diff b1f9bc4..56bc728 univalent_inverse_power_v2.txt
diff --git a/univalent_inverse_power_v2.txt b/univalent_inverse_power_v2.txt
index 6aaf163..56b3ef4 100644
--- a/univalent_inverse_power_v2.txt
+++ b/univalent_inverse_power_v2.txt
@@ -2856,212 +2856,5 @@ formula fails at p = -1. QED
That completes the proof.
-================================================================
-PART 7. WHAT THIS PROOF DOES AND DOES NOT SAY
-================================================================
-
-7.1 WHAT IS PROVED
-
-Exactly one theorem, stated in 0.5: there are constants 0 < eps < 1/4 and
-C < infinity such that
-
- M_{-1}[f'](r) <= C (1-r)^{-1/4+eps}
-
-for every f in S and every 1/2 <= r < 1. The estimate is uniform over
-the whole class, with no hypothesis on the image beyond injectivity and
-the normalization f(0)=0, f'(0)=1. The consequence B_b(-1) < 1/4 is
-strict, so Kraetzer's formula B_b(p) = p^2/4 at p = -1 is false.
-
-7.2 WHAT IS QUOTED, NOT PROVED
-
-Four external facts are used. Each is standard, each is stated in words
-where it is used, and each is genuinely a theorem of others.
-
- C1. Compactness: a normalized family of injective holomorphic
- functions is compact in locally uniform convergence, and limits
- are injective (Montel plus Hurwitz). Used in LEMMA 1.3 for the
- compactness of K.
-
- C2. Sard's theorem in the plane: the set of singular values of a
- smooth map of the plane has area zero. Used in LEMMA 4.5 for
- condition (g2). An elementary self-contained estimate is included
- there, so this is the least "external" of the four.
-
- C3. Two topological facts. (a) A compact one-dimensional manifold
- without boundary is a finite disjoint union of circles; used in
- STEP 2 of LEMMA 4.6. (b) The Hopf tangent-turning lemma: the
- inward normal of a simply closed positively oriented curve has
- winding number one; used in STEP 5 of LEMMA 4.6.
-
- C4. Koebe's quarter theorem with the distortion estimate, in the
- precise form of LEMMA 1.2. Used throughout.
-
-Nothing else is assumed. In particular the geometric function theory
-is confined to C4; the rest of the argument is real analysis, plane
-topology, and probability on a compact space.
-
-7.3 WHAT IS NOT EFFECTIVE
-
-Two things are asserted to exist and are not computed.
-
- * eps. The proof produces a positive gap 1/4 - beta but does not
- bound it away from zero. The reason is structural: the argument in
- Part 6 is a contradiction, so it gives no quantitative control of
- how far beta sits below 1/4. Every step is effective in the weak
- sense that the constants are finite, but none of the constants is
- tracked.
-
- * C. The final constant is C_0 C_b, where C_0 comes from
- PROPOSITION 1.8 and C_b from the root limit. Neither is evaluated.
- The chain of constants in between (C from (1.1), C_1 from STEP 2 of
- PROPOSITION 1.8, C_delta from (3.15), Y_psi from LEMMA 3.4) is
- finite at every stage and enormous if written out.
-
-So the theorem is a strict-inquality statement, not a numerical one.
-
-7.4 WHAT IS NOT SETTLED
-
- * The exponent p = 1 is untouched. Kraetzer's formula is symmetric
- between p = 1 and p = -1, and this proof says nothing about p = 1.
- It would be a mistake to read the failure at p = -1 as evidence
- about the sign of p.
-
- * No other exponent is settled.
-
- * Nothing is said about which maps achieve the extremal growth. The
- proof works with a probability law on the whole class and never
- identifies an extremal map or an extremal domain.
-
- * The distinction between the bounded class S_b and the unrestricted
- class S survives: the bound proved here is uniform over S, hence
- applies to S_b, and B_b(-1) is what it bounds. Makarov's comparison
- theorem identifies the unrestricted spectrum with the maximum of the
- bounded spectrum and the Koebe term 3p-1, and at p = -1 the Koebe
- term is -4, so it plays no role; but at positive exponents the
- distinction is essential and is not addressed here.
-
-7.5 WHERE THE DIFFICULTY ACTUALLY SITS
-
-If a reader wants to know which single step makes the proof work, it is
-STEP 3 of PROPOSITION 5.3, that is, the identity
-
- omega_F J_{F,+} c_F -> the partner's weight,
-
-together with the injectivity of the pairing. Everything before that
-step builds the two objects that must be compared (a law and a pairing);
-everything after it is a box computation and three limits. Mechanically
-it is the change of variables (5.2) that makes the step work; the change
-of variables requires (5.3), which is where the exponent k = (beta+2)/3
-pays off. If one had chosen any other exponent in the potential, the
-two weights in STEP 3 would not have matched and the comparison would
-have failed.
-
-Second in importance is the no-tie condition (g3). Without it the
-pairing would only be non-strict, and the equality analysis of STEP 7 of
-PROPOSITION 5.3 would not close.
-
-
-================================================================
-APPENDIX. NOTATION AND CONSTANTS
================================================================
-A.1 SETS AND SPACES
-
- D the open unit disk, {|z| < 1}.
- H the upper half-plane, {Im z > 0}.
- C the complex plane.
- S univalent f : D -> C with f(0)=0, f'(0)=1.
- S_b the members of S with bounded image.
