> Full-text translation from Chinese. The original remains authoritative for its research claims; this translation is not independent mathematical verification. # NIOT-03 — Unboundedly Expandable Finite Matrices: Dynamic Finite Dimensionality, Sparse Structure, and Task Closure ## Unboundedly Expandable Finite Matrices: Dynamic Finite Dimensionality, Sparse Structure, and Task-Relative Closure **Series:** Numerical Information Object Theory (NIOT) / Numerical Information Object Theory **Paper:** Paper 03 **Author:** Neo.K **Institution:** EveMissLab / 一言諾科技有限公司 **Version:** v0.1 **Date:** 2026-09-22 **Document type:** Dynamic matrix theory / runtime ontology / finite–unbounded bridging paper **Prerequisites:** NIOT-00, NIOT-01, NIOT-02 **Related theories:** ISQL, AIC, BCODR, CAIR, SEDB, dynamic indexing, completion–projection bidirectional semantics **Status:** DRAFT / DYNAMIC MATRIX FOUNDATION / IMPLEMENTATION-READY **Canonical Source:** UTF-8 Markdown **Mathematical source conventions:** Inline math uses only `$...$`, and display math uses only `$$...$$` --- # Abstract NIOT-01 established numerical information objects and the Projection Non-Identity Principle; NIOT-02 further defined information probes as first-class observation operators that refine equivalence classes, compose, admit verification, and have costs. As probes are progressively added, the system immediately faces a central engineering and mathematical question: **How can informational dimensionality continue to increase while every actual execution state remains finite?** This paper proposes the Unboundedly Expandable Finite Matrix (UEFM) as the central dynamic representation of NIOT. Its basic principle is: $$ \boxed{ \forall t,\quad n_t<\infty, } $$ However: $$ \boxed{ \sup_t n_t=\infty. } $$ Here $n_t$ denotes the informational dimensionality activated in epoch $t$. The first expression ensures that the state at any moment can be stored, computed, and verified by a finite machine; the second means that the theory and runtime do not impose a fixed maximum dimension in advance. New probes, relations, indices, spectra, certificates, bridges, or local structures can be added in subsequent epochs. This paper emphasizes that UEFM is not simply "an approximation of an infinite matrix." More precisely, it is a **sequence of dynamic finite structures**: $$ \boxed{ M_0 \rightarrow M_1 \rightarrow M_2 \rightarrow \cdots } $$ Each $M_t$ can change its: - dimensionality; - sparse topology; - index set; - probe basis; - relation graph; - spectral coordinates; - certificate state; - compression state; - local neighborhood. Therefore: $$ M_t \subseteq M_{t+1} $$ is not a necessary condition. More generally: $$ \boxed{ M_{t+1} = \mathcal U \left( M_t, \Delta_t, \Gamma_t \right), } $$ Here $\Delta_t$ denotes new information, while $\Gamma_t$ denotes reindexing, compression, merging, governance, and structural conversion. This paper further defines active dimension, latent dimension, structural rank, probe rank, sparse envelope, dynamic reindexing, matrix folding, task-relative compression, local expansion, frontier cell, closure certificate, and bounded execution budget. Importantly, it separates "unbounded expansion" entirely from "infinite execution": UEFM allows new informational dimensions to exist in the future, while every concrete task can achieve closure under finite conditions. Ultimately, this paper proposes: $$ \boxed{ \text{finite actuality} + \text{unbounded extensibility} + \text{task-relative closure} } $$ as the threefold foundation of dynamic informational structures in NIOT. **Keywords:** Unboundedly Expandable Finite Matrix, Dynamic Finite Dimensionality, Sparse Matrix, Probe Matrix, Task Closure, Dynamic Indexing, Reindexing, Compression, Local Expansion, Numerical Information Space --- # 1. The Problem: Informational Dimensionality Can Increase, but Machines Are Always Finite NIOT-02 has already established: $$ \mathcal P_t = \{ P_1,\dots,P_{n_t} \}. $$ Whenever an effective probe is added: $$ P_{n_t+1}, $$ the system may acquire a new distinguishable dimension. Thus information representation naturally grows: $$ n_t \rightarrow n_{t+1}. $$ But every actual machine has: - finite memory; - finite computation time; - finite storage; - finite bandwidth; - a finite probe budget. Thus one cannot require: $$ n_t=\infty $$ to hold at any actual epoch. The basic problem of this paper is therefore: $$ \boxed{ \text{How can finite executability and unbounded extensibility be retained simultaneously?