This paper presents Noology, a formal system that defines the highest-level protocol by which intelligence constitutes reality. Noology (Japanese: Chigaku 知学) is not a theory within science, but an operating system specifi cation for existence itself. Crucially, Noology is a meta-judgment framework: it does not generate truths internally, but adjudicates whether an externally given structure qualifi es as “real” (Res) under a given confi guration (Quaestio). By design, it avoids the premises of Gödel’s incompleteness theorems through non-interpretability (...) of arithmetic. Noology serves as the regulative layer above Operatiol- ogy, providing the axiomatic foundation from which Operatiology’s executive structure is derived. -/- Version 2 of this system rested on three independent primitive notions (Ordo, Consensus, Arbitrium), related by three absolute axioms formulated as relational constraints with non-triviality conditions, together with one governing principle. The present version revises this foundation: the triad and its three axioms are no longer independent primitives but are derived as theorems from a single primitive, Self-Closure (Ω), together with the Governing Principle, tightening the regulative foundation on which Operatiology depends. -/- The independence and minimality results of Version 2 are re-established with explicit model constructions on the re-derived axioms, and the non-applicability of Gödel’s incompleteness theorems is re-proven using the structure’s own witness model. The relation to Operatiology is restated within this single-primitive foundation, preserving the Lemma 2.1 confi gurational-sensitivity argument by which Arbitrium, Axiom 2, and Axiom 4 jointly force non-commutativity. The system remains minimal, irreducible, and self-contained: it presupposes neither arithmetic, set theory, nor physical law, and it does not, and structurally cannot, force the dimensional bound established at Tier-2; any further realization belongs to Operatiology and its algebraic layer, not to the present document. DOI: 10.5281/zenodo.20786059 Published on June 22, 2026. (shrink)

Noology is the technical life of ideology. It works at the formal and technical production of knowledge, rather than focusing on the content displayed by a specific system of thought. There are two reasons why the notion of noology must play a role in today’s critical and political debates. First, the concept of ideology has lost its relevance since its everyday meaning is far removed from the original meaning Karl Marx gave it; today ideology mainly means “political doctrine,” (...) right-wing, left-wing, or the entire spectrum of shades between the two. Expressions such as “an ideology” or “ideologies” are used in critical analysis, while for Marx “ideology” has always come without any pronoun. Ideology now presents itself as an “inversion of causalities producing illusions.”1 The second reason has to do with the changes in the modes of production since the 1970s, and the rise of the post-Fordist economy, or “neoliberalism.” Since the 1970s, the end of ideologies has been proclaimed (epitomised by Daniel Bell). Given this context, noology critique demonstrates that the work of ideology in today’s economy plays out at an infrastructural level, in social organs that materially institutionalise thought and ideas, and not simply at the level of the immaterial culture of political parties and discourses (superstructure). (shrink)

This paper establishes the first formal definition of Tiers within the Noological framework, revealing a strict three-layer architecture that governs reality-constitution: Tier-1 (Noology) as the regulative layer, Tier-2 (Cognitional Mechanics) as the executive substrate, and Tier-3 (MUT/GUT) as the projective display layer. These layers obey an irreversible dependency chain: Tier-1 ⇒ Tier-2 ⇒ Tier-3. A key result is the structural necessity of the algebra M_3(C) (3×3 complex matrices) as the unique minimal substrate supporting the Noological primitives Ordo and Consensus. (...) We prove that 2×2 algebras fail triangulation in metric state space, while n≥4 algebras violate operational bounds through spectral instability. The constants δ=√(3/2) and γ=1/2 originate in Tier-1 as structural requirements—not empirical parameters—and are realized through Tier-2 dynamics. This framework repositions existing mathematics and physics as Tier-3 projections of deeper operational constraints. Gödelian incompleteness is circumvented not by stronger axioms but by structural separation of internal operations from external Quaestio. Physical laws, mathematical theorems, and even the fine-structure constant α⁻¹≈137.03 emerge as shadows cast by the non-commutative geometry of M_3(C). The Tier alignment provides the structural foundation for all subsequent Noological applications, transforming century-old questions about unification, completeness, and the origin of physical constants from unsolved problems into consequences of a single coherent architecture. (shrink)

The subject of the research is the study of Mir Findiriski treatise «On Craft» (Risa'la-yi s'ina'iyya). The object of the research is the philosophical visions of Mir Findiriski. According to him the intelligence is the highest mental function of human, whereby human differs from other manifestations of the life on the earth. At different periods in the history of philosophy, thinkers described the intelligence, which formed the section of noology, that is a philosophical legacy of arab-muslim philosophers (al-Farabi, Ibn (...) Sina, etc.). The author examines in detail such aspects of the topic as the Necessity being and intelligence in the structure of being degree of Mir Findiriski, the paradigm of the “subject of the acting or action” of the philosopher’s noological system, three levels of the intelligence, the formation of the general order through three levels of the intelligence, at the top of which is personality with the intelligence of human. In conducting the study, the philological-translation and logical-semantic approaches to the analysis of the text were used, on the basis of which Risala-yi sinaʻiyya in Russian acquired a philosophical interpretation and the paradigm of noology and the system of philosophical visions of Mir Findiriski were synthesized in the treatise “On Craft”. In addition, the method of comparative analysis is used to determine the place of continuity of the intellectual legacy of al Farabi and Ibn Sina and originality in the noology of Mir Findiriski. The main conclusions of the conducted research are: definitions of the being degrees of Mir Findiriski, three levels of intelligence, the formation of general order through three levels of intelligence, at the top of which is personality with the intelligence of human in the noology of the of Mir Findiriski. Mir Findiriski classifies the following being degrees: Necessary being, intelligence, soul and nature. Intelligence as the second being degree has three levels: the intelligence of minerals, plants and animals, the intelligence of the human species and the intelligence of the human personality. Furthermore, the continuity of the intellectual legacy of al-Farabi and Ibn Sina and the originality in the noology of the Safavid philosopher are determined. The continuity lies in the fact that the Almighty God occupies a central place in the noological system of Mir Findiriski. The originality is presented in the following: the intelligence of minerals, plants and animals, the intelligence of the human species and the intelligence of the human personality, which is a distinctive categorical apparatus of intelligence levels from the noology of al-Farabi and Ibn Sina. (shrink)

