> 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 03 — Subject–Object–Boundary Geometry ### Subject–Object–Boundary Geometry: From Observer, Perturbing Object and Interface Coupling to Projection Possibility ### Subject–Object–Boundary Geometry: A Triadic Geometric System for Observer, Perturbation Object, Interface Coupling, and Projection Possibility **Document ID:** `PPS-SOBG-03-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 **Status:** `FOUNDATIONAL THEORY / TRIADIC GEOMETRIC STATE MODEL` --- # Abstract Traditional geometric modelling usually treats the object as the primary object of study, while observation position, projection rules, container boundaries, contact interfaces and visibility conditions are relegated to background settings. For problems requiring geometric closure, counterexample search, dynamic projection and the unfolding of possibility spaces, this approach can silently fix variables that actually affect the result. This paper proposes: $$ \boxed{ \textbf{Subject–Object–Boundary Geometry} } $$ abbreviated: $$ \boxed{ \textbf{SOB Geometry}. } $$ Its basic state is not a single object: $$ O, $$ Instead: $$ \boxed{ \Gamma = \langle S, O, \partial \rangle. } $$ where: $$ S = \text{Subject / Observer}, $$ $$ O = \text{Object / Perturbation Object}, $$ $$ \partial = \text{Boundary / Carrier / Interface}. $$ The paper argues that if a geometric conclusion depends on projection, contact, containers, visibility, relative pose or operational methods, then subject, object and boundary should all be explicit state variables; modifying only the object itself is insufficient. The paper further divides: $$ \partial $$ into: $$ \partial_{\rm container}, \quad \partial_{\rm contact}, \quad \partial_{\rm projection}, \quad \partial_{\rm visibility}, $$ and divides Subject into: - pose； - projection convention； - observation direction； - sampling rule； - temporal path； - resolution / tolerance； - query semantics。 The same object: $$ O $$ under different: $$ S $$ or: $$ \partial $$ can therefore produce different effective geometric results: $$ \boxed{ \mathcal R(S_1,O,\partial_1) \ne \mathcal R(S_2,O,\partial_2). } $$ Finally, the paper establishes Triadic Perturbation, Observer Relativity Audit, Boundary Responsibility Graph, Coupled Closure and the Subject–Object–Boundary Certificate, so future 3D-to-2.5D-to-2D dynamic experiments, flexible perturbations and Lebesgue-type geometric closure can operate within the same formal framework. --- # Keywords subject–object–boundary、observer geometry、boundary coupling、projection geometry、interface geometry、triadic state、geometric relativity、visibility boundary、contact boundary、counterexample search、closure audit、interactive geometry --- # 0. Research Position Paper 01 establishes: $$ 3D \rightarrow 2.5D \rightarrow 2D $$ as the unfolding of projection possibilities. Paper 02 establishes: $$ G_t = \Pi_t(T_tM) $$ as continuous geometric experimentation. These two papers nevertheless retain an implicit assumption: > Who is looking? What is being observed? Which boundary determines what can be seen, contacted, contained or projected? Without separating these, $$ \Pi $$ and: $$ T $$ may still silently conceal many background assumptions inside their operators. The paper therefore proposes: $$ \boxed{ \Gamma = \langle S,O,\partial \rangle. } $$ --- # 1. Why Object Alone Is Insufficient Traditional geometric optimization often writes: $$ \min_O F(O). $$ Alternatively: $$ P(O). $$ But if: $$ P $$ actually depends on: - observer； - projection； - container； - visibility； - contact rule； the actual problem should be written as: $$ \boxed{ P(S,O,\partial). } $$ If we modify only: $$ O $$ while holding fixed: $$ S,\partial, $$ the resulting closure is merely: $$ \boxed{ \text{conditional closure under fixed subject and boundary}. } $$ --- # 2. The Triadic State Define: $$ \boxed{ \Gamma = \langle S,O,\partial \rangle. } $$ All three are elements of state spaces: $$ S\in\mathcal S, $$ $$ O\in\mathcal O, $$ $$ \partial\in\mathcal B. $$ The complete state space is therefore: $$ \boxed{ \mathcal X = \mathcal S \times \mathcal O \times \mathcal B. } $$ --- # 3. Subject Is Not a Human Psychological Subject In this paper: $$ S $$ is primarily an operational observer. It does not require: - consciousness; - personality; - feelings; - subjective experience. It requires only: $$ \boxed{ \text{that it determines how geometric information is acquired.} } $$ --- # 4. Subject State Subject can be formalized as: $$ \boxed{ S = ( P, V, \Pi, Q, \Sigma, \tau ). } $$ where: - $P$：observer pose； - $V$：view / observation direction； - $\Pi$：projection / reduction operator； - $Q$：query semantics； - $\Sigma$：sampling / resolution / tolerance； - $\tau$：observation path or time rule。 --- # 5. Observer Pose Most basically: $$ P\in SE(3) $$ or: $$ P\in SE(2) $$ depending on the problem. Different poses: $$ P_1 \ne P_2 $$ Even if: $$ O $$ and: $$ \partial $$ remain unchanged, different poses may still produce: $$ \boxed{ R(P_1,O,\partial) \ne R(P_2,O,\partial). } $$ --- # 6. Projection Convention Subject also includes: $$ \Pi. $$ For example: - orthographic； - perspective； - oblique； - stereographic； - slice； - silhouette； - depth collapse； - visibility projection。 Thus: $$ \boxed{ \text{observer change} \not\equiv \text{camera translation only}. } $$ --- # 7. Query Semantics Furthermore: $$ Q $$ determines what the observer actually asks. For example: $$ Q_1: \text{is it contained?} $$ $$ Q_2: \text{is it visible?} $$ $$ Q_3: \text{is it tangent?} $$ $$ Q_4: \text{is the projected topology unchanged?} $$ The same state: $$ \Gamma $$ may receive different judgments under different queries. Therefore: $$ \boxed{ \text{geometric observation is query-relative}. } $$ --- # 8. Sampling Rule Subject can also carry: $$ \Sigma $$ For example: - angular step； - pixel resolution； - interval precision； - time step； - mesh density； - contact epsilon。 If: $$ \Sigma_1 \ne \Sigma_2, $$ different numerical observations may result. We must therefore distinguish: $$ \boxed{ \text{mathematical state} } $$ and: $$ \boxed{ \text{sampled observation}. } $$ --- # 9. Object State The object: $$ O $$ can be: - rigid body； - convex body； - curve； - deformable body； - articulated body； - multi-body system； - field； - graph-embedded geometry。 A general form is: $$ \boxed{ O = ( G, A, D, L ) } $$ where: - $G$：geometry； - $A$：articulation； - $D$：deformation state； - $L$：labels / provenance。 --- # 10. Object Is Not the Rendered Mesh The object's canonical geometry: $$ O_{\rm can} $$ and its rendering geometry: $$ O_{\rm render} $$ must be distinguished. $$ \boxed{ O_{\rm render} = R(O_{\rm can}). } $$ But: $$ \boxed{ O_{\rm render} \text{ must not become the sole proof source}. } $$ --- # 11. What Is Boundary? In this paper: $$ \partial $$ it is not merely: > the outline of an object. It is: $$ \boxed{ \text{a boundary system determining geometric feasibility, contact, projection, visibility or the observation interface.} } $$ --- # 12. Boundary Layers Define: $$ \boxed{ \partial = ( \partial_C, \partial_K, \partial_P, \partial_V ). } $$ where: $$ \partial_C = \partial_{\rm container}, $$ $$ \partial_K = \partial_{\rm contact}, $$ $$ \partial_P = \partial_{\rm projection}, $$ $$ \partial_V = \partial_{\rm visibility}. $$ --- # 13. Container Boundary $$ \partial_C $$ determines whether: $$ O $$ can be contained. For example: $$ O\subset C. $$ Alternatively: $$ \operatorname{dist}(O,\partial C)\ge0. $$ --- # 14. Contact Boundary $$ \partial_K $$ determines: - contact； - tangency； - penetration； - support； - frictionless interface； - active constraint。 We can define: $$ g_i(S,O,\partial_K)\ge0. $$ If: $$ g_i=0, $$ then active contact is indicated. --- # 15. Projection Boundary $$ \partial_P $$ determines where: $$ \Pi $$ projects. For example, a projection plane: $$ H. $$ or a screen: $$ \Sigma_P. $$ Projection is not an abstract operation without a boundary. --- # 16. Visibility Boundary $$ \partial_V $$ determines: - clipping； - near/far plane； - occlusion； - depth ordering； - visibility mask。 Thus: $$ \boxed{ \text{visible geometry} \ne \text{projected geometry}. } $$ --- # 17. Different Effective Geometries of the Same Object Define the reduction: $$ \mathcal R: \mathcal S \times \mathcal O \times \mathcal B \rightarrow \mathcal Y. $$ then: $$ \boxed{ Y = \mathcal R(S,O,\partial). } $$ If: $$ S_1\ne S_2, $$ we may have: $$ Y_1\ne Y_2. $$ If: $$ \partial_1\ne\partial_2, $$ we may also have: $$ Y_1\ne Y_2. $$ --- # 18. Subject–Object Coupling Subject and Object have a relative geometry: $$ \boxed{ R_{SO}. } $$ For example: $$ R_{SO} = P_S^{-1}P_O. $$ Absolute coordinates are therefore not the only important quantities. Many observations depend only on: $$ \boxed{ \text{relative configuration}. } $$ --- # 19. Object–Boundary Coupling Object and Boundary have: $$ R_{O\partial}. $$ For example: - containment； - contact； - support； - clearance； - penetration。 Thus: $$ \boxed{ O \text{'s geometric role depends on its position relative to the boundary.} } $$ --- # 20. Subject–Boundary Coupling Subject and Boundary also have: $$ R_{S\partial}. $$ For example: - camera relative to the projection plane; - observer relative to the clipping region; - sensor relative to the visibility aperture. Thus: $$ \boxed{ \partial \text{ is not only the Object's boundary; it can also be the Subject's interface.