diff a26c6ee..99b3ba1 diffeomorphic_approx_3d_lp_le2_v2.txt
diff --git a/diffeomorphic_approx_3d_lp_le2_v2.txt b/diffeomorphic_approx_3d_lp_le2_v2.txt
index a1d7dfd..d129d59 100644
--- a/diffeomorphic_approx_3d_lp_le2_v2.txt
+++ b/diffeomorphic_approx_3d_lp_le2_v2.txt
@@ -3102,6 +3102,97 @@ Substituting the preceding face estimates gives (11.11). Allocate
the intermediate errors so their converted sum is below the
prescribed cell error. QED
+11.6 Global choices and proof of Theorem B
+
+We now prove Theorem B. Let f : Omega -> Omega' be a homeomorphism
+in W^{1,p}(Omega; R^3) with 1 <= p <= 2, and eps > 0. We construct
+a locally bi-Lipschitz homeomorphism H : Omega -> Omega' with
+
+ int_Omega (|H - f|^p + |DH - Df|^p) < eps.
+
+Proof of Theorem B. If necessary compose with a reflection in the
+target to fix an orientation convention, and undo it at the end.
+
+Order of the fixed data. Let eps_0 > 0 be a tail tolerance, to be
+sent to zero. Choose the compact rank truncation and its bound M
+(Lemma 11.3). Choose l small enough that every O_M(l) contribution
+in volume, section, and cap regions is below a prescribed auxiliary
+tolerance. Recall gamma = l/K_0.
+
+With M, l fixed, choose the sharp derivative tolerances for line
+and plane tiles (Section 11.4). The speed bounds entering those
+tolerances are fixed in terms of M. When k = p = 2, choose further
+edge subdivision J large enough for the desired boundary closeness.
+Choose eta small enough for the sharp length and area coefficients,
+then tau small enough for all trace errors, including those depending
+on l, eta, J. Next choose delta small enough, refine the compact
+pieces, and choose nested excess neighborhoods so that after
+multiplication by every now-fixed inflation factor, the total excess
+integral is small. These bounds do not depend on the widths of the
+neighborhoods or on the number of compact pieces.
+
+Only now choose h small and make the macro success selections. Fix
+the bounded radial area weights and compact clearance margins
+before choosing the target budget mesh. Choose that mesh using
+Proposition 10.5, Lemma 10.6 (expected budgets), and the length
+and sharp-grid probability bounds. This determines H_0 with the
+required exact affine germs.
+
+The small radial parameters, angular attenuation parameters, and
+vertex contractions are now chosen from the actual finite
+configuration. In kernel blocks choose lambda after the section
+parametrizations and all output maps have been fixed. Finally
+complete the uncontrolled cells using Lemma 11.5 with summable
+positive error prescriptions. The local-tolerance lemma of Section
+7 permits the locally finite fine choices and these summable
+prescriptions.
+
+Derivative comparison. We separate the controlled cells into three
+types.
+
+Full rank. On a full-rank core, H = b_i except on the small
+bounded-derivative caps. On the compact jet set Db_i = Df; on the
+remaining portion the chart derivative is bounded and the excess
+integral is small. Hence the derivative difference is small.
+
+Rank one. On a rank-one core, (11.9) with k = 1 gives closeness to
+D_i with fiber factor h^{3-k} = h^2. The rank-model comparison
+(Lemma 11.3 setup) equates D_i with Df on the compact jet set, and
+the excess-neighborhood bound treats the remaining portion.
+Uncontrolled cells were bounded with summable capacity in Lemma
+11.5.
+
+Rank two. On a rank-two core, (11.9) with k = 2 gives closeness to
+D_i with fiber factor h^{3-k} = h. The section tests (11.10) and
+the rank-model comparison give the same conclusion, with an
+arbitrarily small residual from the excess-neighborhood bounds.
+
+Choosing the finitely many lambda sufficiently small after all other
+data are fixed, we combine the three cases with the outside-core
+integral and obtain
+
+ int_Omega |DH - Df|^p <= C_p eps_0 + o(1). (11.12)
+
+All uncontrolled-cell errors were chosen summably, so the same
+bounds also establish int_Omega |DH|^p < infinity, i.e. H in
+W^{1,p}(Omega; R^3), rather than only local Sobolev regularity.
