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+Bi-Lipschitz Coordinates at Regular Points
+of Noncollapsed RCD Spaces
+A Guided Tour for Young Mathematicians
+
+============================================================
+
+This is a guided tour of a research paper in metric geometry.
+The paper proves that a "regular" point of a certain kind of
+singular curved space always has a small open neighborhood that
+looks, metrically and up to a fixed distortion, like an open
+piece of ordinary flat n-dimensional space. The distortion is a
+constant that depends only on the dimension n, never on the
+particular space or point.
+
+The tour is written for a motivated high-school reader. Every
+technical term is introduced before it is used, the key ideas of
+each argument are explained in plain language, and the precise
+statements are also given so the reader can compare against the
+original. It is honest about what the paper proves from scratch
+and what it uses as a black box from earlier work.
+
+The original paper has a short introduction and four numbered
+sections:
+
+ Section 1. Uniform harmonic coordinates.
+ Section 2. Heat averages and cumulative matrix control.
+ Section 3. Summable directional pinching.
+ Section 4. Renormalization in the space of coordinate values.
+
+The tour follows the paper's own order. A reading guide near the
+end maps each section and each named lemma back to the tour. The
+reader is encouraged to read the tour once for the plot and a
+second time for the estimates.
+
+
+============================================================
+PART I. THE QUESTION AND THE MAIN THEOREM
+============================================================
+
+
+1. WHAT KIND OF SPACE IS THIS ABOUT?
+
+A metric space is a set of points together with a distance
+function d(x,z) that assigns a nonnegative number to each pair of
+points, is zero exactly when the points are equal, is symmetric,
+and obeys the triangle inequality
+
+ d(x,y) <= d(x,z) + d(z,y).
+
+The most familiar example is ordinary flat n-dimensional space
+R^n with d(x,z) = |x - z|. A ball of radius r about a point x is
+
+ B_r(x) = { y : d(x,y) < r }.
+
+Throughout the paper the spaces are:
+
+ complete -- every Cauchy sequence converges (no missing
+ points, no open edge);
+ separable -- there is a countable dense set of points;
+ geodesic -- any two points are joined by a shortest path;
+ measured -- there is a reference measure m on the space,
+ finite on every bounded set, positive on every
+ nonempty open set ("full support").
+
+The measure m is the tool that lets us average functions and talk
+about volume. Concretely we will need: how much volume sits in a
+ball, and how that volume grows with the radius.
+
+
+2. WHERE THE CURVATURE COMES IN
+
+On a smooth Riemannian manifold -- a smooth surface or higher-
+dimensional curved shape whose distances come from a metric
+tensor g, with a notion of length of tangent vectors -- there are
+several notions of curvature. The one that controls volume is
+Ricci curvature. The classical fact (Bishop-Gromov) is:
+
+ if a Riemannian n-manifold has Ricci curvature >= K, then the
+ volume ratio
+
+ r -> m(B_r(x)) / (omega_n r^n)
+
+ is nonincreasing in r, where omega_n is the volume of the unit
+ ball in R^n. In particular the ratio is at most 1.
+
+So negative lower Ricci curvature makes balls grow faster than
+Euclidean balls as the radius increases, and the excess can be
+read off from the shrinking of that ratio at small scales. A
+space in which the ratio is identically 1 near a point is locally
+flat near that point.
+
+The engine of the whole paper is the way this volume comparison
+forces a singular, non-smooth space to look Euclidean on fine
+enough scales, provided its curvature is bounded below.
+
+
+3. METRIC MEASURE SPACES AND THE RCD CONDITION
+
+The trouble is that many interesting limits of Riemannian
+manifolds are not manifolds: they have cone points, edges, or
+lower-dimensional strata. There is no metric tensor to
+differentiate. We need a definition of "Ricci curvature bounded
+below" that makes sense for such a singular metric measure space.
+
+Two equivalent routes were developed:
+
+ (CD) Curvature-Dimension, Lott-Villani and Sturm. The space
+ satisfies CD(K,n) if a certain convexity holds for the entropy
+ functional along optimal-transport geodesics between
+ probability measures. The parameter K is the lower curvature
+ bound and n the dimension bound.
+
+ (Bochner / heat) The same content phrased through the heat
+ flow and a Bochner inequality, by Erbar-Kuwada-Sturm.
+
+The CD condition also allows genuinely non-Riemannian spaces
+(some Finsler-type structures) that behave unlike manifolds. The
+fix is the RCD condition (Riemannian Curvature-Dimension),
+introduced through the work of Ambrosio-Gigli-Savare:
+
+ RCD(K,n) = CD(K,n) plus the requirement that the Cheeger
+ energy is a quadratic form ("Riemannian" or "infinitesimally
+ Hilbertian"). Equivalently, the heat flow is linear and the
+ space admits a Hilbert-space-like first-order differential
+ calculus.
+
+From now on, "the space" means a complete, separable, geodesic
+metric measure space (X,d,m) satisfying RCD(K,n) for some real K
+and some integer n >= 2. On such a space one still has, even
+though there is no coordinate chart:
+
+ - a gradient and a Laplace operator (defined through the
+ quadratic Cheeger energy and its generator);
+ - a carre du champ Gamma(f,g) = <grad f, grad g>, the "dot
+ product of gradients";
+ - a Hessian for nice functions, and a Bochner inequality;
+ - a heat semigroup and a continuous heat kernel.
+
+These are the only analytic tools the proof uses. None of them
+needs a single coordinate chart; that is the whole point.
+
+
+4. NONCOLLAPSED: THE MEASURE REALLY IS n-DIMENSIONAL
+
+A subtle but central hypothesis is noncollapsedness. It says the
+reference measure is exactly the n-dimensional Hausdorff measure
+H^n of the metric, with the usual normalization. In words: the
+space has honest dimension n measured by volume, not a lower
+dimension. In particular the local volume density is at most 1
+everywhere, matching the Bishop-Gromov bound above.
+
+For a limit of Riemannian n-manifolds with lower Ricci bound, the
+noncollapsed case is the one where the limit did not lose a
+dimension: it does not suddenly become a surface or a line
+segment sitting inside R^3. The collapsed case is real and
+important, but this paper is about the noncollapsed case only.
+
+A generous reward for assuming it: by De Philippis-Gigli (2018),
+in a noncollapsed RCD(K,n) space the reference measure equals H^n,
+volume convergence and volume rigidity hold, and the set of
+"Euclidean points" has full measure. That theorem is an input,
+not something reproved here.
+
+
+5. TANGENT CONES AND REGULAR POINTS
+
+To say a space "looks Euclidean near a point" we zoom in. Make the
+distance smaller and smaller:
+
+ fix a basepoint p. For each radius r > 0, form the rescaled
+ space (X, d/r, m/r^n, p): all distances multiplied by 1/r, so
+ what used to be size r now has size 1. Let r tend to 0 and ask
+ what the rescaled spaces converge to.
+
+"Convergence" here is pointed Gromov-Hausdorff (GH) convergence:
+metric spaces can converge to a limit even without sitting in one
+common larger space. The limit of a sequence of pointed metric
+spaces is a new pointed metric space, together with
+almost-isometries on larger and larger balls up to smaller and
+smaller error. In the noncollapsed setting volume convergence
+tracks the limit too.
