> Full-text translation from Chinese. The original remains authoritative for its research claims; this translation is not independent mathematical verification. # Numerical Information Object Theory: From Linear Floating-Point Representation to Unbounded Probe Matrices and the Microscopic World of Numbers ## Numerical Information Object Theory: From Linear Floating-Point Projection to Unbounded Probe Matrices and the Microscopic World of Numbers **Author:** Neo.K **Institution:** EveMissLab / 一言諾科技有限公司 **Document type:** Comprehensive Overview Draft / Internal Research Parent Paper **Version:** v0.1 **Date:** 2026-09-22 **Status:** DRAFT / THEORY INTEGRATION / SERIES-SEED **Canonical Source:** UTF-8 Markdown **Mathematical source conventions:** Inline math uses only `$...$`, and display math uses only `$$...$$` --- # Abstract Modern numerical computation usually treats a numerical value as a terminal quantity in a computational process: a scalar is input, and another scalar is obtained after applying a function or operator. Floating-point systems further engineer this perspective into finite bit patterns, exponents, significands, rounding modes, and exceptional values. This approach is highly successful, but it also creates a long-overlooked epistemic compression: we often mistake "the numerical value currently projected" for "all the information in that numerical state." This paper proposes a working theoretical framework: **Numerical Information Object Theory**. Its core is not the claim that every abstract real number inherently conceals arbitrary infinite information. Rather, it distinguishes an "abstract value" from a "numerical information object in an actual context of representation, generation, measurement, computation, comparison, indexing, relations, history, and certificates." A numerical state usable in computation can be represented as a finite projection of a more complete object: $$ x=\pi_{\mathrm{scalar}}(\mathfrak N_x), $$ Here $x$ is the current scalar projection, while $\mathfrak N_x$ is a numerical information object carrying identity, indices, relations, provenance, generation history, precision, uncertainty, representation, local structure, and directions available for probing. The paper further proposes three central constructions. First, an **information-probe family**. Different probes can extract different information directions from the same numerical information object without first forcing it into another radix or representation format: $$ P_\lambda:\mathfrak N_x\rightarrow \mathcal I_\lambda, \qquad \lambda\in\Lambda. $$ Second, an **Unboundedly Expandable Finite Matrix**. At any actual moment, only a finite information matrix can be maintained: $$ M_t(x)\in\mathcal M_{n_t}, \qquad n_t<\infty, $$ Yet the matrix dimension available for expansion has no fixed upper bound: $$ \sup_t n_t=\infty. $$ This allows the system to retain both engineering-level finite executability and theoretical unbounded extensibility. Third, the **Numerical Identity Unit Principle**. This paper proposes a precise version of "every number is one": it does not assert that different numerical values are equal. Rather, every identifiable numerical object $\mathfrak N_x$ can be regarded as a local information unit preserving its own identity. Indexing, slicing, projection, and probing can change what we see without necessarily destroying its identity as the same object of study. This framework reconnects several existing EveMissLab research lines: ICNS handles the interpretability of numerals; MNVP handles perceptual projections of numerical semantics; CAIR distinguishes authoritative objects from multiple projections; ISQL supplies semantic anchors, addresses, relations, and state operations; Identity Calculus and Active Index Calculus provide slicing, indexing, and identity conservation; BCODR supplies bidirectional responsibility localization, overlap, atomic decomposition, and structural closure; bidirectional completion–projection semantics handles real numbers, floating point, exact representations, and cross-domain bridges. This paper brings these theories together around a common question: $$ \boxed{ \text{Numbers have more than values; a value is only one projection of a numerical information object.