Local nonlinear geometry and Haar realization · Part 2

therefore lie in the inverse-function neighborhood when is sufficiently near. Since uniqueness of the local inverse forces) = focus(). Exceptional nearest choices arbitrarily near identity would contradict this argument. Translation gives a uniform tube about the whole group, including all its components.

Differentiating the inverse gives(matfocus)(1) = proj. Indeed the inverse derivative sends colorMat(), and the exponential derivative discards. Analyticity makes this a strict derivative, which yields, after shrinking the neighborhood for each

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∥mat(u) −mat(v) −projg(X −Y )∥≤ε∥X −Y ∥

for the nearest points of . On the unique tube, unitary norm invariance and uniqueness prove equivariance of focus under left/right group multiplication. Differentiating that equality gives its derivative covariance. These prove the focus properties used in the frame and predictor derivatives from the local group chart and the ordinary inverse function theorem. The source lift uses this actual analytic branch and its radius. Outside it, compactness still gives minimizers but no arbitrary global selection is assigned these uniqueness, analyticity or equivariance properties.

12.8 Analyticity of the actual normalized frame

Fix the period. No volume-independent analytic radius is asserted. For logarithmic input, retain actual patch frame matrices normalized by = 1. At zero all paths are identity; the nonnegative score has identity minimizer of value zero, and uniqueness identifies the selected frame. Its positive cutoffgives an open minimum branch, and selected-minimizer continuity gives with all tags choosing that branch.

The actual self-consistency cloud is the finite analytic expression

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cloudp(v, H) = averageq,o Hq travelv(p →q; o)∗.

The path products are finite products of exponentials and adjoints, so the cloud is analytic jointly in and equals one at (0 1). Continuity places every cloud in the analytic-focus neighborhood; self-consistency says is its nearest group point. Thus the selected frame solves andfocus(cloud) = 0 otherwise. A kernel vector of its derivative has = 0 and = proj(average) for. For the positive number of patch points its average , and is already projected. Hence This finite-dimensional endomorphism is invertible.

Apply the analytic inverse function theorem to ()). The continuous actual solution lies in the local inverse neighborhood; uniqueness identifies it with the analytic implicit branch. This proves analyticity of the actual normalized frame, using the separately retained nearest-point, self-consistency and minimizer arguments. The realized predictor lift composes this same selected operation with readout; it does not enlarge its radius to a uniform-in-volume statement or differentiate an arbitrary fallback branch.

12.9 First derivative of the actual rooted predictor

Let fee) be the tangent link sum of the actual ordered route from to embedded at coarse root, and define tapaverage fee). Differentiating each finite route product gives its color matrix, with reversed links contributing minus signs. Each color component of tap is exactly the scalar path functional) applied to the color-coordinate input. For the actual frame derivative, the cloud derivative at identity is

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averageq dq + colorMat(tapy(p; a)).

The focus derivative is projection onto the Lie algebra, so differentiation gives

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dp = proj(averageq dq) + colorMat(tapy(p; a)).

Since the distinguished frame is identically one, = 0. Subtract its equation to obtain

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dp = colorMat[tapy(p; a) −tapy(p0; a)].

This proves tag independence of the derivative on the common actual branch. For a coarse bond the left frame, length- route and inverse right frame therefore give connector derivative colorMat(bill)), where

bill) = taptap) + linkSum(bondRoute

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−tapy+ei(p; a) + tapy+ei(p0; a).

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The best-predictor branch is open at zero by zero cost and positive cutoff. Every connector is near
one and the weighted connector average equals one at zero. Near zero the actual predictor is focus
of this analytic average, whatever fallback tag was supplied. Its derivative is the weighted sum of
these Lie-algebra connector derivatives, unchanged by the focus projection. Expanding tap and the
original rooted path and probe definitions gives, coordinate by coordinate,
X

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p
weightp billp(a) = probe(a).