+
+Every limit obtained this way is called a tangent of X at p. For
+a smooth Riemannian manifold every tangent is the Euclidean space
+R^n with the flat metric. For a general RCD space a tangent can be
+a cone, or a space with lower-dimensional pieces.
+
+Call p an n-regular point when:
+
+ every pointed GH tangent of X at p is pointed-isometric to
+ (R^n, |.|, 0), the flat Euclidean space based at the origin.
+
+That is the cleanest possible "infinitesimally Euclidean" for a
+point. De Philippis-Gigli proved that in a noncollapsed RCD(K,n)
+space the set of n-regular points has full H^n measure. Full
+measure does not mean every point, and it certainly does not mean
+the regular set is open. Both of those gaps are exactly where the
+difficulty of this paper lives.
+
+
+6. WHAT A BI-LIPSCHITZ CHART IS
+
+A map F : U -> R^n from a subset U of X is bi-Lipschitz with
+constant L >= 1 when, for all x,z in U,
+
+ L^{-1} d(x,z) <= |F(x) - F(z)| <= L d(x,z).
+
+The right inequality says F does not stretch distances by more
+than L; the left says it does not crush them by more than L. A
+bi-Lipschitz map is automatically continuous and injective, and so
+is its inverse on the image, so it is a homeomorphism onto its
+image. It preserves Hausdorff dimension, the metric's notion of
+"size at every scale," and many coarse invariants.
+
+A bi-Lipschitz chart on X is an open U inside X and a
+bi-Lipschitz map F from U onto an open subset V of R^n. Such a
+chart is the strongest "this piece is really n-dimensional and
+metrically finite" statement one can make without any smooth
+structure. The paper proves that each regular point sits inside
+such a chart with L depending only on n.
+
+Two strengths are deliberately NOT claimed:
+
+ - F is not claimed to be smooth, or even differentiable in any
+ classical sense. There is no smooth structure to compare to.
+ - the pullback of the Euclidean metric under F is not claimed
+ to be continuous, or Holder continuous, as a metric tensor.
+ Nearby results (Colding-Naber) show genuine examples where no
+ coordinates make the metric tensor Holder continuous, even
+ though Euclidean tangents exist. So demanding it would be
+ false. Bi-Lipschitz is the honest target.
+
+The distances on both sides of the display are the ambient
+distance d restricted to U. No intrinsic path distance along U is
+substituted: the chart controls the actual distances in the
+whole space between nearby points, which is a genuinely stronger
+and more useful statement.
+
+
+7. THE MAIN THEOREM
+
+Here is the paper's theorem in full.
+
+ Theorem (main). For every integer n >= 2 there is a constant
+ L_n >= 1 with the following property. Let K be a real number and
+ let (X,d,H^n) be a noncollapsed RCD(K,n) space. Let p be a
+ point of X. If every pointed GH tangent of X at p is
+ pointed-isometric to (R^n, |.|, 0), then there exist:
+ - an open neighborhood U of p in X,
+ - an open set V in R^n,
+ - a homeomorphism F : U -> V,
+ such that for all x,z in U,
+ L_n^{-1} d(x,z) <= |F(x) - F(z)| <= L_n d(x,z).
+
+The neighborhood U and the chart F may both depend on the space
+and on p; the constant L_n depends only on the dimension n.
+
+Read it as a sentence: "a single regular point always grows a
+whole open bi-Lipschitz Euclidean patch around itself, with a
+distortion that is uniform in the dimension and nothing else."
+
+
+8. THE FULL-MEASURE ATLAS COROLLARY
+
+Because regular points have full measure, the theorem immediately
+gives a global structure statement.
+
+ Corollary. Under the theorem's hypotheses there is an open set
+ U' containing every n-regular point, with H^n(X \ U') = 0, and a
+ countable atlas of L_n-bi-Lipschitz charts from ambient-open
+ subsets of U' onto open subsets of R^n. Each chart uses the
+ restricted ambient distance, and any two overlapping charts have
+ a transition map that is L_n^2-bi-Lipschitz.
+
+The proof is short. Take the union of all the chart domains from
+the main theorem; it contains every regular point and has full
+measure complement. The union is second countable (the space is
+separable), so a countable subcover exists. On an overlap,
+composing two charts and their inverses multiplies the two
+Lipschitz constants, giving L_n^2. One honest warning: the open
+region U' may contain points that are not themselves regular. The
+corollary does not assert the regular set is open; it only
+produces an open region containing it.
+
+
+9. WHY THIS IS HARD, AND WHAT WAS KNOWN BEFORE
+
+The conjecture is old. Cheeger and Colding, in their 1997 paper on
+the structure of spaces with Ricci curvature bounded below, built
+manifold neighborhoods that are bi-Holder-parametrized and
+explicitly asked (their Theorem 5.14 and Remark 5.15) whether a
+bi-Lipschitz upgrade is possible on quantitatively regular
+Ricci-limit neighborhoods.
+
+Since then:
+
+ - Mondino-Naber (2019) proved countable bi-Lipschitz
+ rectifiability: measurable pieces around almost every point
+ map to measurable Euclidean subsets. This is a measurable
+ almost-everywhere statement, not an ambient-open neighborhood
+ of a prescribed point.
+
+ - Kapovitch-Mondino (2021) proved the whole regular set is
+ contained in an open manifold region with bi-Holder charts.
+ Holder, not Lipschitz, and again not at a prescribed regular
+ point with a fixed distortion.
+
+ - Honda-Zhang (2026) recalled the regular-point bi-Lipschitz
+ conjecture explicitly, in work whose analytic criteria assume
+ a neighborhood-uniform positive lower bound for injectivity
+ radius or a Morrey-type control on Laplacians with a uniform
+ positive Gram determinant. This paper assumes neither.
+
+The obstruction is real and conceptual, not technical. Knowing
+that the tangent cone at p is R^n is an infinitesimal statement
+at one point. A chart has to compare all pairs of points in a full
+neighborhood, across infinitely many scales. A natural attempt is
+to build coordinates from harmonic functions that approximate
+Euclidean ones on finer and finer scales. That works up to a
+point, but the matrix measuring the "size" of the coordinate
+gradients can degenerate as the scale shrinks. Each rescaling uses
+its own normalization matrix, those matrices need not commute, and
+their size may grow without bound. The contribution of this paper
+is a way to live with exactly that degeneration.
+
+Two nearby results calibrate the strength. Colding-Naber (2013)
+give bi-Lipschitz embeddings of bounded pieces of quantitative
+Reifenberg loci into some Euclidean space, possibly of higher
+dimension, and construct examples where Euclidean tangents occur
+but no coordinates have Holder-continuous metric tensor. Neither
+upsets the target: this paper proves an open chart into R^n and is
+silent about Holder continuity of the metric tensor.
+
+
+10. THE SHAPE OF THE PROOF: TWO ESTIMATES AND ONE ITERATION
+
+The proof has three moving parts. The first two are estimates on
+the singular space; the third is a purely Euclidean iteration in
+the space of coordinate values.
+
+ Fix an n-regular point p. Rescale repeatedly by a sequence
+ s_1 > s_2 > ... tending to 0. The rescaled spaces become flatter
+ and flatter. On each we build a column u = (u_1,...,u_n) of
+ harmonic functions that approximate Euclidean coordinates.
