M R4

Prompt

RESEARCH QUESTION (plane topology; chords of an external ray in a disc, separating chords, and the cell of an endpoint (PASSCELL = Lemma C, items C1-C6, of Route I of the L2 route map)). 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, Lemma C; DERIVED there by one agent, UNAUDITED here; lines marked (INFERRED) are my bridging). X is a nondegenerate nonseparating plane continuum, U := C \ X (connected, unbounded). phi: D1 -> U u {infinity} is a conformal map of the open unit disc with phi(0) = infinity. For an angle t, R_t = {phi(s e^{it}) : 0 < s < 1}, run from infinity towards X and parametrised by "time" s increasing to 1. R_t is a simple real-analytic arc, closed in C, missing X, with dist(R_t(s), X) -> 0 as s -> 1. For s in (0,1), rho_s denotes R_t restricted to times <= s: a closed set missing X, properly parametrised, with start R_t(s). - Algebraic intersection (lab PROOF_L2_v5 section 1.5, DERIVED). For a path beta that misses the start v of an access rho and has endpoints off rho: I(beta, rho) := win(beta * delta, v), delta any path in C \ rho from the end of beta to its start. Facts: (I1) additive under concatenation (junction points off rho); (I2) reversing beta changes the sign; (I3) beta inside C \ rho gives 0; (I5) if B is an open convex set not containing v, two paths in B with the same endpoints have the same I. For times s1 < s2 and beta missing R_t(s1), R_t(s2) with endpoints off rho_{s2}: I(beta, R_t[s1,s2]) := I(beta, rho_{s2}) - I(beta, rho_{s1}); it is 0 if beta misses R_t[s1,s2]. - 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; U \ q has exactly these two components. q is an E-crosscut if a tail of R_t lies in Sh(q). For an E-crosscut q, v_q is the first point of cl q met by R_t from infinity, and rho_q := rho_s with R_t(s) = v_q. q is DEEPER than c if q lies in Sh(c). Diameters are of closures. - Scales. z_1 in X; z_2 in X with |z_1 - z_2| < rho/8; sigma := the straight segment [z_1, z_2] (it may alternate between X and U). D := B(z_1, rho) (open disc), B_s := the closed ball cl B(z_1, rho/2), A := B(z_1, 3 rho/2) \ B_s. c is a small E-crosscut with cl c disjoint from B(z_1, 2 rho). (Lab origin: z_1 = h(a), z_2 = h(b), c = (a,b), F an extension of a homeomorphism h of X; none of this is used below.) - (H) = RIG_S(c), in the form used (OPEN as a statement about hosts; here it is the standing hypothesis): for every E-crosscut q with cl q inside A, q deeper than c and cl q disjoint from cl c: I(sigma, rho_q) = -1. (Lab form I(F o c^+, rho_q) = -1; the two agree by (I5), as F(cl c) and sigma lie in the convex ball B_s, which misses v_q.) - Chords. Since R_t is analytic, for all rho outside a countable set the circle |z - z_1| = rho meets R_t([0,T]) in finitely many points, transversally, for every T < 1 (INFERRED reading of the lab's "generic rho"); rho is so fixed. A chord is a component of R_t cap D (an open arc with its two ends on the circle). Chord pi is separating if z_1, z_2 lie in different components of D \ pi, and nu(pi) := I(sigma, pi) in the sense of the I(.,.) definition above (with a segment of sigma, by (I1) and (I5)). "Up to time T": contained in R_t([0,T]). - Standing fact (ACC) (SOURCE-lab, PROOF_L2_v5 Cor 3.7(e) as used in its Lemma 4.1; not re-proved): the principal set of the prime end of R_t is all of X, so R_t accumulates at every point of X. Also diam X > 4 rho (INFERRED: rho is small against the lab scales). Whether C1-C6 need these is part of the task. B. LEMMAS (status; where used) Inputs you may assume without proof (state where each is used). (LP) Lemma P in the form used (the thresholds on every leg; its classical inputs Koebe, Wolff, F. and M. Riesz are the object of the separate task R3_LEMMAP). There is T_1 < 1 such that for every subarc J of R_t whose times exceed T_1 and the last hit of cl c by R_t (this last time exists, R_t is closed and cl c compact), and which crosses the open annulus D \ B_s (from the circle to the sphere bdry B_s or back, inside D \ B_s except for its ends), there are w in J and an E-crosscut q with cl q inside A, cl q disjoint from cl c, cl q cap R_t = {w}, and the tail of R_t after w in Sh(q). (Showing q deeper than c is NOT part of (LP); it is part of C1.) (J) Jordan curve theorem / Schoenflies: a chord is a cross-cut of D, and D \ pi has exactly two components. (Q) In a compact metric space components and quasi-components agree. (K) A compact subset of U meets only R_t([0,T]) for some T < 1. C. THE TARGET Lemma C as the lab states it (b_opus_rig section 4, lines 64-71; DERIVED; transcribed to ASCII; its proof sketches are in brackets). Assume (H). The components of R_t cap D are chords. Generic rho makes finitely many up to any time (analyticity). nu(pi) = I(sigma, pi) is +-1 if chord pi separates z_1, z_2, and 0 otherwise. (C1) Every chord meeting B_s and later than max(T_1, last hit of cl c) has nu = 0. [Its entry and exit legs cross the annulus, so Lemma P gives thresholds on both legs, each with data -1. Chords missing B_s do not cross sigma.] (C2) The set Sep of separating chords is finite and nonempty (sum of nu = -1). It is linearly ordered from z_1 to z_2. Let pi^1 be the member closest to z_1. (C3) No chord at all separates z_1 from pi^1. [A non-separating chord pi has z_1, z_2 on one side. If pi^1 were on the other side it could not separate