} } $$ --- # 21. Triangular Coupling The complete system is not pairwise independent. Define: $$ \boxed{ \mathcal C_{SOB} = ( R_{SO}, R_{O\partial}, R_{S\partial} ). } $$ then: $$ \Gamma $$ may have an effective geometry jointly determined by all three sides. --- # 22. Why Subject, Object and Boundary? Because: $$ S $$ determines how to observe; $$ O $$ determines what to observe; $$ \partial $$ determines what is permitted, contacted, projected, occluded or clipped. Therefore: $$ \boxed{ \text{geometry} = \text{subject} + \text{object} + \text{boundary relation}. } $$ --- # 23. Static SOB State The static state: $$ \Gamma_0 = \langle S_0,O_0,\partial_0 \rangle. $$ has output: $$ Y_0 = \mathcal R(\Gamma_0). $$ --- # 24. Dynamic SOB State The dynamic state: $$ \boxed{ \Gamma(t) = \langle S(t), O(t), \partial(t) \rangle. } $$ has output: $$ \boxed{ Y(t) = \mathcal R( S(t), O(t), \partial(t) ). } $$ Paper 02's dynamic projection is now expanded into triadic form. --- # 25. Perturbation Cannot Be Written Only as $\Delta O$ Traditional stress tests may perform only: $$ O \rightarrow O+\Delta O. $$ This paper requires a more complete form: $$ \boxed{ \Gamma \rightarrow \Gamma' = \langle S+\Delta S, O+\Delta O, \partial+\Delta\partial \rangle. } $$ --- # 26. Three Classes of Perturbation ## Subject perturbation $$ \Delta S. $$ For example: - camera； - projection； - observer motion； - resolution； - query。 ## Object perturbation $$ \Delta O. $$ For example: - rotation； - translation； - bend； - deformation； - topology change。 ## Boundary perturbation $$ \Delta\partial. $$ For example: - container shape； - projection plane； - clipping rule； - contact surface； - visibility aperture。 --- # 27. Triadic Perturbation Space Define: $$ \boxed{ \Delta\mathcal X = \Delta\mathcal S \times \Delta\mathcal O \times \Delta\mathcal B. } $$ The final stress test should explore: $$ \boxed{ (\Delta S,\Delta O,\Delta\partial), } $$ rather than: $$ (0,\Delta O,0). $$ --- # 28. Conditional Closure If we hold fixed only: $$ S=S_0, $$ $$ \partial=\partial_0, $$ and prove for all: $$ O\in\mathcal O $$ : $$ P(S_0,O,\partial_0), $$ we can say only: $$ \boxed{ \text{Object-conditional closure}. } $$ This cannot automatically be promoted to: $$ \boxed{ \text{triadic global closure}. } $$ --- # 29. Observer Relativity Audit If a research conclusion: $$ C $$ holds under: $$ S_1 $$ , we must ask: $$ \boxed{ C(S_1) \overset{?}{=} C(S_2). } $$ This is not physical relativity. Instead: $$ \boxed{ \text{observation-model relativity}. } $$ --- # 30. Observer-Invariant Property Define: $$ P $$ to be observer-invariant if: $$ \boxed{ \forall S\in\mathcal S_{\rm adm}, \quad P(S,O,\partial) = P(O,\partial). } $$ This is a question of provability. Invariance cannot be assumed merely because intuition says the results should be the same. --- # 31. Gauge Subject Some changes of Subject are merely gauge changes. For example: $$ S\rightarrow S' $$ but there exists: $$ g $$ such that: $$ \mathcal R(S',O,\partial) = g\cdot \mathcal R(S,O,\partial). $$ then: $$ S \sim S'. $$ --- # 32. Subject Quotient Define gauge equivalence: $$ \sim_S. $$ The actual observer state can then use: $$ \boxed{ \mathcal S/\sim_S. } $$ to avoid counting the same observational method repeatedly. --- # 33. Object Gauge Similarly: $$ O_1 \sim_O O_2 $$ if the difference is only: - rigid transform； - label change； - parameter chart change； without changing the property under study. --- # 34. Boundary Gauge Boundary may also have: $$ \partial_1 \sim_\partial \partial_2. $$ For example: - coordinate reparameterization； - equivalent clipping representation； - same geometric carrier with different mesh。 --- # 35. Triadic Quotient The actual state space may be: $$ \boxed{ \mathcal X_{\rm phys} = \mathcal X/\sim } $$ where: $$ \sim $$ takes into account simultaneously: - Subject gauge； - Object gauge； - Boundary gauge； - coupled relabeling。 --- # 36. Labeled Geometry If Object carries: $$ L_O $$ and Boundary carries: $$ L_\partial, $$ unmarked geometry and labelled geometry must be distinguished. For example: $$ K $$ and its mirror image may be equivalent as unmarked shapes, But: $$ (K,L) $$ need not be equivalent. --- # 37. Boundary Responsibility Every boundary segment: $$ b_i\subset\partial O $$ can be labelled: $$ \boxed{ \operatorname{Resp}(b_i). } $$ Its responsibility may originate from: - contact witness； - container wall； - support family； - visibility edge； - projection event； - no-slack constraint。 --- # 38. Boundary Responsibility Graph Build a graph: $$ \boxed{ \mathcal G_\partial = (V_\partial,E_\partial). } $$ Its nodes can include: - boundary sector； - witness； - support family； - subject condition； - container condition。 An edge denotes: $$ \boxed{ \text{responsibility relation}. } $$ --- # 39. Boundary Sector Traceability Ideally: $$ \boxed{ b_i \leftarrow c_j \leftarrow w_k } $$ where: - $b_i$：boundary sector； - $c_j$：active contact； - $w_k$：witness / constraint source。 