+
+ Let G = (Gamma(u_j,u_k)) be the n x n matrix of pairwise
+ gradient dot products, the Gram matrix of the coordinate
+ gradients. Let R_i(y) be the average of G over the ball of
+ radius r_i = 2^{-i} centered at y. Let
+
+ M = (1/2) Delta G + kappa G m
+
+ be the matrix-valued Bochner measure (the kappa corrects for the
+ curvature lower bound; the precise formula is below). For a
+ fixed enlargement L > 1 define the positive semidefinite matrix
+
+ D_{i,y} = R_i(y)^{-1} [ r_i^{2-n} M(B_{L r_i}(y)) ] R_i(y)^{-1}
+
+ This D_{i,y} is the local form of the Bochner measure, conjugated
+ twice by the ball Gram so that it lives in coordinate-value units
+ and becomes dimensionless. It is the fundamental small quantity
+ of the construction.
+
+ (Estimate A, cumulative matrix control.) Corrected heat averages
+ of G produce positive definite forms H_i(x), increasing with i.
+ These serve as the "current yardstick" at scale r_i. Section 2
+ proves that the accumulated bounded-overlap sum of the D_{i,y}
+ is comparable to H_i, and proves the crucial reverse comparison
+ for suitably weighted sums. This controls how much the
+ coordinates must be stretched over the whole infinite ladder of
+ scales. The accumulated total may be unbounded; that is allowed.
+
+ (Estimate B, summable directional pinching.) Section 3 proves
+ that, on the same local forms,
+
+ D_{i,y}[ u(x) - u(z) ] <= C r_i^2 ( E_i(x) + E_i(z) ),
+
+ where the defect functions E_i are nonnegative and their total
+ sum over all scales, sup over x, tends to zero:
+ sup_x sum_i E_i(x) = o(1). Here D[xi] = xi^T D xi is the
+ quadratic form. Crucially the sum of the defects, not the sum of
+ the square roots of the defects, is small. That single fact is
+ what lets the infinite iteration in Section 4 converge even
+ though individual per-scale errors need not be summable on their
+ own.
+
+ (The iteration.) Section 4 works on the open coordinate-value
+ set Omega = u(B_1(p)) in R^n. It builds smooth maps
+ h_0, h_1, h_2, ... : Omega -> R^n by
+
+ h_0(P) = P, h_{i+1}(P) = h_i(P) + Dh_i(P)^{-T} v_i(P)
+
+ for smooth covector fields v_i assembled from the D_{i,y}. The
+ leading change in the pullback metric at step i is the positive
+ sum of the local forms D_{i,y}. Differentiating the factor
+ Dh_i(P)^{-T} produces a diagonal second-derivative term; keeping
+ a recurrence for that term is what allows the errors to be
+ summed using sum_i E_i alone. A second estimate on jets of all
+ orders, with factorial growth, keeps the iterates under control
+ despite each step losing one derivative. The limit F = lim h_i
+ o u is the chart.
+
+Four more remarks make the plan honest.
+
+ - The harmonic-coordinate Gram matrices R_i and the heat
+ matrices H_i are only compared up to a multiplicative factor
+ tending to one. They are never assumed bounded or convergent.
+
+ - The matrices at different scales do not commute. Section 2
+ includes a careful matrix-algebra lemma that extracts what is
+ needed (eigenvalue comparisons and summed quadratic-form
+ bounds) without ever commuting two matrices.
+
+ - Every enlargement of a ball is fixed before the late
+ rescaling is chosen. Uniformity statements are always
+ quantified in that order, and "o(1)" always means uniform over
+ the centers and the finitely many scales under discussion.
+
+ - The paper does not reprove the analytic background (Bochner,
+ heat kernel, Poisson regularity, volume rigidity). It cites
+ these and uses them. The list of cited inputs is collected in
+ the reading guide at the end.
+
+
+============================================================
+PART II. THE ANALYTIC TOOLBOX
+============================================================
+
+The proof uses no coordinates on X. It uses five analytic objects
+that exist on every RCD space. Each is covered by a cited theorem,
+used as a black box. This part lists them, defines every symbol,
+and states without proof the inequalities the rest of the tour
+relies on.
+
+
+11. SOBOLEV FUNCTIONS, THE ENERGY, AND HARMONIC FUNCTIONS
+
+On a metric measure space, the role of "functions with one square-
+integrable derivative" is played by the Sobolev space W^{1,2}. The
+Cheeger energy of a function f is
+
+ Ch(f) = (1/2) * inf limit of integral |grad f_k|^2 d m
+
+taken over the standard approximation procedure; in the RCD case
+it is a quadratic form, so it defines a bilinear energy and a
+gradient. Write Gamma(f,g) = <grad f, grad g> for the carre du
+champ (dot product of gradients) and Gamma(f) = Gamma(f,f).
+
+The Laplace operator Delta is the generator of this energy: for
+nice f and every compactly supported Lipschitz test function phi,
+
+ integral <grad f, grad phi> d m = - integral phi * Delta f d m.
+
+A function w is harmonic on an open set if Delta w = 0 there. It
+is a "constant-Laplacian" solution if Delta w = a * m for some
+real constant a. Such functions exist and are as regular as the
+space allows; in particular locally bounded constant-Laplacian
+solutions are locally Lipschitz.
+
+Harmonic functions are the natural candidates for coordinates on
+a singular space, exactly as on a smooth manifold where a harmonic
+coordinate chart is a chart whose coordinate functions are
+harmonic. They are chosen because the Bochner inequality gives
+strong control over the squared gradient Gamma(w) of a harmonic
+function -- the "energy" of the coordinate.
+
+The fundamental approximation statement (Ambrosio-Honda, local
+spectral convergence, 2018) is: on a ball that is close enough to
+a Euclidean ball and whose rescaled curvature is small enough,
+harmonic functions that match prescribed boundary data are close,
+in energy and locally uniformly, to the corresponding Euclidean
+harmonic functions. In particular harmonic functions can be made
+arbitrarily close to the affine coordinate functions x_1,...,x_n
+of R^n on a small ball.
+
+
+12. THE GRAM MATRIX, THE BOCHNER MEASURE, AND PINCHING
+
+Fix a column u = (u_1,...,u_n) of harmonic functions. Its Gram
+matrix is
+
+ G = ( Gamma(u_j,u_k) )_{j,k=1..n} ,
+
+an n x n matrix of functions; G is positive semidefinite and it
+records the squared lengths and pairwise angles of the coordinate
+gradients. Its average over a ball is our scale-by-scale yardstick:
+
+ R_i(y) = average over B_{r_i}(y) of G d m , r_i = 2^{-i}.
+
+If G were Identically the identity matrix we would already have
+Euclidean coordinates. The whole difficulty is that G is only
+close to the identity on small scales, and the closeness can
+degrade as the scale shrinks; R_i(y) need not stay bounded above
+or below.
+
+The tool that measures the size of G is the dimensional Bochner
+inequality, valid on every RCD(K,n) space for test functions. For
+a constant-Laplacian solution w with Delta w = a, and with
+kappa = max{-K, 0}, define the matrix measure
+
+ M(w) = (1/2) Delta Gamma(w) + kappa Gamma(w) m .
+
+Its content: the Bochner inequality says M(w) is a nonnegative
+measure, locally finite, and dominates the squared Hessian:
+
+ M(w) >= |Hess w|^2 m .
+
+Here Hess w is the Hessian of w, defined through the second-order
+calculus of Gigli; its trace is Delta w, because the space is
+noncollapsed and has dimension exactly n almost everywhere. For a
+constant b (a "shift" toward the metric tensor g) one has the
+shifted inequality
+
+ M(w) - (2 b Delta w - n b^2) m >= |Hess w - b g|^2 m .