- K univalent F : H -> C with F(i)=0, F'(i)=1.
- m(z) (z-i)/(z+i), a bijection D -> H; m^{-1}(w) = i(1+w)/(1-w).
- S_0 {z in H : |m(z)| >= 1/2}; its complement is bounded.
- B(z,s) the open Euclidean disk of centre z and radius s.
-
-A.2 QUANTITIES ATTACHED TO A MAP
-
- Q_F 1/F'.
- q_F iy Q_F'/Q_F = -iy F''/F' = a_F + i b_F.
- T_zF the rebasing, (F(z*w)-F(z))/((Im z)F'(z)).
- V_xi^F the potential, y^k/|F(z)-xi|.
- ell log V_xi^F.
- G_F the critical map, F - (iy/k)F'.
- J_F det DG_F / |F'|^2, the signed Jacobian.
- omega_F y^{beta-1}|Q_F|.
-
-A.3 THE AFFINE GROUP
-
- z*w (x+yu) + i(yv) for z=x+iy, w=u+iv.
- z# (-x+i)/y, the inverse of z.
- i the identity element of the group.
- (G1)-(G3) associativity and inverses; Im(z*w)=(Im z)(Im w); the
- height-one elements form a copy of R, the purely imaginary
- elements a copy of (0,infinity).
-
-A.4 OPERATORS AND GROWTH
-
- A_z the positive operator (A_z Phi)(F) = |Q_F(z)| Phi(T_zF).
- L (1/2)(A_{-1/2+i/2} + A_{1/2+i/2}).
- v 2^{-n}, the dyadic spacing at level n.
- u_j -1 + (2j+1)v, the dyadic abscissae.
- rho lim ||L^n||^{1/n}, equal to the exponential growth rate of L.
- beta log_2 rho, so rho = 2^beta; 0 <= beta <= 1.
- k (beta+2)/3, so 3(k-1) = beta-1 and 3/4 <= k <= 1.
-
-A.5 THE LAW AND ITS MOMENTS
-
- P the probability law on K constructed in PROPOSITION 2.7.
- E expectation under P.
- (2.14) E[|Q_F(x+iy)| Phi(T_{x+iy}F)] = y^{-beta} E[Phi(F)].
- (2.20) E|Q_F(x+iy)| = y^{-beta}.
- (2.19) E a_F(i) = -beta, E|q_F(i)|^2 = beta(beta+1).
-
-A.6 GOODNESS AND PAIRING
-
- (g1) all closed positive superlevels of V are compact.
- (g2) the target is a regular value of G_F.
- (g3) distinct preimages of the target have distinct V-values.
- D {(F,z) : J_F(z) > 0, (F,G_F(z)) good}.
- D_N the part of D where 1/N <= y'/y <= N and |x'-x| <= Ny.
- R_F the rank pairing of saddles with peaks (LEMMA 4.7).
- c_F (V(R_F(z))/V(z))^3 on D, 0 off D; c_F > 1 on D.
- B, B' the two boxes {|x|<=S, 1<=y<=Y} and {|x|<=S+NY, 1/N<=y<=NY}.
-
-A.7 CONSTANTS
-
- C the constant of (1.1): |Q_F(x+iy)| <= C/y on |x|<=2, 0<y<=1.
- c 1/4, the constant of (1.2): |F(z)| >= 1/4 when |m(z)|>=1/2.
- C_0 the constant of PROPOSITION 1.8, from the four-arc cover.
- C_1 the bound for |Q_G| on the fixed compact set [-1,1]+i[1/2,4].
- C_delta (2^k/delta)(1+4C_1), the target-disk radius in LEMMA 3.4.
- Y_psi the height bound for the support of (Delta psi) o F.
- C_b the constant in ||L^n|| <= C_b 2^{bn}, valid for b > beta.
- eps 1/4 - b, the gap in the main theorem.
- C C_0 C_b, the constant in the main theorem (unfortunately the
- same letter as in (1.1); the two occurrences are in different
- parts and each is defined where it appears).
-
-A.8 WHERE TO LOOK FOR WHAT
-
- The transfer from the disk to the half-plane: 1.1, FACT 1.1.
- Compactness of K and the two uniform bounds: LEMMA 1.3.
- The rebasing group and its laws: 1.2, LEMMA 1.4.
- The operator L, its dyadic form, its growth rate: 1.3, LEMMAS
- 1.5-1.7.
- The disk means are bounded by ||L^n||: PROPOSITION 1.8.
- The eigenmeasure at the exact rate rho: LEMMA 2.1.
- Horizontal invariance and exact scaling: LEMMAS 2.2-2.6.
- The two moment identities: LEMMA 2.8.
- The weighted area tail: LEMMA 3.3.
- Decay at infinity and horizontal length: LEMMA 3.4.
- Compact positive superlevels: PROPOSITION 3.5.
- The critical map, J_F, and the Hessian: LEMMAS 4.1-4.4.
- Genericity of good targets: LEMMA 4.5.
- Peaks at least as many as saddles: LEMMA 4.6.
- The measurable rank pairing: LEMMA 4.7.
- Change of variables through G_F: PROPOSITION 5.1.
- Goodness at the fixed base point: LEMMA 5.2.
- E J_F(i) <= 0, and the equality case: PROPOSITION 5.3.
- No map has J_F identically zero: LEMMA 6.1.
- The theorem: 6.2.
-
- END