Local nonlinear geometry and Haar realization · Part 3

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because trunks are not hot. Direct rows bound each curl by Eo/
√
direct and noise bounds the
predictor bill by Eo/√penalty. All macro roots occur in the selected neighborhood. Thus

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∥lens(u)e∥≤ZEo,
Z = K| box(macroR)||PlaqAxes|
√
direct
+
1
√penalty.

A clean-patch flattening gauge supplies rate- with its radius factors included, paying the squared-neighborhood absorption in the plate-floor proof. Tag uniformity comes from the actual tree-slice estimate.

12.18 Transported affine rows

Let) be transport along the coordinate-order route from to its centered displacement toward. It ends at and has length at most 4 separation() because each of the four centered coordinate lengths is bounded by that separation. Spin each tangent at parent then apply the cutoffflat row at. The signed row first applies the actual replica tangent transformation, including the background adjoint and minus sign on reversed links. These transformations preserve each color norm and hence the row bound; at = 1 the straightened row is the flat row.

Carrier covariance is , so input and carrier endpoint factors cancel and leave only Ad()). The flat row commutes with this common action. The signed tangent transformation commutes with gauge change after pulling the gauge to the signed sites, giving the corresponding adjoint at the signed row root. A carrier to lies in the queried radius- hull when 4 separation(. On that hull take0. Then replacing exterior links by identity leaves the carrier unchanged. If its chords are at most, carrier chord is at most. Truncating a tangent difference to the required radius- + predictionReach +4 stencil similarly gives row difference at most for local input error

For input bounded by on the required stencil, every carrier changes it by at most 2 colorScale. Thus if 4( + predictionReach +4)

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

The reversed-link tangent change at background differs from its value at identity by at most 2 colorScale and shifts its root by one. Radius- + 1 original data supply radius- bounds after turning; summing reversal and carrier errors gives

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

For a single-link input, signed transport and carrier spin preserve its single-link norm, while parent distance shifts by at most one. The impulse bound is therefore 2 after multiplication by the original exponential distance weight, without smallness of . The room condition gives locality at till using a half hull before turning. The source lift preserves these actual tangent transformations and route endpoints, not a bare signed permutation.

12.19 Affine accuracy on the protected stencil

Grafting on a radius- hull means actual agreement = exp() replica( ) there with Gauge changes preserve it with adjoint-transformed; signed changes retain it on radius because reversed stored roots shift by one. Smaller hulls and larger bounds preserve it.

At a row use = till() and = exp on the hull,. Extend by zero outside to, and put = linearCopy(), both bounded by. For eventually small, the logarithmic product gives linkwise

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

Replica exponential equality is exact. Entry locality identifies actual with entry(exp exp and background with entry(exp exp). Let bound both logarithmic-entry errors against their full linear costs, and let bound the full linear cost of, using the uniform entry-near estimates at enlarged envelope (1 +. Put = linearCost( = linearCost(). Then , and the exact five-term identity is

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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 respective bounds are, with common stencil constant and exponential-chart constant

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q1 = KJH2,
q2 = KHe−width R,
q3 = 2K colorScale BAH2.

They use, in order, the full stencil bound, cutofftail and transported-row comparison with chord at most Locality removes exterior before any estimate. The room inequality 4( predictionReach +4) contains every input. Thus also bounds linearCost(. For rate-, entry-near gives rate1 and rate2. The quadratic, cutoffand radius-weighted quadratic terms each have rate1 under the stated radius/cutoffcontrols. Thus has rate2 andrate1. The lift preserves all five terms and this local grafting contract; horizontal recovery does not license dropping a background transport error or reading unbounded exterior tangents.

12.20 Protected mean: stationary comparison field

Fix scale, geometry, selected rows and forest. In counting coordinates let be the plate, its selected linear form, its bill and the dormant diagonal. The exact pricing identity is priced(. Coordinates are expressed in the fixed orthonormal color basis. Field norms here are supremum norms; color norms are bounded by the sum of absolute color coordinates. A common bounds row sums of, with reach4: the matrix entries of vanish beyond distance

A clean radius-( + 1) hull,, admits an axial gauge with radius- link chord bound

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

The axial-patch bound uses the clean plaquette toll and room temperature comparison. The local logarithm extension then gives a bounded global agreeing there with the gauged background and , of rate2. For the actual chart product) = log(exp exp), inversechart differentiation gives. Analytic Taylor bounds on a fixed ball give exact exponential recovery, error max( and size max(), linkwise. This is not an identification of group elements with tangent vectors. Applying this link by link proves the same uniform bounds for the logarithm of two successive increments.