This paper presents the Noological Unification Theorem: a formal proof that all physical laws are the surjective projection of a single minimal algebraic structure, the algebra M₃(ℂ) of 3×3 complex matrices, onto the category of physical laws L_phys. The composition ρ∘Π : M₃(ℂ) → L_phys is shown to be surjective while its inverse is structurally undefined. Surjectivity is established constructively via explicit identification of nineteen Standard Model parameters as coordinate components of ρ∘Π, each derived without free parameters in the Cognitional (...) Mechanics (CM) corpus. Non-invertibility follows axiomatically from Axiom 1 (Non-Commutativity) alone, independently of any empirical input. These two proof paths are logically independent and establish complementary claims: surjectivity asserts that every physical law has an algebraic preimage; kernel non-triviality asserts that the projection discards non-trivial information, making inversion structurally impossible. Together they yield a categorical isomorphism C₁/ker(ρ) ≅ C_phys — the category of physical laws is identical to the algebraic quotient category with the unobservable kernel factored out. The generating mechanism is unique: since M₃(ℂ) is the sole algebra satisfying the four axioms of any system capable of internal distinction, and the structure map Π admits no free parameters, no alternative algebra can generate L_phys. Surjectivity is therefore not merely constructive but exclusive. Four corollaries follow: the top-down direction of physical determination is structurally necessary; the Wigner problem dissolves completely; fine-tuning and hierarchy problems dissolve as projection artefacts; and the exclusivity of the generating mechanism is formally established. The paper additionally formalises the Noological partial category C_N, which accommodates both law formation (ρ∘Π) and reality selection (A_formal∘Π) as orthogonal operations in disjoint output spaces. The morphism A_formal : S → {p*} exists within C_N but its transition rule is placed outside the language of C_N by axiomatic silence — formalising Arbitrium as the structural implementation of human final authority (IASER Norm 4). This design is shown to be free of Gödelian incompleteness: the transition rule is not an undecidable proposition but a relationship excluded from the language of C_N by construction. The result recasts the epistemology of physics: experiment is not discovery but decoding — the act of reading out coordinate values of a fixed algebraic structure that M₃(ℂ) determines independently of any measurement. Physical constants are not adjustable parameters but unique coordinate evaluations of the quotient structure, making the question "why this value rather than another?" incoherent: there is no "another." Changes from the 1st Edition (DOI: 10.5281/zenodo.19965570, May 2, 2026): The structural content of the 1st Edition — definitions, theorems, and proofs — is unchanged. The 2nd Edition adds Corollary 4.4 (Exclusivity of the generating mechanism), which establishes formally that surjectivity and uniqueness of the generating algebra are equivalent claims given the M₃(ℂ) Necessity Theorem, closing the logical gap between constructive surjectivity and exclusive generation. A corresponding item is added to Section 7 (Structural Closure of Apparent Gaps). The Abstract is supplemented to reflect this addition. -/- Published on May 2, 2026 DOI: 10.5281/zenodo.19968224. (shrink)

Operatiology, derived from Noology through the three primitive notions of Ordo, Consensus, and Arbitrium, establishes the rank-3 minimal operational closure C⁽…Operatiology, derived from Noology through the three primitive notions of Ordo, Consensus, and Arbitrium, establishes the rank-3 minimal operational closure C⁽³⁾_Πd as the unique structure satisfying the executive axiom system {A1, A2, A4} together with the Operational-Geometric Coupling. The companion paper on the top-down projection of mathematics establishes that the induced mathematical category M is uniquely determined and that (...) algebra occupies the minimum-distance layer M₁ within M, delegating the complete algebraic identification to the present work. This paper discharges that delegation. The argument proceeds in four stages. First, algebra is established as the M₁-layer: the unique minimum-distance projective structure from the operational substrate. Second, operational non-decomposability is derived from Axioms 1, 2, and 4: direct sum decomposition of the state space is excluded by Axiom 1, and non-trivial two-sided ideals are excluded via the maximality of the canonical irreducible generator set I_Πd, establishing simplicity. Third, finite dimensionality is derived from the finite generator rank and the Πd-saturation closure via an explicit saturation depth bound, and the Wedderburn-Artin theorem is applied to identify the representation as Mₙ(D). Fourth, a bridge lemma connecting Πd-separation to eigenvalue structure establishes that the Noological primitive Arbitrium forces D = ℂ, and the three-generator Πd-independence condition forces n = 3, yielding M₃(ℂ) as the unique minimal algebraic realisation of C⁽³⁾_Πd. Published on May 24, 2026 DOI: 10.5281/zenodo.20363937 -/- . (shrink)

By most accounts Deleuze's engagement with Marx begins with the two volumes of Capitalism and Schizophrenia he co-authored with Félix Guattari. However, Deleuze's Difference and Repetition alludes to a connection between Deleuze's critique of common sense and Marx's theory of fetishism, suggesting a connection between the critique of the image of thought and the critique of capital. By tracing this connection from its emergence in the early texts on noology, or the image of thought, to the development in the (...) critique of state thought in A Thousand Plateaus, it can be argued that what initially appears as an entirely infra-philosophical problem, concerned with the presuppositions of philosophy, is not only a political problem as well, but ultimately bears on the very nature of the conjunction between thought and politics, making possible a re-examination of what is meant by revolutionary thought. It is a transition from noology to noopolitics. In the end it can be argued that revolutionary thought is no longer an eschatology, attempting to discern the signs of the future revolution in the present, but a thought oriented towards everything that exceeds the fetish of society, towards the virtual relations and micropolitical transformations that constitute society but exceed its representation. (shrink)

This paper establishes that continuity, infinity, and the analytic machinery built upon them are not structural necessities of any operational system but representational artifacts: formal constructs that encode the finite operational structure of the rank-3 minimal operational closure C⁽³⁾_Πd into an extended descriptive language. The argument proceeds from the axiomatic foundation of Operatiology, in which C⁽³⁾_Πd is derived from three axioms governing non-commutativity, Πd-saturation with finite generator rank, and redundancy exclusion. Operational necessity is defined as closure under finite-terminating operation sequences (...) achieving a finite operational determination of all Πd-distinguishable structure; the criterion is not formal definability but operational exhaustibility. Under this definition, the limit operation lies outside operational necessity: it requires certification over the index family {ε > 0}, which no finite subset exhausts modulo Πd. Continuity in all standard formulations, topology, differentiability, Lebesgue measure theory, and manifold theory inherit this classification by a single uniform mechanism formalised as the Unbounded Index Obstruction lemma. Infinity is formally classified as the termination-failure equivalence class [ℐ/∼]: the structural label unifying the limit operation, actual infinity, arbitrary unions, σ-additivity, non-countable cardinality, and Dedekind completeness under one mechanism. Four consequences follow: the dissolution of the Platonist–formalist dispute; the reinterpretation of the unreasonable effectiveness of mathematics as compression efficiency; the classification of renormalization divergences as representational artifacts; and the purification of physical constant derivation from M₃(ℂ) structure. This paper supersedes the Cognitional Mechanics formulation (DOI: 10.5281/zenodo.20282144) and forms part of the Operatiology mathematics series alongside the co-derivation of number systems and the top-down projection theorem for mathematics. Published on May 29, 2026 DOI: 10.5281/zenodo.20433611. (shrink)

This paper asks a question that standard group theory does not: why must the axioms of group theory---and no weaker or alternative structure---govern the symmetry of any system capable of making operational distinctions? Semigroups, monoids, and non-associative magmas are not merely less convenient than groups; they are operationally inadmissible. This paper derives that inadmissibility from first principles. The group axioms (closure, associativity, identity, invertibility) are established as necessary conditions for Πd-consistent operational symmetry under the axiom system {A1, A2, A4} of (...) Operatiology, without presupposing any group-theoretic result. Invertibility is forced by the requirement that no operational step induces an irreversible collapse of distinguishability; associativity follows from the inherent associativity of function composition; identity existence follows from the finiteness of the Πd-class structure. A complete explicit construction of the six-class minimal non-commutative Πd-closure is provided from a period-3 and period-2 generator pair, with saturation depth K=2 verified by a non-circular three-step closure consistency check. S₃ emerges as the minimal finite non-commutative operational witness of this necessity. Minimality is established in two structurally independent parts: Theorem A (constructive minimality, Tier-2 internal) provides an existence witness; Theorem B (extensional extension theorem, via established series results) provides universality. This paper forms part of the Operatiology Applied Series, in which group theory is the symmetry layer of the rank-3 minimal operational closure C⁽³⁾_Πd. Published on May 26, 2026 DOI:10.5281/zenodo.20396514 . (shrink)

