> Full-text translation from Chinese. The original remains authoritative for its research claims; this translation is not independent mathematical verification. # Projection Possibility Space Series ## Paper 04 — Perturbation Objects Beyond Rigidity ### From Rigid and Articulated Bodies to Deformable and Worm-Like High-Freedom Perturbing Objects ### Perturbation Objects Beyond Rigidity: From Rigid Bodies and Articulated Systems to Deformable and Worm-Like High-Degree-of-Freedom Geometries **Document ID:** `PPS-POBR-04-v0.1` **Version:** v0.1 **Date:** 2026-09-21 **Author:** Neo.K **AI Research Collaborator:** Aletheia / ChatGPT, GPT-5.6 Sol **Institution:** EveMissLab / 一言諾科技有限公司 **Series:** Projection Possibility Space / Projection Possibility Space Series **Prerequisites:** Paper 01 — 3D Projection Possibility Space; Paper 02 — Dynamic Projection Geometry; Paper 03 — Subject–Object–Boundary Geometry **Status:** `OBJECT-FREEDOM EXPANSION / PERTURBATION TAXONOMY / COUNTEREXAMPLE FRAMEWORK` --- # Abstract In many geometric stress tests, researchers extensively vary an object's rotation, translation and pose, interpreting this search as sufficient perturbation of the object. If the object is always restricted to a rigid body, however, what is actually explored is only a low-freedom subspace. This paper proposes: $$ \boxed{ \textbf{Perturbation Objects Beyond Rigidity} } $$ and establishes an object-freedom ladder from low to high: $$ \boxed{ O_0 \rightarrow O_1 \rightarrow O_2 \rightarrow O_3 \rightarrow O_4 } $$ where: $$ O_0 = \text{Rigid}, $$ $$ O_1 = \text{Articulated}, $$ $$ O_2 = \text{Smooth Deformable}, $$ $$ O_3 = \text{Locally Bendable Curve / Body}, $$ $$ O_4 = \text{Worm-Like Flexible Object}. $$ The paper's central question is not: > Which kind of object most closely resembles the real world? Instead: $$ \boxed{ \text{As Object's internal degrees of freedom progressively increase, does a geometric possibility space previously regarded as closed remain stable?} } $$ If all higher-freedom objects still collapse to the same finite event classes under the specified query, this provides strong closure evidence. Conversely, whenever a high-freedom perturbation family generates a new projected class, unknown contact topology, unclassified transition or new admissible configuration, a new counter-certificate candidate is obtained. The paper specifically positions worm-like objects as a high-freedom stress-test prototype, not a mathematical model assumed to have been solved. The worm problem itself remains open; the paper extracts only its most important structural feature: $$ \boxed{ \text{internal local bend freedom} } $$ and generalizes it into a geometric adversarial object family. Finally, the paper proposes the Object-Freedom Ladder, Flexibility Escalation Test, Internal Degree-of-Freedom Certificate, Deformation Event Ledger, Worm-Like Stress Family and Object-Class Closure, enabling subsequent 3D-to-2D dynamic projection and subject–object–boundary stress testing to distinguish explicitly between rigid closure and actual high-freedom object closure. --- # Keywords perturbation object、rigid body、articulated body、deformable geometry、flexible curve、worm-like object、internal degrees of freedom、geometric stress test、counterexample generation、shape space、deformation topology、object-class closure --- # 0. Research Position Paper 03 has established: $$ \Gamma = \langle S,O,\partial \rangle. $$ Object: $$ O $$ is explicitly included in the perturbable state. But: $$ \boxed{ \Delta O } $$ itself can still have very different strengths. If: $$ O $$ permits only pose transformations in: $$ SE(3) $$ , we are actually modifying only: - translation； - rotation。 This does not mean: $$ \boxed{ \text{object possibility space has been broadly explored}. } $$ --- # 1. Limits of Rigid Perturbation A rigid body: $$ O_{\rm rigid} $$ keeps: $$ d(p_i,p_j) $$ unchanged. that is: $$ \boxed{ \forall p_i,p_j\in O, \quad d_t(p_i,p_j) = d_0(p_i,p_j). } $$ Its degrees of freedom therefore come only from: $$ SE(3). $$ In three dimensions: $$ \dim SE(3)=6. $$ --- # 2. Six Degrees of Freedom Are Substantial, but Still Only External A rigid body can move: - up / down; - left / right; - forward / backward; - pitch； - yaw； - roll。 