+
+Value convergence. The source self-parametrizations move points only
+inside cells or blocks whose maximal diameter tends to zero. The
+target mesh can be chosen with arbitrarily small value error; the
+target motions (radial, angular, cap) are supported in boxes of
+arbitrarily small diameter in physical coordinates. On any interior
+compact, continuity of f then makes the values converge pointwise.
+Because the domains are bounded and all images lie in a fixed
+bounded target, dominated convergence improves this to L^p
+convergence on Omega.
+
+Combining. The meshes and value prescriptions can be chosen so
+that the value error and the derivative error (11.12) sum to less
+than the prescribed eps. This produces a locally bi-Lipschitz
+homeomorphism H : Omega -> Omega' with H in W^{1,p} and the
+stated W^{1,p} error. Theorem B is proved. QED
+
12. PROOF OF THE MAIN THEOREM
-----------------------------
@@ -3241,263 +3332,3 @@ Vaisala, J.: Lectures on n-Dimensional Quasiconformal Mappings.
END OF PROOF AND REFERENCES.
-11.6 Global choices and proof of Theorem B
-
-We now prove Theorem B. Let f : Omega -> Omega' be a homeomorphism
-in W^{1,p}(Omega; R^3) with 1 <= p <= 2, and eps > 0. We construct
-a locally bi-Lipschitz homeomorphism H : Omega -> Omega' with
-
- int_Omega (|H - f|^p + |DH - Df|^p) < eps.
-
-Proof of Theorem B. If necessary compose with a reflection in the
-target to fix an orientation convention, and undo it at the end.
-
-Order of the fixed data. Let eps_0 > 0 be a tail tolerance, to be
-sent to zero. Choose the compact rank truncation and its bound M
-(Lemma 11.3). Choose l small enough that every O_M(l) contribution
-in volume, section, and cap regions is below a prescribed auxiliary
-tolerance. Recall gamma = l/K_0.
-
-With M, l fixed, choose the sharp derivative tolerances for line
-and plane tiles (Section 11.4). The speed bounds entering those
-tolerances are fixed in terms of M. When k = p = 2, choose further
-edge subdivision J large enough for the desired boundary closeness.
-Choose eta small enough for the sharp length and area coefficients,
-then tau small enough for all trace errors, including those depending
-on l, eta, J. Next choose delta small enough, refine the compact
-pieces, and choose nested excess neighborhoods so that after
-multiplication by every now-fixed inflation factor, the total excess
-integral is small. These bounds do not depend on the widths of the
-neighborhoods or on the number of compact pieces.
-
-Only now choose h small and make the macro success selections. Fix
-the bounded radial area weights and compact clearance margins
-before choosing the target budget mesh. Choose that mesh using
-Proposition 10.5, Lemma 10.6 (expected budgets), and the length
-and sharp-grid probability bounds. This determines H_0 with the
-required exact affine germs.
-
-The small radial parameters, angular attenuation parameters, and
-vertex contractions are now chosen from the actual finite
-configuration. In kernel blocks choose lambda after the section
-parametrizations and all output maps have been fixed. Finally
-complete the uncontrolled cells using Lemma 11.5 with summable
-positive error prescriptions. The local-tolerance lemma of Section
-7 permits the locally finite fine choices and these summable
-prescriptions.
-
-Derivative comparison. We separate the controlled cells into three
-types.
-
-Full rank. On a full-rank core, H = b_i except on the small
-bounded-derivative caps. On the compact jet set Db_i = Df; on the
-remaining portion the chart derivative is bounded and the excess
-integral is small. Hence the derivative difference is small.
-
-Rank one. On a rank-one core, (11.9) with k = 1 gives closeness to
-D_i with fiber factor h^{3-k} = h^2. The rank-model comparison
-(Lemma 11.3 setup) equates D_i with Df on the compact jet set, and
-the excess-neighborhood bound treats the remaining portion.
-Uncontrolled cells were bounded with summable capacity in Lemma
-11.5.