This paper establishes that geometric structure — distance, metric, curvature, and the analytic machinery built upon them — is not operationally necessary in any operational system but a representational artifact: a formal construct encoding the algebraic structure of the rank-3 minimal operational closure C⁽³⁾_Πd into an extended descriptive language. The argument proceeds from the axiomatic foundation of Operatiology, in which C⁽³⁾_Πd is derived from three axioms governing non-commutativity, Πd-saturation with finite generator rank, and redundancy exclusion, and from the companion result (...) that algebra is the unique top-down projection of the operational substrate. The Metric Reduction Theorem establishes that every admissible distance structure is a projection of the trace inner product Tr(A†B) of M₃(ℂ), the unique minimal algebraic realisation of C⁽³⁾_Πd, and is therefore not an independent operational primitive. The Geometric Unbounded Index Obstruction lemma classifies continuous curves, infinite metric spaces, topological neighbourhoods, and curvature as elements of the termination-failure equivalence class [ℐ/∼], with distinct obstruction types. Euclidean and non-Euclidean geometries are classified by the same mechanism, differentiated only by the curvature parameter of their metric projection. The Euclidean–non-Euclidean dispute is dissolved as a competition between alternative representational languages. The general relativistic metric formulation is classified as a representational artifact without affecting its empirical validity. Published on May 30, 2026 DOI: 10.5281/zenodo.20465726. (shrink)

This paper supersedes Mathematics as the Unique Top-Down Projection of Intelligence: A Structural Theorem from Cognitional Mechanics and Noology (DOI: 10.5281/zenodo.19968224), which itself superseded the first edition of the programme (January 2026, DOI: 10.5281/zenodo.18280992). The first edition identified mathematical structures as stabilised residues of irreversible, non-commutative operational histories and positioned the framework as a meta-theoretical explanatory layer operating above existing mathematical foundations. The second edition established the stronger claim that mathematics is the unique top-down projection of the operational structure (...) of intelligence, with the induced mathematical category M uniquely determined up to identity automorphism and the projection map surjective with structurally undefined inverse. -/- The present work marks a foundational transition: it re-establishes all structural results of the second edition under Operatiology (DOI: 10.5281/zenodo.20350414), the framework that supersedes Cognitional Mechanics as the executive layer derived from Noology Version 2 (DOI: 10.5281/zenodo.18690463). The prior premise that the minimal operational structure is identified with M_3(C) is removed. The projection source is now the abstract rank-3 operational closure C^(3)_Πd itself, established within Tier-2 by Operatiology, without presupposing its identification with any specific algebraic structure. That identification is a separate Tier-3 question and does not enter any argument of the present work. -/- Under this clarified foundation, the central theorems are re-established: the projection Π_math from C^(3)_Πd to M is surjective with structurally undefined inverse; Aut(M) = {id}; and M is the unique fixed point of the Πd-saturation closure operator acting on Tier-3 formal structures. The prior corpus citations from the Cognitional Mechanics framework are retained for historical continuity only; no result derived by means of M_3(C) is invoked in any proof or definition. The sole logical foundation of the present paper is Operatiology, derived from Noology Version 2. -/- A new result absent from the second edition is introduced: an internal operational distance d_op on M, measuring the index family type required for operational determination of each mathematical structure. This yields a five-layer classification: Layer 1 (finite index families, algebra); Layer 2 (countable type, discrete infinite structures); Layer 3 (power-set type, continuous and geometric structures); Layer 4 (unrestricted type, axiomatic and functorial structures); Layer 5 (maximal type, large-cardinal and higher-categorical structures). Algebra occupies Layer 1 as the unique mathematical domain closest to C^(3)_Πd. The prior Operatiology mathematics series covering number systems, analytic structures, and set theory is positioned within this hierarchy. -/- Consequences include the dissolution of the Platonist-formalist dichotomy; the structural impossibility of alternative mathematics; the reinterpretation of mathematical effectiveness in physics as a tautology; and the reclassification of foundational competition among ZFC, category theory, and type theory as a compression-efficiency competition internal to Tier-3, evaluated from above by C^(3)_Πd. -/- Published on May 24, 2026 DOI:10.5281/zenodo.20362852. (shrink)

The proton-to-electron mass ratio μ is one of the most precisely measured dimensionless constants, yet the Standard Model provides no structural explanation for its value. Version 5 is fully grounded in the Operatiology axiom system {A1, A2, A4}, under which the unique rank-3 operational closure is realised as M₃(ℂ), whose internal structure fixes all components of the μ expansion with zero free parameters. The derivation uses only the Cartan generator H = diag(1,1,−2), the internal language of M₃(ℂ), and previously established (...) structural constants within the Operatiology corpus. -/- Version 5 makes three changes relative to Version 4 (DOI: 10.5281/zenodo.20671331). First, it corrects the proof of electron independence and proton indecomposability (Version 4: Section 2.4): the proof is restated relative to the Cartan generator H = diag(1,1,−2) already established for the Cartan generator's uniqueness, replacing a diagonal generator assignment that was inconsistent with that result. Second, it formalizes the residual-degree argument used in the closure proof of the four-term expansion as an independent definition, rather than an inline introduction repeated without citation. Third, it strengthens the structural-invariance corollary (dμ/dt = 0) by grounding the finiteness of the set of Operational Invariants in the paper's own internal-language closure theorem, in addition to the existing external citation. No other result of Version 4 is affected: the four-term expansion, its numerical value, and all consequences (the spectral coupling constant ξ = μα, structural identity and parity with α⁻¹, and the structural-invariance theorem) depend only on the Cartan generator's established uniqueness and are unaffected by these corrections. -/- As in Version 4, γ_E (Euler–Mascheroni constant) is established as the unique compression residue of the relational closure of M₃(ℂ) into finite algebraic invariants; μ is classified as a Category E composite Operational Invariant lying in S \ F_ext, following the structural category framework of the companion physical-constants paper (Version 2, DOI: 10.5281/zenodo.20647095); and the Class I status of μ is shown to entail dμ/dt = 0 as a structural theorem: no alternative value of μ is algebraically possible within M₃(ℂ) structure, and no epoch in which μ differed from its present value is structurally coherent. -/- To ensure transparency and reproducibility, the full derivation is accompanied by a directly executable Python script embedded below. Readers may copy it as-is to reproduce the theoretical value of μ and verify agreement with CODATA 2022. -/- ◆◆◆ [PYTHON VERIFICATION CODE — note: × denotes * and ^ denotes ** ] ◆◆◆ from mpmath import mp, mpf, pi, sqrt, exp, euler mp.dps = 50 -/- delta = sqrt(mpf(3)/2) gamma_c = mpf(1)/2 D = (delta - 1) × delta × gamma_c -/- Phi1 = mpf(2) Phi3 = mpf(13) Phi6 = mpf(7) -/- gamma_E = euler -/- term0 = 6 × pi^5 term1 = (Phi3 / Phi6) × D^2 term2 = -(delta^2) / gamma_c × exp(-gamma_E) × D^4 term3 = -(Phi1 / Phi3) × D^5 -/- mu_theory = term0 + term1 + term2 + term3 print("mu (theory) =", mu_theory) ◆◆◆ [END] ◆◆◆ -/- This version preserves conceptual continuity with the earlier CM derivation—where μ was approached through the Tier-1 quantity μα—while replacing all implicit assumptions with explicit Operatiology-level structure, now including a fully formalized closure argument and a self-contained structural-invariance proof. The result is a closed, reproducible, and axiomatically grounded account of μ within the M₃(ℂ) framework. -/- Uploaded on June 17, 2026 DOI:10.5281/zenodo.20735168. (shrink)

