M10r

Prompt

RESEARCH QUESTION (plane topology / dynamics; does non-interleaving (S) force every wall to be dry (ALL-DRY, equivalently SIDE-inf)?). 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 (compressed from the lab notes D2595, D2596, D2599; DERIVED-U there, unaudited here; lines marked (INFERRED) are my bridging). - Zhat := the profinite integers; D: Zhat \ {0} -> E_amb is continuous and even with D(2N) = sigma D(N); E := the cluster set of D at 0 (a Cantor set, the fibre labels), sigma the shift on E. Fibre points p_m(f) (m in Z, f in E), fibres P_m := {p_m(f) : f in E}, K := the closure of the regular points g(z) = (z, (D(z-j))_j), z in Zhat \ Z, and X := K x {0,1}; the address map is pi(g(z)) = z, pi(p_m(f)) = m. The host Y := ([0,1/2] x K)/folds is a planar continuum with q: [0,1/2] x K -> Y the quotient map; dynamics H[t,x] = [2t,2x] (no periodic point), h := H^-1, rays l_e(T) = q(T, 0^e) with H l_e(T) = l_{sigma e}(2T); U := C \ Y. - 2PAGE := a fixed embedding P: X -> S^1 in which the fence and the fold are noncrossing matchings (D1535); a stable cut order <_* on E from depth M_s: every far fibre order is <_* or its reverse. For |n| >= M_0 (far), Hull(P_n^0) is the smallest closed arc of S^1 containing P(P_n x {0}) (length < 1/16), and K_n := {x in K : (x,0) in the preimage of Hull(P_n^0)}; the band n is B_n := q([0,1/2] x K_n). M_s, M_*, T_0 are far thresholds fixed by the host; only their existence is used here. - Folded coordinates: for real u, eta(u) := frac u if frac u <= 1/2, else 1 - frac u; A(u) := floor u if frac u < 1/2, else -1 - floor u. A real t is generic if t is not in Z/2; the slice of Y at depth t is q({eta(t)} x K_A(t)); a section is such a depth T. - A jump J = (e', e) is a pair of consecutive labels of <_*; a label is extreme if it is the min or max of <_*, and two-sided if it is the endpoint of no jump. A slot of band n is an open region of U between two consecutive chords of B_n, where the chord of x in K_n is the arc q([0,1/2] x {x}) (INFERRED). - **Lemma DENS (D2599, as used by Theorem N; DERIVED there).** If x0 = g(z0) is a regular content point of J at a far fibre n0, there are L0 >= 1 and R0 with Hull(P_m^0) inside J_{n0} for every integer m with m = z0 mod L0 Zhat and |m| >= R0. L0 denotes the modulus of this statement and R0 denotes its radius (together the DENS data of x0). T(n0) denotes a section T with A(T) = n0, so T(n0) <= |n0| + 1 (INFERRED). - The location trichotomy of a pass (from the chain record below): IN-SO := the pass lies in J_i's band region at a section T' with T_0 <= T' <= u - M_*/2; IN-DEEP := the pass lies in it at a section T' > u - M_*/2 (at fibre +-A(u), or at T' >= u + M_*/2); OUT := it lies in no band region of J_i at a section >= T_0. NI := the non-interleaving condition of D2596 (also written NI_J): at every far turn n the hulls of P_n^0 and P_{-n}^0 are disjoint (so the wall partner p_{-+n}(e) is not in the closed J-arc at fibre +-n). - SIDE-inf (paraphrase of the lab's residual node): for every jump J with far content, every non-extreme wall label w of J, and all k >= k_0(J), no CH chain of E(h^-k) (defined below) approaches a wall point l_w(t), t generic and far, from inside J_{A(t)}. A4 Crosscut: an open arc q in U whose closure is an arc with distinct ends in X. Sh(q) = bounded component of R^2 \ (X u q), Out(q) = the unbounded one. diam q = diam cl q. A5 (BFMOT 3.6) phi: Riemann map, outside of unit disk onto U u {inf}; ray R_t = phi({r e^(2 pi i t): r > 1}). A chain: disjoint crosscuts Q_i, diam -> 0, Q_j in Sh(Q_i) for j > i; equivalent chains form a prime end E_t. Pr(E_t) = {z: a chain defining E_t converges to z}. The (X,x)-end of R_t (x on R_t): the bounded component of R_t \ {x}; R_t crosses Q essentially iff some (X,x)-end lies in Sh(Q). x in X is accessible iff some arc in U u {x} ends at x. - A prime end E_t with Pr(E_t) = Y is a dense channel. E(h^-k) := the outchannel of h^-k, the unique dense channel which BFMOT Theorem 6.2.1 gives for the map h^-k (Lemma IB* of D2595 says BFMOT applies to every h^-k); BFMOT := the plane-continuum fixed point paper arXiv 1004.0214 (authors Blokh, Fokkink, Mayer, Oversteegen, Tymchatyn, INFERRED from the initials). **Definition.