-
-Rank two. On a rank-two core, (11.9) with k = 2 gives closeness to
-D_i with fiber factor h^{3-k} = h. The section tests (11.10) and
-the rank-model comparison give the same conclusion, with an
-arbitrarily small residual from the excess-neighborhood bounds.
-
-Choosing the finitely many lambda sufficiently small after all other
-data are fixed, we combine the three cases with the outside-core
-integral and obtain
-
- int_Omega |DH - Df|^p <= C_p eps_0 + o(1). (11.12)
-
-All uncontrolled-cell errors were chosen summably, so the same
-bounds also establish int_Omega |DH|^p < infinity, i.e. H in
-W^{1,p}(Omega; R^3), rather than only local Sobolev regularity.
-
-Value convergence. The source self-parametrizations move points only
-inside cells or blocks whose maximal diameter tends to zero. The
-target mesh can be chosen with arbitrarily small value error; the
-target motions (radial, angular, cap) are supported in boxes of
-arbitrarily small diameter in physical coordinates. On any interior
-compact, continuity of f then makes the values converge pointwise.
-Because the domains are bounded and all images lie in a fixed
-bounded target, dominated convergence improves this to L^p
-convergence on Omega.
-
-Combining. The meshes and value prescriptions can be chosen so
-that the value error and the derivative error (11.12) sum to less
-than the prescribed eps. This produces a locally bi-Lipschitz
-homeomorphism H : Omega -> Omega' with H in W^{1,p} and the
-stated W^{1,p} error. Theorem B is proved. QED
-
-12. PROOF OF THEOREM A: FULL DIFFEOMORPHIC APPROXIMATION
--------------------------------------------------------
-
-We now prove Theorem A. Let Omega, Lambda subset R^3 be nonempty
-bounded connected open domains, let 1 <= p <= 2, and let
-f : Omega -> Lambda be a homeomorphism in W^{1,p}(Omega; R^3).
-We must find C^infinity diffeomorphisms f_j : Omega -> Lambda, each
-in W^{1,p}, with
-
- int_Omega (|f_j - f|^p + |Df_j - Df|^p) dx -> 0.
-
-Proof. Fix a sequence eps_j > 0 decreasing to zero (for instance
-eps_j = 2^{-j}).
-
-Step 1 (locally bi-Lipschitz approximants). Apply Theorem B with
-tolerance eps_j/2: get a locally bi-Lipschitz homeomorphism
-H_j : Omega -> Lambda with H_j in W^{1,p} and
-int (|H_j - f|^p + |DH_j - Df|^p) < eps_j / 2.
-
-Step 2 (smoothing the approximant). Apply the smoothing theorem
-(Theorem C, combining Proposition 6.1 for a PL homeomorphism and
-Proposition 7.16 for the reduction from a locally bi-Lipschitz
-homeomorphism to a PL one), with tolerance eps_j/2 and a value
-tolerance small enough that |g_j(x) - H_j(x)| <
-(1/2) dist(H_j(x), R^3 \ Lambda) for every x. Get a C^infinity
-diffeomorphism g_j : Omega -> Lambda with g_j in W^{1,p} and
-
- int (|g_j - H_j|^p + |Dg_j - DH_j|^p) < eps_j / 2.
-
-Step 3 (triangle inequality). Then, by the Minkowski inequality
-applied to the L^p norm of the vector-valued function
-x -> (g_j(x) - f(x), Dg_j(x) - Df(x)),
-
- int (|g_j - f|^p + |Dg_j - Df|^p)^{1/p}
- <= int (|g_j - H_j|^p + |Dg_j - DH_j|^p)^{1/p}
- + int (|H_j - f|^p + |DH_j - Df|^p)^{1/p}
- < 2 (eps_j / 2)^{1/p}. (12.1)
-
-So the W^{1,p} error tends to zero. The maps g_j are C^infinity
-diffeomorphisms from Omega onto Lambda, each in W^{1,p}: their
-integrals against |g_j|^p and |Dg_j|^p are finite, because Omega is
-bounded and their values and derivatives are bounded by the
-construction and the compact exhaustions.
-
-So {g_j}_{j >= 1} is the asserted sequence of Theorem A. QED
-
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