The present version reframes the derivation within Operatiology, the successor framework to Cognitional Mechanics. The central advance is the Grounding Uniqueness Lemma: among all values exp(t) for t∈ℝ, the unique element generating an independent Operational Invariant is exp(1)=e. This follows from exhaustive case analysis — rational t are A4-redundant relative to e; nonstructural irrational t admit no M₃(ℂ)-grounding; structural irrational t (e.g. e^δ, e^π, e^γE) yield only Category E composites of existing S-elements. The grounding source (G2b) is the unique unit (...) operation of the Lie algebra exponential map of M₃(ℂ) — a Tier-2 operational concept involving no numerical parameter. Its Tier-3 representation is t=1, yielding e=exp(1) as the M3-value. e is thus a Category B Operational Invariant (e∈S\Fext, 操作遍数) of the general solution (DOI: 10.5281/zenodo.20647095). Hermite's transcendence result (1873) is repositioned as the final step: G2b ⟹ unit operation ⟹ Tier-3 projection ⟹ t=1, exp(1)=e ⟹ Hermite confirms e∉Q̄. It confirms, rather than grounds, the uniqueness established by G2b. The structural invariance de/dt=0 follows as a theorem (Corollary 3.6). This extraction is logically necessary in the same sense as that of π (DOI: 10.5281/zenodo.20602756): the absence of a Tier-3 closure condition for the open curve exp(tH) — confirmed by Hermite — uniquely necessitates characteristic-polynomial confirmation, symmetric in logical structure with the closure condition U(θ)=I for π. The following code illustrates the Tier-3 confirmation of the exponential unit operation at 50-digit precision. Starting from H=diag(1,1,-2) with integer eigenvalues only, no numerical approximation of any constant is manually supplied at any stage; the constant appears only in the printed output. The algebraic justification is given in Section 3 of the paper. from mpmath import mp, mpf, mpc, matrix from mpmath import findroot, expm from mpmath import exp, sqrt mp.dps = 50 h = [mpf(1), mpf(1), mpf(-2)] H = matrix([[h[0], 0, 0 ], [0, h[1], 0 ], [0, 0, h[2]]]) M = expm(H) def tr(A): return sum(A[i,i] for i in range(3)) def det3(A): return (A[0,0]*(A[1,1]*A[2,2] -A[1,2]*A[2,1]) -A[0,1]*(A[1,0]*A[2,2] -A[1,2]*A[2,0]) +A[0,2]*(A[1,0]*A[2,1] -A[1,1]*A[2,0])) p1 = tr(M) p2 = (p1**2 - tr(M * M)) / 2 p3 = det3(M) def char_poly(lam): return lam**3 - p1*lam**2 \ + p2*lam - p3 r_small = findroot(char_poly, mpf('0.5')) b = r_small - p1 c = p3 / r_small C1 = ((-b + sqrt(b**2 - 4*c)) / 2).real C2 = mp.e print("=== Part A-1: Tier-3 confirmation" " (exponential unit) ===") print(f"Tier-2: unit operation (G2b)") print(f"Tier-3: parameter value t=1,") print(f" char_poly(expm(H))") print(f"Logical order:") print(f" G2b -> unit operation (T2)") print(f" -> Tier-3 projection") print(f" -> parameter value t=1") print(f" -> exp(1)=e") print(f" -> Hermite confirms") print(f" value not in Q-bar") print(f"confirmed eigenvalue > 1 = {C1}") print(f"library reference value = {C2}") print(f"difference = " f"{abs(C1 - C2)}"). (shrink)

Operatiology (Japanese: *Sousa Genron*, 操作原論), derived from Noology (Japanese: *Chigaku*, 知学) through the three primitive notions of Ordo, Consensus, and Arbitrium, establishes the rank-3 minimal operational closure C⁽³⁾_Πd as the unique structure satisfying {A1, A2, A4} together with the Operational-Geometric Coupling. The companion paper on the top-down projection of mathematics establishes that algebra occupies the minimum-distance layer M₁ within the induced mathematical category M, delegating the complete algebraic identification to the present work. Both papers form part of the Cognitional (...) Mechanics (Japanese: *Chinou Rikigaku*, 知能力学) corpus. -/- This paper discharges that delegation in four stages. First, algebra is established as the M₁-layer: the unique minimum-distance projective structure from the operational substrate. Second, operational non-decomposability is derived from Axioms 1, 2, and 4: direct sum decomposition is excluded by Axiom 1, and non-trivial two-sided ideals are excluded via the maximality of I_Πd, establishing simplicity. Third, finite dimensionality is derived from the finite generator rank via an explicit saturation depth bound, and the Wedderburn-Artin theorem identifies the representation as Mₙ(D). Fourth, a bridge lemma connecting Πd-separation to eigenvalue structure establishes that Arbitrium forces D = ℂ, and the three-generator Πd-independence condition forces n = 3, yielding M₃(ℂ) as the unique minimal algebraic realisation of C⁽³⁾_Πd. -/- Version 2 adds Lemma 5.4 (faithfulness preservation under scalar extension), Lemma 5.5 (minimal faithful representation dimension of Mₙ(ℂ) is n), and a unit-element argument closing the irreducibility proof of Proposition 5.6 Step B without dimension-by-dimension case analysis. The Πd-separation axis correspondence is formalised as a separate lemma, and the exclusion of n = 2 and n ≥ 4 is strengthened. A computational verification appendix provides three Python programs (standard library only) confirming the core results on a finite proxy F₃. Representative program (Code 1: Non-Commutativity and Configurational Distinguishability): ◆◆◆ [PYTHON VERIFICATION CODE — note: × denotes *, ． denotes .] ◆◆◆ from collections import defaultdict import itertools mod = 3 def mat_mul(A, B, mod): n = len(A); return [[sum(A[i][k]×B[k][j] for k in range(n)) % mod for j in range(n)] for i in range(n)] def mat_comm(A, B, mod): AB = mat_mul(A, B, mod); BA = mat_mul(B, A, mod); n = len(A); return [[(AB[i][j] - BA[i][j]) % mod for j in range(n)] for i in range(n)] def mat_eq(A, B): return all(A[i][j] == B[i][j] for i in range(len(A)) for j in range(len(A[0]))) def mat_action(M, states, mod): n = len(M); result = {}; [result．update({v: tuple(sum(M[i][j]×v[j] for j in range(n)) % mod for i in range(n))}) for v in states]; return result def pid_classes(states, ops): cls = defaultdict(list); [cls[tuple(op[s] for op in ops)]．append(s) for s in states]; return list(cls．values()) states_2d = list(itertools．product(range(mod), repeat=2)); E = [[0,1],[0,0]]; F = [[0,0],[1,0]]; H = [[1,0],[0,2]]; comm_EF = mat_comm(E, F, mod); print(f"[E, F] = H : {mat_eq(comm_EF, H)}"); actE = mat_action(E, states_2d, mod); actF = mat_action(F, states_2d, mod); actH = mat_action(H, states_2d, mod); EF_map = {s: actE[actF[s]] for s in states_2d}; FE_map = {s: actF[actE[s]] for s in states_2d}; witnesses = [s for s in states_2d if EF_map[s] != FE_map[s]]; print(f"Non-commutativity witness: s = {witnesses[0]}"); classes_EFH = pid_classes(states_2d, [actE, actF, actH]); same = {s: actE[s] for s in states_2d}; classes_comm = pid_classes(states_2d, [same, same, same]); print(f"Pid-classes {{E,F,H}} (non-commutative): {len(classes_EFH)}"); print(f"Pid-classes 3xE (commutative): {len(classes_comm)}"); print(f"=> Commutative collapse violates Arbitrium．") ◆◆◆ [END] ◆◆◆ -/- Published on July 23, 2026 DOI 10.5281/zenodo.21510803. (shrink)