Weights sum to one, retaining both center subtractions. Thus the predictor is identity at zero and its literal derivative is colorMat(probe()). The local analytic frame equation and its normalization are the inputs to this differentiation; the subtraction shows explicitly why the derivative is independent of the frame tag. The horizontal derivative formula proved for the lift recovers precisely this rooted probe, rather than an unrooted or independently averaged substitute.

12.10 Uniform predictor linearization and entry bounds

For fixed patch/kernel let contain every ordered interpatch route, its translated endpoint copy and every length- connector. Equal fields on its rooted links have equal path products, axial/via frames and scores; uniqueness identifies the minimizing frames, while the fallback uses the equal axial frames. Thus actual predictors agree with the original endpoint and connector tags. The maximum route distance of these finitely many roots is predictionReach; agreement on that parent-distance hull suffices.

At fixed period/tags the noise field

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penalty curvature(exp b prediction(exp a, k)−1)
Zk(a, b) =
p

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is analytic near zero, vanishes there and has derivative zlin = √penalty(b −probe(a)). This follows
from the actual predictor derivative, skew-adjoint inverse derivative and curvature’s linear color
reading. Taylor gives a linear bound and quadratic remainder. There are finitely many tags and
bond positions, so take the maximum of their constants and the positive minimum of their radii
on that fixed torus.

For volume uniformity choose a reference coarse side 1 large enough to inject the stencil, with its largest absolute coordinate. At an input bond root, transplant reference fine values by wrap(lift(embed(0))) + embed(

Then set the coarse reference field constantly to the original bond value. Norms do not increase. Copy original endpoint tags to distinct reference sites 0 and its connector tag to their bond. All stencil reads agree for every scalar multiple of the inputs, including repetitions identified by a short original torus. Values agree at one and derivatives agree at zero. The reference Taylor bounds therefore give volume/tag-independent such that

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∥zlin(a, b)∥≤Cz max(∥a∥, ∥b∥),
∥Zk −zlin∥≤Cz max(∥a∥, ∥b∥)2

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inside radius rz. This uses a copied actual branch, not a new minimizer.
The full entry is plotted fine moment direct-summed with Z, and its linear entry is xlin ⊕zlin.
For logarithmic inputs bounded by H, combine the plotted comparison already proved with the
preceding inequality for Z: set M = (Cx + Cz)H and W = CxH widen(R, H, widthtop) + CzH2.
Then each linear entry has norm at most M and each entry-minus-linear-entry has norm at most
W, simultaneously in the two kinds of row. For H of rate −1/2 and R = till(x) they have rates
−1/2, −1. Predictor covariance and curvature conjugation give gauge covariance, and the union of
cutoffstencil, predictor reach and coarse root link gives protected-hull locality. These establish the
entry-near and entry-locality inputs used next.

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A curl costs at most 4∥v∥; path sums cost 4 paths ∥v∥with omitted tail 4 paths ∥v∥e−widthtop R.
√
√
direct∥v∥and Cz∥v∥. Thus Ust = 4(paths +
Direct and noise rows cost 4
direct) + Cz, Kst =
Ust + 4 paths bound full/cutoffrows and their exponentially small difference at every cutoff. The
cutoffbound follows by comparison with the full row and e−widthtop R ≤1. Absolute summability
licenses subtraction.
A single-link impulse at parent distance g has zero cutoffreading if s +
predictionReach +4 < g. For g > predictionReach +5, choosing s = g −predictionReach −5 and
comparing both cutoffs to the full row yields

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ewidthtop g∥FieldR(asingle)∥≤2Kstewidthtop(predictionReach +6)∥a∥.

If is too short the cutoffrow is already zero. Nearby links use the uniform row bound. These bounds also give the spatial impulse estimate used in the plate prices. The lift copies every repeated source value and tag on this stencil, and hence retains both the volume-uniform estimate and its actual locality domain.