+
+The left side is a nonnegative quadratic form in the pair (w,b).
+Polarizing gives a nonnegative bilinear form M(v,w), and matrix
+Cauchy-Schwarz for its density gives, for any Borel set E,
+
+ |M(v,w)|(E) <= sqrt( M[v](E) * M[w](E) ) .
+
+These measure-level statements are the only way the proof uses
+curvature. The point of the shift b is that it lets us center the
+Hessian on a multiple of the metric, which is what "close to
+Euclidean" means at second order.
+
+
+13. THE HEAT SEMIGROUP AND THE TWO HEAT BOUNDS
+
+Parallel to harmonic functions sits the heat semigroup
+
+ (P_t f)(x) = integral p_t(x,z) f(z) d m(z) ,
+
+which solves the heat equation d/dt (P_t f) = Delta (P_t f) with
+P_0 f = f. On an RCD space the heat kernel p_t(x,z) is a
+continuous symmetric function. On a fixed bounded region, for
+0 < t <= 1, the following hold (Jiang-Li-Zhang 2016, Theorem 1.2
+and Corollary 1.2):
+
+ Gaussian upper bound:
+ p_t(x,z) <= C t^{-n/2} exp( - d(x,z)^2 / (C t) ),
+ first spatial gradient bound:
+ |grad_x p_t(x,z)| <= C t^{-(n+1)/2} exp( - d(x,z)^2 / (C t) ),
+ on-diagonal lower bound (when d(x,z) <= A sqrt(t)):
+ p_t(x,z) >= c_A t^{-n/2} .
+
+The constants are uniform on the rescaled balls of the construction
+and depend only on the dimension and the fixed local data.
+
+Why heat instead of plain harmonic averages? Because a heat average
+of the Gram field enjoys a monotonicity in the time variable t, and
+t = r^2 is the scale. Section 2 turns this into matrices H_i(x)
+that are monotone as the scale decreases (H_i <= H_{i+1}), which
+plain ball averages R_i do not have. Monotone yardsticks are what
+make an infinite telescoping sum possible without commuting the
+matrices at different scales.
+
+
+14. RESCALING AND THE SMALL PARAMETERS
+
+The whole proof happens along a sequence of rescalings of a fixed
+n-regular point p. We rescale by a sequence s_1 > s_2 > ... -> 0:
+
+ (X_nu, d_nu, m_nu, p_nu) with d_nu = d / s_nu, m_nu = m / s_nu^n .
+
+Along this sequence, volumes of fixed-radius balls converge to
+their Euclidean values and the lower curvature bound, rescaled,
+tends to 0: kappa_nu -> 0. The spaces become flatter and flatter.
+Two small sequences drive the estimates:
+
+ - kappa_nu, the rescaled curvature lower bound (tends to 0);
+ - o(1), a generic name for any quantity tending to zero, always
+ meant uniformly over the centers and scales under discussion.
+
+Choosing the sequence "late enough" is the recurring order of
+quantifiers: fix all enlargement constants first, then take the
+rescaling late enough that every finiteness threshold holds. This
+is not bookkeeping; it is the only way to keep the constant L_n
+independent of the specific space.
+
+
+15. WHY THE HARD PART IS "IN VALUE SPACE"
+
+Suppose for a moment that the harmonic coordinates u were already
+a bi-Lipschitz chart with constant 1. Then done. The problem is
+the correction: at each scale the coordinates are off from
+Euclidean by something measured by D_{i,y}, and the corrections at
+different scales do not add up trivially.
+
+The paper's discipline is to never correct the coordinates by
+changing them on X. Instead it uses the harmonic map u to push
+everything into the open set
+
+ Omega = u(B_1(p)) inside R^n,
+
+called the space of coordinate values. On Omega the Euclidean
+analysis tools -- ordinary derivatives, ordinary linear algebra,
+invariance of domain -- are all available. The chart is built by
+an explicit smooth iteration on Omega. The geometric content of
+the space is imported only through the local matrices D_{i,y} and
+the heat yardsticks H_i, which are the inputs to the iteration.
+
+That separation is what allows the proof to survive the
+degeneration of R_i: the iteration never differentiates the scale
+metrics H_i and never inverts a pointwise differential of the
+singular map u on X. It only differentiates smooth maps defined
+on the flat model Omega.
+
+
+============================================================
+PART III. PAPER SECTION 1: UNIFORM HARMONIC COORDINATES
+============================================================
+
+Section 1 builds the open coordinate domain, then proves that the
+harmonic coordinates approximate Euclidean coordinates on finer and
+finer scales, and finally that the harmonic map u is a
+homeomorphism onto an open subset of R^n. It is careful to let the
+size of the coordinates degenerate as the scale shrinks; it only
+insists that on each fixed scale the approximation is uniform in
+all centers.
+
+
+16. WHAT SECTION 1 DELIVERS
+
+By the end of the section we have, on a large ball around p in a
+late enough (very flat) rescaling:
+
+ - a column u = (u_1,...,u_n) of harmonic functions that behaves
+ like orthonormal Euclidean coordinates near every point, up to
+ errors tending to zero with the rescaling, uniformly in the
+ center;
+
+ - a positive definite ball-Gram matrix R_i(y) at every dyadic
+ scale r_i = 2^{-i} and center y, whose normalized harmonic
+ map U = R_i(y)^{-1/2}(u-u(y))/r_i is an arbitrarily good model
+ of R^n on a fixed enlargement of B_{r_i}(y);
+
+ - a homeomorphism statement: u is injective, its image is open,
+ and it separates points at a scale proportional to their
+ distance.
+
+The unavoidable gap left open at the end is that R_i(y) may
+degenerate as i grows. Sections 2-4 exist to deal with that gap.
+Section 1 says nothing about a uniform lower bound on R_i; it
+merely records exactly how R_i can vary from scale to scale.
+
+
+17. LEMMA 1.1: THE LOCAL ANALYTIC FACTS
+
+The first lemma is a clearance lemma: it lists every analytic
+fact used later and justifies them from cited input.
+
+ Lemma 1.1 (local analytic facts). Let (Y,d,m) be a noncollapsed
+ RCD(-kappa,n) space and Omega an open subset. A locally bounded
+ solution w of Delta w = a in Omega with a constant is locally
+ Lipschitz and is a local test function; it has a local L^2
+ Hessian; for two such functions Gamma(w_1,w_2) is locally in
+ W^{1,2}, and the Hessian product rule holds locally. The measure
+ M(w) = 1/2 Delta Gamma(w) + kappa Gamma(w) m is locally finite
+ and nonnegative, and for every constant b,
+
+ M(w) - (2 b Delta w - n b^2) m >= |Hess w - b g|^2 m .
+
+ The left side is a nonnegative quadratic form in the pair (w,b)
+ with both entries scaled together.
+
+ Moreover, on fixed interior balls, bounded Lipschitz cutoffs with
+ bounded Laplacian exist, and there are a number d_* > 1 and a
+ constant C such that for every h in W^{1,2}_0(B),
+
+ ||h||_{L^2} + ||h||_{L^{2 d_*}} <= C ||grad h||_{L^2} .
+
+ The constants are uniform on the rescaled balls in use.
