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Prompt

RESEARCH QUESTION (plane topology / conformal mapping; thresholds on every leg of an external ray (LEMMA_P of Route I of the L2 route map: Koebe, Wolff length-area, F. and M. Riesz)). Self-contained. Label every claim PROVED (you prove it), SOURCE (exact reference; unchecked?), LAB (from this prompt, status as given), CONJECTURE, OPEN, or RESTATEMENT (a gap reworded). If a definition is ambiguous, state your reading in one line and continue. Say where your argument stops. A. DEFINITIONS Setting (lab note b_opus_rig, section 4; DERIVED there by one agent, UNAUDITED here; the classical inputs are RECALL, not on disk, which is why this task exists). X is a nondegenerate nonseparating plane continuum, U := C \ X (connected, unbounded). phi: D -> U u {infinity} is a conformal map of the open unit disc onto U u {infinity} with phi(0) = infinity (so phi(z) -> X as |z| -> 1, and phi is bounded on every set {|z| > s} that stays off 0). For an angle t, R_t is the external ray {phi(s e^{it}) : 0 < s < 1}, run from the point of infinity towards X, parametrised by s increasing to 1 ("time" s). - A crosscut q is an open arc in U whose closure is an arc with two DISTINCT endpoints in X. Sh(q) is the bounded component of C \ (X u q), Out(q) the unbounded one. q is an E_t-crosscut ("E-crosscut") if a tail of R_t lies in Sh(q). Diameters are of closures. - For an E-crosscut q, v_q is the first point of cl q met by R_t coming from infinity, and rho_q is the part of R_t from infinity to v_q (so rho_q lies in Out(q) except for its end v_q). For E-crosscuts p, q with disjoint closures one lies in the shadow of the other; q is DEEPER than p if q lies in Sh(p). - Scales (lab, PROOF_L2_v2 3.8). A point z_1 in X, radii 0 < rho_0 small, the disc D_rho := B(z_1, rho), the closed ball B_s := cl B(z_1, rho/2), the annulus A := B(z_1, 3 rho/2) \ B_s. Fixed objects: an E-crosscut c_* , d_* := dist(rho_{c_*}, X) > 0, and a small E-crosscut c far from z_1 with F(cl c) inside B(z_1, rho/8); here F is a continuous extension of a homeomorphism h of X (not needed for Lemma P itself). - RIG_S(c) (the property that uses Lemma P; OPEN, NOT your target): for every E-crosscut q inside A, deeper than c, with cl q disjoint from cl c and F(cl c), one has a_c(q) := I(F o c^+, rho_q) = -1. Lemma P only supplies the crosscuts q to which RIG_S is applied. - Repair R5 (owed by the lab): the thresholds must be taken AFTER the last hit of cl c by R_t; the reason in the lab note ("q small, so c is not in Sh(q)") is to be replaced by a time-order argument. The depth criterion used elsewhere (Sol 6.1, DERIVED-U): an E-crosscut e with cl e disjoint from cl c and diam e < dist(rho_c, X) lies in Sh(c), because rho_c is connected, starts in cl c and reaches infinity without meeting e or X. B. LEMMAS (status; where used) Inputs of Lemma P as the lab writes them (RECALL). You are asked to restate each with exact hypotheses and give a proof or a precise textbook reference (author, book, theorem number), and to say which form Lemma P uses. (K) Koebe. "Hyperbolic length of the subarc >= (rho/2) / (2 max dist(w, X)) -> infinity." Intended form: the hyperbolic density of U (or of D transported by phi) at w is >= 1/(4 dist(w, X)) (Koebe 1/4) up to the normalising factor, and dist(phi(s e^{it}), X) -> 0 as s -> 1. (W) Wolff's length-area lemma. "On the arcs |z - e^{it}| = r (z in D) there is r in a prescribed window (e^2 eps, eps) with the length of the image under phi as small as needed." Intended form (INFERRED reading; the textbook source and number are for you to give): if g is holomorphic on D intersected with the disc B(zeta, sqrt eps_0) and the image has finite area A, then for 0 < eps < eps_0 there is r in (eps^2, eps) with int over the arc gamma_r of |g'| |dz| <= (2 pi A / log(1/eps))^(1/2). (E) Ends of finite-length arcs. "A finite-length image arc has two ends in the limit set." Intended: phi(gamma_r) has finite Euclidean length, hence phi(z) converges to a point of X at each end z -> the endpoints of gamma_r on the unit circle. (R) F. and M. Riesz. "The two ends are distinct." Intended form: a nonconstant bounded holomorphic function on D cannot have the same radial limit on a set of positive measure of the unit circle (uniqueness theorem), applied to the cluster set of phi on the boundary arc cut off by gamma_r. C. THE TARGET Lemma P as the lab states it (b_opus_rig section 4, lines 61-62; DERIVED, classical input; the lab symbol for the disc is written D_rho here). **Lemma P (thresholds on every leg; DERIVED, classical input).