} } $$ --- # 2. Basic Definition of Unboundedly Expandable Finite Matrices ## 2.1 Epoch-wise finite state Let: $$ t\in\mathbb N $$ denote a discrete runtime epoch. Define a finite active index set for each epoch: $$ \boxed{ I_t = \{ i_1,\dots,i_{n_t} \}. } $$ and: $$ \boxed{ n_t=|I_t|<\infty. } $$ --- ## 2.2 Finite matrix state Define: $$ \boxed{ M_t = \left[ m^{(t)}_{ij} \right]_{i,j\in I_t}. } $$ Therefore: $$ M_t \in \mathbb K^{n_t\times n_t} $$ is only the simplest case. More generally, matrix elements may belong to: $$ \mathcal V $$ where $\mathcal V$ may be: - a scalar; - an interval; - a symbolic object; - a relation state; - a certificate; - a sparse object; - a structured record; - a tensor block. --- ## 2.3 Unbounded extensibility Define the central UEFM condition: $$ \boxed{ \forall t,\quad n_t<\infty, } $$ and allow: $$ \boxed{ \sup_t n_t=\infty. } $$ "Allow" is very important here. It does not claim that: $$ \lim_{t\to\infty}n_t=\infty $$ necessarily holds for every task. It means only that the system architecture does not impose a theoretical maximum dimension in advance. --- # 3. Finite Actualization and Unbounded Extensibility This paper calls: $$ \boxed{ \forall t,\quad n_t<\infty } $$ is: the **Finite Actualization Principle**. It calls: $$ \boxed{ \sup_t n_t=\infty } $$ is: the **Unbounded Extensibility Principle**. Together, they form: $$ \boxed{ \text{Finite Actuality} + \text{Unbounded Extensibility}. } $$ --- # 4. UEFM Is Not a Truncation of a Static Infinite Matrix ## 4.1 Static Truncation Model A traditional infinite-matrix approximation may be written as: $$ A_\infty \rightarrow A_n. $$ where: $$ A_n $$ is only a finite truncation of: $$ A_\infty $$ as displayed above. --- ## 4.2 UEFM Does Not Require a Completed Matrix in Advance NIOT's UEFM does not require an already-known complete object: $$ M_\infty $$ as displayed above. A more reasonable formulation is: $$ \boxed{ M_t \rightarrow \text{new distinction} \rightarrow M_{t+1}. } $$ That is: $$ \boxed{ \text{representation grows with discovery}. } $$ --- ## 4.3 Open-ended dimensionality Therefore: $$ n_{t+1} $$ can be determined by new probes, indices, relations, or certificates, rather than a predetermined infinite coordinate set. --- # 5. Probe-induced Matrix ## 5.1 Probe coordinates From NIOT-02: $$ \mathcal P_t = \{ P_1,\dots,P_{n_t} \}. $$ The simplest information vector is: $$ \boxed{ \mathbf p_t(\mathfrak N) = \begin{bmatrix} P_1(\mathfrak N)\\ P_2(\mathfrak N)\\ \vdots\\ P_{n_t}(\mathfrak N) \end{bmatrix}. } $$ --- ## 5.2 Relation matrix To record relationships between probes, define: $$ \boxed{ m_{ij}^{(t)} = R_t \left( P_i(\mathfrak N), P_j(\mathfrak N) \right). } $$ Therefore: $$ \boxed{ M_t = \left[ R_t(P_i,P_j) \right]. } $$ --- ## 5.3 The Matrix Need Not Be Symmetric In general: $$ m_{ij}\neq m_{ji}. $$ For example: - dependency; - derivation; - causal order; - address transition; - a directed certificate; - bridge direction. Thus UEFM permits directed structures by default. --- # 6. Active Dimension and Latent Dimension ## 6.1 Active dimension Define: $$ \boxed{ d_{\mathrm{act}}(t) = n_t. } $$ denotes the informational dimensionality actually materialized at present. --- ## 6.2 Latent dimension Let: $$ \Lambda_{\mathrm{avail}}(t) $$ be candidates for probes / coordinates that are currently discoverable but not materialized. then: $$ \boxed{ d_{\mathrm{lat}}(t) = |\Lambda_{\mathrm{avail}}(t)|. } $$ It may be: - finite; - very large; - unknown; - theoretically unbounded. --- ## 6.3 Active Is Not Total Therefore: $$ \boxed{ d_{\mathrm{act}} \neq d_{\mathrm{total}}. } $$ The current matrix dimensionality is only: $$ \boxed{ \text{current explicit information dimension}. } $$ --- # 7. Structural Rank Traditional matrix rank: $$ \operatorname{rank}(M_t) $$ applies only in a linear-algebra setting. NIOT needs a more general structural rank: $$ \boxed{ \rho_s(M_t) } $$ meaning: > How many structural degrees of freedom in the current active coordinates cannot actually be eliminated or reconstructed from existing rules? In a purely linear scalar matrix, one can set: $$ \rho_s(M_t) = \operatorname{rank}(M_t). $$ But in graph / certificate / symbolic states, a separate definition is needed. --- # 8. Probe Rank For a probe set: $$ \mathcal P_t, $$ Define: $$ \boxed{ \rho_P(\mathcal P_t) } $$ as the size of a minimal probe basis. If: $$ |\mathcal P_t|=100 $$ but only: $$ \rho_P=12, $$ this means that many probes are: - redundant; - derived; - dominated; - equivalent. --- # 9. Matrix Expansion ## 9.1 Additive expansion The simplest case: $$ I_{t+1} = I_t \cup \{i_{n_t+1}\}. $$ Therefore: $$ M_t \mapsto M_{t+1}. $$ --- ## 9.2 Block expansion A block can be added at once: $$ J_t = \{ j_1,\dots,j_k \}. $$ then: $$ \boxed{ I_{t+1} = I_t \cup J_t. } $$ --- ## 9.3 Local expansion Expand only a subregion: $$ U\subseteq I_t. $$ Define: $$ \boxed{ \operatorname{ExpandLocal} ( M_t,U ) \rightarrow M_{t+1}. } $$ This is the main mechanism for avoiding global dimensional explosion. --- # 10. Matrix Contraction UEFM does not require dimensionality to increase monotonically. If some coordinates are: - redundant; - obsolete; - invalid; - revoked; - compressed; they can be removed. Therefore: $$ \boxed{ n_{t+1} < n_t } $$ is entirely admissible. --- # 11. General Update Law This paper adopts: $$ \boxed{ M_{t+1} = \mathcal U \left( M_t, \Delta_t, \Gamma_t \right). } $$ where: - $\Delta_t$: new observations / probes / relations; - $\Gamma_t$: reindex / merge / split / compression / governance / invalidation rules. This is more general than: $$ M_t\subseteq M_{t+1} $$ as displayed above. --- # 12. Sparse Envelope ## 12.1 A Dense Matrix Is Not the Default If: $$ n_t $$ is large, while the actual relation count: $$ |E_t| $$ is much smaller than: $$ n_t^2, $$ a sparse representation should be used. Define the density: $$ \boxed{ \delta_t = \frac{|E_t|}{n_t^2}. } $$ If: $$ \delta_t\ll1, $$ a dense matrix is wasteful. --- ## 12.2 Sparse UEFM Represent it as: $$ \boxed{ M_t \equiv (I_t,E_t,\omega_t). } $$ where: - $I_t$: