[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-85661-en":3,"doc-seo-85661-105":29,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},85661,549758252649,"Ivy","https://ap-avatar.wpscdn.com/avatar/8000253669c5317157?_k=1778319167496531819",8,"Research & Report","Restricted Dynamic Geometric Complexity: Path-Space Reduction and Möbius–Jacobi Response","Structured preconditioners restrict optimization to a small family of positive metrics, yet endpoint condition-number reachability does not capture the geometric effort needed to obtain a useful metric. The framework turns this into a path-space value problem: the restricted dynamic geometric complexity is the least affine-invariant length of an admissible metric path reaching a Hessian-relative generalized-eigenvalue target. Exact min-plus semigroup and Bellman principles arise from path elimination, with fixed-horizon kinetic energy linking complexity to squared action; Tonelli and compact-control results yield existence and a conditional Hamilton–Jacobi layer. Global Hadamard-space results provide smooth intervention-cube branches and coercive Jacobi forms, explicit Green-response laws, and a bordered Jacobi–KKT analysis for hard targets.","arXiv :2607 .07204v2 [math .OC] 13 Jul 2026  \nRESTRICTED DYNAMIC GEOMETRIC COMPLEXITY:  \nPATH-SPACE REDUCTION AND M¨OBIUS–JACOBI RESPONSE  \nZAVIER LI∗  \nAbstract. Structured preconditioners restrict optimization to a small family of positive metrics, but endpoint condition-number reachability does not measure the geometric effort required to reach a useful metric. We formulate this effort as a path-space value problem. Restricted dynamic geometric complexity is the least affine-invariant length of an admissible metric path whose endpoint reaches a Hessian-relative generalized-eigenvalue condition target. Path elimination gives an exact min-plus semigroup and Bellman principle, while fixed-horizon kinetic energy is exactly squared complexity divided by twice the horizon. Explicit Tonelli and compact-control realizations supply existence anda conditional Hamilton–Jacobi layer. The main response result is global on a Hadamard state space: geodesic convexity produces a smooth intervention-cube path branch and a uniformly coercive Jacobi form, while one Green inverse generates the value Hessian, two-sided force-to-curvature bounds, exact M¨obius effects, and arbitrary prescribed finite-order responses. For the hard condition target, a bordered Jacobi–KKT theorem differentiates the moving projection endpoint and multiplier on every regular active spectral stratum; its indefinite inverse also explains why hard-target interactions need not share the unconstrained sign. The theory specializes to affine-invariant positive-definite geometry. A determinant-one two-dimensional diagonal model has an exact target interval, a closed-form forced path, and a strictly negative-definite interaction matrix. A moving diagonal Hessian gives a closedform hard-target projection, multiplier, and pair effects of either sign, while a coordinate-sequential three-dimensional protocol yields an exact path metric strictly larger than the ambient projection distance. Thus the global Green and bordered hard-target responses are explicit laws of restricted metric-path elimination built on Bellman composition.  \nKey words. structured preconditioning, affine-invariant geometry, dynamic programming, Jacobi operator, parametric optimization, M¨obius interaction  \nMSC codes. 49L20, 49K40, 53C23, 65K10, 90C25  \n1. Introduction. Structured preconditioners constrain optimization to a small family of positive metrics. This creates an endpoint question and a path question. The endpoint question asks whether the family can reach a prescribed condition number. The path question asks how much intrinsic metric motion is required to reach it from the current state. Endpoint feasibility alone does not measure this geometric effort: two families may reach the same target but place it at very different affine-invariant distances from the initial metric.  \nWe formulate the path question as a value problem. For a quadratic objective with Hessian H ≻ 0, a metric G ≻ 0 induces the generalized eigenproblem Hv = λGv. In relative coordinates  \nS = H −1/2GH −1/2 ,  \nthe relevant condition ratio is  \nλmax (G−1/2HG −1/2)  \nκgen (H, G) := ~~ ~~ = κ (S)  \nλmin (G−1/2HG −1/2) .  \nThis is the positive generalized-eigenvalue ratio of Hv = λGv, which is invariant under inversion of the relative spectrum. Given a metric family F, an initial state S0 , and K ≥ 1, restricted dynamic geometric complexity DK,F(S0 ;H) is the least affine-invariant length of an admissible metric path that reaches  \nCK = {S ∈ S : κ (S) ≤ K} .  \n∗ Xidian University, Xi’an, China ([zavierli888@gmail.com](zavierli888@gmail.com)).  \n2 Z. LI  \nThe value is +∞ when the target is structurally unreachable. The word complexity refers throughout to minimum intrinsic metric motion (or, after fixing the horizon, its exactly equivalent action value); it carries no iteration-count or wall-clock complexity claim.  \nThe central operation is elimination of an entire future path. If cs,t (x, y) denotes the least action between two state","cbCaij3yHpHK0ZcU","https://ap.wps.com/l/cbCaij3yHpHK0ZcU","pdf",479098,1,32,"English","en",105,"# Introduction\n## Endpoint vs path questions\n## Restricted dynamic geometric complexity as a value problem\n## Path elimination and min-plus semigroup\n## Differential laws via Jacobi operators","[{\"question\":\"What limitation of endpoint condition-number reachability does the document address?\",\"answer\":\"Endpoint feasibility alone can reach the same target while requiring very different intrinsic affine-invariant metric motion. The document separates this geometric effort from mere reachability of a condition-number bound.\"},{\"question\":\"How is restricted dynamic geometric complexity defined in the work?\",\"answer\":\"It is the least affine-invariant length of an admissible metric path that reaches a prescribed set of states with generalized-eigenvalue condition ratio at most K. If the target is structurally unreachable, the value is +∞.\"},{\"question\":\"What role do path elimination and the Bellman principle play?\",\"answer\":\"Eliminating an entire future path yields an exact min-plus semigroup and a Bellman composition law. This provides the global rule governing how least actions concatenate across intermediate states.\"}]",1784205438,81,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":27},"restricted-dynamic-geometric-complexity-path-space-reduction-and-mobiusjacobi-response","",{"@graph":35,"@context":85},[36,53,68],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/restricted-dynamic-geometric-complexity-path-space-reduction-and-mobiusjacobi-response/85661/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-17","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What limitation of endpoint condition-number reachability does the document address?","Question",{"text":75,"@type":76},"Endpoint feasibility alone can reach the same target while requiring very different intrinsic affine-invariant metric motion. The document separates this geometric effort from mere reachability of a condition-number bound.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How is restricted dynamic geometric complexity defined in the work?",{"text":80,"@type":76},"It is the least affine-invariant length of an admissible metric path that reaches a prescribed set of states with generalized-eigenvalue condition ratio at most K. If the target is structurally unreachable, the value is +∞.",{"name":82,"@type":73,"acceptedAnswer":83},"What role do path elimination and the Bellman principle play?",{"text":84,"@type":76},"Eliminating an entire future path yields an exact min-plus semigroup and a Bellman composition law. This provides the global rule governing how least actions concatenate across intermediate states.","https://schema.org",{"og:url":51,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":51},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":20,"doc_module":4,"doc_module_name":45,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":46,"doc_module":4,"doc_module_name":45,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":45,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":45,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":45,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":45,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":45,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":45,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":45,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":45,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":45,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]