m25

Prompt

RESEARCH QUESTION (plane continua / intersection numbers and prime-end dynamics; L-MPN-REL (the signed cancellation of WRAP-MPN, Route I): a coherent all-finite-scale shadow/sign tower with residual +2 satisfies every finite instance of (S); which GLOBAL law of an R-instance does it violate, or can it be realised?). 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 (S) (compressed from lab notes D2608, D2612 and the Opus lane wrapmpn_prove_opus_1102; every item marked DERIVED is the claim of a single agent and is UNAUDITED; lines marked (INFERRED) are my bridging). - X is a nondegenerate nonseparating plane continuum, indecomposable with empty interior, U := C \ X (so U together with infinity is simply connected). h: X -> X is a homeomorphism with h^2 continuum-minimal (cm: no nonempty proper subcontinuum K of X has h^2(K) inside K); m_j := min_x d(x, h^j x) > 0 for j = 1, 2; m := min(m_1, m_2). F: C -> C is a perfect extension of h (continuous, closed, compact fibres). The lab target L2 is that (S) is impossible; we work inside a hypothetical (S). - A crosscut q is an open arc in U whose closure is an arc with two distinct endpoints q^-, q^+ in X. Sh(q) is the bounded component of C \ (X u q). In (S), F has a unique outchannel E_t with external ray R_t, F^2 has a unique outchannel E_s with ray R_s, and Pr(E_t) = Pr(E_s) = X (SOURCE-lab, BFMOT as used in the lab's proof file PROOF_L2_v5). A crosscut q is an E_t-crosscut if a tail of R_t lies in Sh(q); then v_q is the first point of cl q met by R_t from infinity and rho_q the part of R_t from v_q to infinity (the access of q); s_q is the ray depth of v_q. q is deeper than c if q is inside Sh(c). - Intersection number: for a path beta missing v, with ends off rho, I(beta, rho) := winding number about v of beta followed by a path in C \ rho. (I3) a path in C \ rho has I = 0; (I4) I is homotopy invariant rel endpoints in C \ {v}; (I5) two paths with the same endpoints in an open convex set missing v have the same I; (I6) additivity: for a ray split at a point, I(beta, rho) is the sum of the two parts. I_w(beta) denotes I against an access from w in U. Lemma SP (DERIVED): for paths whose endpoints lie in X, I_w depends only on w and is constant on the components of U minus the path. - Thresholds: r' <= m/6; r_1 <= r' with F(B(y, r_1)) inside B(h(y), r') for y in X; delta_0 <= r_1 with F(B(x, delta_0)) inside B(h(x), r_1); r_0 with F(B(y, r_0)) inside B(h(y), r_1). - The node. c = (a, b) is an E_t-crosscut with diam c < delta_0 and access rho_c from v_c. gamma := the segment [h(a), h(b)], inside B(h(a), r_1). An M-piece m is a component of gamma \ X that is not an E_t-crosscut; n_c(m) := I_{v_c}(F o m). L-MPN: n_c(m) = 0 for every M-piece with diam m > eta(c), eta(c) -> 0 as diam c -> 0. WRAP-MPN := the M-pieces with n_c(m) != 0. Fact T1: I_{v_c}(m) = 0 for an M-piece. - L-MPN-REL (the relative form, A164): the signed sum of n_c over the next-scale pieces equals the parent index, uniformly as diam c -> 0; WRAP-MPN is the set of pieces where this bookkeeping can fail. B. LEMMAS (status; where used) Known (DERIVED by one agent, wrapmpn_prove_opus_1102 section 4, UNAUDITED, compressed here). - Base pull-back. For an M-piece m put x^+- := h^-1(m^+-) (both within omega_{h^-1}(omega_h(diam c)) of a) and tau := [x^-, x^+]; let t range over the pieces of tau (crosscuts near a). Then F o tau and m are paths from m^- to m^+ inside B(m^-, r_1), a disk that misses v_c. (PB0) n_c(m) = I_{v_c}(F^2 o tau). (PB1) sum over t of I_{v_c}(F o t) = I_{v_c}(m) = 0. (PB2) sum over t of I_{v_c}(F^2 o t) = n_c(m). Only finitely many terms are nonzero. (Proof: (I5) in the convex disks, plus a sup argument along the compact set of parameters of tau cap X, whose gap-jumps are the I-terms of the pieces.) - A181 (Arena, model unknown, UNAUDITED; free-API check CONFIRMED-WITH-REPAIRS): the channel-coherence law holds in (S): no entangled pair q in Sh(c), c in Sh(q); every index-carrying piece at the next scale is an E_t-crosscut; coherent at every finite scale. Claim 1.1: v_c in Sh(q) iff Sh(c) in Sh(q) (needs a ray-crosscut intersection lemma). - MICROEVAL m21 (5 models, UNAUDITED): under H-EQ, n_c(m) = sum_t (D_2(t) - D_1(t)) as a finite sum of differences; for an E_t-piece D_k(t) = signed intersection of F^k o t