But the object's interior: $$ \boxed{ \text{does not change}. } $$ Therefore: $$ \boxed{ \text{external freedom} \neq \text{internal freedom}. } $$ --- # 3. Internal Degrees of Freedom Let: $$ q_{\rm int} $$ denote the object's internal degrees of freedom. For a rigid body: $$ \boxed{ \dim q_{\rm int}=0. } $$ For an articulated body: $$ \dim q_{\rm int}>0. $$ For a flexible curve or continuum body: $$ \boxed{ \dim q_{\rm int} \rightarrow \infty } $$ in a continuous model, the freedom may even be that of a function space. --- # 4. A More Complete Form of Object State Write Paper 03's Object: $$ O $$ as: $$ \boxed{ O = ( G, q_{\rm ext}, q_{\rm int}, \mathcal K, L ). } $$ where: - $G$：canonical geometry； - $q_{\rm ext}$: external pose; - $q_{\rm int}$: internal degrees of freedom; - $\mathcal K$：constitutive / deformation constraints； - $L$：labels / provenance。 --- # 5. Object Freedom Ladder The paper defines: $$ \boxed{ \mathcal O_0 \subset \mathcal O_1 \subset \mathcal O_2 \subset \mathcal O_3 \subset \mathcal O_4 } $$ as an object-freedom ladder for research. --- # 6. Level 0 — Rigid Object $$ \boxed{ O_0 = \text{Rigid Object}. } $$ Features: - internal distances fixed； - topology fixed； - shape fixed； - only global transform。 State: $$ q \in SE(3). $$ --- # 7. Uses of a Rigid Object It is best suited to testing: - orientation； - placement； - projection； - occlusion； - support direction； - container fit。 but not suited to testing: $$ \boxed{ \text{shape adaptation}. } $$ --- # 8. Level 1 — Articulated Object $$ \boxed{ O_1 = \text{Articulated Object}. } $$ It consists of multiple rigid links: $$ L_1,\ldots,L_n $$ and joints: $$ J_1,\ldots,J_m $$ . --- # 9. Articulated Coordinates In general: $$ \boxed{ q_{\rm int} = (\theta_1,\ldots,\theta_m). } $$ Each: $$ \theta_i $$ can control one joint. Therefore: $$ \dim q_{\rm int}=m. $$ --- # 10. Articulated Objects Can Already Generate New Topology Events Even if every link is rigid, joint motion can generate: - folding； - self-contact； - branch occlusion； - projected crossing； - reachability change； - container passage。 Therefore: $$ \boxed{ \mathfrak P(O_1) \supsetneq \mathfrak P(O_0) } $$ in general. --- # 11. Level 2 — Smooth Deformable Object $$ \boxed{ O_2 = \text{Smooth Deformable Object}. } $$ For example: - elastic sheet； - soft body； - spline surface； - deformable convex body； - finite-element body。 --- # 12. Deformation Field We can write: $$ \boxed{ x \mapsto x+u(x), } $$ where: $$ u: O \rightarrow \mathbb R^d. $$ If deformation varies with time: $$ u=u(x,t). $$ --- # 13. Internal Degrees of Freedom of a Deformable Object Discrete model: $$ q_{\rm int} \in \mathbb R^N. $$ continuum model： $$ \boxed{ q_{\rm int} = u(\cdot) } $$ is a function. Configuration space thus progressively moves from finite to infinite dimensions. --- # 14. Level 3 — Locally Bendable Curve / Body $$ \boxed{ O_3 = \text{Locally Bendable Object}. } $$ Its emphasis is not merely the ability to deform. Instead: $$ \boxed{ \text{local shape can change semi-independently along the object}. } $$ --- # 15. Curve Representation For a curve: $$ \gamma: [0,L] \rightarrow \mathbb R^d. $$ its local shape is determined by: $$ \gamma(s) $$ . If it varies with time: $$ \boxed{ \gamma = \gamma(s,t). } $$ --- # 16. Tangent Field Define: $$ T(s,t) = \frac{\partial_s\gamma}{\|\partial_s\gamma\|}. $$ Local bending directly changes: $$ T(s,t). $$ --- # 17. Curvature Field In the plane: $$ \boxed{ \kappa(s,t) } $$ describes local curvature. It is therefore no longer a few joint angles, but an entire curvature field: $$ \kappa: [0,L] \rightarrow \mathbb R. $$ --- # 18. Worm-Like Object The paper defines, for research: $$ \boxed{ O_4 = \text{Worm-Like Flexible Object}. } $$ Its necessary condition is not biological realism. Instead: 1. longitudinal ordering； 2. local bending； 3. distributed internal degrees of freedom； 4. possible self-approach / self-contact； 5. shape adaptation under environment。 --- # 19. Why Worm-Like? Worm-like is a working name. It indicates: $$ \boxed{ \text{that any intermediate region of the entire object may participate in deformation.} } $$ rather than changes occurring only at: - endpoints； - fixed joints； - one global curvature parameter； . --- # 20. The Boundary Between the Worm Problem and This Paper This paper does not claim: $$ \boxed{ \text{that the worm problem has been solved.} } $$ Nor does it claim: $$ \boxed{ \text{that a particular worm configuration solves a particular geometric problem.} } $$ The paper extracts worm-like systems only as: $$ \boxed{ \text{high-internal-freedom perturbation family}. } $$ --- # 21. Flexible Curve Constraints A worm-like curve can have: ## Arc-length constraint $$ \boxed{ \|\partial_s\gamma\|=1. } $$ ## Curvature bound $$ \boxed{ |\kappa(s)|\le\kappa_{\max}. } $$ ## Thickness $$ \boxed{ d(\gamma(s),\gamma(t)) \ge r_{\rm thick} } $$ for non-neighbouring parameters. --- # 22. Different Worm Models We can distinguish: $$ W_0 $$ no curvature bound; $$ W_1 $$ bounded curvature； $$ W_2 $$ bounded curvature + thickness； $$ W_3 $$ self-contact allowed； $$ W_4 $$ active growth / contraction； $$ W_5 $$ branched flexible body。 --- # 23. Worm-Like Is Not the Same as an Arbitrary Curve If an arbitrary: $$ \gamma(s), $$ is permitted without restriction, the model may be too powerful. Every study must therefore declare: $$ \boxed{ \mathcal K_{\rm worm}. } $$ the deformation contract. --- # 24. Perturbation Strength Define an abstract freedom index: $$ \boxed{ F(O). } $$ It can consist of: - number of DOFs； - local controllability； - topology freedom； - curvature freedom； - self-contact freedom； . In general, we seek: $$ F(O_0) < F(O_1) < F(O_2) < F(O_3) < F(O_4). $$ --- # 25. The Freedom Index Is Not a Single Natural Constant This paper does not claim: $$ F $$ has a unique standard. It merely requires researchers to state explicitly: > To what extent has Object actually been relaxed in this round? --- # 26. Object-Class Inclusion Ideally: $$ \mathcal O_0 \subseteq \mathcal O_1 \subseteq \cdots $$ In practice, however, different representations may not be completely nested. More generally, therefore, write: $$ \boxed{ \mathcal O_i \rightsquigarrow \mathcal O_{i+1} } $$ this indicates: > The next level permits more internal variation. --- # 27. Geometry Image For an Object family: $$ \mathcal O_i, $$ under fixed: $$ S,\partial $$ define its observable image: $$ \boxed{ \mathfrak P_i = \mathcal R( S, \mathcal O_i, \partial ). } $$ --- # 28. Freedom Expansion Question The central question is: $$ \boxed{ \mathfrak P_0 \overset{?}{=} \mathfrak P_1 \overset{?}{=} \cdots } $$ If all higher-freedom objects still yield only the same classifications, rigid analysis may suffice. --- # 29. Strict Expansion If there is: $$ Y \in \mathfrak P_{i+1} $$ But: $$ Y \notin \mathfrak P_i, $$ then: $$ \boxed{ \mathfrak P_i \subsetneq \mathfrak P_{i+1}. } $$ This means closure at the previous level cannot be directly promoted. --- # 30. Query-Relative Collapse Even if geometries differ, they may be indistinguishable for a specified query: $$ Q $$ . For example, all flexible states under: $$ Q=\text{projection width} $$ may collapse to the interval already known for rigid bodies. We therefore define: $$ \boxed{ \mathfrak P_i^Q. } $$ --- # 31. Object-Class Closure If: $$ \boxed{ \mathfrak P_i^Q = \mathfrak P_{i+1}^Q } $$ with a theorem / certificate, we can say: $$ \boxed{ \text{class }i \text{ is }Q\text{-complete relative to class }i+1. } $$ --- # 32. This Is Stronger Than Testing Many Flexible Objects Actual closure requires proving: $$ \boxed{ \forall O\in\mathcal O_{i+1}, \quad Q(O) \in \mathfrak P_i^Q. } $$ rather than sampling. --- # 33. Flexibility Escalation Test This paper proposes: $$ \boxed{ \textbf{Flexibility Escalation Test}. } $$ Workflow: $$ O_0 \rightarrow O_1 \rightarrow O_2 \rightarrow O_3 \rightarrow O_4. $$ At every escalation, ask: $$ \boxed{ \text{Did a new geometric event class appear?} } $$ --- # 34. Escalation Stop Rule If: $$ \mathcal O_i $$ has been proved, for: $$ Q $$ to cover: $$ \mathcal O_{i+1}, $$ escalation can stop. Otherwise, continue. --- # 35. Internal Perturbation For a flexible object: $$ O(s), $$ a perturbation can be written as: $$ \boxed{ O_\varepsilon(s) = O(s) + \varepsilon v(s). } $$ where: $$ v(s) $$ is a local perturbation field. --- # 36. Rigid Motion Is a Special Perturbation Field An infinitesimal rigid motion: $$ v(x) = a+\omega\times x. $$ is only a very small subspace of all: $$ v $$ . Thus: $$ \boxed{ \text{rigid perturbation} \subset \text{general perturbation field}. } $$ --- # 37. Localized Perturbation We can make: $$ v(s) $$ nonzero only for: $$ s\in I $$ . Only a short section is therefore bent. This is precisely a degree of freedom that a rigid body can never test. --- # 38. Localized Counterexample If a theorem is stable under rigid perturbations, but there exists a local: $$ v(s) $$ that causes its conclusion to fail, then: $$ \boxed{ \text{rigid closure was false as a flexible-object closure}. } $$ --- # 39. Deformation Modes Implementation can use a mode basis: $$ \boxed{ v(s) = \sum_{k=1}^N a_k\phi_k(s). } $$ For example: - Fourier； - spline； - wavelet； - eigenmode； - local bump basis。 --- # 40. Mode Ladder Start with a small: $$ N $$ . Progressively: $$ N rightarrow 2N \rightarrow 4N. $$ If new modes cease to generate new events, we obtain empirical saturation. --- # 41. Empirical Saturation Is Not a Theorem Even if: $$ N=10^6 $$ produces no new event, this still does not directly imply that: $$ N=\infty $$ produces none either. We need: - compactness； - approximation theorem； - Lipschitz control； - interval enclosure； - finite-mode completeness theorem。 --- # 42. Deformation Energy A regularizer can be added: $$ \boxed{ E_{\rm bend} = \int_0^L \kappa(s)^2ds. } $$ or: $$ E_{\rm stretch}. $$ This is not for physical realism. It controls the deformation family. --- # 43. Constraint-Relative Flexibility Different: $$ \mathcal K $$ produce different object spaces. Therefore: $$ \boxed{ \mathcal O_{\rm flex} = \mathcal O_{\rm flex}(\mathcal K). } $$ --- # 44. Self-Contact Flexible objects often exhibit: $$ \gamma(s_1) = \gamma(s_2). $$ This generates: - loops； - folds； - knots； - projection ambiguity； - visibility changes。 --- # 45. Self-Intersection Is Not Necessarily Permitted We must distinguish: ```text SELF_CONTACT_ALLOWED SELF_INTERSECTION_ALLOWED SELF_INTERSECTION_FORBIDDEN THICK_BODY ZERO_THICKNESS_CURVE ``` --- # 46. Topological Freedom If topology change is permitted: $$ \mathcal O $$ expands substantially again. For example: - split； - merge； - branching； - tearing。 Paper 04 primarily studies, by default: $$ \boxed{ \text{shape deformation without topology change} } $$ unless otherwise stated. --- # 47. Articulation vs Deformation Articulation： $$ \boxed{ \text{piecewise rigid}. } $$ Deformation： $$ \boxed{ \text{material geometry itself changes}. } $$ The two must not be confused. --- # 48. Curve vs Body Worm-like representations can be divided into: ## Centerline Model $$ \gamma(s). $$ ## Tube Model $$ \boxed{ W_r = \{x: d(x,\gamma)\le r\}. } $$ ## Full Body Model Use a deformable volume directly. --- # 49. Advantages of the Tube Model It introduces thickness: $$ r>0. $$ and thus avoids some nonphysical solutions of zero-thickness curves. Computation is more expensive, however. --- # 50. Amplification of Degrees of Freedom After Projection A small local bend: $$ \delta\gamma $$ under a particular projection angle may produce a large 2D boundary change. Thus: $$ \boxed{ \text{small 3D internal change} \not\Rightarrow \text{small 2D projected change}. } $$ --- # 51. Projection Sensitivity Define: $$ \boxed{ S_{\Pi} = \frac{ d_{2D}( \Pi(O), \Pi(O+\delta O) ) }{ d_O(O,O+\delta O) }. } $$ Regions with high: $$ S_\Pi $$ are especially suited to stress testing. --- # 52. Flexible Hotspot If a local region: $$ s^\star $$ is particularly sensitive with respect to the projected result, mark it: $$ \boxed{ \texttt{FLEX-HOTSPOT}. } $$ --- # 53. Curvature Hotspot If: $$ |\kappa(s)| $$ approaches: $$ \kappa_{\max}, $$ it may be an admissibility boundary. Therefore: $$ \boxed{ |\kappa| = \kappa_{\max} } $$ is also an event. --- # 54. Length Constraint Event If extension is constrained by: $$ L\le L_{\max}, $$ then: $$ L=L_{\max} $$ is an active boundary. --- # 55. Flexible Object Event Types The paper recommends at least: ```text BEND_LIMIT STRETCH_LIMIT SELF_CONTACT SELF_INTERSECTION FOLD UNFOLD CURVATURE_SIGN_CHANGE PROJECTED_CROSSING VISIBILITY_SWAP CONTAINER_CONTACT UNKNOWN_DEFORMATION_CLASS ``` --- # 56. Deformation Event Ledger Every event retains: ```text object_class deformation_model internal_state event_type event_parameter subject_state boundary_state projected_state certificate ``` --- # 57. Internal State Hash For a flexible object, hashing only a mesh frame is insufficient. Hash the canonical deformation state: $$ q_{\rm int}. $$ For a continuum: - spline coefficients； - basis coefficients； - control points； - analytic function； - interval enclosure。 --- # 58. Worm-Like Stress Family Define: $$ \boxed{ \mathcal W(\mathcal K,L,r) } $$ to be a family of worm-like objects. where: - $\mathcal K$：curvature / regularity contract； - $L$：length； - $r$：thickness。 --- # 59. Worm-Like Projection Family In the SOB context: $$ (S,\partial) $$ : $$ \boxed{ \mathfrak P_{\rm worm} = \mathcal R( S, \mathcal W, \partial ). } $$ --- # 60. Worm-Like Stress Question The actual question is: $$ \boxed{ \mathfrak P_{\rm worm} \overset{?}{\subseteq} \mathfrak P_{\rm known}. } $$ rather than: > Does the worm resemble a candidate solution? --- # 61. Worm-Like as Adversarial Object Its value is: $$ \boxed{ \text{maximize internal geometric freedom while retaining a controlled contract}. } $$ It is therefore particularly suited to finding: - unknown projection； - hidden passage； - new contact； - unexpected support pattern。 --- # 62. The Unsolved Worm Problem Does Not Prevent Its Use as a Stress-Test Model An object family need not first have been fully solved analytically to serve as: $$ \boxed{ \text{adversarial generator}. } $$ as long as: - its state can be generated; - its constraints can be verified; - its result can be replayed. --- # 63. Research Separation We must distinguish: $$ \boxed{ \text{Worm Problem Proper} } $$ and: $$ \boxed{ \text{Worm-Like Stress Object}. } $$ The former is an independent mathematical problem. The latter is a research tool. --- # 64. Difference Between a Long Curve and a Worm If a long curve has a fixed shape: $$ \gamma_0 $$ and undergoes only rigid motion, it still belongs to: $$ O_0. $$ A worm-like curve instead has: $$ \boxed{ \gamma(s) \text{ itself varies}. } $$ --- # 65. A Global Curvature Parameter Is Still Insufficient If we allow only: $$ \kappa(s)=c, $$ we have merely a constant-curvature family. Its degrees of freedom still number only: $$ 1. $$ A genuinely worm-like model must permit: $$ \kappa=\kappa(s). $$ --- # 66. Piecewise Curvature Model We can use: $$ \kappa(s) = \kappa_j $$ taken constant on: $$ I_j $$ . If it is divided into: $$ N $$ segments, then: $$ q_{\rm int} \in \mathbb R^N. $$ this is a very practical finite approximation. --- # 67. Segment Refinement Progressively: $$ N=4,8,16,32,\ldots $$ observe whether new events occur. --- # 68. Deformation Convergence If: $$ O_N \rightarrow O_\infty $$ converges under a metric, we must check whether: $$ \mathcal R(O_N) \rightarrow \mathcal R(O_\infty) $$ is continuous. If the projection encounters a topology event, it may be discontinuous. --- # 69. Event-Aware Convergence We therefore cannot consider only: $$ d(O_N,O_{N+1}) \rightarrow0. $$ We must also consider: $$ \boxed{ \text{event graph stabilizes?} } $$ --- # 70. Event-Graph Saturation Define: $$ \mathcal G_N $$ to be the event graph under the approximation with: $$ N $$ degrees of freedom. If: $$ \mathcal G_N \cong \mathcal G_{2N} $$ holds over several successive levels, Possible labels: ```text EVENT-GRAPH-SATURATED-PILOT ``` but this is still not a theorem. --- # 71. Flexible Object Counterexample A counterexample should retain at least: $$ \boxed{ C_{\rm flex} = ( O, q_{\rm int}, \mathcal K, S, \partial, Y ). } $$ --- # 72. Why Retaining Only the Final Shape Is Insufficient The same final shape may be reached through different deformation paths. If the path contains: - self-contact； - forbidden crossing； - invalid stretch； it may not be admissible. We must therefore retain: $$ \boxed{ \text{deformation history}. } $$ --- # 73. Path-Admissible Object We say that: $$ O_1 $$ is admissibly reachable from: $$ O_0 $$ if there exists a path: $$ \gamma_O(t) $$ satisfying: $$ \gamma_O(0)=O_0, $$ $$ \gamma_O(1)=O_1, $$ and obeying throughout: $$ \mathcal K. $$ --- # 74. Reachability Differs from Feasibility The final state: $$ O_1 $$ may itself be admissible, but no admissible path may exist. Thus: $$ \boxed{ \text{state feasible} \neq \text{path reachable}. } $$ --- # 75. Does the Study Need Reachability? It depends on the query. If only a set of shapes is studied, reachability may be unnecessary. If dynamic operations / physical reachability are studied, it is needed. --- # 76. Object-Class Ontology Recommended canonical types: ```text RIGID ARTICULATED DEFORMABLE LOCALLY_BENDABLE WORM_LIKE TOPOLOGY_CHANGING FIELD_OBJECT ``` --- # 77. Object Freedom Certificate Define: $$ \boxed{ C_O } $$ It contains at least: ```text object_class canonical_geometry external_dof internal_dof_model constraint_contract topology_policy self_contact_policy thickness basis / discretization precision state hash ``` --- # 78. Flexible Certificate If: $$ q_{\rm int} $$ is a finite vector, retain it directly. If it is a function, retain: - formula； - spline； - coefficients； - interval bounds； - reconstruction hash。 --- # 79. Deformation Verifier Check: 1. internal constraints； 2. length； 3. curvature； 4. thickness； 5. self-contact policy； 6. topology； 7. path admissibility； 8. projection result。 --- # 80. Mutation Test Deliberately: - exceed the curvature bound; - change the length; - remove a control point; - introduce self-intersection; - violate thickness. If the verifier still reports PASS, the object constraints have not actually been verified. --- # 81. Low-Freedom Models Remain Highly Valuable This paper does not reject rigid bodies. On the contrary: $$ \boxed{ O_0 } $$ is usually the cheapest and most understandable first level. Only when: $$ O_0 $$ is near closure is there reason to escalate. --- # 82. Lazy Flexibility Activation We can therefore design: $$ \boxed{ \text{Rigid} \rightarrow \text{Articulated} \rightarrow \text{Flexible} } $$ escalating only when necessary. This resembles lazy dimension activation in a proof runtime. --- # 83. Flexibility Necessity Marker If a residual can be approached