them.] (C4) Let Omega_T be the component of z_1 in D minus the chords up to time T, and Y := the intersection over T of cl Omega_T. Y is a continuum containing z_1 and pi^1. (C5) Let K(z_1) be the component of X cap cl D containing z_1. For every eps > 0 there is y in U cap D, off R_t, lying in every Omega_T, with dist(y, K(z_1)) < eps. [K is a quasi-component, so there is a relatively clopen A0 in X cap cl D with K in A0 in the eps/2-neighbourhood of K. G := N_delta(A0) cap cl D, delta small, so that the frontier of G in cl D misses X. Y is connected and contains z_1: either Y lies in G, and then pi^1 lies in G, or Y meets the frontier of G. Either way Y has a point w in U within eps of K. A small ball about w in U meets finitely many chords, so it has finitely many pieces off the chords; one piece meets Omega_T for cofinal T, hence lies in all of them (it is connected, off the chords, and the Omega_T are nested). Take y there.] (C6) For every E-crosscut q outside D: I([z_1, y], rho_q) = 0. [z_1 and y are joined in Omega_T, T the time of v_q, by a path off rho_q; then (I3) and (I5).] Use in the lab's Theorem R* (Step 1; NOT your target): C5 and C6 are applied at BOTH z_1 and z_2, inside the same disc D (so pi^2 := the member of Sep closest to z_2, cells Omega^2_T of z_2, K(z_2)), to get y_1, y_2 with I([y_1,y_2], rho_g) = I(sigma, rho_g) = -1 for the E-crosscuts g near X cap bdry B(z_1, 5 rho/4), and y_1 != y_2. D. ASK Decide PASSCELL = Lemma C. Either (A) prove C1-C6 from scratch from (H), (LP), (J), (Q), (K) and the definitions, for both z_1 and z_2 inside the same D, with every quantifier explicit and every standing fact located; or (B) find the step that needs an extra hypothesis, with an explicit counter-configuration (X, R_t, z_1, z_2, rho, c) satisfying everything stated above in which a claim fails; or (C) repair with a minimal extra hypothesis and say whether Theorem R* Step 1 still follows. Say in particular: (1) why nu(pi) is +-1 or 0 as stated (the definition through I(beta, rho_{s2}) - I(beta, rho_{s1}) uses only (I1)-(I5)); (2) in C1, how q is shown deeper than c (a time-order argument: the part of R_t after the last hit of cl c lies in one component of U \ cl c), and why the entry and exit legs can be chosen inside the open annulus, so that I(sigma, rho_q) at the exit threshold minus the entry threshold equals nu(pi); (3) in C2, which threshold witnesses sum nu = -1 and why one exists AFTER every nonzero chord: does this need (ACC), and is a chord that straddles the time max(T_1, last hit of cl c) handled; (4) what happens if R_t never leaves D after some time (is that excluded by diam X > 4 rho and (ACC)?); (5) in C5, the finiteness of the pieces (choice of the ball radius), the case w on the circle, and whether the case "Y inside G" gives y off R_t; (6) in C6, the cases v_q on the circle or q meeting D; (7) every use of CP1 or of the laws V1 = -1, V2 in {0,-1} (INFERRED: none, the value -1 enters only through (H)), and every classical or RECALL input. E. CONTROLS (all must hold) Control 1, vacuity (INFERRED by me, to be checked): X a closed disc, z_1, z_2 in X, R_t a straight ray landing at p far from B(z_1, 2 rho), c a small E-crosscut at p. No E-crosscut lies in A, so (H) holds vacuously, sigma lies in X and misses R_t, so every nu is 0 and Sep is empty. Say which of C1-C6 fail and what exact hypothesis (return of R_t to A, i.e. ACC at z_1 or something weaker) the correct statement needs. Control 2, several separating chords: Sep with three nested chords with nu = +1, -1, -1 in order from z_1 to z_2, two of them before the last hit of cl c. Check items C2 to C5 for both z_1 and z_2 (pi^1 != pi^2, disjoint cells, y_1 != y_2) and that C1 does not claim nu = 0 for them. Control 3, accumulation at z_1: infinitely many chords converging to z_1 (R_t accumulates at z_1 inside D, as ACC allows). Check C4 and C5: here Y contains X-points, the cell shrinks to K(z_1), and the finiteness argument in C5 must use the analytic ball argument, not a global finiteness. Control 4, say where planarity and where "U connected, simply connected in the sphere" are used (INFERRED: planarity in (J), the winding calculus and the separation of z_1 from z_2 by a chord; connectedness of U in (I5) and the access calculus); name a nonplanar or separating-X variant in which the argument breaks. F. DEAD ENDS (not progress) Proving Lemma P (task R3_LEMMAP); proving (H) = RIG_S, RIG, Lemma AB or Theorem R* beyond Step 1's use of C5, C6; the segmentwise vanishing of far data (FAILED, see GATE_LEDGER case RIG/END/Theorem R: only the aggregate sum vanishes); using the Denjoy germ of b_opus_rig section 6 as if it were an L2 system. G. FORMAT Line 1 exactly: VERDICT: PASSCELL PROVED (C1-C6 for both z_1 and z_2, hypotheses listed) | CORRECT WITH REPAIRS (list) | FAILS (configuration, failing step, extra hypothesis) | 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: treating Lemma P as unproved here (assume it in the stated form); using the value -1 of a_c(q) through any law other than (H); a cell argument that never uses planarity or that U is connected; claiming sum nu = -1 without exhibiting a threshold q AFTER every nonzero chord; taking "finitely many chords" for a ray that accumulates at z_1 without the analyticity argument; leaving the case that R_t eventually stays in D unexamined