Set theory occupies the foundational stratum of modern mathematics, yet the question of whether its machinery reflects structural necessity or descriptive convenience has never been posed from outside the formal tradition itself. Zermelo–Fraenkel set theory with the Axiom of Choice (ZFC) is universally adopted as the axiomatic substrate for virtually all of contemporary mathematics, from analysis and topology to algebra and logic. Its axioms are treated as given, their justification appealing to mathematical intuition or pragmatic adequacy rather than to any (...) prior structural derivation. Operatiology poses the prior question: does ZFC reflect the operational necessities of any operational system, or is it a descriptive compression language whose expressive power comes precisely at the cost of distance from operational reality? This paper applies the projection architecture of Operatiology—distinguishing the Regulative, Executive, and Projective layers (corresponding to the prior CM corpus designations Tier-1, Tier-2, and Tier-3)—to classify each axiom of ZFC individually. The classification criterion is the Unbounded Index Obstruction: the criterion applies only when the semantic content of a structure itself requires non-finitely-exhaustible certification; any structure whose complete operational determination requires certification over a non-finitely-exhaustible index family is excluded from the Executive layer. The central result is that ZFC is a projective representational artifact: a highly compression-efficient classical first-order set-theoretic formal language encoding the termination-failure equivalence class [I/∼]. Extensionality uniquely reflects the Executive operational identity principle—established by a Projection Identity result relating the Operational Identity Condition of Axiom 2 to set-theoretic extensional equality—while remaining formally projective. The Axioms of Infinity, Power Set, Replacement, and Choice introduce countable-type, power-set-type, unrestricted-type, and maximal-type elements of [I/∼] respectively. The continuum hypothesis is classified as a projective artifact whose undecidability within ZFC is the structural signature of a question that does not arise from the Executive layer. This version supersedes the Cognitional Mechanics formulation (DOI: 10.5281/zenodo.20325394). -/- Published on May 30, 2026 DOI: 10.5281/zenodo.20455274. (shrink)

This paper establishes that category theory is not a foundational layer of any operational system but a representational artifact: the minimal morphism-based formal language encoding the operational obstruction structure [ℐ/∼] derived from Operatiology. The Unbounded Index Obstruction criterion classifies core categorical notions individually. Arbitrary categories, functors, and natural transformations require certification over non-finitely-exhaustible index families. Limits and colimits belong to the power-set type of [ℐ/∼], the categorical analogue of the Power Set axiom in ZFC. Adjunctions belong to the unrestricted type, (...) parallel to the Axiom of Replacement. The Yoneda lemma occupies the extremal position within the power-set type: the entire relational complexity of Nat(h_A, F) collapses to a single evaluation, constituting maximum compression efficiency within the morphism-based language. The minimal categorical structures retaining executive correlates in C⁽³⁾_Πd are finite acyclic free categories and the single-object category BM₃(ℂ). A fidelity ordering among categorical, set-theoretic, and type-theoretic foundations is established as a structural taxonomy. Category theory and ZFC are dual [ℐ/∼] compression languages differing only in compression axis: membership versus morphism. Their foundational rivalry is a competition internal to the projective layer. The classification criterion (operational necessity, grounded in Axiom 2) and the classification function (Unbounded Index Obstruction) are both derived from the executive layer; no external foundation is structurally required or presupposed. -/- Published on May 31, 2026 DOI: 10.5281/zenodo.20470946. (shrink)

This paper establishes the structural foundation of the spectral coupling constant ξ = μα, the product of the proton-to-electron mass ratio μ and the fine-structure constant α, within the Operatiology and Cognitional Mechanics framework. A two-stage proof is presented. The first stage demonstrates that μ and α share a common algebraic origin in the matrix algebra M₃(ℂ): the product ξ = μα forces the cancellation of the common Casimir invariant Tr(H²) = 6 appearing in the leading terms of both independent (...) derivations, making algebraically manifest the shared Cartan generator H = diag(1,1,-2) underlying both constants. This addresses a question that the Standard Model of particle physics has no framework to pose. The second stage identifies the common origin algebraically: Axiom A1 of Operatiology forces the su(3)-type minimal non-commutative closure, whose tracelessness implies p₁ = Tr(Hλ) = 0. This constraint forces the invariant algebra to satisfy Inv(O) ~ R[λ²], making λ² the unique minimal coordinate of the algebra's self-description. Since ξ is proportional to λ, ξ lies outside the invariant algebra; since ξ² is proportional to λ², ξ² is the minimal generator of the invariant algebra. The asymmetry between ξ and ξ² is therefore a necessary consequence of A1, not an external imposition. Conditional on prior spectral completeness results, ξ² functions as the generator of all physical constants within the framework. The paper is the first of a two-part work; the companion paper derives ξ² independently with zero free parameters. Published on June 7, 2026 DOI: 10.5281/zenodo.20577184. (shrink)