only through internal deformation, mark it: $$ \boxed{ \texttt{FLEX-NECESSARY}. } $$ then escalate the object class. --- # 84. Flexibility Escalation Cost At every escalation: - search cost increases; - verifier cost increases; - the state space grows; - visualization cost increases. A scheduler is therefore needed. --- # 85. Object-Class Scheduler Define: $$ \operatorname{Cost}(O_i) $$ and: $$ \operatorname{ExpectedNovelty}(O_i). $$ Choose: $$ \boxed{ i^\star = \arg\max_i \frac{ \operatorname{ExpectedNovelty}(O_i) }{ \operatorname{Cost}(O_i) }. } $$ --- # 86. Counterexample-First Escalation If the goal is the final stress test, there is no need to explore all flexible states evenly. Prioritize: - high sensitivity； - near critical boundary； - contact-rich； - projection-degenerate； - topology-switch-prone。 --- # 87. Active Flex Regions Define: $$ \boxed{ \mathcal A_{\rm flex} } $$ to be the high-information region in the object's internal state. It can be detected through: - curvature hotspots； - contact hotspots； - projected novelty； - verifier slack； . --- # 88. Flexible Object–Subject Coupling A deformation that is invisible to: $$ S_1 $$ , for: $$ S_2 $$ may nevertheless generate a new projection for this subject. Thus: $$ \boxed{ \mathfrak P_{\rm flex} } $$ must therefore be placed back into the SOB context. --- # 89. Flexible Object–Boundary Coupling The container can constrain: $$ \gamma(s). $$ and even induce deformation. Therefore: $$ \boxed{ q_{\rm int} = q_{\rm int}(S,\partial) } $$ may be a coupled variable. --- # 90. Passive vs Active Deformation ## Passive Deformation is caused by boundary / contact. ## Active The object actively changes: $$ q_{\rm int}. $$ The two should be recorded separately. --- # 91. Controlled Worm If a controller can determine each segment's curvature: $$ u(s,t), $$ the worm-like object can become: $$ \boxed{ \text{active adversarial geometry agent}. } $$ --- # 92. AI-Controlled Flexible Object AI policy： $$ \pi_A $$ can output: $$ u_t(s). $$ Goal: $$ \boxed{ \max \text{projection novelty}. } $$ --- # 93. This Is Not Letting AI Play with Animation Instead: $$ \boxed{ \text{AI searches an internal-deformation function space}. } $$ Animation is merely its visual projection. --- # 94. Flexible Search Objective We can set: $$ J = \lambda_1N_{\rm unknown} + \lambda_2N_{\rm transition} + \lambda_3S_\Pi - \lambda_4E_{\rm deform}. $$ --- # 95. Multiscale Flexible Search First, coarse: $$ N=8. $$ Then, on a hotspot: $$ N=64. $$ Then, locally: $$ N=512. $$ This is more efficient than globally high resolution. --- # 96. Finite Closure of a Flexible Space To actually prove: $$ \mathcal O_{\rm flex} $$ closure， we can attempt: - compactness； - interval function enclosure； - curvature-cell partition； - spline coefficient boxes； - convexity； - monotone functional bounds。 --- # 97. Function-Space Box If: $$ \kappa(s) = \sum a_i\phi_i(s), $$ we can construct interval boxes for: $$ a_i $$ . Then perform: $$ \boxed{ \text{finite coefficient-space closure}. } $$ --- # 98. Basis Completeness Requires a Separate Proof Finite coefficient closure is valid only on: $$ \operatorname{span}\{\phi_1,\ldots,\phi_N\} $$ . To extend to the full function space, we need an approximation remainder: $$ r_N. $$ --- # 99. Remainder Certificate If we can prove: $$ \boxed{ \|r_N\| \le \varepsilon_N } $$ and the query is Lipschitz with respect to the object: $$ |Q(O)-Q(O_N)| \le L_Q\varepsilon_N, $$ the finite model can be extended to the continuum. --- # 100. When Topology Events Break Lipschitz Control Near: $$ E_{\rm topology}, $$ the Lipschitz constant may become very large. These regions must be treated separately. --- # 101. Worm-Like Event Stratification We can stratify the flexible state space as: $$ \boxed{ \mathcal W = \bigcup_\alpha \mathcal W_\alpha } $$ Each: $$ \mathcal W_\alpha $$ has a fixed: - self-contact graph； - curvature saturation pattern； - projection topology； - boundary contact graph。 --- # 102. An Infinite Flexible Space Is Not Untreatable Infinite-dimensional does not mean impossible to study. It can be compressed through: $$ \boxed{ \text{finite structural strata} } $$ . --- # 103. If It Ultimately Collapses to a Few Event Classes we obtain a very important conclusion: $$ \boxed{ \text{huge internal state space} \rightarrow \text{small event space}. } $$ This is one of the phenomena most desired from the final stress test. --- # 104. If It Does Not Collapse That is valuable too. It means: $$ \boxed{ \text{the original finite closure lacked a layer of freedom.