The linear forest response spring has zero trunk values and coordinate bound on unit input

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C0 = Response. total(j, η, C) + eη.

Sum the coordinate bound over colors on unit input, rescale by for= 0, and use linearity for = 0. This givesspring(, where

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

Signed changes preserve this bound and commute with linearCopy; the latter follows either from the copy formula or by differentiating the exact identity for its exponentiated replicas. For the same axial gauge, let) spring(). It remains zero on trunks with size at most. Gauge covariance identifies the selected path, away from seeds, with request in the gauged background plus its actual oriented row on spring.

On the common clean hull the exact stationary cancellation is

linearCost(linearCopy(linearGuide() + uncutField(spring(

Indeed the stationary shifted field is linearCopy plus spring, and linearity gives its defining guide. Subtracting this identity leaves four errors: base-stroke linearization, guide linearization, background-oriented versus flat truncated row, and truncated versus uncut row. The first two envelopes have rate1 from the earlier affine/guide estimates. The explicit remaining bounds and the padded-inverse mean calculation follow below. The source lift uses this same gauge, forest and hull for every row and column; the stationary cancellation is performed before norms or coordinate reduction. The last two stationary comparison errors are

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U1 = 2Kst colorScale(B + 1)CchCsw2,
U2 = KstCswe−width till(x).

Here bounds chord(exp. The first error has rate1 because it is quadratic in with only the radius factor; the second has rate1 by cutoffdecay. Thus has rate1 and bounds each path of outside the 20 seed neighborhood on its common radius-2 agreement hull. Let be the nonsymmetric inverse-column pad, and. Symmetry of gives, and therefore proves exactly

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μ −v = E⊤v −P ⊤(Av + b).

Only column window of is read at coordinate. If the residual is bounded there by and , the proved window bounds give

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|μi −vi| ≤payE M + scraps(B)W.

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For any row, | P
j Mijtj| ≤K P
j |Mij| when |tj| ≤K on its nonzero entries. This uses no residual
bound outside that support. Outside the 11B seed neighborhood Sv = 0: trunks have zero com-
parison field and every other coordinate is active. Pricing and the transpose row price then give
|(Av + b)i| ≤Y scale A0 when paths within distance B have the A0 bound. Similarly local input
error U gives path error at most dim(Color) scale−1 Y U on its radius-B coordinate neighborhood.

For a path root outside 30, flatten its clean radius-(8 + 1) hull on radius 8 and use one comparison for all tested coordinates. Its paths within 6 obey. All column windows needed within lie within 3 and remain outside the 11 exclusion, so the residual bound above applies throughout each window. Therefore, with

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W = Csw,
M = Y scale A0,
U = payE M + scraps(B)W,

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Emean = A0 + dim(Color) scale−1 Y U,

we haverealPath. The scraps estimate at any prescribed negative power gives rate 1 for and then for. Arbitrarily negative-power scraps bounds pay the error, with eventual nonnegative envelopes. The same seed condition selects the root. 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 bound used here follows from the injective cover-box gauge, 16(2 times face sum, bounded by 16(2PlaqAxes when 2 and. The lift retains the single comparison field and the actualmean; symmetrizing the covariance does not symmetrize this mean or enlarge its protected stencil.

12.21 Local normalization differences

The source condensation is [joint(joint(2 on the same window. This subsection derives the remote normalization estimate used below from the local form comparison and the already established inverse-window bounds. Blank geometry retains forest, scale, windows and tags, with identity background and empty seeds; nontrunks are hot and all rows selected.

For positive plates on a common window, put) for 1. The intensity is the distinguished diagonal of . If absolute entries of there are at most0, its census∆= ballCost( bounds the difference row sums by . Suppose interpolation inverse row sums are at most = payE and row price at most + 1. The exact resolvent identity

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g−1
Z −g−1
A = g−1
Z (gA −gZ)g−1
A

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gives price C2
invd. At the same interpolation value s, use the principal-window inverses throughout
and decompose the intensity difference as

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(fA −fZ)g−1
A −fZ(g−1
Z −g−1
A ).