indices; - $E_t$: active relations; - $\omega_t$: edge / cell payload. Thus, in engineering terms, UEFM can be closer to: $$ \boxed{ \text{sparse dynamic graph with matrix semantics}. } $$ --- # 13. Matrix / Graph Duality A UEFM can be represented simultaneously as: $$ M_t $$ or: $$ G_t=(V_t,E_t). $$ where: $$ V_t=I_t. $$ The matrix perspective facilitates: - linear operators; - algebra; - GPU; - tensorization. The graph perspective facilitates: - sparse traversal; - provenance; - locality; - adaptive expansion. This paper therefore does not regard matrix form as the only physical implementation. --- # 14. Matrix / Hypergraph Extension If a relation involves: $$ k>2 $$ coordinates, a pairwise matrix may be insufficient. One can therefore establish the hypergraph: $$ \boxed{ \mathcal H_t = (V_t,\mathcal E_t) } $$ as displayed above. Conceptually, UEFM permits: $$ \boxed{ \text{matrix} \leftrightarrow \text{graph} \leftrightarrow \text{hypergraph} } $$ as different projections. --- # 15. Dynamic Reindexing ## 15.1 Indices Can Change Let: $$ \rho_t: I_t \rightarrow I_t'. $$ Define: $$ \boxed{ M_t' = \rho_t(M_t). } $$ This is reindexing. --- ## 15.2 Identity preservation If only addresses change, canonical object identity should be preserved: $$ \boxed{ \iota(M_t') = \iota(M_t). } $$ This continues AIC. --- # 16. Split A coordinate: $$ i $$ may be discovered, by a new probe, to contain: $$ i_1,i_2,\dots,i_k. $$ Define: $$ \boxed{ \operatorname{Split}(i) = \{i_1,\dots,i_k\}. } $$ This is an important operation in making a view microscopic. --- # 17. Merge If: $$ i_1,\dots,i_k $$ if equivalent for the current task, one can define: $$ \boxed{ \operatorname{Merge} ( i_1,\dots,i_k ) = \hat i. } $$ Require preservation of: $$ \boxed{ \text{merge certificate}. } $$ --- # 18. Split / Merge Are Not Naturally Inverses of Each Other Usually: $$ \operatorname{Merge} \circ \operatorname{Split} $$ may involve information loss. A certificate is therefore needed: $$ \boxed{ \varepsilon_{\mathrm{merge}}. } $$ --- # 19. Matrix Folding When several coordinates share the same structural pattern, one can perform: $$ \boxed{ \operatorname{Fold} ( M_t ) \rightarrow \widehat M_t. } $$ Folding is not deletion; it establishes: - an equivalence class; - a representative; - multiplicity; - an unfolding map. --- # 20. Matrix Unfolding If a fold preserves enough information, one can perform: $$ \boxed{ \operatorname{Unfold} ( \widehat M_t ) \rightarrow M_t. } $$ If exact reconstruction is impossible, one must mark: $$ \boxed{ \text{lossy fold}. } $$ --- # 21. Compression ## 21.1 Task-relative compression Define: $$ \boxed{ C_{\mathcal T} : M_t \rightarrow \widehat M_t. } $$ Require: $$ \boxed{ \operatorname{Close}_{\mathcal T}(M_t) = \operatorname{Close}_{\mathcal T}(\widehat M_t). } $$ equivalence at least with respect to task-required invariants. --- ## 21.2 Lossless and Lossy If: $$ M_t $$ can be fully reconstructed: $$ \boxed{ C_{\mathcal T} \text{ is lossless}. } $$ Otherwise: $$ \boxed{ C_{\mathcal T} \text{ is lossy}. } $$ --- # 22. Compression Certificate Every lossy compression must record: $$ \boxed{ \mathcal C_{\mathrm{comp}} = \langle \text{preserved}, \text{discarded}, \text{error}, \text{task} \rangle. } $$ This is compatible with MNVP's loss-budget idea, but applies to runtime states rather than visual projections. --- # 23. Frontier Coordinates Define: $$ \boxed{ F_t \subseteq I_t } $$ as the frontier. The frontier denotes: - unresolved; - expandable; - high information gain; - structurally ambiguous; - near a certificate boundary. --- # 24. Interior Coordinates If a coordinate is already: - stable; - closed; - compressed; - low in uncertainty; it can be placed in: $$ \boxed{ K_t = I_t\setminus F_t. } $$ called the closed interior. --- # 25. Frontier-First Expansion The next round's probes act preferentially on: $$ F_t. $$ Therefore: $$ \boxed{ P_{t+1} = \operatorname{Select} ( F_t ). } $$ This can greatly reduce global expansion cost. --- # 26. BCODR Frontier A BCODR-type system can directly mark an overlap / unresolved responsibility region as: $$ \boxed{ F_t^{\mathrm{BCODR}}. } $$ then expand at higher resolution only along the surviving spine. --- # 27. Local Matrix For: $$ U\subseteq I_t, $$ define the principal / induced local state: $$ \boxed{ M_t[U]. } $$ Many expensive computations need to run only on: $$ M_t[U] $$ as displayed above. --- # 28. Local Closure Define: $$ \boxed{ \operatorname{Close}_{\mathcal T} ( M_t[U] ) = 1. } $$ means that the local region has achieved closure for the task. The overall matrix may not yet have achieved closure. --- # 29. Hierarchical Closure One can define: $$ U_1 \subset U_2 \subset \cdots \subset I_t. $$ If: $$ \operatorname{Close}_{\mathcal T}(M_t[U_k])=1, $$ some outer layers may not need expansion. --- # 30. Epoch Every structural update produces: $$ \boxed{ \mathsf E_t. } $$ An epoch should record: - the matrix version; - active indices; - active probes; - the update rule; - provenance; - the certificate state; - the closure state. --- # 31. Persistent Identity Across Epochs If: $$ M_t $$ and: $$ M_{t+1} $$ are merely different states of the same numerical information system, then: $$ \boxed{ \iota(M_t) = \iota(M_{t+1}). } $$ unless a canonical fork occurs. --- # 32. Canonical Fork If an