** Let R_t be the conformal external ray. There is T_1 such that every subarc of R_t after time T_1 crossing the annulus D_rho minus B_s contains a point w with a small E-crosscut q through w inside the annulus, crossing R_t once at w. *Lab proof.* Koebe: hyperbolic length of the subarc >= (rho/2) / (2 max dist(w, X)) -> infinity. Wolff's length-area inequality on the arcs |z - e^{it}| = r (RECALL; Wolff's lemma quoted in a_opus_rln l.66) gives an r in the window with image of length as small as needed. The ends of a finite-length arc exist. They are distinct by F. and M. Riesz (RECALL). The shadow contains the tail. Where it is used (lab Lemma C, item C1, in the form needed; same note, section 4): a chord pi of R_t cap D_rho (a component) that meets B_s, entered and left after the last hit of cl c and after T_1. Its entry leg and its exit leg both cross the annulus D_rho \ B_s. Lemma P gives on each leg a small E-crosscut q (inside A, deeper than c, far from c and F(c), crossing R_t exactly once, at a point w of the leg) to which RIG_S is applied, so that a_c(q) = -1 on each leg. Hence what is needed from Lemma P, in the form used, is: for every subarc J of R_t after time T_1 that crosses D_rho \ B_s there is w in J and an E-crosscut q such that (i) cl q is inside A and misses B(z_1, rho/4); (ii) cl q misses cl c and F(cl c) and q lies in Sh(c); (iii) q meets R_t exactly once, at w, transversally in the sense that the tail of R_t after w lies in Sh(q) and the part before w in Out(q); (iv) diam q is below any prescribed bound eta > 0 (so that q is small in the sense of RIG_S, say diam q < delta_0 and q in Sh(c_*)). D. ASK Decide Lemma P. Either (A) prove it: state (K), (W), (E), (R) in exact form with hypotheses, give a proof of each or an exact textbook citation, then derive (i) to (iv) with every quantifier explicit (T_1 may depend on rho, c, eta, X, phi); or (B) find where the thresholds fail: an explicit planar X, a ray, an annulus and a leg on which no q with (i) to (iv) exists, or an explicit gap in one of the four inputs as the lab applies it; or (C) repair with a stated extra hypothesis and say whether the other lab steps still hold. Say in particular: how a window of radial scales (eps^2, eps) is placed inside the subarc J so that the point w lies in the middle part of J and q stays inside A (the lab leaves this out); why the Wolff area A is finite and uniform in eps (phi is unbounded near 0, and the arcs are centred at e^{it}); why (iii) holds (the arc gamma_r meets the radius once; give the argument that the component of D cut off by gamma_r, containing the tail of the radius, maps onto Sh(q) and not onto Out(q)); and why the repair R5 holds without using smallness of Sh(q) (note that diam Sh(q) need not be small when the impression of the prime end E_t is large, e.g. a Knaster or planar odometer end). E. CONTROLS (all must hold) (C1) Take X = a closed disc and R_t a straight ray: all four inputs are elementary, and a direct computation of q must agree with the general proof. (C2) Take X with a prime end E_t whose impression is an arc (a topologist's sine curve): check that Sh(q) is large while diam q is small, and that Lemma P still gives (i) to (iv). (C3) A leg that runs inside A close to the outer circle of A (it touches the edge of the annulus): the proof must still give w with q inside A, or say exactly how rho or the annulus is shrunk. (C4) Each input must be checked for the exact normalisation of phi: phi(0) = infinity, not phi(0) = a finite point. F. DEAD ENDS (not progress) Proving RIG_S or any statement about a_c(q) (the data of F o c), proving Lemma C or Theorem R* beyond Lemma P, re-deriving Lindelof's theorem for CP1. G. FORMAT Line 1 exactly: VERDICT: LEMMA P PROVED (inputs proved or exactly quoted) | CORRECT WITH REPAIRS (list) | FAILS (configuration, failing step) | OPEN. Then SUMMARY (<= 100 words), the argument (<= 2,000 words), STILL OPEN EXACTLY. Short of effort: line 1 + summary; an honest OPEN is valid. No finite computation as proof; no theorem beyond its hypotheses; UNKNOWN if unsure a cited result exists. RED FLAGS: quoting Koebe, Wolff or Riesz only by name; giving a Wolff statement whose area bound is not finite or not uniform in the scale; a crosscut whose two ends coincide (closed curve); a crosscut q with c in Sh(q); a point w of R_t near the edge of the annulus, so that q leaves it; claiming q crosses R_t once without checking that the arc and the ray are transverse