finite stencil.

The axial patch estimate follows by lifting the torus patch to its radius-r cover box. The axial gauge construction bounds the total edge chord by 16(2r+1) times the face-chord sum. If every face chord is at most q, the sum is at most (2 |PlaqAxes| q. Descend the gauge to the torus using injectivity on the box when 2. Both endpoints of every radius-d bond lie in that box when, and each individual edge chord is bounded by the total nonnegative sum. Thus the bound is 16(2 |PlaqAxes| q, as used above.

Local normalization differences

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The condensation letter at a fine link e is

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condensation(e) = [joint(Ablank, window; e) −joint(A, window; e)]/2.

The blank geometry has the same forest, scale, windows and tags, but background 1 and empty seeds. Its hot links are exactly the nontrunks, and all its rows are selected. No second forest or tag choice is made.

We first compare joint diagnostics on the same finite window U. For positive plates A,Z let and), with 01, and similarly for Z. The window intensity is, evaluated at its distinguished diagonal coordinate. Suppose all entries of AZ on U have absolute value at most0. The census bounds∆= ballCost(, so the row sum of and of is at most. Let bound both inverse row sums uniformly along this interpolation, and let Q+1 bound the row sum of

The resolvent identity gives

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g−1
Z −g−1
A = g−1
Z (gA −gZ)g−1
A ,

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so its row sum is at most C2
invd. Decomposing the difference of the two intensities into (fA −
fZ)g−1
A −fZ(g−1
Z −g−1
A ) yields the diagonal bound

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pulseHead(e0) = [Cinv + (Q + 1)C2
inv] · ∆e0.

All inverse matrices here are the principal-window inverses at that same s. Integrating from 0 to 1 and summing over the color coordinates of e gives

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|joint(A; e) −joint(Z; e)| ≤dim(Color)pulseHead(e0).

Continuity in s, supplied by positivity of the interpolated principal matrices, justifies subtraction of these integrals. The same calculation without taking a difference bounds each individual joint diagnostic by dim(

To apply the local difference bound, take e outside the 30B seed neighborhood and flatten its background in a single radius-6B patch. If the resulting link chords are at most, compare each actual gauged form entry to the blank one by testing on a single coordinate impulse. Trunk columns are zero in both forms. A nontrunk column in the needed neighborhood is hot in both geometries. Rows farther than B/4 from that column vanish in both; all other rows are selected by the seed margin. The transported-row comparison already proved therefore gives

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|Dgauged(p, i) −Dblank(p, i)| ≤swell,
swell = scale · Kst · 2 colorScale ·(B + 1)h0.

The dormant diagonal terms of their plates agree in this neighborhood. Expand entrywise using. The two transpose row sums are at most Y, so

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|Agauged(i, j) −Ablank(i, j)| ≤2swellY =: repair

throughout every window read at e. This is an entrywise comparison only on the patch, with the census factor retained in pulseHead.

Gauge invariance of joint follows by conjugating the plate by the block diagonal orthogonal color action, which preserves every link window. The same conjugation preserves its principal inverse, interpolation intensity and color trace at e, hence its integrated diagnostic. Thus the gauged plate can replace the actual plate in the condensation difference.

Flattening gives) with w of rate1/2. Hence repair has rate1/2, including its radius factor. The window census ∆has rate zero. The envelope

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Econd(x) = dim(Color)pulseHead(repair(x))

therefore has rate1/2 and bounds the absolute condensation outside the 30B seed neighborhood; dropping the factor 1/2 only enlarges the bound. This derives the remote normalization estimate used below from the local form comparison and the already established inverse-window bounds.

Exact locality of the cut gaps

Every walk of length at most R reads only the radius-R hull of its starting point, in any projection for which neighboring sites have projected separation at most one. This includes stored negative links: their positive-branch root is one step behind the current point, still paid by the step being read. Induction on the walk proves equality of its transport for fields agreeing on that hull.

A retained plotted slot has its transport length at most R, and its final plaquette needs four more steps. Thus plotted fields agree under radius R+4 agreement. The cutoffF,I,N kernel rows are banded at radius R and their entries read only the radius-R coarse hull about the row root. For the fine F row this follows by lifting its designated route in the replica; its endpoint projects to the coarse column, and the route-length cutoffbounds that projected separation. The lane rows use the same argument on their coarse route.

For a banded row ), its transpose at output j reads input rows within distance R. Agreement on radius 2R about j makes each such row entry equal; input values need agree only at those rows. Likewise w is unchanged by this background replacement on that hull, with

the entire arbitrary array w held fixed. The inner row coefficients agree individually, so no locality assumption on w is needed.

Apply this to lGap. The fine plotted inputs at relevant rows require at most R further travel plus their four plaquette steps; the noise input additionally queries its actual predictor through predictionReach. The coarse plotted term needs R+4. Therefore common fine/coarse agreement on

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D ≥3R + 4 + predictionReach

at the tested anchor makes the whole lGap equal, with the same actual predictor tags on both sides. For dGap the three banded row-square operators are unchanged on radius 2R; hence the same D suffices for every fixed input array w. These prove the local gap identities used in the chart transfer, without changing any clipping indicators in an unrelated data array.