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This gives diagonal bound

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

Positivity of the interpolated principal matrices gives continuity along, justifying subtraction of the two integrals; so integration and color summation yield joint(joint( dim(Color) pulseHead(). Each individual diagnostic is bounded by dim(Color)( For outside the 30 seeds, flatten once on radius 6. Single-coordinate impulses compare the gauged and blank forms: trunk columns vanish in both, nearby nontrunk columns are hot in both, rows farther than4 vanish in both, and all remaining rows are selected by the seed margin. The transported-row bound gives entry error swell = 2 scale colorScale(. Dormant diagonals agree. Expanding in, with transpose row prices , gives plate entry error repair = 2 swell on every read window.

Block-diagonal orthogonal gauge action preserves whole-link windows, principal inverses, interpolation intensity and its color trace. Hence joint is gauge invariant. Flattening gives = with rate radius and census factors have rate zero. Therefore = dim(Color) pulseHead(repair) has rate2 and bounds remote condensation, enlarging its half factor harmlessly. The lift retains whole-color windows and their common forest/tags, so this determinant diagnostic transport is justified by the actual trace conjugation rather than entrywise gauge invariance.

12.22 Exact cut-gap locality and a reliable cutofflogarithm

For a projection in which neighboring sites have projected separation at most one, a walk of length at most reads only its radius- projected hull; the stored positive root of a negative link is paid by that step. The fine plotted inputs at relevant rows require at most further travel plus their four plaquette steps; the noise input additionally queries its actual predictor through predictionReach. Induction proves equality of transport under hull agreement. A plotted slot needs + 4, while cutoff rows are banded at radius and read their radius- background hull. For the fine row, lift its designated route in the replica: its endpoint projects to the coarse column and its cutoffbounds the projected separation. The lane rows use their coarse routes in the same way. Consequently a transpose reads input rows within and its coefficients agree under radius-2 background agreement. The same holds for with arbitrary fixed, since coefficients agree individually, without a locality assumption on

The plotted, noise-predictor and coarse terms therefore give exact lGap locality when

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D ≥3R + 4 + predictionReach,

with identical actual predictor tags. The three row-square terms of dGap agree already under 2 background agreement, so this same suffices for every fixed input array. These prove the local gap identities used in the chart transfer, without changing any clipping indicators in an unrelated data array.

Choose a smooth profile valued in [0 1], equal to one on the inner unit interval and zero beyond its prescribed outer interval. The inner-share operation evaluates it at (chord( smooth squared matrix norm. Define

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cutLogd(U) = χinnerShare(d, chord(U)) log(mat(U)).

The cutoffequals one on the inner shell and is zero beyond 2. For local constants, analytic matrix radius and inverse exponential color radius, take

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

For 0, every point with chord at most 2 obeys all three chart conditions and its logarithm has norm at most. Nonzero cutoffvalues imply chord strictly below 2. The ordinary logarithm is only required to have its chart properties on this latter shell. Applying the inverse identity to ) log andlog proves conjugation covariance and inversion oddness there. The invariant cutoffextends both to all group arguments. Its global size is at most 2, and on chord at most it is the genuine inverse chart. Smoothness follows from analytic composition on the outer shell and an identically zero neighborhood outside it; continuity implies Borel measurability. No regularity of an exterior logarithm choice is used. Readout preserves the hull, tags and squared-chord cutoff together, so these exact identities apply to the realized cut gaps and later signals on precisely the same domains.

12.23 Sector signals and a common grafted chart

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Fix forest F, coarse V , tags and frozen values Y . Hot links are nontrunks whose coarse endpoints
avoid the 10B seed dome. Input puts exp(ue) replica(V )e on hot links, frozen Y on other nontrunks
and prescribed tree values on trunks. It remains on the same slice for all u. Define small(e, U)
as the cutofflogarithm of Ue replica(V )−1
e
at radius width t.
Saved is this logarithm on frozen
nontrunks and zero elsewhere; lens is u on hot links and zero elsewhere. Signal is saved plus lens,
hence zero on trunks.

Shell access at width equal to shellPrice makes the discrepancy chord at most width under Interior, uniformly over forests and actual inputs. Taking large puts width fiery in the reliablelog radius; the constants are chosen once after. On hot links the grafting identity is definitional; on trunks it is the replica/tree identity; on frozen nontrunks the uncut logarithm recovers the discrepancy. Saved is globally bounded by 2 width. Thus Interior throughout a radius- hull, hot tangent bound and 2 width prove grafting there. Outside the (11 + 1) seed neighborhood, every nontrunk in that hull is hot: otherwise an endpoint touches the 10 dome and its parent is within one further site, contradicting the separation. Thus grafting follows directly without Interior for any nonnegative tangent bound. These are the required packet-annulus and optional remote-letter constructions, respectively.