+
+In plain terms: constant-Laplacian solutions are as smooth as the
+space allows, the Bochner measure M is a genuine nonnegative
+measure even when Hess w is only an L^2 object, the shifted
+inequality always centers correctly, and the Dirichlet boundary
+problem has coercivity (a Sobolev inequality with exponent
+2 d_* > 2). The proof assembles facts from Bacher-Sturm,
+Kell, Jiang, De Philippis-Gigli, Han, Gigli, Connell et al.,
+Mondino-Naber, and Hajlasz-Koskela; the paper makes a point of
+handling dimension n = 2 as well by choosing d_* larger than both
+n and 2.
+
+Two subtleties worth noticing:
+
+ - The Hessian inequality for M is stated as an inequality of
+ measure-valued quadratic forms. There is no pointwise Hessian
+ to compare against; the "square" |Hess w - b g|^2 lives as a
+ density of a nonnegative measure through polarization. Section
+ 2 will use this heavily.
+ - The cutoffs with bounded Laplacian exist only on fixed
+ interior scales. They cannot be made uniform over all radii.
+
+
+18. LEMMA 1.2: AN EXPANDING ALMOST EUCLIDEAN REGIME
+
+The second lemma sets up the actual sequence of spaces used
+throughout. It is a compactness-and-rigidity statement.
+
+ Lemma 1.2 (expanding almost Euclidean regime). Fix an
+ n-regular point p of the space in the main theorem. There are
+ rescalings
+
+ (X_nu, d_nu, m_nu, p_nu) = (X, d/s_nu, m/s_nu^n, p),
+
+ numbers kappa_nu decreasing to 0, and radii T_nu increasing to
+ infinity with kappa_nu T_nu^2 -> 0, such that the following hold.
+ Each X_nu is RCD(-kappa_nu, n). For a sequence delta_nu -> 0,
+ every ball with center in B_{64 T_nu}(p_nu) and radius at most
+ 64 T_nu satisfies
+
+ | m_nu(B_r(y)) / (omega_n r^n) - 1 | <= delta_nu ,
+ d_GH( closure B_r(y) , closure B_r(0^n) ) <= delta_nu r .
+
+ There are harmonic columns u_nu on B_{32 T_nu}(p_nu) converging
+ on every fixed ball locally uniformly and strongly in W^{1,2} to
+ Euclidean coordinates, and
+
+ |grad u_nu| <= C_n on B_{16 T_nu}(p_nu) .
+
+Two things are happening, both worth stating plainly.
+
+ - Because p is n-regular, the rescalings converge to flat R^n.
+ Noncollapsed volume convergence then identifies the limit
+ measure with H^n. The convergence of volumes is uniform over
+ the center within the working ball; the proof is by
+ contradiction, using that pointwise convergence plus
+ compactness of centers yields a uniform rate.
+ - The radii T_nu are chosen to grow to infinity slowly (slowly
+ compared to how fast the rescalings flatten). So on the
+ expanding balls of radius T_nu, the space is very nearly
+ Euclidean with rapidly shrinking error.
+
+The uniform almost-Euclidean statement is extracted from volume
+rigidity (De Philippis-Gigli): tiny volume excess forces tiny
+Gromov-Hausdorff distance. Bishop-Gromov then traps the volume
+ratio between its model-maximal value and 1, giving the display.
+
+Harmonic coordinate columns are delivered by the local spectral
+convergence of Ambrosio-Honda, applied on fixed enlargements and
+then combined by a diagonal subsequence. The gradient bound
+|grad u_nu| <= C_n is dimensional: the coordinate functions have
+size O(r) on a ball of radius r, and the interior gradient estimate
+carries the matching factor r^{-1}.
+
+From here on the index nu is suppressed. A statement like "for all
+large enough rescalings" is part of the meaning of every o(1).
+
+
+19. DYADIC RADII AND THE BALL-GRAM MATRIX
+
+With the rescalings fixed, set
+
+ r_i = 2^{-i}, i = 0, 1, 2, ... ,
+ R_i(y) = average over B_{r_i}(y) of G d m ,
+
+where G = (Gamma(u_j,u_k)) is the Gram matrix of the harmonic
+column. R_i(y) is the ball Gram matrix; it is the yardstick at
+scale r_i and center y. Its inverse R_i(y)^{-1} shows up in the
+"normalized" coordinate map
+
+ U_{i,y} = R_i(y)^{-1/2} (u - u(y)) / r_i .
+
+Roughly: subtract the coordinate value at the center, rescale the
+ball of radius r_i to unit size, and correct the target by the
+inverse square root of the average Gram matrix. U_{i,y} is the
+coordinate map that "would be Euclidean if the space were flat and
+R_i were perfectly Euclidean". Lemma 1.3 makes this precise.
+
+
+20. LEMMA 1.3: UNIFORM HARMONIC NORMALIZATION
+
+This is the technical heart of Section 1. It says the normalized
+harmonics are almost Euclidean on every fixed enlargement of every
+ball, uniformly in the center, at every dyadic scale.
+
+ Lemma 1.3 (uniform harmonic normalization). For every pair of
+ fixed constants A,C >= 1, the matrices R_i(y) are positive
+ definite for y in B_A(p) and all i >= 0. In the rescaled ball
+ B_{C r_i}(y) the map U_{i,y} is, after an orthogonal change of
+ target coordinates, an o(1) approximation to Euclidean
+ coordinates, uniformly and strongly in interior W^{1,2}. Its
+ gradients are uniformly bounded on each fixed interior
+ enlargement, and its Gram matrix converges strongly in every
+ finite L^p on that enlargement to the identity Id.
+
+ For every fixed C >= 1, if d(y,z) <= C r_i and
+ C^{-1} r_i <= r_j <= C r_i, then
+
+ (1 - o(1)) R_i(y) <= R_j(z) <= (1 + o(1)) R_i(y) .
+
+ For every a > 0, after choosing a sufficiently late rescaling
+ and a constant C_a depending only on a,
+
+ C_a^{-1} 2^{-2a|i-j|} R_j(y) <= R_i(y) <= C_a 2^{2a|i-j|} R_j(y) .
+
+ In particular, since R_0(y) = Id + o(1) on fixed balls,
+
+ c_a r_i^{2a} Id <= R_i(y) <= C Id .
+
+The first sentence gives positivity. The second is the core:
+U_{i,y}, viewed on a fixed enlargement of its own ball of radius
+r_i after rescaling to unit size, looks like the identity
+coordinate map. The word "uniformly" is essential: the error tends
+to zero at once over every center y and every scale i in a fixed
+ball. The third sentence (nearby-R) says that Gram averages on
+nearby balls of comparable radius agree up to a factor 1 + o(1).
+The fourth (slow-R) says that between two dyadic scales i and j,
+R can grow or shrink by at most a controlled exponential factor
+2^{2a|i-j|}, with arbitrary prescribed a > 0 once the rescaling is
+late enough. The final line tracks R_0(y) = Id + o(1): the scale-0
+Gram average is almost Euclidean, and back-propagating using slow-R
+gives a lower bound r_i^{2a} Id on R_i.
