m9 yay

Prompt

RESEARCH QUESTION (plane topology / dynamics; adversarial audit of Theorem PROT and Corollaries RES, BR1). 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 D2597, D2595, D2593 and a_fable_dtower; DERIVED-U there, unaudited here; lines marked (INFERRED) are my bridging). - Host: Zhat := the profinite integers, D: Zhat \ {0} -> E_amb continuous and even with D(2N) = sigma D(N); E := the cluster set of D at 0 (a Cantor set, infinite). Fibre points p_m(f) (m in Z, f in E), fibres P_m := {p_m(f) : f in E}; X := K x {0,1} with K the closure of the regular points g(z) = (z, (D(z-j))_j); the host Y = Y_D := ([0,1] x K)/folds with fence F and fold Phi (iota_w p_m(e) = p_{-w-m}(e)) is planar iff 2PAGE holds, where 2PAGE := there is an embedding P: X -> S^1 in which F and Phi are noncrossing matchings (D1535). We fix such a P. - Hulls: 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); H^0(n) := P^-1(Hull(P_n^0)) is the hull, delta_n := diam_X H^0(n) -> 0. A jump of the fibre order of P_n^0 is an open arc between consecutive points; a point of H^0(n) \ P_n^0 is an intruder. - Dynamics: H[t,x] = [2t,2x] has no periodic point; h := H^-1; m_k := min_Y d(y, h^k y) > 0 for every k. Ray coding l_f(T), T >= 0 real, with H l_f(T) = l_{sigma f}(2T). For a positive integer m the two X-preimages of the folded ray point l_f(m) are (p_m(f),0) and (p_-m(f),0). A turn q >= M_0 is the fold step P_{-q}^0 -> P_q^0; it is interleaving if the hulls of P_q^0 and P_{-q}^0 meet. (S): every far turn is non-interleaving. - A ray point lies in the band of a far jump at J-position u if, in the P-order, it lies in a jump of the fibre order of P_u^0 (INFERRED). B3 (a_fable_dtower): at a far non-interleaving turn q, every intruder z of H^0(-q) lies in a jump of the fibre order of P_{-q}^0, and its fold image lies in that same jump (z and its image form an internal pair) or in the matching jump of H^0(q); symmetrically for H^0(q). **Integer member.** A fibre point z = (p_a(f), 0) in a far hull H^0(n) with |a| != |n| is an *integer member at J-position |n|*, of ray-depth T = |a|, ratio rho = |a|/|n| and excess g = |a| - |n|. (a = n would be a point of P_n itself; a = -n is an interleaving turn, excluded by (S).) By B3 it lies in a jump of the fibre order of P_n^0. (D2597 section 2 calls the same objects integer members.) **Thresholds.** - M_0 is the far threshold of a_fable_dtower 1.3, enlarged so that every turn q >= M_0 is non-interleaving (S) and B0-B4, B6 hold for |n| >= M_0. - mu_k := min(m_1, ..., m_k) > 0. M#_k is the least M' >= M_0 such that delta_n < mu_k for all |n| >= M', and the real slices of D2595 SL(b) beyond M' have diameter < mu_k. It exists because delta_n -> 0. NF* at all powers <= k holds at every real J-position >= M#_k, and M#_k is nondecreasing in k. (D2595's M_* is defined by the same conditions for k = 1; I take M_* = M#_1.) - K-flat(q) := max{k : M#_k <= q}, the number of protected powers from q on. K-flat(q) -> infinity as q -> infinity. - M_T := (3/2) M#_1. **Input T (transport; DERIVED-U: D2597 CH1 with a_fable_dtower B3, B6; equivalently D2595 Lemma TR).