} } $$ The purpose is not to complete all formalization at once, but to establish a parent framework that can later be divided into a paper series, engineered, verified, and made compatible with existing mathematics. **Keywords:** Numerical Information Objects, Floating Point, Information Probes, Unboundedly Expandable Finite Matrices, Numerical Identity, ISQL, ICNS, MNVP, CAIR, Active Index Calculus, BCODR, Cross-Domain Bridges, Information Spectra, Representation Projections --- # 1. The Problem: Mistaking the Projection for the Whole ## 1.1 The Implicit Model of Traditional Numerical Computation Traditional computation is usually written as: $$ x\xrightarrow{f}y. $$ Here $x$ and $y$ are treated as complete numerical objects, and $f$ is a function mapping one value to another. On a finite machine, this perspective is implemented as: $$ \text{bit pattern} \rightarrow \text{floating-point value} \rightarrow \text{operation} \rightarrow \text{floating-point value}. $$ This model is sufficiently effective for many engineering tasks, but it leaves much information outside the value, for example: - how this value was generated; - which model or measurement chain produced it; - which semantic domain it belongs to; - whether it is a version field rather than a real number; - its precision and resolution; - which other values share its provenance; - whether it is a projection of a dynamic state; - whether its correspondence in another representation domain is exact, approximate, a quotient, or a completion; - whether it carries an identifiable structural role that has not yet been explicitly marked. Thus what genuinely needs correction is not the floating-point format itself, but the following default identification: $$ \boxed{ \text{current scalar representation} = \text{whole numerical object}. } $$ This paper rejects that default. --- ## 1.2 The First Distinction: Abstract Values and Numerical Information Objects This paper does not claim that a purely abstract number $x\in\mathbb R$ automatically contains arbitrary additional information without any external structure. It claims that a number in actual use is often not a bare value, but exists within an identifiable structure. Define a numerical information object: $$ \boxed{ \mathfrak N_x = \left\langle \iota_x, v_x, \Sigma_x, R_x, A_x, H_x, P_x, Q_x, C_x, \mathcal B_x \right\rangle. } $$ where: - $\iota_x$: identity, the identity of the numerical object; - $v_x$: the value layer, the current or canonical value; - $\Sigma_x$: representation / semantic schema; - $R_x$: relation structure; - $A_x$: address / index structure; - $H_x$: history / derivation; - $P_x$: provenance; - $Q_x$: quality / uncertainty / precision state; - $C_x$: constraints / context; - $\mathcal B_x$: the bridge set, containing bridge relations to other representation or completion domains. The user usually sees only: $$ \pi_{\mathrm{scalar}}(\mathfrak N_x)=x. $$ Thus the scalar is no longer treated as the whole, but as: $$ \boxed{ x = \text{a projection of } \mathfrak N_x. } $$ --- # 2. Informational Differences: Existence, Distinguishability, Addressability, and Computability Are Different ## 2.1 Existing Information Is Not Necessarily Usable Information A numerical state may already have a structural difference that the system has not yet defined as an operable object. This paper distinguishes four layers: $$ \boxed{ \text{latent} \rightarrow \text{distinguishable} \rightarrow \text{addressable} \rightarrow \text{computable}. } $$ ### 2.1.1 Latent Information A difference already exists in provenance, relations, history, representation, or a neighborhood, but has not yet become an explicit field. ### 2.1.2 Distinguishable Information A test, rule, probe, or semantic condition exists that can determine that two states differ in that direction. ### 2.1.3 