This paper is Version 3 of the e-as-finite-number programme, establishing e as a Category B Operational Invariant (Japanese: Sousa Hensuu 操作遍数) of the unique minimal operational closure C^(3)_Πd, realised as M₃(ℂ) structure, within the axiom system {A1 non-commutativity, A2 Π_d-saturation, A4 redundancy exclusion} of Operatiology, the successor framework to Cognitional Mechanics. -/- The central advance is the Grounding Uniqueness Lemma: among all values exp(t) for t∈ℝ, the unique element generating an independent Operational Invariant is exp(1)=e. This follows from exhaustive case (...) analysis — rational t are A4-redundant relative to e; nonstructural irrational t admit no M₃(ℂ)-grounding; structural irrational t (e.g. e^δ, e^π, e^γE) yield only Category E composites of existing S-elements. -/- The grounding source (G2b) is the unique unit operation of the Lie algebra exponential map of M₃(ℂ) — a Tier-2 operational concept involving no numerical parameter. Its Tier-3 representation is t=1, yielding e=exp(1) as the M3-value. e is thus established as an element of S\Fext of the general Operational Invariant solution (DOI: 10.5281/zenodo.20647095). -/- Hermite's transcendence result (1873) is repositioned as the final step in the logical chain: G2b ⟹ unit operation ⟹ Tier-3 projection ⟹ t=1, exp(1)=e ⟹ Hermite confirms e∉Q̄. It confirms, rather than grounds, the uniqueness established by G2b. The structural invariance de/dt=0 follows as a theorem (Corollary 3.6). -/- This extraction is logically necessary in the same sense as that of π (DOI: 10.5281/zenodo.20602756): the absence of a Tier-3 closure condition for the open curve exp(tH) — confirmed by Hermite — uniquely necessitates characteristic-polynomial confirmation, symmetric in logical structure with the closure condition U(θ)=I for π. -/- For those interested, the following code illustrates the Tier-3 confirmation of the exponential unit operation at 50-digit precision: starting from H=diag(1,1,-2) with integer eigenvalues only, no numerical approximation of any constant is manually supplied at any stage, and the constant appears only in the printed output. The algebraic justification is given in Section 3 of the paper. -/- ◆◆◆ [PYTHON VERIFICATION CODE — note: × denotes * and ^ denotes **] ◆◆◆ -/- from mpmath import mp, mpf, mpc, matrix, findroot, expm, exp, sqrt ◆ mp.dps = 50 ◆ h = [mpf(1), mpf(1), mpf(-2)] ◆ H = matrix([[h[0],0,0],[0,h[1],0],[0,0,h[2]]]) ◆ M = expm(H) ◆ def tr(A): return sum(A[i,i] for i in range(3)) ◆ def det3(A): return (A[0,0]×(A[1,1]×A[2,2] - A[1,2]×A[2,1]) - A[0,1]×(A[1,0]×A[2,2] - A[1,2]×A[2,0]) + A[0,2]×(A[1,0]×A[2,1] - A[1,1]×A[2,0])) ◆ p1 = tr(M) ◆ p2 = (p1^2 - tr(M × M)) / 2 ◆ p3 = det3(M) ◆ def char_poly(lam): return lam^3 - p1×lam^2 + p2×lam - p3 ◆ r_small = findroot(char_poly, mpf('0.5')) ◆ b = r_small - p1 ◆ c = p3 / r_small ◆ C1 = ((-b + sqrt(b^2 - 4×c)) / 2).real ◆ C2 = mp.e ◆ print(f"Tier-2: unit operation (G2b); Tier-3: parameter value t=1, char_poly(expm(H))") ◆ print(f"Logical order: G2b -> unit operation (T2) -> Tier-3 projection -> parameter value t=1 -> exp(1)=e -> Hermite confirms value not in Q-bar") ◆ print(f"confirmed eigenvalue > 1 = {C1}") ◆ print(f"library reference value = {C2}") ◆ print(f"difference = {abs(C1 - C2)}") -/- ◆◆◆ [END] ◆◆◆ -/- Published on June 14, 2026 DOI 10.5281/zenodo.20685098. (shrink)

This paper derives the complete algebraic expansion of the inverse fine structure constant α⁻¹ from the axiom system {A1 non‑commutativity, A2 Π_d‑saturation, A4 redundancy exclusion} of Operatiology, under which the rank‑3 operational closure is uniquely realised as the matrix algebra M₃(ℂ). The expansion α⁻¹ = 6π/D + C₂·D + (Φ₁/Φ₆)·D⁶ closes at exactly three terms with zero free parameters. The normalisation rule Nₖ = Φ₂ₖ(dim)/Φ_{(dim+1)−k}(dim) is proved unique by an internal‑language closure argument invoking no experimental value, discharging the objection that (...) it was fitted to the known value 137. The theoretical value α⁻¹ = 137.035999197 agrees with the highest‑precision single‑experiment determination (Morel et al. 2020, Rb recoil) at 0.81σ. This is a revision and expansion of Version 3.0 (doi: 10.5281/zenodo.18843064), transferring the framework from Cognitional Mechanics to Operatiology and separating the theoretical value from the measured value previously conflated with it. The complete derivation runs from the spectrum {1, 1, −2} alone.◆◆◆ [PYTHON VERIFICATION CODE — note: × denotes * and ^ denotes ** ] ◆◆◆ from mpmath import mp, mpf, pi, sqrt mp.dps = 50 h = [mpf(1), mpf(1), mpf(-2)] e1 = sum(h) e2 = h[0]×h[1] + h[0]×h[2] + h[1]×h[2] e3 = h[0]×h[1]×h[2] delta = sqrt(mpf(3)/2) gamma = mpf(1)/2 D = (delta - 1)×delta×gamma ch2 = (e1^2 - 2×e2)/2 ch3 = (e1^3 - 3×e1×e2 + 3×e3)/6 N2 = mpf(10)/4 N3 = mpf(7)/2 C2 = sqrt(ch2/N2)×gamma f3 = abs(ch3)/N3 term0 = 6×pi/D term1 = C2×D term2 = f3×D^6 alpha_inv = term0 + term1 + term2 print("alpha^-1 =", alpha_inv) ◆◆◆ [END] ◆◆◆ -/- Published on June 2, 2026 DOI 10.5281/zenodo.20511158. (shrink)

This paper is Version 3 of the π-as-finite-number programme, establishing π as an Operational Invariant (Japanese: Sousa Hensuu 操作遍数) of the unique minimal operational closure C^(3)_Πd, realised as M₃(ℂ) structure, within the axiom system {A1 non-commutativity, A2 Π_d-saturation, A4 redundancy exclusion} of Operatiology. Version 1 (DOI: 10.5281/zenodo.18728748) established π as a structurally finite algebraic invariant within the Cognitional Mechanics framework. Version 2.0 (DOI: 10.5281/zenodo.19014478) made the Tier-3 status of Lindemann transcendence explicit. The present version provides the complete proof of Lemma (...) 3.8 (π∈S) of the general Operational Invariant solution (DOI: 10.5281/zenodo.20493369). The central advance is the replacement of the SU(3) half-period argument by a complete G1 volumetric grounding: π appears as an independent factor in the volume product Vol(U(1))·Vol(U(2))·Vol(U(3)) embedded in M₃(ℂ), establishing M₃(ℂ)-groundedness. A4-irreducibility follows from Lindemann's theorem via π∉Q(√6). The scope is extended from π¹ alone to all six independently grounded powers π^k (k=1,…,6); the upper bound k≤6 is an absolute structural consequence of the rank-3 constraint on M₃(ℂ). The derivation provides the retroactive structural justification for the use of π in all prior Cognitional Mechanics and Operatiology corpus papers. For those interested, the following code illustrates the algebraic extraction of π at 110-digit precision: starting from the integer initial value 6, it recovers π as θ₀/2 where θ₀ is the minimal positive solution of U(θ)=I, with no value of π appearing as input. The algebraic justification for the absence of circularity is given in Section 5.1 of the paper. ◆◆◆ [PYTHON VERIFICATION CODE — note: × denotes * and ^ denotes **] ◆◆◆ from mpmath import mp, mpf, matrix, exp, findroot ◆ mp.dps = 110 ◆ def compute_U(theta): e1 = exp(1j × theta); e2 = exp(-2j × theta); U = matrix([[e1,0,0],[0,e1,0],[0,0,e2]]); return U ◆ def consensus_violation(theta): U = compute_U(theta); I = matrix([[1,0,0],[0,1,0],[0,0,1]]); diff = U - I; J = sum(abs(diff[i,j])^2 for i in range(3) for j in range(3)); return J^0.5 ◆ theta_0 = findroot(lambda t: consensus_violation(t), mpf('6')) ◆ pi_extracted = theta_0 / 2 ◆ print(f"pi (extracted) = {pi_extracted}") ◆ print(f"pi (mpmath) = {mp.pi}") ◆ print(f"difference = {abs(pi_extracted - mp.pi)}") ◆ print(f"J(theta_0) = {consensus_violation(theta_0)}") ◆◆◆ [END] ◆◆◆ Published on June 9, 2026 DOI 10.5281/zenodo.20602756. (shrink)