12.11 Shell comparison and the tree-slice floor

A differentiable group curve with ̇(0) = colorMat() satisfies chord( because chord is the Frobenius distance from identity divided by colorScale andcolorMat( colorScale vanishes on forest trunks, exp() remains on the exact tree slice. Near zero the actual frame branch has positive mass and usable tags. Its nonlinear shell comparison with coarse field one gives

chord( port(parent()) + chord(

This upper bound is valid in the internal-block case as well, where adding the last nonnegative term is harmless. Bound the port by the full macro-box plaquette sum, retaining periodic repetitions, divide by and take the finite-sum limit. Plaquette and predictor derivatives give

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p
∥curl(w)wrap z+embed(parent(e)),p∥+ ∥probe(w)parent(e),axis(e)∥.
∥we∥≤C
X
X

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z∈box(macroR)

This uses the nonlinear shell comparison, not a newly chosen linear gauge.

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Let N = | box(macroR)|, m = |PlaqAxes|, q = L4. Macro-sum Cauchy–Schwarz costs Nm.
For each fixed offset z, the map (x, p) 7→(wrap z + embed(x), p) from coarse site and plaquette
type is injective, so summing squares costs at most the full fine sum; summing offsets adds N.
Each coarse site has q children and four outgoing fine axes. Square the local link estimate using
(a + b)2 ≤2a2 + 2b2. Therefore
X
e
∥we∥2 ≤β
X
p
∥curl(w)p∥2 + γ
X
b
∥probe(w)b∥2,
β = 8C2qN 2m,
γ = 2q.

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Different offsets need not be injective against each other. If the fine bed has curl floor a > 0 and
penalty κ > 0, its vertical tree-chart Hessian dominates a P ∥curl w∥2 + κ P ∥probe w∥2. Linear
inclusion identifies this Hessian with the full germ on the collected field zero on trunks. Hence

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η = min(a, κ)
1 + β + γ > 0

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gives the floor η P
free ∥ve∥2, since ηβ ≤a and ηγ ≤κ. This unfolds the fixed-block counting-
coordinate Hessian floor. The nonlinear shell comparison supplying C remains the stated input.
Here “vertical” is the source’s free tree-chart variation, not the additional frame-fibre kernel of the
source–readout realization. Keeping those spaces distinct preserves both this positive floor and the
exact redundant-frame null directions.

12.12 Uniform cutoffs, continuity and minimizer uniqueness

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The finite-pair small-transport constant d > 0 depends only on G.
Choose frame cutoffat
most d/ colorScale. On an all-minimum patch use its center axial frame a and pair transports
Wxyk = ax travel(x →y, k)a−1
y . Their matrix distance from identity is at most d. Transforming a
competing frame by bxa−1
x
identifies pair cost and original frame score. Every competitor can be
left normalized without changing cost. Compactness of the finite product of G with its center fixed
gives a normalized minimizer, pair uniqueness gives uniqueness modulo common left multiplication,
and center value one removes it. Thus the cutoffis independent of patch order and size.
For real weights wi summing to one and m = P
i wi mat(fi), completion of the square gives

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2 repDim
X
i
wi price(fiu−1) =
X
i
wi∥mat(fi)∥2 −∥m∥2 + ∥m −mat(u)∥2.

Hence predictor minimization is nearest-point minimization. For nonnegative weights the squared-norm inequality for a weighted average gives the explicit bound

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∥m −mat(u)∥2 ≤2 repDim
X
i
wi price(fiu−1).

If this right side is strictly below, the centroid lies in the nearest-point tube and has a unique nearest point. Cutoffat most(2 repDim), where is the focus tube radius, ensures uniqueness; this choice also depends only on

A closed minimizer relation with compact has continuous arbitrary selections at each singleton fibre: for an open neighborhood of its unique value, the projection of closed and omits that input. Its complement forces every selected minimum into . For frames this relation is the intersection of the closed center-normalization condition and the score inequalities against all normalized competitors; each score is a continuous finite path expression. Predictor relations use the corresponding cost inequalities. Strict finite cutofftests make the branches open. This supplies the continuity used in the analytic-frame theorem.