Addressable Information The difference has a stable index, label, path, schema, or semantic anchor. ### 2.1.4 Computable Information The difference can enter operators, matrices, graphs, constraints, routing, certificates, or a Runtime. These four layers cannot simply be equated. --- ## 2.2 The Earlier Insight from ISQL ISQL-like systems can extract addresses, anchors, relations, and state fragments from complex semantic states not because they create that information out of nothing, but because: $$ \boxed{ \text{existing differences} + \text{identification rules} + \text{addressing mechanisms} \Rightarrow \text{operable information}. } $$ Thus Numerical Information Object Theory transfers this idea to the numerical domain: $$ \boxed{ \text{numerical micro-information} \neq \text{merely a collection of added metadata}. } $$ More importantly, it establishes: $$ \boxed{ \text{a method for identifying which structures merit becoming computable dimensions.} } $$ This is the role of information probes. --- # 3. Information Probes: Entering Micro-Information Space Without First Rewriting the Number ## 3.1 The Basic Definition of a Probe Let: $$ \Lambda $$ be the probe-index space. For any: $$ \lambda\in\Lambda, $$ Define: $$ \boxed{ P_\lambda: \mathfrak N_x \rightarrow \mathcal I_\lambda. } $$ $P_\lambda$ need not return a scalar; it may also return: - a category; - a subgraph; - a local matrix; - an interval; - a topology signature; - a support pattern; - a provenance chain; - a relation neighborhood; - a spectral coordinate; - a certificate state; - an uncertainty object; - an address set. Thus a probe is not merely "reading a field" in the usual sense. It can also be a structural discriminator. --- ## 3.2 Probes Versus Representation Transformations An older strategy may be: $$ x_{10} \rightarrow x_2 \rightarrow x_{16} $$ observing structure by changing representations. This remains an effective method, but it is only a special case of probing. Numerical Information Object Theory allows: $$ \boxed{ \text{keep identity} + \text{change observation operator}. } $$ That is: $$ \mathfrak N_x \xrightarrow{P_\lambda} \mathcal I_\lambda $$ without requiring: $$ \mathfrak N_x \rightarrow \mathfrak N_x' $$ a complete representation transformation first. Thus radix conversion shifts from being the primary method to: $$ \boxed{ \text{one probe family among many}. } $$ --- ## 3.3 Probes Can Be Local For a large numerical information object, not all information needs to be read at once. Let: $$ U\subseteq\mathfrak N_x $$ denote the current local domain of study. Then one can define: $$ P_{\lambda,U}(\mathfrak N_x). $$ This allows: $$ \boxed{ \text{local numerical microscopy}. } $$ what this paper calls "entering the microscopic world of numbers." Here "microscopic" means not more decimal places, but: $$ \boxed{ \text{finer information decomposition of the same numerical identity.} } $$ --- # 4. Spectral Information Discrimination ## 4.1 From Binary Probes to Spectral Probes If a probe answers only: $$ 0/1, $$ it can indicate only whether a difference exists. Yet much numerical information has continuous or multiscale structure. Thus define a spectral probe: $$ \boxed{ P_\lambda(\mathfrak N_x), \qquad \lambda\in\Lambda. } $$ Here $\lambda$ may represent: - scale; - frequency; - semantic depth; - relation strength; - uncertainty level; - topological radius; - temporal depth; - derivation depth; - precision scale; - observation orientation. A numerical object then has an information spectrum: $$ \boxed{ \mathcal S_x = \left\{ (\lambda,P_\lambda(\mathfrak N_x)) : \lambda\in\Lambda \right\}. } $$ --- ## 4.2 The Spectrum Is Not Decorative If different $\lambda$ produce different structures for the same object, its information should no longer be compressed into a single scalar summary. For example: $$ P_{\lambda_1}(\mathfrak N_x) \neq P_{\lambda_2}(\mathfrak N_x) $$ may be not noise, but: $$ \boxed{ \text{scale-dependent informational structure}. } $$ This creates a natural interface between ISQL's spectral