batch8

Prompt

R147_LTDUNB_prompt.txt 100 % RESEARCH QUESTION (PAL folded suspensions / unbounded leaf-time distortion). 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 PRE-DISPATCH CERTIFICATE (batch 8, owner microeval channel). (C1) R2 / NoTCM_{h=0}, GAP-LTD-UNB, route DISCHARGE / CHANGE-MAP. For every planar nonseparating PAL fold and zero-entropy H with sup_{s,u}|c(s,u)-u|=infinity, force a periodic end, reduce to bounded displacement preserving the end question, or build a planar PAL H with unbounded displacement and no periodic end. YES eliminates/reduces this sector; NO defeats this periodic-end route, not automatically PPF on the whole host. (C2) Controls: finite-end const3/silver force a periodic end without a displacement bound. K_Toe has unbounded rank but M_ht=1; its bounded-displacement closure cannot be imported here. Global controls and first exits appear below. (C3) New: Argon95 Base-Jump Identity works before its bounded-displacement height/measure estimate. Test cocycle straightening or variable-time returns at this exact boundary. (C4) Argon96 owns bounded-displacement L_disp (G3, perfect-end builder, prover); COH-PAL owns commuting cohomological construction. This infinite-sup task is distinct. A34/A35, ds_R146T, batch7 R140-R146 and launched web jobs are excluded by the writer's queue/preflight checks. NoTCM_{h=0} means no nondegenerate planar tree-like PPF tcm homeomorphism has zero entropy. Definitions: a continuum is a nonempty compact connected metric space; subcontinua are closed and connected; nondegenerate means >1 point. Planar means embeddable in R^2, nonseparating means connected plane complement. Tree-like means admitting onto epsilon-maps to finite trees for every epsilon>0 (each fibre diameter <epsilon). PPF means H^n(x)!=x for all x,n>=1. Continuum-minimal (cm) means no proper nonempty invariant subcontinuum; tcm means every positive power is cm. Minimality of point orbits is stronger. Indecomposable means not a union of two proper subcontinua; HI means every subcontinuum is indecomposable. Monotone maps have connected fibres; light maps have zero-dimensional fibres. h_top denotes topological entropy. A reversible minimal Cantor system consists of a Cantor space K, a homeomorphism tau with every orbit dense, and an involution i0 with i0 tau i0=tau^{-1}. Its suspension is S=(R x K)/((t+1,x)~(t,tau x)); write [t,x], Phi_u[t,x]=[t+u,x], R[t,x]=[-t,i0 x], Y=S/<R>, and Q:S->Y. The flow is free. Each flow line contains at most one R-fixed point. The intrinsic end set is E=Q(Fix R); it is compact and H-invariant for every homeomorphism H of Y. Every H has two lifts commuting with R; F denotes the lift preserving flow orientation. Its time cocycle is defined by F(Phi_u s)=Phi_{c(s,u)}F(s), with c(s,0)=0, c(s,u+v)=c(s,u)+c(Phi_u s,v); u->c(s,u) is increasing and onto. Define c^{(m)} by F^m(Phi_u s)=Phi_{c^{(m)}(s,u)}F^m(s). PAL means that there exist clopen first-return partitions P_n={tau^l B_{n,j}:1<=j<=J_n,0<=l<h_{n,j}} of K, refining and generating its topology, with bases B_n=disjoint union_j B_{n,j} decreasing to {x*}; i0 B_n=B_n; and phi_{n,j}(y)=tau^{-h_{n,j}}i0(y) preserves B_{n,j}. Heights h_{n,j} are the actual first return times to B_n. Put p*=[0,x*], e*=Q(p*), V_n=E intersect Q({[0,y]:y in B_n}), D_ht(n)=|{h_{n,j}:1<=j<=J_n}|, M_ht=liminf_n D_ht(n). For e outside V_n its unique fixed lift p_e is a tower midpoint; j_n(e) denotes that tower, L_n(e)=h_{n,j_n(e)}/2, and q_n(e)=Phi_{-L_n(e)}p_e lies in the base section. PAL supplies a finite-star inverse-limit tree-like certificate; it does NOT by itself supply a planar embedding. These are LAB-DERIVED audited inputs H0ABS_v3 / TCMH0_v3 (repairs closed 2026-11-06); no independent reconstruction is claimed here. The supremum sup_{s,u}|c(s,u)-u| is over S x R, unlike the automatically bounded sup_s d(Fs,s). A straightening Psi(s)=Phi_{a(s)}s must be a homeomorphism with continuous a, compatible with R; a roof change must be continuous positive. Prove that conjugacy/clock changes preserve the host, end action and relevant boundedness; boundedness in an unrelated clock is insufficient. B. LEMMAS (status; where used) KNOWN: Argon 94 Thm B2 (LAB-DERIVED-UNAUDITED) defines, when H^k e*!=e*, T_n(k)=c^{(-k)}(F^k p*,L_n(H^k e*)). Bilateral midpoint returns imply: if distinct fixed nonzero k1,k2 satisfy liminf_n |T_n(k1)-T_n(k2)|<infinity, then H^{k1-k2}e*=e*. With sup|c-u|<=C this yields bounded-height-diversity closure p(e*)<=M_ht+1. The last implication is unavailable when C=infinity. Live source update (16:0x, p61106A94AUD cross-family audit): B1-B2's return argument passes under the displayed PAL/lift hypotheses, but the article has GAP at B5(iv). Its local strips do not give compatible planar embeddings; 'chainable implies at most two intrinsic ends' is false. K_Toe is only an abstract calibration, with symbolic repairs owed; never assume its planarity/nonchainability. Argon95's similar planar-builder claim is also unestablished. These builder defects do not refute the conditional return identity. The commuting COH-PAL lane now owns TRANSVERSE-KERNEL COH-PAL; no repeat requested here. Argon 95 Thm C1(ii)-(iii) (LAB-DERIVED-UNAUDITED, not audited here): for m!