For a finite pair family set term() = price( ) and average over the three indices for the cost. For self-consistency, reversal exchanges and applies the mirror permutation to, preserving the sum. Thus reversal symmetry pairs the off-diagonal terms when one frame coordinate is varied; diagonal terms vanish because = 1. Varying only changes the total cost by twice the off-diagonal row change. Global minimality therefore minimizes the off-diagonal

row. Adding the frozen self term, nonnegative and zero at, proves whole-row minimality. The normalized whole row is the weighted squared-distance problem to , up to a positive constant and a competitor-independent term. Completing its square identifies as nearest to feedback( If, identity-array comparison bounds mean squared pair separation by 4. After normalizing so identity is nearest to its centroid, a suitable index gives centroid distance at most , and feedback differs from the centroid by at most, hence lies within 3 of identity. Choose the focus two-point radius for error 14 and = min(4). Multiply each of two minimizing

arrays on the left by the inverse of a nearest point to its centroid. This preserves minimization and makes identity nearest to both centroids. With = max, their feedback difference at a maximizing index obeys. Subtracting centroid difference gives error at most by factoring each term as ( 1). Both centroid-minus-identity vectors are normal, so proj. Since 3, self-consistency and the focus estimate apply to both feedbacks and imply proj

Thus = 0. Undoing normalization proves common-left-multiple uniqueness; center normalization makes it literal uniqueness. The source lift retains this group freedom separately from its redundant presentation frames and does not extend the singleton-fibre claims to arbitrary nonunique branches.

12.13 Finite-pair alignment and route census

For reversal-symmetric pair transports and a minimizing array, put income( average chord() and = average income(). Identity-array comparison bounds minimum price by, since chord squared is at most twice chord. Factoring the pair separation through its discrepancy and squaring gives

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averagex,y ∥hx −hy∥2
F ≤8 colorScale2 m.

Indeed the squared triangle inequality bounds each squared pair chord by twice the squared discrepancy plus 4 chord(); averaging with minimality gives the factor eight. Set

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centroid(h) = averagex mat(hx),

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feedback(h)x = averagey,k mat(hyW −1
xyk).

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Direct subtraction gives ∥feedback(h)x −centroid(h)∥F ≤colorScale income(x). By self-consistency
each hx is nearest to its feedback. Choose a nearest centroid point b and replace h by b−1h. Identity
is then nearest to the centroid, whose distance from identity is at most
√
8 colorScale2 m, using a
suitable index and the squared-average inequality.
Choose the focus comparison radius δ for error coefficient one, σ = (δ/2)2/(8 colorScale2) and
a = δ/(2 colorScale).
If m < σ and income is below a, both feedback and centroid lie in the
comparison domain. The centroid lies within δ/2 of identity and feedback within δ. The focus
comparison and projection norm at most one give

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∥mat(hx) −I∥≤2∥feedback(h)x −centroid(h)∥,

and hence chord(2 income(). Remaining indices use 2) income(). Averaging these two index classes gives the constant max(2, choose = 1; the universal chord bound gives 2. Undoing the common normalization and using conjugation invariance therefore gives an aligning element with

average chord( = max(2

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No individual-transport smallness is required for this averaged statement.
For a radius-r box of N sites, replacing selected coordinates of x by those of y gives dp(x, y).
Swapping those coordinates is an endpoint-pair involution, hence P
x,y f(dp(x, y)) = N P
x f(x).
Swapping a site coordinate with an independent height likewise gives
X
x,height
f(updatej(x, height)) = (2r + 1)
X
x
f(x).