semantics, existing multiscale research, and Numerical Information Object Theory. --- # 5. Unboundedly Expandable Finite Matrices ## 5.1 Basic Definition At each time or research stage $t$, the system actually maintains only a finite probe set: $$ \Lambda_t = \{ \lambda_1,\ldots,\lambda_{n_t} \}. $$ where: $$ n_t<\infty. $$ Define a finite information vector: $$ \mathbf p_t(x) = \begin{bmatrix} P_{\lambda_1}(\mathfrak N_x)\\ P_{\lambda_2}(\mathfrak N_x)\\ \vdots\\ P_{\lambda_{n_t}}(\mathfrak N_x) \end{bmatrix}. $$ When coupling relations between probe outputs must be recorded, construct: $$ \boxed{ M_t(x) = \left[ m_{ij}^{(t)}(x) \right]_{1\le i,j\le n_t}. } $$ called an: $$ \boxed{ \text{Unboundedly Expandable Finite Matrix} } $$ provisionally translated as: **Unboundedly Expandable Finite Matrix**. --- ## 5.2 Simultaneous Finiteness and Unboundedness At every moment: $$ \boxed{ n_t<\infty. } $$ Yet over the entire extensible research process: $$ \boxed{ \sup_t n_t=\infty. } $$ These two statements do not contradict each other. The former ensures: - storability; - computability; - verifiability; - deployability; - execution on a finite machine. The latter ensures: - no maximum information dimension is frozen in advance; - new probes can continue to be added; - new relations can be discovered; - new representations can be connected; - new theories need not rewrite the entire underlying object. Therefore: $$ \boxed{ \text{finite actualization} + \text{unbounded extensibility}. } $$ is an important engineering principle of this theory. --- ## 5.3 A Dynamic Matrix Is Not Merely Appending Fields The weakest version may be: $$ M_t\subseteq M_{t+1}. $$ Yet a fuller version allows: - adding dimensions; - merging dimensions; - reindexing; - compression; - local expansion; - removing redundant probes; - adjusting couplings; - reconfiguring matrix topology. Thus, more generally, write: $$ \boxed{ M_{t+1} = \mathcal U \left( M_t, \Delta_t, \Gamma_t \right), } $$ where: - $\Delta_t$: new probes, observations, and certificates; - $\Gamma_t$: rules for reindexing, compression, coupling, and governance; - $\mathcal U$: the matrix-update operator. --- # 6. Information-Gap-Driven Probe Selection ## 6.1 Expansion Should Not Be Blindly Infinite If probes can be added without bound, the most direct risk is: $$ \text{probe explosion}. $$ Thus an information gap must be defined: $$ \boxed{ G_t = \mathcal G \left( M_t, \mathcal O_t, \mathcal T \right), } $$ where: - $M_t$: the existing information matrix; - $\mathcal O_t$: current observations or certificates; - $\mathcal T$: the research objective. Then use: $$ \boxed{ Q_{t+1} = \Phi(G_t,M_t) } $$ to select the probes most worth using in the next round. The overall process: $$ \boxed{ M_t \rightarrow G_t \rightarrow Q_{t+1} \rightarrow \Delta M_{t+1} \rightarrow M_{t+1}. } $$ This makes unbounded expansion not aimless growth, but: $$ \boxed{ \text{gap-directed expansion}. } $$ --- # 7. BCODR as a Numerical Micro-Information Localizer ## 7.1 From Geometric Proof Space to General Information Space The core of BCODR is not the circle itself, but: $$ \boxed{ \text{bidirectional expansion} \rightarrow \text{overlap} \rightarrow \text{atomic decomposition} \rightarrow \text{role separation} \rightarrow \text{structural closure}. } $$ Thus it can be abstracted into two information fields: $$ \mathcal C^+_x, \qquad \mathcal C^-_x. $$ which can respectively represent: - supporting evidence; - excluding evidence; - forward derivation; - backward certificates; - candidate families; - constraint families; - generation fields; - reconstruction fields. The key region is: $$ \boxed{ \mathcal C^+_x \cap \mathcal C^-_x. } $$ --- ## 7.2 Overlap Does Not Mean Identity An important lesson from BCODR-LUC is: $$ \boxed{ \text{same container overlap} \neq \text{same point identity} \neq \text{same mechanism identity}. } $$ This should directly become a safeguard in Numerical Information Object Theory. Thus probe results must also distinguish: $$ \text{projection overlap}, $$ $$ \text{point alignment}, $$ $$ \text{seed alignment}, $$ $$ \text{structural alignment}. $$ A numerical object appearing close under two probes does not mean that they capture the same information mechanism. --- ## 7.3 Structural Responsibility Rather Than Merely Finding an Answer Traditional computation often asks: $$ \text{what is }x? $$ A BCODR-style method can instead ask: $$ \boxed{ \text{which substructure is still responsible for uncertainty?