This makes: $$ \boxed{ \text{shape explanation} } $$ more than a visual description. --- # 40. No-Slack Boundary in SOB Language If a boundary sector: $$ b $$ has no: - contact； - support； - visibility； - containment； - other active responsibility， an inward perturbation may exist: $$ b \rightarrow b-\varepsilon n. $$ If the objective improves and feasibility remains unchanged, the original state is not extremal. Thus: $$ \boxed{ \text{extremal boundary} \Rightarrow \text{responsibility coverage}. } $$ --- # 41. Visibility Responsibility Some 2D boundaries are not the object's actual boundary. They may merely be: $$ \boxed{ \text{visibility boundary}. } $$ For example, a silhouette. The following labels must therefore be assigned: ```text PHYSICAL_BOUNDARY CONTACT_BOUNDARY SILHOUETTE_BOUNDARY CLIPPING_BOUNDARY PROJECTION_BOUNDARY ``` Otherwise a render edge can easily be mistaken for a mathematical edge. --- # 42. Contact Responsibility A contact boundary: $$ b_c $$ is generated by: $$ g_i=0 $$ . If: $$ \lambda_i>0 $$ is the corresponding dual multiplier, we can retain: $$ \boxed{ (b_c,g_i,\lambda_i). } $$ This can connect directly to LESR-type contact pressure. --- # 43. Subject-Induced Boundary Some boundaries are generated entirely by Subject. For example, a camera clipping plane: $$ \partial_V(S). $$ Thus: $$ \boxed{ \partial \text{ can depend on }S. } $$ More generally: $$ \partial = \partial(S,O). $$ --- # 44. Object-Induced Boundary Similarly: $$ \partial_K $$ may be generated by an object pair. For example, a Voronoi / contact bisector. Thus: $$ \boxed{ \partial \text{ need not be an exogenous variable.} } $$ --- # 45. Endogenous Boundary Let: $$ \boxed{ \partial = B(S,O). } $$ The actual state can then be reduced to: $$ \Gamma = \langle S,O,B(S,O) \rangle. $$ For auditing, however, we still retain: $$ \partial $$ as an explicit state. --- # 46. Exogenous Boundary If a container: $$ C $$ is specified externally by the problem, then: $$ \partial_C $$ is exogenous. Boundary should therefore be labelled: ```text ENDOGENOUS EXOGENOUS DERIVED GAUGE ``` --- # 47. Equivalence of Moving the Subject and Moving the Object In some cases: $$ S\rightarrow gS $$ is equivalent to: $$ O\rightarrow g^{-1}O. $$ Thus: $$ \boxed{ \text{move observer} \sim \text{inverse-move object} } $$ is valid only when: - projection operator covariance； - boundary covariance； - no external anisotropy； hold. --- # 48. Why This Equivalence Often Fails If the boundary is fixed: $$ \partial=\partial_0, $$ then: $$ g^{-1}O $$ may change: $$ R_{O\partial}. $$ Thus: $$ \boxed{ \text{observer/object duality breaks under fixed boundary}. } $$ This is one reason Boundary cannot be ignored. --- # 49. Triadic Symmetry Let: $$ g $$ act simultaneously: $$ g: (S,O,\partial) \mapsto (gS,gO,g\partial). $$ If: $$ \mathcal R(gS,gO,g\partial) = g'\mathcal R(S,O,\partial), $$ the system has triadic covariance. --- # 50. Partial Symmetry If the action is applied only to: $$ O $$ and not to: $$ S,\partial, $$ symmetry may fail. Therefore: $$ \boxed{ \text{Object symmetry} \neq \text{system symmetry}. } $$ --- # 51. The SOB Version of a Mirror Test An ordinary mirror test performs only: $$ O \rightarrow \sigma O. $$ An SOB mirror test should examine: $$ \boxed{ (S,O,\partial) \rightarrow ( \sigma_S S, \sigma_O O, \sigma_\partial\partial ). } $$ and determine whether: $$ \mathcal R $$ is covariant. --- # 52. Why Labelling Systems Affect Mirror Tests If: $$ B_3 $$ and: $$ B_5 $$ are different labels, after reflection it may be necessary to map: $$ B_3 \leftrightarrow B_5. $$ Without a label map, the result will be misclassified as: $$ \boxed{ \text{asymmetry}. } $$ --- # 53. Subject Label Subject may also have an orientation label. For example: ```text LEFT_HANDED_CAMERA_FRAME RIGHT_HANDED_CAMERA_FRAME ``` When moving across runtimes: $$ \boxed{ \text{coordinate handedness} } $$ must therefore be explicitly retained. --- # 54. Boundary Orientation Boundary can also have an orientation: $$ n. $$ Reflection changes orientation. The transformation of: $$ n $$ in contact equations must therefore be performed as well. --- # 55. The Observer Is Not an Omniscient Viewpoint If a proof system fixes: $$ S=S_{\rm global} $$ as an omniscient observer, its result may merely be: $$ \boxed{ \text{global-observer model result}. } $$ If the actual system has only partial visibility, we must separately consider: $$ S_{\rm partial}. $$ --- # 56. Partial Observation Subject may acquire only: $$ \mathcal O_S(X) $$ rather than the complete: $$ X. $$ Therefore: $$ \boxed{ Y = \mathcal O_S( S,O,\partial ). } $$ This connects to perception / sensing problems. --- # 57. Multiple Subjects More generally, there may be: $$ S_1,\ldots,S_n. $$ forming: $$ \boxed{ \Gamma = \langle \{S_i\}, O, \partial \rangle. } $$ --- # 58. Collective Observation If each: $$ S_i $$ sees only a partial state, but their combined set can reconstruct: $$ O, $$ then: $$ \boxed{ \text{collective observability} } $$ holds. This can support future geometric audits with multiple cameras, AIs and agents. --- # 59. Subject Disagreement If: $$ Y_i = \mathcal R(S_i,O,\partial) $$ disagree, we should not immediately judge: $$ \boxed{ \text{which one is wrong.