A positive straight segment’s chord is bounded by its whole internal line sum, by the product triangle inequality and then enlargement to the full line; for a negative segment reverse the walk and use inversion invariance. Appending an axis adds this line cost at). The two census identities bound its total by + 1) edgeSum(). Induction yields length( + 1) edgeSum( with constant four for all axes. The source lift retains these endpoint-pair sums, roots and route multiplicities, so the averaged alignment is not replaced by an unsupported pointwise bound.

12.14 Predictor queries and synchronization

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For connector values Yp with nonnegative weights summing to one, take q = xy−1 and τ =
P
p weightp chord(x−1Ypy). Cyclic invariance identifies it with the discrepancy from q. For m =
P
p weightp mat(Yp), one has ∥m−mat(q)∥≤colorScale τ. In the best branch its output b is nearest
to m, so
∥mat(b) −mat(q)∥≤∥mat(b) −m∥+ ∥m −mat(q)∥≤2 colorScale τ.

In the fallback branch its minimum price is at least predictorCut; since price = chordchord, comparison with givespredictorCut. Since 0chord2, the universal chord bound therefore proves in both branches

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chord(x−1 predict(Y )y) ≤2 max(1, predictorCut−1)τ.

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This includes the actual fallback output.
The radius-r route census gives P
p,q chord(travel(p →q; o)) ≤4N(2r + 1) edgeSum(U) and
hence N meanPairChord ≤4(2r + 1) edgeSum(U); the census follows by charging each coordinate
segment to its internal positive edges, as in the route-count argument above. In the minimum
frame branch, after gauge H, the array bp = framep H−1
p
minimizes the corresponding pair en-
ergy; gauge covariance and center normalization preserve competitors.
Alignment gives c with
averagep chord(bpc−1) ≤C meanPairChord. Cyclic invariance and the route bound then give

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X
p
chord(c−1 framep H−1
p ) ≤4C(2r + 1) edgeSum(H · U).

In the fallback branch positive frame mass makes the axial tag eligible. Its normalized route gauge has edge sum at most axialFactor(2 + 1) faceSum(). Set. Factoring links gives

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edgeSum(g · 1) ≤edgeSum(d · U) + edgeSum(H · U). Pure-gauge travel is gpg−1
q . Choose a root q
whose endpoint sum is at most the average, and apply the route bound. With c = gq this gives

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X
p
chord(c−1 framep H−1
p ) ≤4(2r + 1)
 edgeSum(H · U)
+ axialFactor(2r + 1) faceSum(U)
.

Thus = max(4 4) pays both synchronization branches, using the finite-pair cluster inequality and route census as its two inputs.

The patch weight is the distribution of independent uniform small-box rolls. Conditional on all but the first roll, at most one first roll yields any specified sum. Its box has points, so weight1. Coincident roll sums are not assumed distinct. The source lift retains this probability census and actual fallback tags alongside the connector readout; no successful minimizing branch replaces a fallback contribution.

12.15 One macro gauge and predictor price

The planting argument uses the axial estimate already proved and the following frame synchronization inequality. Synchronization on a usable radius- patch gives a constant with frame-error sum at most + 1)[edgeSum() + axialFactor(2 + 1) faceSum()]. The constant depends only on the group. For smoothing index, put and + axialFactor). The macro radius is patch radius plus 4, so 2 macroR +1 Apply the four-axis axial construction to the whole macro box, giving one gauge with edge sum at most port. All patches centered at, lie there. Nonnegativity bounds their sums by the macro sums. Applying synchronization to each with that same gives and error sum at most port.

For adjacent roots, a connector factors into left frame error, length- transport in and inverse right error. If frame-error sums are at most port and macro edge sum at most port, the sum of transport chords over starting points is at most times that edge sum. Each step translation is injective and stays in the macro box; repeated torus edges still count as separate steps. The weight bound weight1 gives) port. The query estimate bounds the chord by 2 max(1 predictorCut. Multiply by and use 0 to obtain

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L2 chord(p−1
y Qy,ipy+ei) ≤2 max(1, predictorCut−1)(2K + T) port .