} } $$ The computational result of a numerical information object may therefore be not a scalar, but: $$ \boxed{ \text{surviving informational spine}. } $$ This expands numerical computation from single-value output into a traceable responsibility structure. --- # 8. Every Number Is One: The Numerical Identity Unit Principle ## 8.1 The Proposition Is Not "All Numbers Are Equal" This paper proposes: > Every number is one. But this statement must be formalized to avoid misreading it as: $$ 1=2=3. $$ The actual claim is: $$ \boxed{ \forall x, \quad \mathfrak N_x \text{ can be treated as one identity-preserving informational unit.} } $$ Here "one" means not the numerical value $1$, but a: $$ \boxed{ \text{unit of identity}. } $$ --- ## 8.2 The Numerical Version of Identity Calculus The basic idea of Identity Calculus is: $$ \text{slicing} \neq \text{splitting the underlying object}. $$ For numerical information objects, one can define: $$ \boxed{ d_I(\mathfrak N_x) = \{ (\mathfrak N_x,i) : i\in I \}. } $$ Each $(\mathfrak N_x,i)$ is a differently indexed view of the same numerical identity. Forgetting the index: $$ \boxed{ \int_{\mathrm{id}} (\mathfrak N_x,i) = \mathfrak N_x. } $$ Therefore: $$ \boxed{ \int_{\mathrm{id}}\circ d_I = \operatorname{id}_{\mathfrak N_x}. } $$ This supplies an identity-conservation principle for numerical micro-slicing. --- ## 8.3 The Active-Index Version When indices are not merely passive, but can be moved, combined, and readdressed: $$ \boxed{ \widehat{\mathcal E}_{i\rightarrow j} : (\mathfrak N_x,i) \mapsto (\mathfrak N_x,j). } $$ then: - probes; - indices; - addresses; - matrix coordinates; - semantic anchors; can become different layers of the same operational family. This supplies an operational theory of "entering a number's interior" without requiring destruction of its identity. --- # 9. Repositioning Floating Point ## 9.1 Floating Point Is No Longer the Endpoint Traditional perspective: $$ \text{real-world state} \rightarrow \text{float} \rightarrow \text{calculation}. $$ This paper rewrites it as: $$ \boxed{ \mathfrak N_x \xrightarrow{\pi_{\mathrm{float}}} f_x. } $$ where: $$ f_x $$ is only a projection on a finite machine. Therefore: $$ \boxed{ f_x \neq \mathfrak N_x. } $$ --- ## 9.2 Completion, Projection, and Bridges Existing bidirectional completion–projection semantics has shown that different representation domains cannot naively be treated as reversible transformations within the same set. Let a cross-domain bridge be: $$ \mathfrak B: X \rightsquigarrow Y. $$ A complete bridge must record at least: $$ \boxed{ \mathfrak B = \left( X, Y, \llbracket-\rrbracket_X, \llbracket-\rrbracket_Y, B, \overline B, \kappa, \varepsilon, \mathsf{Exc}, \mathsf{Exec} \right). } $$ Therefore: $$ \mathbb R \xrightarrow{\text{rounding}} \mathbb F $$ is not: $$ \mathbb R \cong \mathbb F. $$ Conversely, returning from floating point to fuller semantics is not simply taking an inverse either. One must know: - the original domain; - rounding; - precision; - the enclosure; - exceptions; - semantic interpretation; - execution context. --- ## 9.3 From Floating-Point Research to Numerical Microscopy The earlier question in cross-radix or cross-representation research was: > Can another representation reveal structures ignored by the original representation? This paper generalizes that question to: > Without first changing the complete representation, can a probe directly identify a structural direction? Formalized as: $$ \boxed{ \mathfrak N_x \xrightarrow{P_\lambda} \mathcal I_\lambda. } $$ Thus floating point is not merely an error-engineering problem; it also becomes: $$ \boxed{ \text{a major instance of projection theory.