} } $$ First inspect: - projection difference； - visibility； - resolution； - query semantics； - boundary access。 --- # 60. Observation Equivalence Define: $$ S_1 \equiv_{O,\partial,Q} S_2 $$ If: $$ \boxed{ \mathcal R_Q(S_1,O,\partial) = \mathcal R_Q(S_2,O,\partial). } $$ This is a task-relative observer equivalence. --- # 61. Boundary Equivalence Similarly: $$ \partial_1 \equiv_{S,O,Q} \partial_2 $$ if they produce the same result for the specified query. --- # 62. Object Equivalence $$ O_1 \equiv_{S,\partial,Q} O_2 $$ if the observer cannot distinguish them under the specified boundary and query. --- # 63. Triadic Equivalence Is Not a Simple Product of Component Equivalences It is possible that: $$ S_1\not\equiv S_2, $$ $$ O_1\not\equiv O_2, $$ But: $$ \boxed{ (S_1,O_1,\partial_1) \equiv (S_2,O_2,\partial_2). } $$ because the differences cancel each other. --- # 64. Counterfactual SOB Test For a known state: $$ \Gamma $$ generate a counterfactual: $$ \Gamma'. $$ For example, modify only: $$ S. $$ If the conclusion changes abruptly, this indicates: $$ \boxed{ \text{that the result is observer-sensitive}. } $$ Modify only: $$ \partial. $$ If the conclusion changes, this indicates: $$ \boxed{ \text{that the result is boundary-sensitive}. } $$ --- # 65. Sensitivity Tensor Locally, we can define: $$ \boxed{ D\mathcal R = \left[ \frac{\partial\mathcal R}{\partial S}, \frac{\partial\mathcal R}{\partial O}, \frac{\partial\mathcal R}{\partial\partial} \right]. } $$ This is a conceptual representation. For nonsmooth events, use: - directional derivative； - generalized derivative； - finite difference； - event-based sensitivity。 --- # 66. Responsibility Sensitivity If a boundary sector's responsibility: $$ R_i $$ is extremely sensitive to: $$ S $$ , then: $$ \boxed{ \text{it may be an observation-induced boundary}. } $$ If it is sensitive to: $$ O $$ , it may be: $$ \boxed{ \text{object-driven}. } $$ If it is sensitive to: $$ \partial $$ , then: $$ \boxed{ \text{container/interface-driven}. } $$ --- # 67. The Subject–Object–Boundary Classification Table Every event / boundary / shape feature can be labelled: ```text SUBJECT_DOMINANT OBJECT_DOMINANT BOUNDARY_DOMINANT COUPLED UNKNOWN ``` This assists subsequent closure. --- # 68. Coupled Feature If a feature: $$ F $$ appears only when: $$ S=S^\star, $$ $$ O=O^\star, $$ $$ \partial=\partial^\star $$ hold jointly, then: $$ \boxed{ F \text{ is a triad-coupled feature}. } $$ --- # 69. Three Responsibilities of Closure ## Subject Closure $$ \boxed{ \mathcal S_{\rm adm} } $$ has it been covered? ## Object Closure $$ \boxed{ \mathcal O_{\rm adm} } $$ has it been covered? ## Boundary Closure $$ \boxed{ \mathcal B_{\rm adm} } $$ has it been covered? --- # 70. Coupled Closure Even if all three are individually finite: $$ |\mathcal S|<\infty, $$ $$ |\mathcal O|<\infty, $$ $$ |\mathcal B|<\infty, $$ their combination: $$ |\mathcal S| |\mathcal O| |\mathcal B| $$ may still be large. Actual closure should therefore exploit: - symmetry； - equivalence； - monotonicity； - responsibility pruning； - event decomposition。 --- # 71. False Closure Example Suppose we test only: $$ S_0, $$ $$ \partial_0 $$ and find no counterexample for many: $$ O_i $$ . This does not imply: $$ \boxed{ \forall S,O,\partial, P(S,O,\partial). } $$ because a counterexample may occur only under: $$ S_1 $$ or: $$ \partial_1 $$ . --- # 72. SOB Counterexample A complete counterexample should retain: $$ \boxed{ C_{\rm x} = ( S^\star, O^\star, \partial^\star, Q^\star, Y^\star ). } $$ not merely: $$ O^\star. $$ --- # 73. Why This Matters for 3D-to-2D In dynamic projection: $$ S $$ controls: - camera； - projection； - sampling。 $$ O $$ controls: - 3D source object； - deformation； - motion。 $$ \partial $$ controls: - screen； - clipping； - contact； - visibility。 Thus Paper 02's: $$ G_t = \Pi_t(T_tM) $$ should more completely be written as: $$ \boxed{ G_t = \mathcal R( S_t, O_t, \partial_t ). } $$ --- # 74. The SOB Interpretation of 2.5D 2.5D state： $$ Y^{2.5D} $$ does not retain only depth. It should also retain: ```text subject_id object_id boundary_id source_primitive visibility_reason projection_reason contact_reason ``` Every 2D feature can thus be traced backward. --- # 75. SOB Controls in an Interactive UI Three control panels can be built: ## Subject Controls - camera； - projection； - observation path； - resolution。 ## Object Controls - transform； - articulation； - deformation； - material geometry。 ## Boundary Controls - container； - slice plane； - clipping； - support plane； - visibility mask。 --- # 76. Why the Three Control Groups Cannot Be Mixed If the UI offers only: ```text Move Object ``` the user may not know that: $$ \boxed{ \text{changing the camera} } $$ can actually generate an entirely different family of 2D states. Similarly: $$ \boxed{ \text{changing the boundary} } $$ may expose a new configuration more readily than changing the object. --- # 77. An AI Agent's SOB Action Space An agent action can be written as: $$ \boxed{ a_t = ( a_t^S, a_t^O, a_t^\partial ). } $$ An agent does not manipulate only the object. It can choose perturbations in three directions: - observer； - object； - boundary； . --- # 78. Counterexample Policy Define: $$ \pi: \Gamma_t \mapsto a_t. $$ If the goal is to find unknown classes: $$ \boxed{ \pi^\star = \arg\max_\pi \mathbb E[ N(Y_t) ]. } $$ where: $$ N $$ is the novelty / counterexample score. --- # 79. The Subject Itself Can Also Be Attacked The final stress test cannot assume that: $$ S $$ is always correct. We can deliberately: - shift camera； - perturb projection matrix； - lower resolution； - change sampling phase； - change observation order。 to see whether a theorem-like conclusion is merely an observer artifact. --- # 80. Boundary Must Also Be Attacked We can deliberately: - deform container； - change contact surface； - move clipping plane； - introduce aperture； - perturb projection plane。 If the conclusion is unstable, we have found: $$ \boxed{ \text{boundary dependence}. } $$ --- # 81. Object Attack Is Only One Third Complete adversarial geometry is therefore: $$ \boxed{ \text{Attack} = \text{Attack}_S + \text{Attack}_O + \text{Attack}_\partial. } $$ --- # 82. SOB Certificate Define: $$ \boxed{ C_{\rm SOB} } $$ Minimum contents: ```text subject_state object_state boundary_state query projection/reduction operator gauge conventions labels responsibility graph event ledger result precision policy verifier hashes ``` --- # 83. Subject Certificate Subject retains at least: - pose； - projection matrix / analytic operator； - handedness； - clipping； - sampling； - query； - time path。 --- # 84. Object Certificate Object retains at least: - canonical geometry； - transform； - deformation； - articulation； - labels； - topology。 --- # 85. Boundary Certificate Boundary retains at least: - geometric carrier； - orientation； - contact rule； - visibility rule； - projection interface； - provenance。 --- # 86. Coupling Certificate Also retain: $$ \boxed{ R_{SO}, R_{O\partial}, R_{S\partial}. } $$ to prevent three separate files from failing to reconstruct the actual relationships. --- # 87. A Verifier Should Not Assume the Three Components Are Independent If a verifier checks only that: $$ S, $$ $$ O, $$ $$ \partial $$ are individually admissible, but does not check: $$ \mathcal C_{SOB}, $$ it may still accept an incorrect state. --- # 88. Boundary Mutation Test Deliberately mutate: - normal； - container size； - clipping； - contact threshold。 If the result is unchanged when theory says it should change, the verifier may not actually be reading the boundary. --- # 89. Subject Mutation Test Deliberately mutate: - handedness； - camera； - projection； - query。 If the verifier still returns the same result, check: $$ \boxed{ \text{whether Subject has actually been ignored.} } $$ --- # 90. Object Mutation Test Deliberately mutate: - coordinate； - edge； - articulation； - deformation。 This is the most traditional test. It cannot, however, replace the preceding two. --- # 91. The Triadic Completeness Question The final question is: $$ \boxed{ \text{Is the tested triadic state family complete?} } $$ that is: $$ \mathcal X_{\rm tested} \overset{?}{=} \mathcal X_{\rm admissible}. $$ --- # 92. Completeness Need Not Come from Enumeration It can rely on: - quotient； - symmetry； - interval covering； - monotonicity； - topology classes； - deformation retraction； - finite event classes。 --- # 93. Subject Compression If many: $$ S $$ are equivalent for the query, compress them to: $$ [S]. $$ --- # 94. Object Compression If many: $$ O $$ are equivalent in the specified SOB context, compress them to: $$ [O]. $$ --- # 95. Boundary Compression Similarly: $$ [\partial]. $$ --- # 96. Triadic Class The final classification unit can be: $$ \boxed{ [\Gamma] = [(S,O,\partial)]. } $$ --- # 97. Closure Graph Build: $$ \boxed{ \mathcal G_{\rm SOB}. } $$ node： $$ [\Gamma_i]. $$ edge： - subject perturbation； - object perturbation； - boundary perturbation； - coupled transition。 --- # 98. Edge Typing Possible labels: ```text S-EDGE O-EDGE B-EDGE SO-EDGE SB-EDGE OB-EDGE SOB-EDGE ``` This directly indicates the source of a transition. --- # 99. Critical Triadic Event If an event occurs only when: $$ S,O,\partial $$ are modified simultaneously, mark it: $$ \boxed{ E_{SOB}. } $$ This is the type most easily missed by single-variable scans. --- # 100. The SOB Mapping in Lebesgue-Type Problems For a geometric covering experiment, we can set: ## Subject $$ S = \text{projection / observation / support-query frame}. $$ ## Object $$ O = \text{witness or perturbation object}. $$ ## Boundary $$ \partial = \text{candidate container / support / contact boundary}. $$ --- # 101. Container and Object Roles Must Not Be Confused In some problems: $$ O $$ is the witness placed inside the container. In other experiments: $$ O $$ can also be a container candidate. The role should therefore be determined by: $$ Q $$ . A geometric object must not be confused with its role in the problem. --- # 102. Role Map Define: $$ \boxed{ \rho_{\rm role}: \text{entity} \rightarrow \{ S,O,\partial \}. } $$ The same geometric entity can change roles between experiments. --- # 103. Interchanging Subject and Object Some problems permit: $$ S \leftrightarrow O. $$ For example, an object-mounted camera. This is not an automatic symmetry. We must check whether: $$ \mathcal R $$ is covariant. --- # 104. Object–Boundary Duality Some container problems allow us to exchange viewpoints between: $$ O $$ and: $$ \partial_C $$ . For example: > Can the object enter the container? and: > Can the container enclose the object? Their meanings are close, but their mathematics need not be identical. --- # 105. Boundary Is Not a Passive Background This is one of the paper's most important ideas: $$ \boxed{ \partial \text{ can actively generate geometry}. } $$ For example: - silhouette； - slice； - contact； - clipping； - constrained motion。 --- # 106. Subject Is Not a Passive Background Either Similarly: $$ \boxed{ S \text{ can actively generate geometry}. } $$ By changing only the camera: $$ O $$ can remain entirely stationary while still generating a family: $$ G_t. $$ --- # 107. Object Is Not the Only Source Either Therefore: $$ \boxed{ \text{geometric possibility} \neq \text{object possibility only}. } $$ Instead: $$ \boxed{ \text{triadic possibility}. } $$ --- # 108. Triadic Possibility Space Define: $$ \boxed{ \mathfrak P_{\rm SOB} = \mathcal R( \mathcal S_{\rm adm} \times \mathcal O_{\rm adm} \times \mathcal B_{\rm adm} ). } $$ This is the paper's core space. --- # 109. Projected Possibility Space If we ultimately care only about two dimensions: $$ \boxed{ \mathfrak P_{2D} = \Pi( \mathfrak P_{\rm SOB} ). } $$ Paper 01's 3D-to-2D possibility expansion is now extended to a triadic source space. --- # 110. Omitted Possibilities If research takes only: $$ S=S_0, $$ $$ \partial=\partial_0, $$ what is actually explored is: $$ \boxed{ \mathfrak P_{2D}^{(0)} = \Pi( \{S_0\} \times \mathcal O \times \{\partial_0\} ). } $$ In general: $$ \mathfrak P_{2D}^{(0)} \subseteq \mathfrak P_{2D}. $$ --- # 111. Missing Possibility Region Define: $$ \boxed{ \Delta\mathfrak P = \mathfrak P_{2D} \setminus \mathfrak P_{2D}^{(0)}. } $$ One important goal of the final stress test is: $$ \boxed{ \text{search }\Delta\mathfrak P. } $$ --- # 112. Why Object Enumeration Alone Is Insufficient Even if: $$ \mathcal O $$ is scanned densely, as long as: $$ S $$ and: $$ \partial $$ are incorrectly fixed, we may still miss: $$ \Delta\mathfrak P. $$ --- # 113. Research and Public Explanation Share the Same Model The SOB model is also suited to visualization. The left side can show: ```text Subject Object Boundary ``` and the right side: ```text Projected Result Responsibility Event ``` This is not an additional model for public explanation. It is the public projection of the same canonical state. --- # 114. A Blender / Engine SOB UI Three layers of gizmos can be designed: ```text S: camera/projection gizmo O: object/deformation gizmo B: boundary/interface gizmo ``` Every operation is recorded as: $$ \Delta S, \Delta O, \Delta\partial. $$ --- # 115. Replay An SOB interaction history: $$ H $$ can be retained as: $$ \boxed{ H = \{ (\Delta S_t, \Delta O_t, \Delta\partial_t) \}_{t}. } $$ The same experiment can therefore be replayed completely. --- # 116. An AI Observer Can Also Be Included in Subject In future, AI can serve as: $$ S_{\rm AI}. $$ Its observation policy: $$ \pi_{\rm obs} $$ also becomes part of the subject state. --- # 117. Human and AI Subjects Need Not Be the Same A human: $$ S_H $$ may use: - visual projection； - semantic labels。 AI： $$ S_A $$ may directly read: - exact coordinates； - graph； - interval certificate。 