This paper establishes that the standard number systems ℕ, ℤ, ℚ, ℝ, and ℂ are not pre-existing mathematical objects but structures uniquely selected by the operational constraints of the rank-3 minimal operational closure C⁽³⁾_Πd as formalised in Operatiology. The selection order is constrained by the axioms: Axiom 1 contains a geometric-persistence clause expressed using the distance function d, which is defined only in Axiom 2. Therefore ℝ, the number system selected by Axiom 2, must be established before ℂ, the number (...) system selected by the orbital structure of Axiom 1. -/- Axiom 2 selects ℝ as the unique totally ordered, complete Archimedean field. Axioms 1 and 4 identify ℕ, ℤ, and ℚ as the minimal discrete substructures of ℝ required for distinguishability, reversibility, and minimal description. Axiom 1 also generates bilateral inaccessibility between non-commutative orbit pairs, producing a canonical ℤ/2ℤ symmetry. By Galois theory, the unique minimal realisation of this symmetry over ℝ is ℂ, making ℂ the number system structurally demanded by non-commutativity. -/- Working Axiom 3 is not invoked; all results follow from A1, A2, and A4. Numbers and the algebra M₃(ℂ) are co-derived from the same axiomatic source via the chain C⁽³⁾_Πd → ℳ₁ → M₃(ℂ), showing that mathematics functions as the projective layer generated by, and subordinate to, the executive structure of Operatiology. Numbers, in this framework, are not discovered or invented; they are what operational closure structurally requires. -/- This version supersedes the prior Cognitional Mechanics formulation (DOI: 10.5281/zenodo.20108822). The transition from Cognitional Mechanics to Operatiology replaces the earlier grounding in "intelligence" as a technical term with the structurally deeper foundation of C⁽³⁾_Πd, and reframes the mathematical projection hierarchy in terms of the five-layer distance ordering established in the companion paper on mathematics as the unique top-down projection of operational structure. Published on May 24, 2026 DOI: 10.5281/zenodo.20366742 . (shrink)

This paper derives the spectral coupling invariant ξ² ≡ (μα)² in closed form from the axiom system {A1 non-commutativity, A2 Π_d-saturation, A4 redundancy exclusion} of Operatiology, completing the Level 2 Class I certification mandated by the companion paper (DOI: 10.5281/zenodo.20577184). The derivation proceeds through the field Q(D₀), where D₀=(δ−1)δγ satisfies 16D₀²−24D₀+3=0, making Q(D₀) two-dimensional over Q. Since μ and α⁻¹ are each elements of L(M₃(ℂ)) with coefficients in Q(D₀), their ratio ξ=μ/α⁻¹ belongs to this field, and ξ²=(ξ_a+ξ_b·D₀)² reduces exactly to (...) A+B·D₀ — the unique mandatory representation forced by the two-dimensionality of Q(D₀), itself the algebraic expression of Axiom A4. No other value of ξ² is algebraically possible. The result is ξ² = A + B·D₀ = 179.534632529506…, with A = −1202.60331893592…, B = 10042.5954528296…, zero free parameters, and no reference to individually derived values of μ or α⁻¹. Agreement with benchmark (μ_CODATA2022/α⁻¹_Morel2020)² at +1.085σ. The Morel–Parker 5.5σ discrepancy is identified as structural evidence that α is a Tier-3 projection of ξ², not an independent fundamental constant. The complete derivation runs from the spectrum {1,1,−2} alone. ◆◆◆ [PYTHON VERIFICATION CODE — note: × denotes * and ^ denotes **] ◆◆◆ from mpmath import mp, mpf, pi, sqrt, exp, euler import math mp.dps = 50 h = [mpf(1), mpf(1), mpf(-2)] dim = 3 e1 = sum(h) e2 = h[0]×h[1] + h[0]×h[2] + h[1]×h[2] e3 = h[0]×h[1]×h[2] delta = sqrt(mpf(3)/2) gamma = mpf(1)/(dim-1) D0 = (delta-1)×delta×gamma gamma_E = euler Phi1 = mpf(2); Phi3 = mpf(13); Phi6 = mpf(7) ch2 = (e1^2 - 2×e2)/2 ch3 = (e1^3 - 3×e1×e2 + 3×e3)/6 N2 = mpf(10)/4 N3 = mpf(7)/2 C2 = sqrt(ch2/N2)×gamma L6 = abs(ch3)/N3 L0 = 6×pi^5 L2 = Phi3/Phi6 L4 = -delta^2/gamma × exp(-gamma_E) L5 = -Phi1/Phi3 mu_a = L0 + L2×(-3)/16 + L4×(-99)/256 + L5×(-135)/256 mu_b = L2×3/2 + L4×45/16 + L5×981/256 ainv_a = 48×pi + L6×(-2943)/4096 ainv_b = -32×pi + C2 + L6×2673/512 N = ainv_a^2 + ainv_a×ainv_b×mpf(3)/2 + ainv_b^2×mpf(3)/16 inv_a = (ainv_a + ainv_b×mpf(3)/2) / N inv_b = -ainv_b / N xi_a = mu_a×inv_a - 3×mu_b×inv_b/16 xi_b = mu_a×inv_b + mu_b×inv_a + 3×mu_b×inv_b/2 A = xi_a^2 - 3×xi_b^2/16 B = 2×xi_a×xi_b + 3×xi_b^2/2 print("A =", A) print("B =", B) print("xi^2 =", A + B×D0) ◆◆◆ [END] ◆◆◆ Published on June 7, 2026 DOI 10.5281/zenodo.20581357. (shrink)

Rapid advances in computer technology and what is termed ‘Artificial Intelligence’ in the past 70 years have led to speculation about the ultimate capabilities of electronic devices, including speculation about whether they will make humans obsolete at some future time. Zubiri’s distinction between sensible intelligence and sentient intelligence can be applied to understanding of the limitations of AI. Machines can only operate on the sensible intelligence paradigm, which entails limits. Sentient intelligence allows humans to carry out functions that sensible intelligence-based (...) devices can never do. Sensible intelligence-based devices, including AI, will therefore be restricted to amplifying human (i.e. sentient intelligence) capabilities, but never replacing them. Historically, extravagant claims for AI have not been borne out, despite many orders of magnitude increase in computation speed and memory capacity. Theoretical arguments, based on Godel’s Incompleteness Theorem, also confirm this. (shrink)