For clean coarse background on radius + 1, flatten to exp in one gauge, rate2. Grafting on radius with rate-2 tangent bound transforms the fine field to exp(Ad() times the replica of the gauged coarse field. Product-triangle bounds then give gauged fine chords by), because replica links are identity or coarse links. The single rate- envelope bounds all chart data eventually, with no exterior condition. Restricting from4 allows affine accuracy. The enlarged envelope supplies both its background-log and tangent bounds. All entry, replica and straightened-row terms rotate by the same root adjoint, transporting size rate2 and error rate1 back to the original fields.

The identity cover field has frame, connectors and predictor all identity with strict minimum tests. Continuity on these branches, simultaneously over finite axes/tags, gives a product neighborhood for any chord tolerance. This proves the finite-stencil branch activation used in the local chart-mark argument without requiring any tags or defaults away from the finite hull. Choose the finite set of restricted cover links and a common positive chord radius meeting each coordinate neighborhood; chord-to-matrix conversion uses the fixed factor colorScale. Let be the largest coordinate-order route length from zero to a root in. Expanding any torus input at the embedded coarse root sends each required cover link to a fine link whose coarse-parent distance is at most that route length, hence at most. Parent-distance chord bounds therefore place the expanded field in this same neighborhood, uniformly in period, with actual endpoint tags. The actual connector equals the cover test connector with its actual two endpoint tags, so its predictor meets the chosen tolerance. Every fallback tag has the same best value there, and normalized site/pin masses are one, their unnormalized values being reciprocal tag counts.

Choose eventually2 andmin(4). Gauge invariance preserves masses; wire discrepancy is at most coarse chord plus predictor chord A four-link plaquette in the unchanged hull has chord at most 4. Thus both actual wire and fine plaquette meet the desired small-curvature threshold. The uniform branch-cutoffand predictor estimates in this chapter give this finite-stencil estimate. The same chart proves that the actual local branch tags and holonomies lie in any prescribed small neighborhood eventually. The lift retains the same sector, frozen data and tag census; this local activation does not specify branch outcomes outside the finite hull.

12.24 Global and localized mismatch bounds

Let a cut transported row and full flat have row/column prices at most, difference prices , and input comparisons. The exact ordered decomposition

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F ⊤f −F ⊤
0 t = F ⊤(f −t) + (F −F0)⊤t

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and the column-sum estimates give error 2C2hq0 at every output coordinate. Applying the same
estimate to f = Fw, t = F0w gives ∥F ⊤Fw −F ⊤
0 F0w∥≤2C2∥w∥∞q0. Choose C to dominate the
fine and coarse plotted bounds, predictor-noise bound and all three kernel bounds, intersect chart

radii and take = min( width width = widen(). Noise’s quadratic remainder uses The uncut flat cross-roll identity cancels the lGap linear terms, leaving two transpose/input errors and one coarse plotted error; the full diagonal-roll identity cancels the three flat row squares against. Hence

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∥lGap ∥≤(4C2 + C)hq0,
∥dGap(w)∥≤6C2∥w∥∞q0.

The common 6 pays both, including cutoffand transport errors for every0, short periods and winding routes. The local transfer described next uses exactly these same gaps. For local chord bounds in one common fine/coarse gauge on radius satisfying

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d ≥3R + 4 + predictionReach,

the bounded logarithm extension gives global agreeing with the actual fine/coarse links there and bounded by. Let = max(1) and requiremin(). Exact gap locality identifies the global comparison with the actual tested gaps. Gauge covariance rotates lGap, and rotates both dGap and its arbitrary input, preserving its independent bound. Because 1 and both widening summands are nonnegative,

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widen(R, Xh, τ) = (1 + R)Xh + e−τR ≤X widen(R, h, τ).

Thus, enlarging the global constant gives

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∥lGap ∥≤CX2h widen(R, h, τ),
∥dGap(w)∥≤CX2q widen(R, h, τ).