The two can produce different observation representations for the same: $$ O,\partial $$ . --- # 118. Multi-Representation Consistency If different Subject representations all claim to describe the same state, a mapping is required: $$ \boxed{ \phi_{HA}: Y_H \leftrightarrow Y_A. } $$ This can be used in a cross-observer audit. --- # 119. Subject Should Not Contaminate Canonical Truth Subject's representation: $$ Y_S $$ is not canonical object truth. Therefore: $$ \boxed{ \text{Observation} \neq \text{Object}. } $$ --- # 120. Boundary Should Not Be Mistaken for Object Either For example, the silhouette boundary: $$ \partial_{\rm sil} $$ is not the physical object's boundary. The canonical ontology must therefore distinguish them. --- # 121. SOB Ontology Recommended minimum categories: ```text SUBJECT OBJECT BOUNDARY OBSERVATION PROJECTION CONTACT VISIBILITY RESPONSIBILITY EVENT CERTIFICATE ``` --- # 122. Triadic Error Taxonomy Errors can be classified as: ```text SUBJECT_ERROR OBJECT_ERROR BOUNDARY_ERROR COUPLING_ERROR PROJECTION_ERROR OBSERVATION_ERROR GAUGE_ERROR LABEL_ERROR ``` This is more diagnostic than: ```text geometry wrong ``` . --- # 123. Closure Status Every research node can retain: ```text SUBJECT_CLOSURE OBJECT_CLOSURE BOUNDARY_CLOSURE COUPLING_CLOSURE PROJECTION_CLOSURE ``` Only when all are sufficient is the following permitted: ```text TRIADIC_CLOSURE ``` --- # 124. Conditions for Triadic Closure Formally: $$ \boxed{ \forall (S,O,\partial) \in \mathcal X_{\rm adm}, \quad P(S,O,\partial). } $$ If the theorem itself is invariant with respect to: $$ S $$ or: $$ \partial $$ we can first prove invariance and then quotient it out. --- # 125. A Better Closure Strategy Rather than brute-force traversal of: $$ \mathcal S \times \mathcal O \times \mathcal B. $$ Instead: $$ \boxed{ \text{prove invariance} \rightarrow \text{quotient gauges} \rightarrow \text{classify events} \rightarrow \text{attack coupled residuals}. } $$ --- # 126. Counterexample-First Strategy If closure is nearly complete, first search for: $$ \boxed{ E_{SOB} } $$ rather than continuing to repeat known object-only perturbations. A coupled event is the most likely place for the original model to have omissions. --- # 127. Relation to the Next Paper Paper 03 establishes: $$ \Gamma = \langle S,O,\partial \rangle. $$ For now, however: $$ O $$ is still a general object. The next paper will specifically relax: $$ O $$ in its internal degrees of freedom: $$ \boxed{ \text{Rigid} \rightarrow \text{Articulated} \rightarrow \text{Deformable} \rightarrow \text{Locally Bendable} \rightarrow \text{Worm-like}. } $$ The next paper is therefore: $$ \boxed{ \textbf{Paper 04 — Perturbation Objects Beyond Rigidity}. } $$ --- # 128. The Paper's Main Propositions ## Proposition A $$ \boxed{ \text{Geometric conclusions can depend jointly on Subject, Object, and Boundary.} } $$ ## Proposition B $$ \boxed{ \text{Object-only perturbation does not establish triadic closure.} } $$ ## Proposition C $$ \boxed{ \text{Boundary is an active geometry-generating component, not merely background.} } $$ ## Proposition D $$ \boxed{ \text{Observer configuration must be auditable whenever projection or visibility matters.} } $$ ## Proposition E $$ \boxed{ \text{A complete counterexample must preserve }(S,O,\partial)\text{ provenance.} } $$ ## Proposition F $$ \boxed{ \text{The true projected possibility space is the image of the triadic admissible state space.} } $$ --- # 129. Non-Claims This paper does not claim: 1. that every mathematical geometry problem needs Subject; 2. that every boundary must be modelled independently; 3. that observer relativity is the same as physical relativity; 4. that Subject must have consciousness; 5. that Boundary must be a physical interface; 6. that merely adding Subject, Object and Boundary proves global closure; 7. that a triadic model is necessarily more effective than every binary model; 8. that all subject differences are substantive; 9. that all object / boundary roles cannot be exchanged; 10. that any currently unsolved geometric problem has been solved by the SOB framework. --- # 130. Conclusion This paper proposes: $$ \boxed{ \Gamma = \langle S,O,\partial \rangle } $$ as the basic state of a geometric experiment. This model corrects a common but hidden assumption: > Geometric results are determined only by the object. In fact, when a problem involves: - projection； - visibility； - contact； - containment； - support； - observation； its result may depend on: $$ \boxed{ \text{who observes} + \text{what is observed} + \text{which boundary mediates the relation}. } $$ Complete perturbation must therefore be: $$ \boxed{ (\Delta S,\Delta O,\Delta\partial), } $$ rather than: $$ (0,\Delta O,0). $$ This upgrades the final stress test from: > We changed the object many times. to: > Have we actually attacked the coupled space of the observation model, object degrees of freedom and boundary conditions? This is also the common foundation for subsequent flexible perturbations, worm-like objects, projection-induced hidden geometry and adversarial closure. --- # Canonical Source Declaration This file is the UTF-8 Markdown canonical source for `PPS-SOBG-03-v0.1`. The only mathematical delimiters used are `$...$` and `$$...$$`. The paper strictly distinguishes: - Subject； - Object； - Boundary； - Observation； - Projection； - Responsibility； - Gauge； - Triadic Closure。 ===END===