For = 2 max(1 predictorCut), each gauged predictor link costs port. Endpoint factors cancel around the coarse plaquette, yielding chord(face port. For any , the factorization inequality

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chord(ab(cd)−1) ≤chord(ac−1) + chord(bd−1)

follows by factoring into one discrepancy and a conjugate of the other. Iteration with inversion invariance on the reversed links gives chord(facechord(face)+fuel( ), where fuel retains all four wire discrepancies. The lift uses one common macro witness and these same endpoint gauges; independent favorable gauges on separate patches would not establish this joint cancellation.

12.16 Nonlinear shell-link comparison

On the actual forest slice close a fine link by the reverse tree hike from its first coarse root and the forward hike to its second coarse root. Both hike transports are identity, so this walk reads), has

length at most 8+1, and has the two embedded coarse parents as endpoints. Lift the whole walk to the cover starting at zero over its first root; every point lies in the same macro box. Planting there supplies with edge sum at most port and frame-error sum at most port. At the distinguished patch point the frame is one, giving chord((centerRoot( port. The adjacent- port, where = 2 max(1 predictorCut root predictor bound is chord(

The triangle inequality along all occurrences of the lifted walk bounds its gauged chord by at most port,. The period condition 100 macroRfineSide makes wrapping injective on the macro box. If coarse parents agree, the lifted endpoint is zero, the replica comparison link is identity and conjugation gives the bound directly. If parents differ, the centered parent-hop identity makes the second parent the positive coarse neighbor, and injectivity gives the endpoint centerRoot(), including across a periodic seam. Factoring the predictor through the same bounds its gauged chord by port, . Compare the fine link and predictor in this same macro gauge and use the triangle inequality. Therefore

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chord(U(x, i)Q−1) ≤(u + v) port .

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Comparison to the desired coarse link V adds exactly its wire discrepancy with coefficient one.
The constant is uniform in period and forest on the room domain. The room hypotheses supply
its size assumptions, so the same constant is available in the local driving inequality. The source
lift retains that exact forest slice, macro gauge and wrapping hypothesis, so the shell estimate
used by local driving has the same domain and coefficient; it is not inferred from separate local
gauges at the walk endpoints. Explicitly, let D be this forest-slice room domain and let f(U, V ) be
the left-hand discrepancy, with the replica interpretation when both parents agree. The recovered
scalar is ef = f ◦π on π−1(D), and the port and wire budgets are pulled back by the same map.

Thus the bound pulls back pointwise with the identical constants and one. Evaluation at the section recovers the original inequality because. Endpoint frame factors on the lifted walk cancel internally and leave precisely its endpoint transforms; their chord is unchanged by the corresponding unitary conjugation. This establishes recovery with the same periodic injectivity and forest hypotheses.

12.17 Local driving from selected motions

Let lens() be the active fine tangent. At coarse, current is the square root of the sum of squared lens norms within the radius-3 fine-link neighborhood, and response is the square root of the sum of squared selected-motion norms within radius macroR +2. Nonnegativity of all squared terms bounds each summand norm by the corresponding root sum. For a hot link with parent, a gauge making background chords at most on radius 2 permits the transported-to-flat comparison. Hotness puts sufficiently far from the seed set, and the room radius keeps this entire row-root neighborhood clear, so every required row is selected in identity orientation and its motion norm is bounded by response. All required tangent values lie in current’s radius-3 neighborhood. Gauge covariance and norm preservation therefore give each relevant row obeys

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∥FieldR(Ad(z) lens(u))∥≤Eo,

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Eo = response(o, u) + 2Kst colorScale(B + 1)h current(o, u).

Let be the common local comparison constant for the actual predictor tags, supplied by the gauge-flattening/tree-slice construction. Its link estimate boundsby times the macro-box sum of curl norms plus the predictor-bill norm. Apply it to) lens(), zero on trunks