} } $$ --- # 10. A Unified Interface for ICNS, MNVP, CAIR, ISQL, AIC, and BCODR ## 10.1 ICNS: What This Number Actually Is ICNS addresses: $$ \boxed{ \text{surface numeral} \neq \text{semantic numeral object}. } $$ Its responsibility is to turn bare numerals into decidable, parsable, and comparable numerical semantics. In this paper, ICNS can be regarded as: $$ \boxed{ \mathfrak N_x \text{ one of its semantic entry layers.} } $$ --- ## 10.2 MNVP: How We See It MNVP handles: $$ \text{Semantic Numerical Object} \rightarrow \text{Perceptual Representation}. $$ Thus, in this paper: $$ \boxed{ \mathrm{MNVP} \approx \text{projection-to-human-perception layer}. } $$ Numerical Information Object Theory addresses the more upstream question: $$ \boxed{ \text{what can be projected at all?} } $$ --- ## 10.3 CAIR: A Projection Is Not the Authoritative Object The core principle of CAIR is: $$ \boxed{ \text{projection} \neq \text{canonical authoritative object}. } $$ This paper extends that principle into the numerical domain: $$ \boxed{ x = \pi(\mathfrak N_x) \neq \mathfrak N_x. } $$ Thus CAIR and this paper share a deeper structure: $$ \boxed{ \text{one canonical object} \rightarrow \text{many legitimate projections}. } $$ --- ## 10.4 ISQL: Addressing, Relations, and State Operations ISQL turns: - anchors; - relations; - addresses; - states; - semantic paths; into computable objects. This paper adopts the same idea, promoting internal numerical information from: $$ \text{latent distinction} $$ to: $$ \text{addressable distinction}. $$ Therefore: $$ \boxed{ \mathrm{ISQL} \text{ can serve as the semantic addressing layer for numerical information objects.} } $$ --- ## 10.5 AIC: An Index Is Not an Annotation, but a Degree of Freedom Active Index Calculus supplies: $$ \text{index} \rightarrow \text{computational degree of freedom}. $$ Thus probes, matrix coordinates, and semantic addresses can be unified further: $$ \boxed{ P_\lambda \leftrightarrow i_\lambda \leftrightarrow a_\lambda. } $$ This gives numerical micro-information space an operable geometry. --- ## 10.6 BCODR: Multi-Directional Information-Responsibility Localization BCODR supplies: $$ \boxed{ \text{bidirectional structural localization}. } $$ This paper treats it as a particularly valuable special strategy for probe orchestration: $$ \boxed{ P_+, P_- \rightarrow \text{overlap field} \rightarrow \text{atomic responsibility} \rightarrow \text{surviving spine}. } $$ --- # 11. A Unified Abstraction Compress the theories above into a process: $$ \boxed{ \text{Numeral} \xrightarrow{\mathrm{ICNS}} \mathfrak N_x \xrightarrow{\mathrm{ISQL/AIC}} \text{addressable information structure} \xrightarrow{\mathrm{Probe}} M_t(x) \xrightarrow{\mathrm{BCODR/Spectral}} \text{localized informative structure} \xrightarrow{\mathrm{CAIR}} \text{canonical state} \xrightarrow{\mathrm{MNVP}} \text{human-perceptual projection}. } $$ Floating point, decimal, binary, graphics, vectors, matrices, and symbolic representations can all be merely different projections or bridge targets. Therefore: $$ \boxed{ \text{numerical computation} \rightarrow \text{computation in numerical information space} } $$ is the main paradigm shift proposed in this paper. --- # 12. Dynamic Numerical Information Space ## 12.1 Local Information Neighborhoods Define a local neighborhood for each numerical information object: $$ \mathcal U_r(\mathfrak N_x). $$ $r$ need not be a Euclidean distance; it may also be: - semantic distance; - provenance distance; - derivation depth; - relation hops; - scale differences; - certificate distance. Thus the numerical micro-world need not be restricted to a real-number line. --- ## 12.2 Dynamic Neighborhoods As new probes are added: $$ \mathcal U_t(\mathfrak N_x) $$ may change. This means: $$ \boxed{ \text{We are not changing numerical identity; we are changing the observable information neighborhood.} } $$ --- ## 12.3 A Nonlinear Numerical World The traditional scalar view: $$ \cdots