} } $$ The mathematics must be revisited and repaired. --- # 105. Implications for Lebesgue-Type Problems If the mathematics currently lists only a few geometric possibilities, we can ask: > Do these possibilities hold only for rigid witnesses / rigid placements? Then escalate Object level by level. --- # 106. Do Not Simply Assume Flexible Witnesses Are Admissible Whether: $$ O_4 $$ is permitted depends on the original problem's admissible class. If a witness in the original theorem must be rigid, worm-like objects can serve only as: $$ \boxed{ \text{metamodel stress test}, } $$ not directly as counterexamples to the original problem. --- # 107. Two Stress-Test Roles ## In-Domain Stress The object remains within the original problem's admissible class. A counterexample found here can directly challenge the theorem. ## Out-of-Domain Stress Deliberately relax the object. Use it to test: $$ \boxed{ \text{classification robustness / hidden assumption}. } $$ It cannot directly serve as a theorem counterexample. --- # 108. This Distinction Is Crucial Otherwise, one might say: > I broke the problem with an inadmissible object, so the original proof is wrong. This does not follow. --- # 109. Domain Tag Every object class should be labelled: ```text IN_DOMAIN OUT_OF_DOMAIN RELAXED_DOMAIN UNKNOWN ``` --- # 110. Relaxation Test If: $$ O_4 $$ is a relaxation of the original problem, it is still valuable. If even the relaxed problem leaves only the same geometric classes, the original problem is more likely to be stable. --- # 111. Relaxation Counterexample If the relaxed domain yields a new class: $$ Y^\star, $$ ask: $$ \boxed{ \text{can }Y^\star \text{ be approximated by in-domain objects?} } $$ If so, it may reveal a genuine gap. --- # 112. Projection Approximation Even if the flexible object is inadmissible, its 2D projection: $$ Y^\star $$ may be approximable by an admissible rigid / piecewise-rigid sequence. This is worth pursuing. --- # 113. Counterexample Lifting Workflow: $$ \boxed{ \text{relaxed counterexample} \rightarrow \text{attempt in-domain lifting} } $$ If successful, promote it to an actual counterexample. --- # 114. Counterexample Compression After a high-freedom flexible state yields an unknown class, try reducing its degrees of freedom: $$ O_4 \rightarrow O_3 \rightarrow O_2 \rightarrow O_1 \rightarrow O_0. $$ Find the lowest class that still preserves the counterexample. --- # 115. Minimal Counterexample Object Class Define: $$ \boxed{ i^\star = \min \{ i: \exists O\in\mathcal O_i \text{ generating counterexample} \}. } $$ This tells us: > At which layer of freedom did closure first break? --- # 116. This Is More Informative Than Merely Knowing a Counterexample Exists because it locates: $$ \boxed{ \text{responsible freedom}. } $$ --- # 117. Freedom Responsibility Graph Build: $$ \boxed{ \mathcal G_F } $$ node： - rigid motion； - joint； - local bend； - curvature mode； - self-contact。 edge： - event causality； - dependency。 --- # 118. Flexible Counterexample Explainability Ideally, we should finally be able to say: > It is not simply that flexibility caused a problem; a particular local bending mode opened a particular projection transition. --- # 119. Flexible Object Public Visualization This is also well suited to animation. For example, display simultaneously: - 3D worm； - curvature heatmap； - contact points； - 2D projection； - event graph。 --- # 120. 2.5D Is Even More Important for Flexible Objects Different body segments may have overlapping projections. 2.5D must retain: - segment id； - depth； - local curvature； - deformation mode； - visibility。 --- # 121. Blender / Engine Implementation Role Blender is well suited to: - curve； - armature； - lattice； - geometry nodes； - deformable visualization； - camera projection。 But the canonical flexible state should not exist only in: ```text modifier stack ``` . --- # 122. Canonical Flex State Retain separately: ```text centerline control points joint parameters curvature coefficients constraint bounds topology policy ``` --- # 123. Render Approximation render curve： $$ \gamma_{\rm mesh} $$ is merely: $$ \boxed{ R(\gamma_{\rm canonical}). } $$ --- # 124. Cross-Runtime Replay Ideally, use: $$ \boxed{ \text{Blender curve} \leftrightarrow \text{independent numerical curve runtime} } $$ for cross-verification. --- # 125. Object-Freedom Dashboard At every stress test, display: ```text OBJECT CLASS INTERNAL DOF CONSTRAINT CONTRACT EVENT TYPES SEEN UNKNOWN CLASSES COUNTEREXAMPLES CLOSURE STATUS ``` --- # 126. Closure Status Ladder ```text RIGID-CLOSED ARTICULATED-CLOSED DEFORMABLE-PILOT LOCALLY-BENDABLE-OPEN WORM-LIKE-OPEN ``` Prevent a single statement: ```text OBJECT SPACE CLOSED ``` from conflating different degrees of freedom. --- # 127. Paper 04's Main Methodological Statement $$ \boxed{ \text{Do not ask only whether the object can move.