+
+The proof is a compactness induction. Saying it is "almost
+Euclidean at scale r_i" means: after normalizing by an invertible
+matrix T, the rescaled map T(u - u(y))/r_i lies within a small
+sup distance of the standard Euclidean coordinates on the ball of
+radius 2. The proof shows this property improves strictly under
+passing from scale r to a fixed fraction b = 2^{-m} of r:
+
+ - On an exactly Euclidean ball, applying interior harmonic
+ derivative estimates to V - x, one gets that the rescaled
+ normalized map is strictly closer to Euclidean after
+ contracting by a dyadic factor b chosen small enough -- its
+ derivative at the center is within O(tau) of the identity, and
+ its Taylor remainder is controlled by the Hessian estimate.
+ - On an almost-Euclidean ball in the sequence, the same
+ improvement propagates by contradiction. If it failed, rescale
+ the failing ball to unit size; compactness of rescaled
+ harmonic maps on almost-Euclidean spaces (Ambrosio-Honda)
+ yields convergence of the normalized harmonic to a Euclidean
+ harmonic column; that limit has the strict improvement
+ property; contradiction.
+ - The induction runs on the finitely many dyadic residue
+ classes modulo m, so every dyadic scale is reached.
+
+The "mean Gram matrix of a normalized column is Id + O(tau)" step
+is what lets the proof replace an inductively chosen correction
+matrix T with the canonical R_i(y)^{-1/2}; the two differ by an
+orthogonal matrix times an O(tau) error, which is absorbed.
+
+Fixed interior enlargements are handled by fixing the largest
+enlargement first and running the induction on an "ancestor" ball
+large enough to contain it; the number of contraction steps between
+the ancestor and any descendant is bounded by a fixed integer, so
+all constants survive.
+
+The nearby-R estimate is the same argument run on a single
+enlargement containing both balls. The slow-R estimate is
+nearby-R iterated along adjacent scales, doubling the tolerance
+once per step and choosing the per-step tolerance so that the
+factor after |i-j| steps is at most 2^{2a|i-j|}.
+
+
+21. LEMMA 1.4: SMALL NORMALIZED BOCHNER MASS
+
+ Lemma 1.4 (small normalized Bochner mass). Uniformly for y in
+ B_A(p) and all i >= 0, for fixed A,C >= 1,
+
+ r_i^{2-n} M( B_{C r_i}(y) ) <= o(1) R_i(y) .
+
+The Bochner measure of a ball of radius comparable to r_i, scaled
+by r_i^{2-n} and conjugated by R_i(y)^{-1}, is small compared to
+the identity.
+
+The proof is a one-step rescaling exercise with the local
+calculus, and it is short enough to keep in view.
+
+ - In units r_i, apply Lemma 1.1 to a scalar combination of
+ U = U_{i,y}. The rescaled Bochner matrix is
+
+ M_{r_i}(U) = r_i^{2-n} R_i(y)^{-1/2} M(u) R_i(y)^{-1/2},
+
+ with curvature term kappa r_i^2.
+ - Pick a nonnegative cutoff phi, equal to 1 on B_C and supported
+ on a larger fixed ball, with uniformly bounded gradient and
+ Laplacian.
+ - Integrate the definition of M against phi. The curvature term
+ is kappa r_i^2 * (integral of phi Gamma(b.U)), and the
+ harmonic-energy term is (1/2) integral (Gamma(b.U) - |b|^2)
+ Laplacian(phi). Both terms tend to zero uniformly by Lemma
+ 1.3 (normalized Gram -> Id; rescaled curvature -> 0).
+ - Positivity of the Bochner measure bounds its mass on the
+ inner ball by what the cutoff saw. Polarizing over the fixed
+ finite dimension gives the matrix inequality.
+
+There is no subtlety about singular parts: the proof only uses M
+on Lipschitz cutoffs, and M is a nonnegative measure there.
+
+
+22. LEMMA 1.5: OPEN HARMONIC COORDINATES
+
+ Lemma 1.5 (open harmonic coordinates). For every fixed A >= 1,
+ the map u is a homeomorphism from B_A(p) onto an open subset of
+ R^n once the rescaling is late enough. There is a fixed c > 0
+ such that, whenever the balls involved stay in a fixed interior
+ region,
+
+ { P : |P - u(x)|_{R_i(x)^{-1}} < c r_i } is contained in
+ u( B_{r_i/2}(x) ) .
+
+ For every fixed D >= 1, if D^{-1} r_i <= d(x,z) <= D r_i, then
+
+ | |u(z) - u(x)|_{R_i(x)^{-1}} - d(x,z) | <= o_D(1) r_i .
+
+The first display is the crucial "ellipsoid containment": the image
+of a ball of radius r_i/2 contains a small ellipsoid, in the
+R_i(x)^{-1}-norm on coordinate values, of radius c r_i. The second
+display is the "separation-scale" estimate: for two points at a
+comparable dyadic distance, the coordinate difference in the
+R_i(x)^{-1} norm equals the true distance up to a relative error
+tending to zero.
+
+The proofs are pleasingly geometric.
+
+ - Separation-scale is just Lemma 1.3: at a dyadic scale
+ comparable to d(x,z), U_{i,x} is close to Euclidean
+ coordinates, and comparing z to x through it gives the
+ display.
+ - Injectivity follows immediately: two distinct points in
+ B_A(p) at dyadic scale comparable to their distance have
+ coordinate values distinct, because the error term is a
+ fraction of the distance.
+ - Ellipsoid containment is proved by an iterative preimage
+ argument on the space X. Start at x_0 = x with a target P in
+ the ellipsoid, and a dyadic radius rho_0 = r_i.
+ Approximate surjectivity of the normalized Euclidean
+ coordinate map provides x_{j+1} within (c + eta) rho_j of
+ x_j, with the normalized distance of P from u(x_{j+1}) at most
+ eta rho_j; then rho_{j+1} = rho_j / 4, and the induction
+ continues. The total walked distance is at most (c + eta) *
+ (sum of rho_j) <= rho_0 / 2, so all centers stay in working
+ range. Completeness gives a limit point x_* with u(x_*) = P,
+ and the upper bound R_i <= C Id converts the
+ R_i(x)^{-1}-norm error into an ordinary coordinate residual
+ going to zero.
+ - Open image: apply ellipsoid containment locally at each point
+ to get an open image. Homeomorphism: a continuous injection
+ onto an open subset of a proper space is a local homeomorphism,
+ and local injectivity plus continuity of the inverse (from the
+ separation estimate) makes it global.
+
+Two remarks round out Section 1.
+
+ - No openness of the set of exact regular points is used. The
+ proof runs entirely on the level of a fixed point p and the
+ local flatness around it; it produces an open ball around p
+ after the fact.
+ - The scale of the working region is expanding: it grows with
+ T_nu. That is why Section 2-4 can afford fixed enlargements
+ and still demand uniform constants.
+
+
+============================================================
+PART IV. PAPER SECTION 2: HEAT AVERAGES AND CUMULATIVE
+ MATRIX CONTROL
+============================================================
+
+Section 1 left open the possibility that the ball Gram matrices
+R_i degenerate. Section 2 replaces them, for the purpose of the
+proof, with heat averages that are monotone in the scale, and then
+proves a cumulative comparison: a bounded-overlap sum of the local
+Bochner matrices D_{i,y} is comparable to the heat yardstick Q_i,
+and -- crucially -- the heat yardstick is bounded by a weighted
+sum of those same matrices. That reverse direction is what Section
+4 needs to conclude that the iteration is stable.
+
+
+23. WHAT SECTION 2 DELIVERS
+
+Three matrices appear and must not be confused:
+
+ - R_i(x): the plain ball average of the Gram matrix G at scale
+ r_i. No monotonicity in i.