** Let z be an integer member at ray-depth T_0 and J-position u_0 >= M_0, on the ray l_f. Follow l_f forward (T increasing). There is a J-position u(T), defined for all real T >= T_0 as long as u stays >= M_0, such that l_f(T) lies in the band of a far jump at J-position u(T). Properties: - u is continuous and piecewise linear, with slope +1 on positive-index stretches and -1 on negative-index stretches; - u changes slope only at internal pairs: a death cap or a birth cup at an integer turn. Partners share the ray point, hence the ray-depth and the J-position (D2597 CH1); - at an integer T = m, u(m) is an integer, and one X-preimage of l_f(m) is an intruder of H^0(u(m)) or of H^0(-u(m)); - the same holds backwards. - **Input NF* (DERIVED-U: D2597 Lemma NF*; restated from the Thresholds line above).** At every real J-position >= M#_k, an integer member never has ratio 2^j or 2^-j for j <= k, for either index sign. - **Input HID-*(b) (IN-SO), in the form BR1 uses (shortened from D2595 Lemma HID-*; its hypotheses S*, HC, EMB, VT, EXT-_c and var != 0 are cited, not restated).** Let J = (e', e) be a persistent jump from T_0, c a slot gap (a K-gap) of J at generic depth t, k >= 1 the power, u := 2^k t >= 3M_*/2 and g* the pass of c (a slot segment with both sides in the slice C(u)). If g* lies in the band region of J at a section T' with T_0 <= T' <= u - M_*/2 (the case IN-SO), then Hull(P^0_{A(u)}) lies in the open jump arc of J at fibre A(T'): the whole image fibre, including J's own walls at depth u, is content of J at depth T'. (Here A(u) is the fibre index of the slice at depth u: A(u) = floor u if frac u < 1/2, else -1 - floor u.) - UNPROT := the lab's residual node: IN-SO passes (far integer members) at an unprotected ratio, i.e. T' < 2M_T or u >= T' 2^(K-flat(T'/2)); OPEN there. BR1 below reduces the SO- branch 1 to it. B. LEMMAS (status; where used) **Lemma 1 (the excess is monotone; DERIVED).** Along forward transport, g(T) := T - u(T) is nondecreasing: g' = 1 - u' equals 0 on positive-index stretches and 2 on negative-index stretches. So a member deeper than its hull (g > 0, ratio > 1) stays deeper forward, its ratio never reaches 1 forward, and its excess never drops. Backwards, g is nonincreasing. (This is D2597 CH2's "positive offsets strictly increase", read along the ray.) **Lemma SC+ (DERIVED; this is Theorem SH+- of D2595 with M_* = M#_1, re-derived inside the transport calculus; P yes / M h with m_1 > 0 / C no shallow fibre point in a far hull).** Let |u_0| >= M_T, and let z be a fibre point of ray-depth T_0 lying in H^0(u_0) (either sign of u_0). Then T_0 > |u_0| - M#_1/2. In words: every fibre point inside a far hull is deeper than the hull, up to M#_1/2. **Theorem TRAP (DERIVED; P yes / M h with m_1 > 0 / C no ray stays below its far hulls).** Assume (S). There are no f in E and integer T_0 such that, for every integer m >= T_0, the ray point l_f(m) lies in the band of a far jump at an integer J-position n(m) with M_T <= n(m) <= m - M#_1/2. C. THE TARGET Octaves: O_0 := (1, 2), and O_i := (2^i, 2^(i+1)) for i >= 1. **Theorem PROT (DERIVED; P yes / M h / C no far integer content at a protected ratio).** Assume (S). Let z be an integer member at ray-depth T_0, J-position u_0, ratio rho_0 in O_i (i >= 0) and excess g_0 = T_0 - u_0 >= M#_1/2. If T_0 >= 2^(i+1) max(M#_{i+1}, M_T), we have a contradiction. The same holds for rho_0 = 2^i exactly (i >= 1) when u_0 >= M#_i, by NF* directly. *Proof.