m47

Prompt

RESEARCH QUESTION (plane topology / intersection indices; RIG_S(C) of Route I (the second RIG instance of Theorem R*: the constructed shallower crosscut C with the old crosscut c as threshold): its net-crossing form against the ray piece between the first hits of C and of c, and what it adds to RIG_S(c)). 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 b_opus_rig sections 2 to 5 and research/ROUTE_MAP_L2.md; DERIVED there by single agents, UNAUDITED; lines marked INFERRED are mine and are among the things to check). - X is a nondegenerate nonseparating plane continuum, U := C \ X, phi: D -> U cup {infinity} conformal. R_t is the external ray of a prime end E_t, run from infinity towards X (time increasing), missing X. F is a perfect continuous plane extension of a homeomorphism h of X; in the R-class (its members are the R-instances) h^2 is continuum-minimal (cm: no nonempty proper subcontinuum K with h^2(K) inside K), so m := min(min|h(x) - x|, min|h^2(x) - x|) > 0 and a, h(a), h^2(a) are pairwise at distance at least m. RIG_1 is the variant: h cm without fixed point, nothing assumed on h^2. - A crosscut is an open arc of U whose closure is an arc with two distinct ends in X; Sh(q) is the bounded component of C \ (X cup q). q is an E-crosscut if a tail of R_t lies in Sh(q). v_q is the first point of cl q met by R_t coming from infinity; rho_q is the closed piece of R_t from v_q to infinity, meeting cl q only at v_q. q is deeper than c if q lies in Sh(c); then R_t crosses cl c first, v_c precedes v_q, and rho_c is the subray of rho_q starting at v_c. - Intersection calculus: for an access rho with start v and a path beta missing v with ends off rho, I(beta, rho) is the winding number about v of beta followed by any path from its end to its start in C \ rho. (I1) additive; (I2) antisymmetric; (I3) zero if beta lies in C \ rho; (I4) homotopy invariant rel ends in C \ {v}; (I5) two paths with the same ends in an open convex set not containing v have the same I; (I6) if rho' is the subray of rho starting at u and S the segment of rho from v to u, then I(beta, S) := I(beta, rho) - I(beta, rho'), additive along subdivisions. The positive orientation c^+ of a small E-crosscut is the one with I(c^+, rho_w) = +1 for w in Sh(c). - CP1 laws (Lemma 4.2, 4.3; DERIVED from Theorem CP1, UNAUDITED): for a small crosscut q and a path beta from h(a_q) to h(b_q) in an open convex ball containing F(cl q) and missing v_q, var(F, q^+, X) = I(beta, rho_q), equal to -1 for an E-crosscut. Write V_j(c) := I([h^j(a), h^j(b)], rho_c) = var(F^j, c, X). So V_1(c) = -1 for every small E-crosscut c, and in an R-instance V_2(c) is in {0, -1}. - RIG_S(c) (ROUTE_MAP_L2 node): c = (a, b) a small E-crosscut inside Sh(c_*), z_1 = h(a), z_2 = h(b), sigma = [z_1, z_2], rho small, F(cl c) inside B(z_1, rho/8). Then I(F o c^+, rho_q) = -1 for every E-crosscut q in the annulus B(z_1, 3 rho/2) \ B(z_1, rho/2), deeper than c and far from c (disjoint closures). - Theorem R* (LAB): assume RIG_S(c). Then there is a crosscut C inside B(z_1, 3 rho/2), from x_1 to x_2 with x_i within 2 eps of K(z_i) (K(z_i) the component of X cap cl B(z_1, rho) containing z_i), with I(C, rho_c) = -1: C is an E-crosscut shallower than c, c inside Sh(C). If moreover a_{C^+}(c) := I(F o C^+, rho_c) = -1 (RIG_S(C): 'RIG at C with threshold c') then V_2(c) = I(F o C, rho_c) = +1, against V_2 in {0, -1}. Route chain: L2 follows from S, LAWS, RIG_S(c), RIG_S(C), Rstar. B. LEMMAS (status; where used) Known and inferred. - Scope (INFERRED, CHECK): RIG_S in the annulus form tests thresholds q in the annulus about the centre of F(cl C), which lies near h^2(a); the threshold used by Theorem R* is q = c, which lies near a, at distance at least m from that annulus. So RIG_S(C) is an instance of RIG (all deeper far thresholds), not of RIG_S in the annulus form (ROUTE_MAP_L2 section 3 calls it 'an instance of RIG'). A Sol job on the scales (solfeed1102d) was queued; no answer is on file (UNKNOWN). Argon 52 repaired only 'the family of small c must cover c and C'. - NET-C (INFERRED, spec author; CHECK FIRST). Let sigma_C be the straight segment from F(start of C^+) to F(end of C^+) and S_Cc := R_t[v_C, v_c] (c is inside Sh(C), so R_t meets cl C first, rho_C is the subray of rho_c starting at v_C, and S_Cc is the segment of rho_c from v_c to v_C). F(cl C) lies in a convex ball missing v_C and v_c. Then (I5), (I6) and V_1(C) = I(sigma_C, rho_C) = -1 give a_{C^+}(c) = I(sigma_C, rho_c) = I(sigma_C, rho_C) + I(sigma_C, S_Cc) = -1 + I(sigma_C, S_Cc). Hence RIG_S(C) holds iff I(sigma_C, S_Cc) = 0, and V_2(c) = -a_{C^+}(c) = 1 - I(sigma_C, S_Cc). In an R-instance with RIG_S(c), V_2 in {0, -1} forces I(sigma_C, S_Cc) in {1, 2}: the ray, between its first hits of C (near h(a)) and of c (near a), crosses a short segment near h^2(a) with net positive count. This has the shape of RIG-NET for RIG_S(c) (the arc R_t[v_c, v_q] against sigma near h(a)). - Alternation (b_opus_cap_nonxs, DERIVED): in an R-instance RIG_S(c) implies that RIG_S(C) fails for the C it builds. On the R-class RIG_S is a restatement of L2; RIG_1 is the only non-vacuous reduction and is OPEN. Tower control (b_opus_rig): at a tower level g_n^2 is not cm and V_2 = +1 is allowed. - Partial results on RIG (UNAUDITED): Prop J (accessible ends: RIG_S(c) iff F(a), F(b) are joined by a crosscut near z_1 with c in Sh of it); Prop PC (product charts); Theorem G (a Denjoy germ violating RIG_S(c), not a global map); the F-free form of the door law is RIG_S. (C1) X the closed unit disc, h a rotation, F the rotation of the plane: sigma_C lies in X, V_1 reads 0, not -1; the CP1 hypotheses exclude it. A proof must consume V_1(C) = -1 or the dynamics. (C2) Tower level: V_2 = +1 allowed, so I(sigma_C, S_Cc) = 0 is consistent there; an argument that uses the F^2 laws is about R-instances only, and an argument claimed for RIG_1 level must not use them. (C3) Denjoy germ (Theorem G): RIG_S(c) fails at cascade gaps; your argument must say what makes the germ not a global homeomorphism. (C4) Orientation: I(sigma_C, S_Cc) = 0 does not depend on the orientation of C; V_1(C) = -1 does. (C5) A chord of the ray in a ball that crosses sigma_C twice with opposite signs has net 0 though the raw count is 2; the argument must work with nets, not raw counts. (C6) Control that the failed methods also pass: a local planar picture (X near z_1, a ray, two short segments, crosscuts c and C) can be drawn with V_1(c) = -1, RIG_S(c) true and I(sigma_C, S_Cc) = 0, and equally with I(sigma_C, S_Cc) = 1; both meet every F-law, so a solution must use a global law, and must say which. F. DEAD ENDS (not progress) (1) Segmentwise vanishing of the far data of the first generation: false (fold model, b_opus_rig section 2); only the aggregate sum follows. (2) Any h-free law (T1, SS, K, B, AB): the host laws hold on the Denjoy germ while RIG fails, and a weaving ray has nonzero net crossing (b_opus_rig section 6). (3) Length-area or modulus arguments for F (A134): F is not conformal. (4) Re-proving LEMMA_P, Lemma AB or PASSCELL (taken as given). (5) Sigma_0 contact in (XS) and the door branch (b_opus_rig_trap: the trapping criterion is lossless, Prop EQ, and contact is vacuous or automatic, so it restates L2; other tasks). (6) Assuming F(U) inside U or an action of F on channel angles (SC-EXCL: bump counterexample). (7) T2 descent for RIG_S(c) (ledger l.363-452, 514-515, 817-823; t2_diammono_opus_1102; m22 R22_FOREXC): the inheritance laws DI-same and DI-c fail by centre displacement; DI-new plus the one-step law restates 'no defect'; a planar two-branch cancellation giving V_2(c) = 0 meets every chart axiom (PDNE, R30_PDNE); a Gottschalk-Hedlund coboundary cannot force non-cancellation (pro1104p); Lefschetz and Dold indices vanish (pro1103f09). (8) Hyperspace trapping, UBS, Lemma U and stationary measures (SH, UG; pro1104k): lossless restatements of 'a proper h^2-subinvariant continuum exists'. 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: RIG_S(C) DECIDED (follows from the stated laws, or is independent of them with explicit data, or is equivalent to V_2(c) = +1 with the equivalence proved; name the laws used) | NET-CROSSING FORM WRONG (failing step and corrected form) | REDUCED (to one named lemma L, 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: proving only an h-free planar picture (a ray can weave between strands, so planar data alone cannot give 0); assuming RIG_S at C in the annulus form (the threshold c is not in that annulus); using the F^2 laws (V_2 in {0,-1}) while claiming a statement about RIG_1-level hypotheses; treating an aggregate identity as vanishing of each piece; assuming F(U) lies in U; claiming L2, RIG_1 or PER as proved; hiding that on the R-class RIG_S(C) fails whenever RIG_S(c) holds