A reliable cutoff logarithm

Choose a smooth profile with values in [0,1], equal to one on the inner unit interval and zero beyond its prescribed outer interval, and set

chord( log(mat(

Here innerShare evaluates at (chord(, so it uses a smooth squared matrix norm. Its nonzero values imply chord(. The ordinary logarithm is only required to have its chart properties on this latter shell.

The local logarithm estimates give 0 with exp(log andlog chord( when chord(. Choose also a matrix radius t where log is analytic and a color radius R where log(exp. Set

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r = 1
2 min(s, t/ colorScale, R/C), γ = C.

For every 0, the entire chord shell satisfies all three chart conditions. In particular the logarithms there have norms. Applying log(exp to)log U and log U, whose norms are unchanged, and using exponential conjugation and inversion, proves log()log U and log(log U on the shell.

On chord( the profile equals one, so the cutofflogarithm is the actual inverse chart. At every U its norm is: if the profile vanishes this is immediate, and otherwise use its value 1 and the shell logarithm bound. Chord conjugation and inversion invariance extend the two equivariance identities to all U, splitting into the zero and nonzero profile cases.

For any smoothly varying matrix-valued group input, the cutofflogarithm is smooth. Inside the closed outer shell this is composition with the analytic log and the smooth squared-norm profile. Outside it, continuity of chord gives a neighborhood where the profile vanishes identically. The same argument proves continuity, hence Borel measurability, as a function on G. Thus no regularity of the chosen logarithm outside its chart is used.

The sector input and its grafted signal

Fix a forest F, coarse background V, tags k and frozen slice values Y. A fine link is hot when it is not a trunk and neither of its coarse endpoints touches the 10B seed dome. Define input() by putting exp()(replica( on hot links, the fixed Y values on other nontrunks, and the prescribed tree value on trunks. This is on the same tree slice for every u.

Let small() be the cut logarithm of (replica( at radius widtht. Define saved to be this logarithm on frozen nontrunks and zero elsewhere; lens() is u on hot links and zero elsewhere. The signal is saved+lens(). Thus signal equals u on hot links and zero on trunks.

Shell access gives two facts, uniformly over every forest and actual input: a reliable logarithm on the chord shell of radius widtht, and the shell displacement bound chord( replica( whenever the local Interior condition holds. It is available by choosing width equal to the previously proved shellPrice and then taking x large enough that widthfiery lies in the reliable logarithm radius. Its constants are chosen once after L.

On a link satisfying that Interior condition,) = exp(signal()) replica( ). For a hot link this is its definition. On a trunk both the signal and the replica trunk increment are zero or identity as appropriate. On a frozen nontrunk the shell bound makes the cutofflogarithm uncut and its exponential recovers the actual discrepancy, giving the same equality. The reliable cut logarithm also supplies the all-argument bound

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∥savede∥≤γ · 2(width · t).

Hence local Interior conditions for every link of a radius-B hull, tangent bounds at its hot links, and imply the grafted condition on that entire hull with Frozen links use the saved bound; hot links use the given tangent bound.

There is a simpler remote version requiring no Interior premise. If the hull center is outside the (11B+1) seed neighborhood, every nontrunk in its radius-B hull is hot: otherwise one endpoint would touch the 10B dome, and its parent lies within one further site, contradicting the separation. Only hot links and trunks occur in that hull. The defining identities above and hot tangent bound therefore prove grafting directly for any These are the required and distant grafted constructions used in the packet annulus and the optional remote letters, respectively.

A common chart on the grafted hull

Suppose the coarse background V is clean on radius d+1 and U is grafted to V by a tangent field bounded by) on radius d, with H of rate1/2. Flatten V there to exp b in gauge g, with ) of rate is the exponential chord constant, the gauged coarse links have chords v. Gauge covariance of grafting writes the gauged fine links as exp() times the replica of the gauged coarse field. Their chords are therefore at most v by the product triangle inequality; replica links are either 1 or a coarse link. The single envelope

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W = H + v + DchH + Dchv

bounds b, both kinds of link chord, and H itself, and has rate1/2. All chart radii are met eventually. This is the common coarse/fine local chart; it imposes no condition outside the hull.

Apply affine accuracy in this gauge after restricting the hull from B/2 to B/4. The enlarged envelope W supplies both its background-log and tangent bounds. Entry covariance, replica covariance and covariance of the straightened row rotate the entire affine difference by the same root adjoint. Norm preservation then transports the estimate back to the original fields. This proves the strokes bounds M of rate1/2 and E of rate1 used in the soft remainder argument.

The same chart proves that the actual local branch tags and holonomies lie in any prescribed small neighborhood eventually. Given 0, the finite-stencil branch estimate supplies a radius r and reach k such that fine links of chord imply site mass 1, pin mass 1, and predictor chord 2. Choose x so that2 andmin(4). Apply this estimate to the gauged fine field. Site and pin masses are gauge invariant. The actual wire discrepancy is also invariant and