** Let J = (e', e) be a jump, w in {e', e} a wall label, and n a far fibre. - (J, w) is **wet at n** if (p_n(w), 0) is a limit of points of X cap J_n (content of J accumulates at the wall from inside J). - Otherwise (J, w) is **dry at n**. Then P(p_n(w), 0) is an endpoint of a gap of P(X) lying in J_n. A strip is dry at both walls. - **ALL-DRY** := every (J, w) is dry at every fibre n with |n| >= N_*, where N_* is the far threshold of Lemma CH (A(t) for t >= M_s, 2t >= 3M_*/2). **The chain (Lemma CH of D2595).** g_i are slot segments at a generic far depth t inside jumps J_i, with g_i -> p = l_e(t) and var(H^k, g_i) = -1. The passes g*_i are slot segments of gamma_{g_i} with both sides in C(u), on the access of g_i between g_i and g_{i_0}, and g*_i -> H^k p = l_{sigma^k e}(u). Location trichotomy (HID-*): - IN-SO: g*_i is in J_i's band region at a section T' with T_0 <= T' <= u - M_*/2; - IN-DEEP: the same at T' > u - M_*/2, i.e. at fibre +-A(u), or at T' >= u + M_*/2; - OUT: g*_i is not in the band region of J_i at any section >= T_0. **Definition.** N(J,u): there is T' in [T_0, u - M_*/2] such that Hull(P^0_{A(u)}) lies in J's open jump arc at fibre A(T'). B. LEMMAS (status; where used) **Lemma AD (address form of wetness; DERIVED; P none / M none / C equivalence).** (J, w) is wet at n for w = e or w = e' iff there are content points y_i of J at n with pi(y_i) -> n in Zhat. So J is dry at both walls at n iff J's content offsets {pi(y) - n} at n stay at positive Zhat-distance from 0. **Lemma WR (wetness is a run invariant; DERIVED from B0, B1, B3).** If the fibres m and m' are consecutive steps of the ray with no unclean turn between them (a fence step, an iota_1 step, or a clean iota_0 turn), then (J, w) is wet at m iff it is wet at m'. So wetness is constant on runs and can change only at unclean turns. **Lemma WA (an inside approach needs a wet wall; DERIVED).** If a chain of crosscuts of any prime end converges to p = l_w(t) (A(t) = n, t generic) with all its elements in slots of band n inside J_n, then (J, w) is wet at n. **Lemma TWO-SIDE (DERIVED + SOURCE + RECALL; P yes / M none (any dense channel) / C side of approach).** Let E_t be a prime end of U with Pr(E_t) = Y (a dense channel). Let n be far, let x in K_n be a point whose chord in band n is accumulated, inside Hull(P_n^0), by chords of X-points on one side A of x, and let p = q(eta, x) with eta in (0, 1/2). Then E_t has a defining chain of crosscuts converging to p, each lying in a slot of band n between two chords of points on side A. In particular, **at a wall that is wet at n, every dense channel approaches p = l_w(t) from inside J, for every generic t with A(t) = n.** **Lemma CH (D2595; DERIVED + RECALL; black box).** Take k >= 1, a generic far t >= M_s with u = 2^k t >= 3M_*/2, and e in E with e and sigma^k e both non-extreme. Put p := l_e(t). Then: (i) there is a chain (g_i) defining E(h^-k) such that each g_i is a slot segment at depth t inside a jump J_i of the fibre order, and g_i -> p; (ii) every J_i is persistent from M_s; (iii) (EXT-) holds for g_i, i >= i_0; (iv) var(H^k, g_i) = -1 for i large, so HID-* applies to every g_i; (v) for fixed large i_0 and i >= i_1(i_0), the pass g*_i lies on the part of the access of g_i between g_i and g_{i_0}, and g*_i -> H^k p. It is called a CH chain. **Corollary 2S (two-sided e; DERIVED).** Suppose e is two-sided in <_* (it is the endpoint of no jump). Then: - N(J_i, u) fails for all large i; - for all large i, g*_i lies in J_i at no section with A(T') = A(u); - hence IN for large i <=> the deep configuration of A(d), with J_i in place of J (and (NI) at \|A(u)\| for the mirror case). **Theorem N (DERIVED; P yes / M h^-k / C IN-SO for a slid chain).