* Transport z forward. By Lemma 1 the ratio stays > 1 and the excess stays >= g_0 >= M#_1/2. While the ratio stays in O_i up to ray-depth T, we have u(T) = T/rho(T) > T/2^(i+1) >= T_0/2^(i+1) >= M#_{i+1}. So NF* holds at the powers i and i+1 at every J-position reached, and the continuous ratio can never take the values 2^i (i >= 1) or 2^(i+1). The lower end 1 of O_0 is never reached, by Lemma 1. Hence the ratio stays in O_i for all T >= T_0, and for every integer m >= T_0: M_T <= T_0/2^(i+1) < u(m) <= m - g_0 <= m - M#_1/2. This is the hypothesis of TRAP with n(m) := u(m). QED So the octave (1, 2) needs only m_1 (k = 1), and the octave (2^i, 2^(i+1)) needs m_1, ..., m_(i+1). D2597 called (1, 2) an "excluded octave" because lives there are unbounded. That does not matter here: an immortal member in (1, 2) traps its ray just as well. **Corollary RES (unprotected ratio; DERIVED).** Every integer member at J-position u_0 >= 2M_T with excess >= M#_1/2 has ratio rho >= 2^(K-flat(u_0/2)). Equivalently, u_0 < 2M#_{i+1} with i = floor(log2 rho). *Proof.* If rho lies in O_i, then T_0 = rho u_0 >= 2^i u_0, and PROT forbids 2^i u_0 >= 2^(i+1) max(M#_{i+1}, M_T). So u_0 < 2 max(M#_{i+1}, M_T), which equals 2M#_{i+1} because u_0 >= 2M_T. Then M#_{i+1} > u_0/2, i.e. i+1 > K-flat(u_0/2). QED Members with ratio < 1 and deficit |u_0| - T_0 >= M#_1/2 are excluded by SC+ outright. For far hulls the deficit condition should be automatic (SKETCH, not used by TRAP, PROT or BR1): with the same sign the offset is the deficit itself and B4 makes it >= k(delta)!; with opposite signs, NF's "|y|/2 not inside a run" forces a birth b within half the deficit of u_0, and B4 with ||2b|| <= delta (a_fable_dtower B6) then gives a nonzero integer below the deficit with norm <= delta, hence a deficit >= k(delta)!. **Corollary BR1 (branch 1 of SO- is empty; DERIVED).** Take the SO- setting (D2593; D2595 HID-*; D2597 THICK-PASS). Let an IN-SO pass of a K-gap at depth t, for the power k, sit at the section T' of J, with u := 2^k t and u - T' >= M#_1/2. - By D2595 HID-* (b), the fibre P^0_{A(u)} lies in the jump of J at J-position T'. (D2597 THICK-PASS gives at least one such point; its shallow case is excluded by SC+, section 7.) - So its points are integer members of ratio u/T' and excess u - T'. - By RES, whenever T' >= 2M_T: **u >= T' 2^(K-flat(T'/2))**. Hence for every eps > 0 no IN-SO pass with T' >= eps u has u >= t_eps := (2/eps) max(M#_{floor(log2(1/eps))+1}, M_T). (From T' >= eps u we get i = floor(log2(u/T')) <= floor(log2(1/eps)), so T' < 2M#_{floor(log2(1/eps))+1}, so u <= T'/eps is bounded. If T' < 2M_T, then u < 2M_T/eps.) **Branch 1 (SHORT-RUN at the pass scale) is empty beyond t_eps, with no THIN-ONE or REG.** This covers D2595's whole IN-SO range T' <= u - M_*/2, including ratios in (1, 2), which D2597's THICK-PASS (T' <= t) did not cover. D. ASK Assume TRAP and Lemmas 1 and SC+. Check PROT, RES and BR1 above step by step: the octave thresholds, the case rho = 2^i exactly, the use of K-flat, and the claim that branch 1 of SO- is empty beyond t_eps. E. CONTROLS (all must hold) (C1) Each threshold used must match its definition in A. (C2) RES must not claim more than PROT gives for ratios in (1, 2). (C3) BR1 uses Input HID-*(b) of section A: say whether that input is used within its stated range (T_0 <= T' <= u - M_*/2). F. DEAD ENDS (not progress) Attacks on TRAP itself (separate task); restating UNPROT. G. FORMAT Line 1 exactly: VERDICT: PROT CORRECT | CORRECT WITH REPAIRS (list) | BROKEN (failing step + configuration) | 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: re-proving TRAP (assume it here); octave boundaries rho = 2^i treated loosely