update produces two incompatible but admissible states: $$ M_{t+1}^{(A)}, \qquad M_{t+1}^{(B)}, $$ one can establish: $$ \boxed{ \operatorname{Fork}(M_t). } $$ One cannot pretend there is only one history. --- # 33. Matrix Provenance Each cell / edge: $$ m_{ij}^{(t)} $$ should have traceable: $$ \boxed{ \operatorname{Prov} ( m_{ij}^{(t)} ). } $$ Possible sources include at least: - a direct probe; - a derived probe; - an import; - a bridge; - an inference; - a certificate; - a manual assertion. --- # 34. Cell State Each cell can be labeled: ```text KNOWN UNKNOWN INFERRED DERIVED OBSERVED NOT_APPLICABLE WITHHELD REVOKED STALE ``` Thus the matrix stores not only values, but also epistemic states. --- # 35. An Unknown Cell Is Not Zero This is a very important protective condition: $$ \boxed{ \text{UNKNOWN} \neq 0. } $$ Likewise: $$ \boxed{ \text{missing edge} \neq \text{zero relation}. } $$ --- # 36. Explicit Sparsity Thus a sparse representation must distinguish: $$ \boxed{ \text{structural zero} } $$ and: $$ \boxed{ \text{unobserved}. } $$ --- # 37. Dynamic Dimension Informational dimensionality should be written as: $$ \boxed{ n = n ( \mathfrak N, t, \mathcal T, \mathcal P_t ). } $$ Therefore: $$ n $$ is not a fixed constant. --- # 38. Dimension Growth Rate Define: $$ \boxed{ g_t = n_{t+1}-n_t. } $$ If: $$ g_t>0, $$ this is expansion. If: $$ g_t<0, $$ this is contraction / compression. --- # 39. Relative Growth Define: $$ \boxed{ r_t = \frac{ n_{t+1}-n_t }{ \max(1,n_t) }. } $$ This can monitor dimensional explosion. --- # 40. Probe Growth and Matrix Growth Need Not Be Equal Even if: $$ k $$ probes are added, it need not be that: $$ n_{t+1}=n_t+k. $$ Because: - Probes may be redundant; - Multiple probes may map to one coordinate; - A probe may produce a block; - A probe may cause a split; - A probe may trigger a merge. --- # 41. Expansion Operator Define: $$ \boxed{ \mathcal E: (M_t,Q_{t+1}) \mapsto M_{t+1}. } $$ where: $$ Q_{t+1} $$ is the probe plan selected by NIOT-02. --- # 42. Contraction Operator Define: $$ \boxed{ \mathcal C: (M_t,\tau) \mapsto \widehat M_t. } $$ where: $$ \tau $$ is the task / cost / redundancy threshold. --- # 43. Reindex Operator Define: $$ \boxed{ \mathcal R: (M_t,\rho) \mapsto M_t'. } $$ --- # 44. Repair Operator When a matrix has: - a stale cell; - a conflict; - a certificate revocation; - an invalid bridge; one can use: $$ \boxed{ \mathcal F: M_t \mapsto M_t^{\mathrm{repaired}}. } $$ --- # 45. Update Algebra Thus the runtime's basic operators can be written as: $$ \boxed{ \mathfrak A_M = \langle \mathcal E, \mathcal C, \mathcal R, \mathcal F, \operatorname{Split}, \operatorname{Merge}, \operatorname{Fold} \rangle. } $$ --- # 46. Update Non-Commutation In general: $$ \mathcal C\circ\mathcal E \neq \mathcal E\circ\mathcal C. $$ Likewise: $$ \operatorname{Merge}\circ\operatorname{Split} \neq \operatorname{Split}\circ\operatorname{Merge}. $$ Thus update order is a first-class issue. --- # 47. Update Trace Every epoch should preserve: $$ \boxed{ \mathcal T_t = \left( M_t, \mathcal O_t, M_{t+1} \right) } $$ where $\mathcal O_t$ is this round's operation sequence. --- # 48. Matrix History The complete history: $$ \boxed{ \mathcal H_M = \{ \mathcal T_0, \mathcal T_1, \dots \}. } $$ This provides a foundation for replay, rollback, and audit. --- # 49. Rollback If epoch: $$ t+1 $$ is judged invalid, one can: $$ \boxed{ M_{t+1} \rightarrow M_t. } $$ But external side effects must be handled separately. --- # 50. Snapshot For any epoch: $$ t, $$ establish: $$ \boxed{ S_t = \operatorname{Snapshot}(M_t). } $$ It can serve as: - a proof checkpoint; - a publication artifact; - a test fixture; - a cross-AI handoff. --- # 51. Matrix State Hash If integrity is needed: $$ \boxed{ h_t = H ( \operatorname{CanonicalSerialize}(M_t) ). } $$ This lets different runtimes check that they have the same state. --- # 52. Matrix Closure ## 52.1 Absolute Closure Is Not Required This paper does not require: $$ M_t $$ to achieve universal completeness of all information. --- ## 52.2 Task closure For a task: $$ \mathcal T, $$ Define: $$ \boxed{ \operatorname{Close}_{\mathcal T}(M_t)=1. } $$ if: - the required invariants are decided; - critical uncertainty is below the threshold; - certificate obligations are closed; - the frontier no longer contains task-critical unresolved regions; - the expected utility of new probes is below the threshold. --- # 53. Closure Certificate Define: $$ \boxed{ \mathcal C_{\mathrm{close}} = \langle \mathcal T, t, I_t, \mathcal P_t, F_t, \varepsilon_t, \text{reason} \rangle. } $$ --- # 54. Closed Is Not Complete $$ \boxed{ \operatorname{Close}_{\mathcal T}=1 } $$ means only: $$ \boxed{ \text{sufficiency for task }\mathcal T\text{.} } $$ It does not mean: $$ \boxed{ \text{completeness for all future questions.