+
+ - Q_i(x): the heat average of G at time r_i^2, plus a small
+ correction. Positive definite. Monotone: Q_i >= Q_{i+1}.
+
+ - H_i(x) = Q_i(x)^{-1}: the inverse. Monotone in the other
+ direction: H_i <= H_{i+1}.
+
+At the end of Section 2 we have a constant C_* (depending only on
+n and on the fixed enlargement constants) such that for every
+testing center x and every integer k >= 0:
+
+ Id + sum_{i<k} Z_i(x) <= C_* H_k(x),
+ H_k(x) <= C_* ( Id + sum_{i<k} S_i(x) ),
+
+where Z_i is the unweighted sum of the local D-forms at scale i
+(over the active centers of the patch) and S_i is a weighted sum
+with weights in [0,1] at least one of which equals one and has a
+covering ball. The second inequality is the deep one; without it
+nothing stops H_k from running off to infinity faster than the
+local mass accounts for.
+
+Five smaller lemmas feed into these two inequalities. The order
+below is the paper's order.
+
+
+24. THE HEAT YARDSTICK, BUILT CAREFULLY
+
+Fix a cutoff chi with the following properties:
+
+ chi = 1 on B_T(p) ,
+ supp chi inside B_{2T}(p) ,
+ |grad chi| <= C / T ,
+ |Delta chi| <= C / T^2 .
+
+Here T = T_nu is the expanding radius from Lemma 1.2. Extend chi G
+by zero outside its support. The distributional Laplacian of chi G
+is then the compactly supported matrix measure
+
+ Delta(chi G) = 2 chi M - 2 kappa chi G m + E_cut m ,
+ E_cut = (Delta chi) G + 2 Gamma(chi, G) .
+
+The first two terms on the right are the "true Bochner" content of
+the chip cutoff; E_cut is a cutoff error supported in the annulus
+between radii T and 2T. Because the cutoff lives far from any
+testing center (distance at least T/2), heat averaging will make
+E_cut negligible. That is the point of the expanding domain T_nu.
+
+Given a parameter lambda > 0 that will tend to 0, define
+
+ Q_i(x) = P_{r_i^2}(chi G)(x) + lambda r_i^2 Id ,
+ H_i(x) = Q_i(x)^{-1} .
+
+Q_i is positive definite because the heat average P_{r_i^2}(chi G)
+is a positive semidefinite matrix and the lambda r_i^2 Id term is
+strictly positive. H_i is also positive definite.
+
+The correction term lambda r_i^2 Id is a technical necessity: it
+absorbs the curvature contribution and the cutoff error uniformly,
+without which Q_i need not be positive definite in the strict
+sense that later matrix inequalities need.
+
+
+25. LEMMA 2.1: KERNEL DIFFERENCES AND MEASURE SOURCES
+
+The proof needs two comparisons of heat kernels: a short-time
+bound on the average of the kernel over a ball, and a long-time
+bound on how much that average differs from a point evaluation.
+
+ Lemma 2.1 (kernel differences and measure sources). Fix
+ C_0 >= 1, points x,y in a fixed testing region, and
+ d(x,y) <= C_0 r with 0 < r <= 1. Put
+
+ K_t(z) = average over B_r(y) of p_t(a,z) d m(a) ,
+ D = d(x,z) .
+
+ After enlarging fixed constants:
+
+ K_t(z) <= C r^{-n} exp( - c D^2 / r^2 ) for 0 < t <= r^2 ,
+ | K_t(z) - p_t(x,z) | <=
+ C r t^{-(n+1)/2} exp( - c D^2 / t ) for r^2 <= t <= 1 .
+
+ If f in W^{1,2} cap L^inf has compact support and its Laplacian
+ is a finite signed measure mu of compact support, then
+
+ P_t f - P_s f = integral_s^t P_tau mu d tau , 0 < s < t .
+
+The lemma is two-sided and the proof is worth summarizing because
+the whole section's difficulty is concentrated here.
+
+Short-time bound. If z is close to the averaging ball, symmetry and
+total heat mass bound the average of the kernel by one; dividing by
+volume gives C r^{-n}. If z is far, the Gaussian upper bound alone
+gives C t^{-n/2} exp(-cD^2/t); for t <= r^2 and D/r bounded below,
+its small-time power is absorbed in the exponential.
+
+Long-time bound. For a in B_r(y), d(a,x) <= (C_0+1) r. Integrate
+the first spatial gradient estimate along a minimizing curve from
+x to a; the inequality d(b,z)^2 >= D^2/2 - d(b,x)^2 absorbs the
+variation of the Gaussian along the curve. This gives the bound
+pointwise for almost every fixed source pole z. To extend to every
+pole, use symmetry and continuity of the heat kernel: the full-
+measure set of "good" poles is dense, and passing to the limit
+with relaxed Gaussian constants gives the inequality at every z.
+No gradient representative is ever paired with a singular measure:
+all the integration-by-parts is done against the smooth heat
+kernel at positive time.
+
+Semigroup identity against a measure. Write the distributional
+Laplacian of f as mu; test against positive-time heat regularizations
+of compactly supported test functions, integrate by parts using
+symmetry, and use the local W^{1,2} regularity of p_t at positive
+time. The result is the displayed identity, valid in distributions
+and then pointwise by continuity of both sides.
+
+
+26. LEMMA 2.2: MONOTONE HEAT GRAM MATRICES
+
+ Lemma 2.2 (monotone heat Gram matrices). There is a real
+ lambda = lambda_nu > 0, tending to 0 along the rescalings, such
+ that at all fixed testing centers and for 0 < t <= 1,
+
+ P_t(chi M) <= d/dt( P_t(chi G) + lambda t Id )
+ <= 3 P_t(chi M) + C Id .
+
+ With Q_i and H_i defined as above,
+
+ (1 - o(1)) R_i(x) <= Q_i(x) <= (1 + o(1)) R_i(x) ,
+ Q_0(x) = Id + o(1) ,
+ H_i(x) <= H_{i+1}(x) <= (1 + o(1)) H_i(x) .
+
+ The nearby-center and slow-growth estimates for R_i (Lemma 1.3)
+ hold for Q_i as well, and for every prescribed a > 0, once the
+ rescaling is late enough,
+
+ Q_j(x) <= C_a 2^{2a(i-j)} Q_i(x) (j <= i) ,
+ || H_i(x) || <= C_a 2^{2a i} .
+
+The derivative estimate is the analytical content. Differentiating
+P_t(chi G) and using the semigroup identity gives
+
+ d/dt P_t(chi G) = 2 P_t(chi M) - 2 kappa P_t(chi G)
+ + P_t E_cut .
+
+Adding lambda Id and using the chip cutoff estimates:
+
+ - The cutoff error P_t E_cut is uniformly small: E_cut has total
+ variation at most C T^{n-2}, its support is a distance at least
+ T/2 from fixed testing centers (for T large along the
+ expansion), and the Gaussian estimate gives
+
+ sup_{0<t<=1} || P_t E_cut(x) ||
+ <= C T^{n-2} sup_{0<t<=1} t^{-n/2} exp(-c T^2 / t) = o(1) .
+
+ - The curvature contribution has norm at most 2 kappa ||G||_inf
+ on the fixed testing region.