with the ray segment R_t[v_c, v_t]; D_k(t) != 0 iff the ray's visits to the image ball separating the image endpoints are unbalanced. The cancellation across pieces is OPEN (= DETOUR). - A188 (Arena, model unknown, UNAUDITED, Tier 0 CANDIDATE): coherence does NOT force cancellation: an explicit coherent all-finite-scale shadow/sign tower carries residual +2 and satisfies every finite Tier-0 instance of (S); realisation by one (S)-host is OPEN. A178's earlier +2 germ was inadmissible (its v_c lay in X at Scale 1). C. THE TARGET L-MPN-REL in (S), equivalently: no coherent tower with nonzero residual is realised by an (S)-host. D. ASK Part 1 (the tower's anatomy). Reconstruct the simplest coherent tower with residual +2 from the description above (state the scale-n data: shadows, signs, which pieces are E_t, the ray visits that produce the residual via the m21 criterion). Say which finite instances of (S) it satisfies and why coherence holds at every scale. Part 2 (global laws). Test the tower against laws that are NOT finite-scale: (a) cm of h^2 (does the residual force a proper subcontinuum K with h^2(K) inside K, e.g. a limit of the pieces carrying the residual?); (b) uniqueness of the outchannel E_t and Pr(E_t) = X (does the residual need a second ray visiting the image balls infinitely often?); (c) positive displacement m (do the image balls B(m^-, r_1), B(h(m^-), r') shrink coherently?); (d) compactness: is the scale-n residual stable under passing to the limit, or does it need pieces whose diameters do not tend to 0 (violating eta(c) -> 0)? Prove one exclusion or show each passes. Part 3 (builder). If Part 2 finds no violated law, give the strongest realisation you can (an explicit plane picture of two scales with F defined near X, or an inverse-limit construction) and the first property you cannot verify. Each claim labelled PROVED, SOURCE (exact reference), LAB, DERIVED, CONJECTURE, OPEN or RESTATEMENT. UNKNOWN if unsure a cited result exists. E. CONTROLS (all must hold) (C1) The termwise identity of m21 must be used as a finite sum of differences, not two separate series. (C2) Periodic control (by hand): X = [-1,1] x {0}, h(x,0) = (-x,0) has a fixed point and h^2 = id is not cm; it lies outside (S) and only fixes sign conventions. (C3) A tower that is coherent only up to scale n for each n is not a realised host. F. DEAD ENDS (not progress) (1) Coherence alone (A188's tower). (2) Lemma LV at every piece (the ray may enter the image balls). (3) A178's germ (inadmissible at Scale 1). (4) Renaming the cancellation as "no WRAP-MPN piece exists" (RESTATEMENT). RESTATEMENT GATE (mandatory; answered BEFORE the headline, in its own section). If your verdict reduces the node N to a residual lemma L (with or without other proved pieces), you must give ALL of: (G1) the exact statement of L and of N side by side, with the same standing hypotheses; (G2) a proof that L implies N (the reduction); (G3) a reason L is STRICTLY weaker than N: either an explicit configuration (or consistent abstract data meeting every standing law) where L holds and the conclusion of N is not yet available, or the named hypothesis of L that the negation of N does NOT supply; (G4) the line of your argument where L's own hypothesis (the defect, the witness, the scale) is actually used; if it is never used, L is a restatement. If G3 or G4 fails, the headline word is RESTATEMENT, not REDUCED, and the answer reports the proved building blocks only. A headline REDUCED without G1-G4 counts as a failed answer. G. FORMAT Line 1 exactly: VERDICT: TOWER EXCLUDED (a named global law of (S) - cm of h^2, uniqueness of E_t, positive displacement, compactness/realisation - is violated by every coherent tower with nonzero residual; proof) | TOWER REALISED (explicit planar data or a construction realising a +2 tower in one (S)-host, Tier 1 rules) | REDUCED (to one named lemma, gate G1-G4 passed) | PARTIAL (one named sub-claim settled) | 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 a finite-scale tower as one infinite (S)-host; using coherence as if it implied cancellation (it does not: the +2 tower is coherent); using an extension of h to a plane homeomorphism; summing detour series separately when only their differences have finite support; calling the tower a counterexample to L2 or PER