=0 and e outside V_n union H^{-m}V_n, delta_n^{(m)}(e)=L_n(H^m e)-c^{(m)}(p_e,L_n(e)), F^m(q_n(e))=Phi_{delta_n^{(m)}(e)}q_n(H^m e). If for fixed m and B<infinity there are n_i->infinity and eligible varying e_i with |delta_{n_i}^{(m)}(e_i)|<=B, then F^m p*=p* and H^m e*=e*. Both base points tend to p*; a bounded-time subsequence gives the same R-fixed point on one free leaf. This identity/implication does not assume bounded displacement. In contrast the estimate |delta-(h_target-h_source)/2|<=|m|C, positive-mass height-cluster closure (Thm C2), and its dispersion residual require C<infinity. Argon 95's L_disp has perfect ends, uniform delta divergence, and dispersed height classes on every invariant measure/subsystem, within that bounded sector. Do not apply its measure criterion here without proving a replacement. Bounded rank J_n of these self-reversed PAL partitions along a subsequence gives finite E, hence periodic ends for arbitrary H (Argon93, CWR with repairs owed). This is not a theorem about arbitrary topological rank. The new target has infinite E/unbounded J_n; do not repeat the finite-end case. Zero entropy supplies no bounded time deviation. C. THE TARGET OPEN: GAP-LTD-UNB as quantified above. Seek either (a) a conjugate/return-clock reduction which preserves planar Y, zero entropy, PAL applicability, and periodic-end equivalence and genuinely gives a bounded cocycle; (b) a direct bounded-delta subsequence forced for some fixed m; or (c) an explicit planar PAL H with sup|c-u|=infinity and no periodic end. Any weaker builder is a calibration with its first GAP, not a negative answer to the full target. D. ASK Locate the first step beyond Argon 95 that can actually control variable return time. Prove your sharpest claim. If using straightening, write the transfer/roof equation and prove positivity, continuity, bijectivity, R-compatibility and what happens to c and E. If using the Base-Jump Identity, fix m and B before choosing n_i and e_i; show why the same m recurs. If building, give exact reversible tower/bond data and time change, prove all-level coherence and the embedding, calculate the cocycle and end action, and distinguish no periodic end from no periodic point. A local interval perturbation, finite tower table or reparametrization with no global inverse is not the required object. Conclude with the exact remaining hypothesis if only a restricted theorem is obtained. Research, not audit. Use UNAUDITED inputs conditionally as ASSUMED-name or re-prove them. Label new results DERIVED/COMPUTED/CONJECTURE/FAILED/OPEN; give dependencies, weakest row and insight. Check hypotheses, degeneracies, quantifiers and limits. A reduction needs implication and strictness proofs or is RESTATEMENT. Builders need all-level coherence, continuity, inverse and planar embedding proofs or GAPs. No novelty/VERIFIED/CLOSED claims. No helper agents/sessions; checks are not independent-agent checks. Computations are optional: exact arithmetic, independent verifier/negative tests, <=2 workers, <2 GB/process; one foreground heavy run after memguard OK, z3 memory_max_size=2000 before building and timeout<=600000 ms/check. E. CONTROLS (all must hold) Global controls (SOURCE-lab facts, not independently reconstructed): the pseudo-arc has the fixed point property, never a PPF control. The Knaster bucket-handle shift fixes 0hat=(0,0,...), so is not cm. SPIRAL(3) and the pseudo-circle separate the plane. On S^1, z -> -z is fixed-point free with period 2, not PPF. Irrational circle rotation is PPF but not tree-like; irrational disk rotation fixes its centre. State the first exit of every added control; topology without the required dynamics is not a target-class example. PAL controls: const3 a->aba,b->aaa and silver a->aba,b->a have J_n=2 and three ends in the lab calibration. Finite E forces a periodic end for every H; Argon93's closure is CWR with counting/source repairs owed, not VERIFIED. K_Toe (Argon94 Proposition B5, UNAUDITED) has J_n=2^n, equal heights h_n=9 product_{a=1}^{n-1}(2^{a+1}+3), M_ht=1 and uncountably many ends; its claimed planarity is UNAUDITED. The bounded-displacement theorem applies only with that extra bound. Argon95's UNAUDITED K_var has a unique isolated end, fixed by every H; K_var^perf has perfect ends but a common prefix fixes the distinguished point under sliding-block centralizers. Neither is PPF. The planar Cantor fan with H[c,t]=[Tc,t] fixes its apex and invariant truncations, failing PPF/tcm. F. DEAD ENDS (not progress) Do not repeat: Argon 93 finite-rank proof; Argon 94 bounded-height/bounded-displacement pigeonhole; Argon 95's L_disp escapes or Argon 96's three jobs; weak-mixing inverse-product PAL builders with nonzero fold cohomology; assuming that zero entropy implies equicontinuity or a constant roof; choosing m depending on n and then declaring periodicity. A finite-star inverse limit proves tree-likeness, not planarity. A bounded continuous point displacement along leaves is different from uniform bounded deviation of the time cocycle. No ambient extension is supplied by the intrinsic H. G. FORMAT Line 1 exactly: VERDICT: PROVED | COUNTEREXAMPLE-CANDIDATE | PARTIAL | OPEN-OBSTRUCTION | RESTATEMENT. 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: confusing bounded point displacement with bounded time cocycle; importing bounded-displacement estimates; L_disp duplication; no-periodic-end versus PPF Zobrazuje se R147_LTDUNB_prompt.txt.