} } $$ --- # 55. Reopen If a new task appears in the future: $$ \mathcal T', $$ the old closure: $$ \operatorname{Close}_{\mathcal T}=1 $$ does not prevent: $$ \boxed{ \operatorname{Reopen}_{\mathcal T'}(M_t). } $$ --- # 56. Closure Is Relative Thus closure is: $$ \boxed{ \text{task-relative} + \text{epoch-relative} + \text{probe-relative}. } $$ --- # 57. Bounded Execution Budget An actual runtime must have a budget: $$ \boxed{ B = \langle B_{\mathrm{time}}, B_{\mathrm{memory}}, B_{\mathrm{compute}}, B_{\mathrm{probe}} \rangle. } $$ --- # 58. Budget-Constrained Expansion Define: $$ \boxed{ \operatorname{Expand}_B } $$ permits only: $$ K(\mathcal E) \le B. $$ --- # 59. Graceful Stop If closure is not achieved but the budget is exhausted, one must not falsely claim PASS. Output: ```text STATUS: INCOMPLETE_UNDER_BUDGET CLOSED: false REMAINING_FRONTIER: ... NEXT_RECOMMENDED_EXPANSION: ... ``` --- # 60. Complexity The simplest dense matrix has: $$ O(n_t^2) $$ storage. Sparse matrix: $$ O(n_t+|E_t|). $$ Therefore: $$ \boxed{ \text{sparsity is essential for scalable UEFM}. } $$ --- # 61. Incremental Complexity Updates should not recompute the entire matrix every time. If only: $$ \Delta I_t $$ and local edges: $$ \Delta E_t, $$ are updated, the cost should be kept, as far as possible, at: $$ \boxed{ O ( |\Delta I_t| + |\Delta E_t| ) } $$ or near it. --- # 62. Locality Principle If a new probe affects only a local region: $$ U\subset I_t, $$ it must not force global recomputation unless the dependency graph indicates a need. --- # 63. Dependency Closure If coordinate: $$ i $$ changes, find: $$ \boxed{ \operatorname{Desc}(i) } $$ all downstream dependencies. Recompute only: $$ \operatorname{Desc}(i). $$ --- # 64. SEDB-Type Sparse Persistence UEFM is particularly well suited to: $$ \boxed{ \text{sparse entity / relation store}. } $$ rather than necessarily materializing the entire matrix in RAM. Therefore: $$ M_t $$ can be regarded as a logical matrix, with a sparse database used for storage. --- # 65. Virtual Matrix Define: $$ \boxed{ \widetilde M_t } $$ as a virtual matrix. Some cells: $$ m_{ij} $$ are not computed until queried. This is: $$ \boxed{ \text{lazy materialization}. } $$ --- # 66. On-Demand Cell Only when a probe queries: $$ (i,j), $$ does one execute: $$ \boxed{ m_{ij} = \operatorname{Materialize}(i,j). } $$ --- # 67. Cached Cell After computation, one can cache: $$ \boxed{ \operatorname{Cache}(m_{ij}). } $$ and invalidate it if dependencies change. --- # 68. Matrix Projection The complete matrix can be projected into: - a scalar summary; - a graph; - a heatmap; - a spectral view; - a proof spine; - a local block. Define: $$ \boxed{ \Pi_\alpha(M_t). } $$ --- # 69. Projection Non-Identity for Matrix Likewise: $$ \boxed{ \Pi_\alpha(M_t) \neq M_t. } $$ This continues NIOT-01. --- # 70. Matrix Identity Unit Every epoch state: $$ M_t $$ can serve as: $$ \boxed{ \mathbf 1_{M_t} } $$ a versionable identity unit. The entire temporal sequence: $$ \{M_t\} $$ can in turn form a higher-level identity. --- # 71. Nested Identity Therefore: $$ \boxed{ \mathbf 1_{\mathfrak N} \supset \mathbf 1_{M_t} \supset \mathbf 1_{M_t[U]}. } $$ Here $\supset$ denotes identity containment, not a mandatory set-theoretic relation. --- # 72. Hierarchical Matrix UEFM may have: $$ \boxed{ M_t^{(0)}, M_t^{(1)}, \dots, M_t^{(L)}. } $$ Different levels represent different resolutions. --- # 73. Coarse-to-Fine Expansion First use: $$ M_t^{(0)} $$ at coarse resolution. Only the frontier proceeds to: $$ M_t^{(1)}. $$ then, when necessary, to: $$ M_t^{(2)}. $$ --- # 74. Multi-Resolution Matrix Therefore: $$ \boxed{ \mathcal M_t = \{ M_t^{(\ell)} \}_{\ell=0}^{L_t}. } $$ can be regarded as a multi-resolution UEFM. --- # 75. Spectral Matrix If the probe parameter: $$ \lambda $$ exists, one can define: $$ \boxed{ M_t(\lambda). } $$ Different $\lambda$ correspond to different informational scales. --- # 76. Spectral Transition If: $$ M_t(\lambda^-) $$ and: $$ M_t(\lambda^+) $$ differ in topology / rank / active set, then: $$ \boxed{ \lambda } $$ is a spectral transition. --- # 77. Matrix Fiber For a matrix projection: $$ \Pi(M_t)=Y, $$ one can define: $$ \boxed{ F_Y^\Pi = \{ M:\Pi(M)=Y \}. } $$ This continues the fiber structure of NIOT-01. --- # 78. Matrix Refinement After adding a probe or splitting a coordinate: $$ \boxed{ F_{t+1} \subseteq F_t. } $$ means, in terms of equivalence classes, that matrix distinguishability has increased. --- # 79. Matrix Degeneracy If several coordinates remain completely isomorphic over a long period, there may be: $$ \boxed{ \text{structural degeneracy}. } $$ Folding / quotienting is then reasonable. --- # 80. Matrix Quotient Define equivalence: $$ i\sim j $$ if they are indistinguishable for task $\mathcal T$ under all relevant probes. then: $$ \boxed{ M_t/\sim } $$ can serve as a compressed representation. --- # 81. Quotient Certificate One must record: - the equivalence rule; - the task; - omitted distinctions; - reversibility; - an expansion map. --- # 82. Dynamic Quotient As new probes are added, an old quotient class can split: $$ [i] \rightarrow [i_1]\cup[i_2]. $$ Thus a quotient is not permanently fixed. --- # 83. Matrix Entropy For a partition / state distribution, one may choose to define: $$ \boxed{ H(M_t). } $$ But this paper does not mandate Shannon entropy as the sole information measure. --- # 84. Matrix Information Gain Adding a new probe: $$ P $$ one can define: $$ \boxed{ \Delta I_t(P) = I(M_{t+1})-I(M_t). } $$ The particular $I$ can be defined according to the task. --- # 85. Utility of Expansion Define: $$ \boxed{ U(\mathcal E) = \alpha\Delta I - \beta K - \gamma R. } $$ where: - $\Delta I$: information gain; - $K$: cost; - $R$: risk. --- # 86. Expansion Policy An AI controller may choose: $$ \boxed{ \mathcal E_{t+1} = \arg\max_{\mathcal E} U(\mathcal E). } $$ --- # 87. Expansion Horizon Short horizon: $$ H=1. $$ Long horizon: $$ H>1. $$ Multi-step matrix expansion can be planned. --- # 88. Expansion Tree Candidate future states form: $$ \boxed{ \mathcal T_M. } $$ Each node is a possible matrix state. --- # 89. Pruning Low-utility branches can be pruned: $$ \boxed{ \operatorname{Prune} ( \mathcal T_M ). } $$ --- # 90. BCODR on Expansion Tree One can use: - outward candidate growth; - inward constraint growth; to perform BCODR on the expansion tree. Thus BCODR acts not only on geometric proof space, but also on matrix future space. --- # 91. Matrix Responsibility Spine After bidirectional localization and pruning, there may remain: $$ \boxed{ S_M } $$ namely the matrix responsibility spine. Meaning: > Which coordinates / blocks / probes still genuinely bear unresolved task responsibilities? --- # 92. Global Matrices and Local Matrices A large system can have: $$ M_t^{\mathrm{global}} $$ and: $$ M_{t,k}^{\mathrm{local}}. $$ Local matrices can be expanded independently by agents / subsystems. --- # 93. Merge of Local Matrices If several local matrices return: $$ M_{t,1},\dots,M_{t,k}, $$ one needs: $$ \boxed{ \operatorname{Reconcile} ( M_{t,1},\dots,M_{t,k} ) } $$ rather than a direct union. --- # 94. Conflict Matrix If local results conflict, one can establish: $$ \boxed{ C_t } $$ specifically storing: - a conflict pair; - the source; - the version; - the certificate; - the unresolved reason. --- # 95. Secondary Verification Matrix For high-risk results, one can establish: $$ \boxed{ V_t } $$ independently of the primary matrix. This continues BCODR-LUC's: ```text STATIC_FORMAL_FACTS LOCAL_COMPUTE_RESULTS SECONDARY_VERIFICATION ``` layered approach. --- # 96. Three-Layer Matrix Architecture Thus one can define: $$ \boxed{ \mathcal M_t = \left( M_t^{\mathrm{formal}}, M_t^{\mathrm{compute}}, M_t^{\mathrm{verify}} \right). } $$ The three layers must not be substituted for one another. --- # 97. Formal Layer Stores: - theorems; - exact rules; - proved invariants; - definitions. --- # 98. Compute Layer Stores: - numerical output; - heuristic output; - simulation; - optimizer results. --- # 99. Verification Layer Stores: - replay; - independent reproduction; - interval audits; - formal checks; - cross-tool validation. --- # 100. Promote Rule For a computed result to enter a formal / canonical state, it must undergo: $$ \boxed{ \operatorname{Promote} } $$ with a certificate. --- # 101. Demote Rule If the certificate becomes invalid: $$ \boxed{ \operatorname{Demote} } $$ downgrade the old canonical fact to pending / revoked. --- # 102. Matrix Governance Each update may have permissions: - read; - probe; - expand; - compress; - reindex; - promote; - revoke. This is compatible with CAIR governance. --- # 103. Matrix Schema Minimal logical schema: ```json { "matrix_id": "niot:matrix:...", "epoch": 12, "object_id": "niot:number:...", "indices": [], "cells": {}, "edges": [], "probes": [], "frontier": [], "certificates": [], "closure": {}, "history": [] } ``` --- # 104. Canonical Serialization If a hash is needed: - UTF-8; - stable key order; - canonical numerical representation; - explicit UNKNOWN; - explicit sparse zeros; - deterministic index order. --- # 105. One Should Not Directly Rewrite Canonical State Using Floats If a cell originates from an exact rational, it should not be automatically converted to a float in canonical serialization. This continues the bridge semantics that NIOT-06 will address in detail. --- # 106. Matrix MVP The first minimal experiment can involve only: 1. one numerical information object; 2. 5 initial probes; 3. a sparse relation store; 4. adaptive probe expansion; 5. redundancy compression; 6. task closure; 7. a replay log. There is no need to implement the entire theory at the outset. --- # 107. MVP Example Given: $$ x=0.5. $$ Initially: $$ I_0 = \{ \text{value}, \text{representation}, \text{precision} \}. $$ Therefore: $$ n_0=3. $$ Adding a provenance probe: $$ n_1=4. $$ After adding a derivation probe, provenance is found to be derivable, without increasing the independent dimension: $$ n_2=5, \qquad \rho_P=4. $$ Then compress: $$ n_3=4. $$ Thus: $$ \boxed{ 3 \rightarrow 4 \rightarrow 5 \rightarrow 4 } $$ is entirely admissible. "Expansion" in UEFM is not monotonically increasing dimensionality. --- # 108. Second Example: Local Expansion If only: $$ i_2 $$ has a conflict in its associated certificate, one need not expand all of: $$ I_t $$ as displayed above. Perform only: $$ \boxed{ M_t[\mathcal N(i_2)]. } $$ --- # 109. Third Example: Spectral Expansion For a scale probe: $$ P_\lambda, $$ a transition appears only at: $$ \lambda\in[\lambda_a,\lambda_b] $$ as displayed above. Thus adaptive expansion adds spectral coordinates only in that region. --- # 110. Fourth Example: BCODR Responsibility Spine The outward probe finds: $$ S_+. $$ The inward certificate probe finds: $$ S_-. $$ Only blocks near: $$ S_+\cap S_- $$ are marked as frontier. Thus expensive local matrices are materialized only in the overlap responsibility region. --- # 111. Differences between UEFM and Traditional Matrices Traditional matrices usually assume: - fixed dimensionality; - a fixed index set; - fixed element types; - fixed semantics. UEFM allows: $$ \boxed{ \text{dimension} + \text{index} + \text{semantics} + \text{topology} } $$ all to be updated dynamically. --- # 112. Differences between UEFM and Databases Databases store data. UEFM describes: $$ \boxed{ \text{how distinguishable informational dimensions are dynamically composed into computable states.