+
+ - Choose lambda >= (that small bound) + 2 kappa ||G||_inf + a
+ positive number tending to zero. Then the sum of the last three
+ terms on the right is nonnegative and bounded above, giving
+ the two-sided estimate displayed.
+
+The comparison (1-o(1))R_i <= Q_i <= (1+o(1))R_i is a comparison of
+relative, not just absolute, sizes. Fix x and i, conjugate by
+R_i(x)^{-1/2}: on every fixed ball B_{L r_i}(x) the conjugated
+Gram field converges in normalized L^1 to Id (Lemma 1.3), so the
+heat-kernel-weighted error tends to zero. The tails -- the shell
+with radius 2^m r_i up to a distance and beyond -- are computed by
+slow growth of the conjugated average (bounding it by C 2^{2am})
+against the Gaussian decay of the kernel; both contributions are
+summable in m uniformly in i. The missing-heat-mass correction
+from (1 - chi) is bounded the same way using total heat mass one.
+Then lambda r_i^2 R_i(x)^{-1} is bounded by C_a lambda r_i^{2-2a}
+Id = o(1) Id, which is why a is chosen below 1.
+
+Monotonicity: from the derivative estimate, d/dt(P_t(chi G) +
+lambda t Id) >= 0, so Q_i is decreasing in i (larger i = smaller
+r_i^2 = earlier time). Therefore H_i = Q_i^{-1} is increasing in
+i. Relative comparison with R_i and nearby-R of Lemma 1.3 propagate
+the adjacent-scale upper bound.
+
+The slow-growth estimate for Q_i and H_i follows by iterating the
+adjacent-scale estimate, with the per-step tolerance chosen after
+the prescribed a. Note ||H_i|| grows like 2^{2 a i} -- this is
+allowed, and Section 4 never says otherwise. The reverse comparison
+in Lemma 2.6 is exactly what quantifies how bad this growth can
+possibly be.
+
+
+27. LEMMA 2.3: THE DYADIC HEAT COMPARISON
+
+This lemma produces the small series F_i that will appear
+everywhere afterwards and states two inequalities: a decrement
+estimate for Q_i - Q_{i+1} and a bilinear one relating R_i - Q_i
+to F_i. Both are indispensable for the linear algebra in Section
+2.6.
+
+ Lemma 2.3 (dyadic heat comparison). Fix A, C_0 >= 1 with
+ d(x,y) / r_i <= C_0, and fix a bounded region for x. There is a
+ fixed exponent gamma > 0 (taken to be 1/2), with the following
+ properties for all sufficiently late rescalings. Set
+
+ F_i(x) = 2^{-gamma i} Id + sum_{j=0}^{i}
+ 2^{-gamma(i-j)} N_j(x) ,
+ N_j(x) = r_j^{2-n} M( B_{r_j}(x) ) .
+
+ For the prescribed enlargement A,
+
+ c_A r_i^{2-n} M( B_{A r_i}(x) ) <= Q_i(x) - Q_{i+1}(x)
+ <= C F_i(x) .
+
+ For d(x,y) <= C_0 r_i and v,w in R^n,
+
+ | v^T ( R_i(y) - Q_i(x) ) w |
+ <= C |v|_{Q_i(x)} sqrt( F_i(x)[w] ) .
+
+The series F_i(x) is a damped sum of the local Bochner masses
+N_0,...,N_i at all coarser scales, with the dyadic ratio 2^{-gamma}
+as damping, plus an explicit background term 2^{-gamma i} Id to
+guarantee positivity at every scale. Section 3 will show that the
+individual N_j are summable in a strong sense; Section 4 only
+needs to know that such an F_i exists.
+
+The content of the lemma, phrase by phrase:
+
+ - The left side of the decrement estimate says that the decrement
+ Q_i - Q_{i+1} is at least as large as the local Bochner mass
+ A r_i inside the ball of radius A r_i. This is the lower bound
+ that lets the sum of decrements dominate the sum of local
+ masses later.
+
+ - The right side of the decrement estimate says the decrement
+ cannot be much bigger than F_i. Small F_i means the yardstick
+ barely changes between consecutive scales.
+
+ - The bilinear estimate says that the ball average R_i(y) at a
+ nearby center differs from the heat yardstick Q_i(x) only by
+ something of size |v|_{Q_i} sqrt(F_i). This is the "relative
+ comparison" -- magnitudes of operators are compared, not just
+ norms -- and it is what makes the linear-algebra estimates of
+ Section 2.4-2.6 work.
+
+Proof sketch. Write r = r_i, suppress x, and let A f denote the
+average of f over B_r(y). The exact semigroup identity for the
+chip cutoff reads
+
+ R_i(y) - Q_i(x) = A P_1 f - P_1 f(x) - lambda r^2 Id
+ - integral_0^{r^2} (A P_t S) d t
+ - integral_{r^2}^{1} ( A P_t S - P_t S(x) ) d t ,
+
+with f = chi G and S = Delta f. The first two lines are "endpoint
+heat regularizations"; the third is "short-time", where the
+kernel-based approximations do all the work; the fourth is
+"long-time", where the difference between average and point
+evaluation is controlled by the first spatial gradient of the
+kernel.
+
+The local source chi M is decomposed into contributions by time
+block I_q = [r_{q+1}^2, r_q^2]. On each block, integrate the two
+kernel comparisons of Lemma 2.1 across the ball and against the
+measure M, and observe that the Gaussian at scale r_q is constant
+on shells of radius between 2^{h-1} r_q and 2^h r_q, with a
+coefficient k_h = 2^{(n-2)h} exp(-c 4^{h-1}) that is summable
+against 2^h. The sum over blocks and shells reverses into a single
+sum that gives both decrement estimates.
+
+The exterior of B_1(x) is controlled by the raw-mass-growth
+estimate
+
+ M(B_s(x))[v] <= C (s^{n-2} + kappa s^n) |v|^2 ,
+
+applied shell by shell, and by the exponentially decaying
+integrated kernel bounds outside B_1(x). The curvature source is
+bounded by total heat mass one at short time and by
+integral_r^2^1 r t^{-1/2} d t at long time; both together are
+bounded by C r. The cutoff source E_cut is handled by the Gaussian
+decay far from x.
+
+With all those contributions collected into the raw bilinear
+estimate
+
+ |v^T(R_i(y)-Q_i(x))w| <= C r |v||w|
+ + C sum_{j=0}^i 2^{-(i-j)} sqrt( N_j[v] N_j[w] ) ,
+
+the normalized bilinear comparison follows from slow growth of
+N_j relative to Q_i and weighted Cauchy-Schwarz. The rest of
+Section 2 only ever uses the normalized form.
+
+
+28. A SMALL INDEXING REMARK
+
+Sometimes the mass is taken on B_{A r_i}(y) for a fixed A > 1,
+not the unit ball. The paper records how to write such an enlarged
+mass back in terms of the notation, losing only a dimension-
+dependent factor:
+
+ r_j^{2-n} M(B_{A r_j}(x)) <= 2^{h(n-2)} N_{j-h}(x) ,
+ h = ceil( log_2 A ) ,
+
+with the h < small-index cases absorbed in the initial-scale bound
+C_A Id. A center shift bounded by C_0 r_j only increases A by C_0.
+So fixed enlargements do not change the summability structure of
+F_i. A separate remark adds: the matrices D_{i,y} below carry two
+Gram inverses; their comparison requires the argument of the next
+three lemmas, not just this bookkeeping.