The source lift preserves this fixed arbitrary input during background replacement. In particular changing exterior links is not assumed to regenerate the same thinned data array: when is the actual thinned data, changing the links outside the hull need not preserve that entire array. Only its separately retained supremum bound is used.

12.25 Quartic curvature deficit

For every , not only near identity, the Lie projection satisfies(mat( colorMat(curvature()). Subtract the skew part (mat(mat(2 from mat(. The remainder is self-adjoint, orthogonal to every skew-adjoint Lie matrix by cyclicity of trace in the real Frobenius pairing. The surviving Lie projection of the skew part is the defining curvature. Pythagoras therefore gives exactly

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colorScale2[chord(U)2 −∥curvature(U)∥2] = ∥mat(U) −1 −Pg(mat(U) −1)∥2.

Focus fixes nearby group matrices and has derivative at identity. Its ambient analytic remainder bounds the last norm bymat( for that norm at most. Squaring and restoring colorScale gives, with colorScale

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chord(U)2 −K chord(U)4 ≤∥curvature(U)∥2.

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For δ = min(r/ colorScale,
p
1/(4K)), chord at most δ gives K chord(U)2 ≤1/4 and therefore
curvature square at least 3/4 chord square.
Constants depend only on the group and precede
block-length selection. This is a group-point restriction of an ambient analytic estimate, not a
global configuration logarithm. The lift preserves both the global projection identity and the local

smallness domain under conjugation, with the same physical color normalization. For the source– readout correspondence, conjugation by a group matrix is an orthogonal action on the ambient real Frobenius space and preserves its Lie subspace. Therefore it commutes with, and preserves the normal-projection norm in the exact deficit identity. The framed plaquette readout is conjugate to the physical plaquette by its base frame, so the deficit, quartic bound and chord threshold recover unchanged. Evaluation on the identity-frame section returns the physical equality. Thus the local estimate uses precisely the same group-only constants and smallness condition after realization.

12.26 Soft curvature and noise remainders

) = [chord(curvature(0. On a retained fine plaquette within its clipping threshold, expansion gives

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2
X
i∈Bool
∥entryface,i(U)∥2.
crown = direct flaw(face U) + 1

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The guide cross term and square complete both bill squares, bill being entry minus guide. Thus
workF is direct times flaw plus those two half-squares.
Without a retention premise, workI is
penalty times flaw(V Q−1) plus its single bill half-square, with actual Q = prediction(U, k) and the
same guide.
Affine accuracy bounds entry by M of rate −1/2 and bill-minus-reply by E of rate −1; sub-
tracting the same guide leaves that affine error unchanged. The coarse maximum bounds guide by
A fiery, so bill has size m = M + A fiery. Factoring squared norms gives
 1

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2∥v∥2 −1
2∥w∥2 ≤E(2m + E)/2 =: Z

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for ∥v∥≤m, ∥v −w∥≤E; Z has rate −3/2. Local branch activation puts the actual fine and
noise holonomies in the quartic-deficit neighborhood. On this neighborhood the earlier bounds are
chord(g)2 −J chord(g)4 ≤∥curvature(g)∥2 and 3
4 chord(g)2 ≤∥curvature(g)∥2. A scaled curvature
bound ∥
√
k curvature(g)∥≤M then gives chord at most 2M/
√
k and flaw at most (J/2)(2M/
√
k)4.
Consequently

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X = direct(J/2)(2M/
√
direct)4,
Y = penalty(J/2)(2M/
p
penalty)4

have rate2. The envelope of rate2 bounds each soft energy minus its affine reply half-square sum. The factor two pays the two fine coordinates; noise has one. Readout preserves the exact square completion and the retained/clipped fine-plaquette domain before using these bounds; the unconditional noise identity is kept distinct.

12.27 Outer corrections and light annulus bounds

On a clean radius-(4 + 1) hull, the flattened guide has zero linear part and error of rate 1, and gauge covariance preserves its norm, so its half-square workN is bounded by2, of rate 2. For, toll at most temp makes clipping exactly one, including at zero. Its exact formula gives workC nonpositive, and the quartic estimate gives absolute bound (direct2)(2 fiery) since temp is at most 2 fiery, again rate

Use the same clean grafted local gauge for the mismatch terms, with common rate- 2 envelope For = widen() the local gap bounds givelGap and dGap(vision) fiery, using the actual maximum bound on vision. Cauchy–Schwarz bounds each inner product by and each half-inner-product by