} } $$ We must also ask: $$ \boxed{ \text{Can the object change internally?} } $$ --- # 128. A Second Methodological Statement $$ \boxed{ \text{Rigid stress testing probes pose space;} } $$ $$ \boxed{ \text{flexible stress testing probes shape space.} } $$ --- # 129. A Third Methodological Statement $$ \boxed{ \text{Worm-like objects are useful because local deformation can generate possibilities that no finite rigid pose sweep can create.} } $$ --- # 130. Integration with Paper 03 The complete state: $$ \Gamma = \langle S,O,\partial \rangle. $$ Object can now be written as: $$ \boxed{ O = O_i(q_{\rm ext},q_{\rm int}). } $$ Thus: $$ \boxed{ \Gamma_i = \langle S, O_i(q_{\rm ext},q_{\rm int}), \partial \rangle. } $$ --- # 131. Dynamic Flexible SOB State Over time: $$ \boxed{ \Gamma(t) = \langle S(t), O(q_{\rm ext}(t),q_{\rm int}(t)), \partial(t) \rangle. } $$ This is the future fully specified animation stress-test state. --- # 132. The Next Paper's Question Even after relaxing Object, a deeper question remains: > Might the types we currently see in 2D be classifications arising only from direct parameterization in 2D? that is: $$ \boxed{ \text{Higher-dimensional motion may generate hidden lower-dimensional geometry.} } $$ The next paper is therefore: $$ \boxed{ \textbf{Paper 05 — Projection-Induced Hidden Geometry}. } $$ --- # 133. The Paper's Main Propositions ## Proposition A $$ \boxed{ \text{Rigid pose space is only a low-internal-freedom subset of object possibility space.} } $$ ## Proposition B $$ \boxed{ \text{Increasing internal degrees of freedom can create new geometric event classes.} } $$ ## Proposition C $$ \boxed{ \text{A worm-like object is best treated as a controlled high-freedom adversarial geometry family, not as a presumed solution.} } $$ ## Proposition D $$ \boxed{ \text{Object-class closure must be stated relative to a query and an admissible deformation contract.} } $$ ## Proposition E $$ \boxed{ \text{A flexible counterexample should be compressed to the minimal responsible freedom class when possible.} } $$ ## Proposition F $$ \boxed{ \text{The worm problem itself remains open; its unresolved status does not prevent worm-like models from serving as stress-test generators.} } $$ --- # 134. Non-Claims This paper does not claim: 1. that the worm problem has been solved; 2. that a worm-like object is the final answer to any particular geometric problem; 3. that greater flexibility is closer to the actual solution; 4. that rigid-body testing has no value; 5. that every arbitrary curve is an admissible object; 6. that all function-space deformations admit finite closure; 7. that finite-mode saturation is a continuum theorem; 8. that an out-of-domain flexible counterexample directly refutes the original theorem; 9. that Blender deformation can directly serve as proof; 10. that more internal degrees of freedom necessarily yield a counterexample. --- # 135. Conclusion The paper advances object perturbation from: $$ \boxed{ \text{pose change} } $$ to: $$ \boxed{ \text{shape-space change}. } $$ Its freedom ladder: $$ \boxed{ \text{Rigid} \rightarrow \text{Articulated} \rightarrow \text{Deformable} \rightarrow \text{Locally Bendable} \rightarrow \text{Worm-Like} } $$ provides a geometric stress architecture that can be escalated level by level. Its core is not: > Every problem must be turned into a worm. Instead: $$ \boxed{ \text{When closure is nearly complete, we must know how many internal degrees of freedom our Object model restricts.} } $$ If a high-freedom object family ultimately still collapses to the same few event classes, that is a very strong result. If it generates new unknown classes, that is equally valuable. It means: $$ \boxed{ \text{the actual gap in the original closure has been found.} } $$ Thus the research value of worm-like perturbation does not lie in whether it is the answer, but in: $$ \boxed{ \text{whether it can reveal candidate answers or counterexamples invisible to rigid bodies.} } $$ --- # Canonical Source Declaration This file is the UTF-8 Markdown canonical source for `PPS-POBR-04-v0.1`. The only mathematical delimiters used are `$...$` and `$$...$$`. The paper strictly distinguishes: - rigid motion； - articulation； - deformation； - local bending； - worm-like flexibility； - in-domain stress； - relaxed-domain stress； - counterexample generation； - theorem-level closure。 The worm problem itself explicitly remains `OPEN`. ===END===