** Assume the D2596 one-sided test: a CH chain (g_i) of E(h^-k) at a generic far depth t approaches p = l_e(t) from inside the jump J = (e', e) (all g_i in J_n, n = A(t)), with e and sigma^k e non-extreme. Let x0 = g(z0) be a regular content point of J at a far fibre n0 with section T(n0) >= T_0, and let (L0, R0) be its DENS data. If 2^(k-1) >= L0 and 2^k (|n| - 1) >= R0 + |n0| + M_* + 1, then there is a generic s with A(s) = n such that N(J, 2^k s) holds for the height-slid chain (g_i') at depth s. By D2596 Theorem A(a), every pass of (g_i') is IN-SO. C. THE TARGET Known (D2603, D2604): ALL-DRY <=> SIDE-inf <=> the residual node DRY_0 of the chart branch of RIG-THICK. Known (D2609, DERIVED, unaudited): DRY_0 is NOT provable from the 2PAGE order plus all h-free laws that ignore NI, even with SC+ in its strictest form: an explicit recursive convex-block 2PAGE order (Construction R, below) has a wall wet at every fibre. The first law it violates is NI. In every convex-block order NI fails cofinally: ### 2.3 Theorem R (DERIVED; P yes / M none / C an explicit 2PAGE order) For every depth-only Z_2 host and every anchor system with (TOP-A), the relation above is a compact linear order on K whose order topology is the topology of K, and iota_0 and iota_1 are noncrossing. Hence the F-ribbon circular order on X (K x {0} in order, then K x {1} reversed) is a 2PAGE order, and Y_D is planar. **Lemma INT (interleaving in convex-block orders; P yes / M none / C exact interleaving set).** In every Construction R order, the far turn q >= 1 is interleaving iff q = 2^r and B_{r+1}'s anchor is not in the gap of P_0 (read cyclically) that contains B_r's anchor. **Corollary INT.** If phi(r+1) = sigma phi(r) with sigma a fixed-point-free homeomorphism of E (all doubling hosts D(2N) = sigma D(N); on Y_2, sigma = +1), then (TOP-A) puts the anchors of B_r and B_{r+1} into different gaps for all large r. So **every far turn q = 2^r interleaves, and (S) fails, for every convex-block order, wet or dry.** COMPUTED: the interleaving turns with q <= 40 are exactly {1, 2, 4, 8, 16, 32}. **Corollary OCT.** If the turn 2^R interleaves, hull(P_n) supseteq Psi^{-1}(P_{2^R}) = P_{n -+ 2^{R+1}}, where the sign is chosen towards 0. The ratio is 2^{R+1}/|n| - 1. It lies in the protected octave (1,2) whenever |n| > 2^{R+1}/3, for example for n = 3*2^k (ratio 5/3), cofinally. COMPUTED: 14 such fibres with |n| <= 20, all at the predicted index. D. ASK Decide whether NI (non-interleaving at every far turn) forces ALL-DRY. Either (A) prove SIDE-inf from NI together with Lemmas AD, WR, WA, TWO-SIDE, CH, Corollary 2S and Theorem N as stated, saying exactly where NI enters; or (B) give an explicit 2PAGE order on a doubling host, necessarily with non-convex blocks, that satisfies NI at every far turn and has a wall that is wet at infinitely many far fibres, and check it against every lemma above; or (C) reduce the question to one named, exactly stated lemma that is strictly weaker than ALL-DRY. A renamed gap is a RESTATEMENT: say so. E. CONTROLS (all must hold) (C1) Your argument must break on Construction R exactly where NI fails there (the turns q = 2^r). (C2) It must not prove ALL-DRY for a nonplanar order (drop the 2PAGE noncrossing condition and check that a step fails). (C3) If you build an order, give the first 8 fibres explicitly and verify NI on them. F. DEAD ENDS (not progress) A static proof without NI (D2609); convex-block constructions (Corollary INT); integer-depth-only versions of SC+ (D2610 uses a real-slice law). G. FORMAT Line 1 exactly: VERDICT: ALL-DRY PROVED (using NI) | COUNTEREXAMPLE (explicit order; NI holds and some wall is wet) | REDUCED (to a named lemma) | 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: a static proof that never uses NI (known impossible, see the statements below); convex-block orders (they violate NI by Corollary INT); treating a finite truncation as the infinite host; assuming ALL-DRY or its negation