} } $$ A database can be a storage backend, but is not the complete theory. --- # 113. Differences between UEFM and Knowledge Graphs Knowledge graphs excel at relation representation. UEFM additionally emphasizes: - finite epochs; - unbounded dimensionality; - probe activation; - task closure; - compression; - matrix / graph duality; - dynamic rank. --- # 114. Differences between UEFM and Tensors A tensor usually has fixed axes in advance. UEFM axes themselves can be: - discovered; - split; - merged; - reindexed; - revoked. --- # 115. Differences between UEFM and Infinite Matrices Infinite matrix: $$ A=(a_{ij})_{i,j\in\mathbb N}. $$ UEFM does not require all: $$ a_{ij} $$ to be defined in advance. Rather: $$ \boxed{ \text{coordinates themselves can emerge}. } $$ --- # 116. Differences between UEFM and Streaming Streaming usually adds: $$ \text{rows over time}. $$ UEFM can add: - rows; - columns; - axes; - relations; - semantics; - index topology. Thus it is not an ordinary append-only stream. --- # 117. Five Central Invariants of UEFM ## Invariant A — Finite epoch $$ \boxed{ n_t<\infty. } $$ ## Invariant B — Explicit identity $$ \boxed{ \iota(M_t) } $$ can be tracked. ## Invariant C — Unknown explicitness $$ \boxed{ \text{UNKNOWN} \neq 0. } $$ ## Invariant D — Update traceability $$ \boxed{ M_t \rightarrow M_{t+1} } $$ must be traceable. ## Invariant E — Closure relativity $$ \boxed{ \operatorname{Close}_{\mathcal T} } $$ must carry a task. --- # 118. Three Kinds of "Unboundedness" That Must Not Be Confused ## 118.1 Unbounded Dimensionality $$ \sup_t n_t=\infty. $$ ## 118.2 Unbounded Time A runtime can theoretically keep updating. ## 118.3 Unbounded Value Domain The value domain of matrix elements may itself be unbounded. These three are not the same concept. --- # 119. This Paper Requires Only the First The necessary UEFM condition is only: $$ \boxed{ \text{dimension may remain open-ended}. } $$ It does not require: - tasks never to stop; - an unbounded value domain; - infinite actual memory. --- # 120. Unbounded Expansion and Completion Semantics If some theoretical objects require completion, UEFM can preserve: - finite prefixes; - approximants; - certificates; - bridge states. But completion itself is addressed by NIOT-06. --- # 121. Matrix Completion Is Not Mathematical Completion In this paper: $$ \operatorname{Close}_{\mathcal T} $$ is task closure. It is not: - metric completion; - Cauchy completion; - algebraic closure. These must not be conflated. --- # 122. Dynamic Matrix Theorem 1 For any UEFM sequence: $$ \{M_t\}, $$ if every epoch satisfies: $$ n_t<\infty, $$ each epoch can be represented by a finite description. Therefore: $$ \boxed{ \text{unbounded extensibility does not imply infinite current storage}. } $$ --- # 123. Dynamic Matrix Theorem 2 If: $$ \sup_t n_t=\infty, $$ there is no single fixed finite dimension: $$ N $$ such that: $$ n_t\le N $$ for all $t$. This is merely a direct consequence of unboundedness. --- # 124. Dynamic Matrix Theorem 3 If a task $\mathcal T$ achieves, at epoch $T$: $$ \operatorname{Close}_{\mathcal T}(M_T)=1, $$ then even if: $$ \sup_t n_t=\infty, $$ that task can stop in finite time. Therefore: $$ \boxed{ \text{unbounded model capacity} \not\Rightarrow \text{nonterminating task}. } $$ --- # 125. Dynamic Matrix Theorem 4 If compression: $$ C_{\mathcal T} $$ preserves task invariants, then: $$ \operatorname{Close}_{\mathcal T}(M_t) = \operatorname{Close}_{\mathcal T}(C_{\mathcal T}(M_t)). $$ Thus task-relative compression does not invalidate closure. --- # 126. Open Problem: Minimal Dynamic Basis Does there exist: $$ \boxed{ \mathcal B_t } $$ such that: $$ |\mathcal B_t| = \rho_P(\mathcal P_t) $$ while minimizing the current information representation? This is an important subsequent question. --- # 127. Open Problem: Dynamic Rank Curvature If: $$ \rho_s(M_t) $$ produces different local histories along different probe paths, does there exist: $$ \boxed{ \text{rank-path curvature} } $$ This merits study. --- # 128. Open Problem: Optimal Expansion Strategy For a given budget: $$ B, $$ how can one maximize: $$ \operatorname{ClosureProbability} $$ or minimize: $$ \operatorname{ExpectedRemainingGap}? $$ This will connect to the AI-Native Runtime. --- # 129. Open Problem: Formal Semantics of Unbounded Matrices Should UEFM ultimately be formalized as: - an inductive family; - a filtered colimit; - a direct system; - a dynamic category; - a sparse sheaf; - an evolving hypergraph; remains to be compared in subsequent work. --- # 130. Possible Interface with Category Theory If each epoch is an object: $$ M_t, $$ and an update is a morphism: $$ u_t: M_t \rightarrow M_{t+1}, $$ then: $$ \boxed{ M_0 \xrightarrow{u_0} M_1 \xrightarrow{u_1} M_2 \rightarrow \cdots } $$ forms a directed system. But because updates may contract / merge / fork, it need not be a simple inclusion system. --- # 131. Differences from a Direct Limit Only if all updates are compatible embeddings can a direct limit be considered. A general UEFM does not guarantee this. Therefore: $$ \boxed{ \text{UEFM} \neq \text{automatic direct-limit construction}. } $$ --- # 132. Interface with ISQL ISQL can provide: - index identity; - a semantic address; - a relation key; - state serialization; - a dynamic link. Therefore: $$ \boxed{ \mathrm{ISQL} \approx \text{UEFM address / relation substrate}. } $$ --- # 133. Interface with AIC AIC provides: $$ \widehat{\mathcal E}_{i\to j}. $$ Thus reindexing, navigation, splitting, and merging can acquire mathematical semantics. --- # 134. Interface with BCODR BCODR provides: $$ \boxed{ \text{frontier localization}. } $$ UEFM provides: $$ \boxed{ \text{localized dynamic state}. } $$ Combining them: $$ \boxed{ \text{Expand only in the matrix regions that genuinely bear unresolved responsibilities.} } $$ --- # 135. Interface with CAIR Matrix updates should not directly modify canonical state. One can use: $$ \boxed{ M_t \rightarrow \Delta M \rightarrow \text{validate} \rightarrow M_{t+1}. } $$ This continues the CAIR candidate-diff architecture. --- # 136. Interface with MNVP UEFM is a machine / reasoning state. MNVP converts: $$ \Pi_\alpha(M_t) $$ into: - a table; - a graph; - a heatmap; - a spectral visualization; - a local view. --- # 137. Interface with NIOT-04 The next paper will address: $$ \boxed{ \text{Spectral discrimination and multi-probe coupling.} } $$ In NIOT-04, UEFM will be upgraded to: $$ \boxed{ M_t(\lambda_1,\dots,\lambda_k). } $$ and study: - spectral coordinates; - cross-probe coupling; - phase transitions; - multi-scale structure; - information resonance; - degeneracy. --- # 138. The Most Important Conceptual Correction in This Paper Do not interpret: $$ \boxed{ \text{Unboundedly Expandable Finite Matrix} } $$ as: > A very large matrix with columns continually added. More precisely: $$ \boxed{ \text{A dynamic state system that is finite at any actual moment, but whose coordinates, dimensionality, indices, topology, certificates, and representation can be reorganized as new information arrives.} } $$ --- # 139. Threefold Principle Finally, this paper condenses UEFM into three statements. First: $$ \boxed{ \forall t,\quad \dim M_t<\infty. } $$ Second: $$ \boxed{ \sup_t\dim M_t=\infty } $$ as a condition permitted by the architecture. Third: $$ \boxed{ \exists T<\infty \text{ such that } \operatorname{Close}_{\mathcal T}(M_T)=1 } $$ holds for concrete tasks that admit finite closure. Therefore: $$ \boxed{ \text{Finite Actuality} + \text{Unbounded Extensibility} + \text{Finite Task Closure}. } $$ --- # 140. Conclusion NIOT-01 lifts numerical values from scalars to identity-bearing numerical information objects. NIOT-02 turns informational distinctions into probe-driven distinguishability. This paper further answers: > When distinguishable informational directions continually increase, how should this information exist in finite machines? The answer is not: $$ \boxed{ \text{Establish a genuinely infinite matrix.} } $$ Rather: $$ \boxed{ M_0 \rightarrow M_1 \rightarrow M_2 \rightarrow \cdots } $$ Every: $$ M_t $$ is finite. But in the future one can: - add probes; - split coordinates; - establish new relations; - introduce new spectra; - add certificates; - reindex; - compress; - reopen; - fork. Therefore: $$ \boxed{ \text{being finite now} \neq \text{being constrained to a fixed finite size in the future.} } $$ This is precisely the core of UEFM. It provides a genuinely executable carrier for "microscopic numerical worlds": rather than materializing all possible information at once, it materializes only the finite structure currently valuable at each task, epoch, and local frontier. Thus: $$ \boxed{ \text{microscopic numerical worlds} } $$ do not require infinite RAM. They require: $$ \boxed{ \text{unbounded extension rules} + \text{a finite current state} + \text{provable stopping conditions}. } $$ The next paper: $$ \boxed{ \textbf{NIOT-04 — Spectral Discrimination and Multi-Probe Coupling.} } $$ will address a deeper question: $$ \boxed{ \text{When information is not unidirectional but forms spectra along scale, frequency, semantic depth, and multi-probe relationships, how does a matrix represent coupling, phase transitions, and multi-scale structure?} } $$ --- # Canonical Source Declaration This file is the UTF-8 Markdown canonical source for `NIOT-03-v0.1`. The only mathematical delimiters are `$...$` and `$$...$$`. This paper formally defines: - UEFM; - finite actualization; - unbounded extensibility; - active / latent dimension; - structural rank; - probe rank; - sparse envelope; - expansion; - contraction; - split; - merge; - folding; - compression; - frontier; - local closure; - task closure; - update trace; - matrix provenance; - dynamic reindexing. This paper does not claim that: - the actual dimensionality of every task necessarily tends to infinity; - UEFM must use dense matrices; - UEFM is equivalent to a traditional infinite matrix; - task closure is equivalent to mathematical completion; - all updates commute; - all compression is lossless; - all current propositions have been verified in Lean / Coq. Subsequent NIOT-04 should use this paper as the dynamic carrier foundation for spectral matrices and multi-probe coupling. ===END===