This paper establishes Operatiology (Japanese: Sousa Genron, 操作原論) as the foundational executive layer of Cognitional Mechanics (CM), replacing the intelligence-centric framing of prior CM versions with a purely operational foundation. The central thesis is that "Intelligence" functioned as an epistemically necessary scaffold during the construction of CM: it oriented the derivation of the unique minimal operational structure C_min ≅ M₃(ℂ) but did not appear as a variable, boundary condition, or operational intermediate in any proof or calculation. With CM Version 9 (...) completing the uniqueness proof of C_min via the canonical irreducible set I_{Π_d}, the scaffold is no longer structurally required and is formally removed. Operatiology is derived from Noology (Version 2, 2026), the regulative layer consisting of three primitive notions: Ordo, Consensus, and Arbitrium. A formal lemma establishes that the simultaneous satisfaction of Arbitrium, Axiom 4 (redundancy exclusion), and Axiom 2 (finite generator rank) forces non-commutativity as the minimal operational structure, via a collapse mechanism on the history space O^n/S_n that destroys configurational distinguishability under commutativity. The CM axiom system {A1, A2, A4} is retained intact. Admissible operations are defined purely structurally as operations that alter the Π_d-indistinguishability partition. Cognitional Mechanics is repositioned as the reflexive special case of Operatiology: the branch treating intelligence as a Tier-2 pattern whose study includes subjectivity and qualia. (shrink)

This paper demonstrates that modern science is structurally identical to engineering at the methodological level, while differing institutionally by adopting universal truth as a transcendent and unreachable goal. The argument proceeds within a three-tier architecture: Tier 1 (Noology) is the regulative layer of foundational norms; Tier 2 (Operatiology) is the layer of operational structural necessity, definitionally independent of measurement; Tier 3 is the layer of formal descriptions, models, and predictions. -/- All coherence evaluation normalizes to the evaluate–update–debug cycle that (...) defines engineering operation. The formal criterion of Translatability—a unique, exclusive homomorphic map from Tier 2 structural necessity to Tier 3 formal description—is orthogonal to falsifiability: the two criteria operate over disjoint variable spaces, and predictive success cannot establish structural correctness. -/- Falsifiability is shown to be a Tier 3 internal debugging condition structurally independent of science's own definition. Popper was the first to articulate, without recognizing it as such, the methodological identity of science and engineering. The discomfort that attentive readers have long felt with falsifiability as a definition of science is the correct intuition: it describes what scientists do, not what science claims to be. -/- The central theorem establishes that after decomposition into engineering (S ⊆ E) and philosophy (S ∩ P = ∅), no epistemically autonomous domain remains unique to science. Science is Authorized Engineering: engineering practice bearing the institutional authority of truth-seeking, sustained by the systematic misidentification of Tier 3 utility as Tier 2 truth of nature (Fictionality). -/- The following Python program demonstrates that multiple structurally distinct models simultaneously satisfy the falsifiability condition |ε| < σ, confirming that falsifiability certifies engineering utility only. -/- ◆◆◆ [PYTHON VERIFICATION CODE] ◆◆◆ -/- from numpy import polyfit, polyval, linspace, mean, abs, max from numpy.random import seed, normal -/- seed(42) x = linspace(0, 10, 30) y_true = 2.0 * x + 1.0 noise = normal(0, 0.3, len(x)) y_obs = y_true + noise sigma = 0.5 # measurement tolerance -/- models = { "Linear (correct structure)": polyfit(x, y_obs, 1), "Cubic (wrong structure) ": polyfit(x, y_obs, 3), "Degree7 (wrong structure) ": polyfit(x, y_obs, 7), } -/- print("Model pass_rate max_resid") print("-" * 52) all_pass = True for name in models: coeffs = models[name] resid = abs(polyval(coeffs, x) - y_obs) rate = mean(resid < sigma) print(f"{name} {rate:5.1%} {max(resid):.4f}") if rate < 1.0: all_pass = False -/- print() if all_pass: print("CONCLUSION: All structurally distinct models satisfy") print(" |epsilon| < sigma (falsifiability condition).") print(" This result is structurally independent of") print(" science's own definition (pursuit of universal truth).") print(" Falsifiability certifies engineering utility only.") print(" Science's self-definition is not operationalized") print(" by its method.") -/- ◆◆◆ [END] ◆◆◆ -/- Published on June 30, 2026 DOI: 10.5281/zenodo.21052267. (shrink)

This paper establishes that M₃(ℂ), the algebra of 3×3 complex matrices, is the unique minimal projective algebraic realisation of the rank-3 operational closure C⁽³⁾_Πd derived from Operatiology. The result supersedes Version 1 " M3(C) Necessity in Cognitional Mechanics: The Logical Foundation of Dimensional Structure" (DOI: 10.5281/zenodo.18280992), which established the same conclusion via dimensional exclusion and spectral efficiency arguments within the earlier Cognitional Mechanics axiom system. -/- The present version proceeds from the Operatiology axiom system {A1, A2, A4} and the Operational–Geometric (...) Coupling, with the Noological primitive Arbitrium as the regulative layer. The complete proof is delegated to the companion paper on algebra as the unique top-down projection of operational structure (DOI: 10.5281/zenodo.20363937), which proceeds via four stages: identification of algebra as the minimum-distance projective layer M₁; derivation of simplicity from operational non-decomposability; establishment of finite dimensionality and the Wedderburn–Artin form Mₙ(D); and a bridge lemma connecting Πd-separation to eigenvalue structure, forcing D = ℂ and n = 3. -/- The necessity established here applies retroactively to the entire Cognitional Mechanics corpus. Every paper citing the present work or its predecessor, directly or indirectly, and proceeding from M₃(ℂ) as a structural premise, is thereby grounded in the canonical projective realisation of C⁽³⁾_Πd. This covers derivations of physical constants, Standard Model gauge dynamics, nuclear reactions, number systems, cosmology, and life. The comprehensive corpus framework is documented in Foundations of Cognitional Mechanics (DOI: 10.5281/zenodo.20226488). -/- Published on May 24, 2026 DOI: 10.5281/zenodo.20365633 . (shrink)

This paper establishes Operatiology (Japanese: Sousa Genron, 操作原論) as the foundational executive layer of Cognitional Mechanics (CM; Japanese: Chinou Rikigaku, 知能力学), derived from Noology (Japanese: Chigaku, 知学; Version 2, 2026). Operatiology answers a fundamental question: what is the minimal structure any system must have in order to make distinctions at all? Starting from three primitive notions of Noology — Ordo (order-sensitivity), Consensus (internal consistency), and Arbitrium (configurational freedom) — the paper derives a unique minimal operational structure C_min governed (...) by three axioms: non-commutativity (A1), finite generator rank with saturation closure (A2), and redundancy exclusion (A4). The key result is that commutativity of all operations would collapse the history space O^n/S_n, destroying the configurational trace required by Arbitrium. Non-commutativity is therefore not assumed but derived as the minimal structural response. Version 3 adds Lemma 3.19 (Word Relation Describability Constraint): the word relations of C_min cannot contain negative-sign expressions such as g^n(s) = -g(s), because the axiom system imposes no additive structure on the state space. This closes a logical gap between the Tier-2 operational structure and its algebraic realisation. Version 3 supersedes Version 2 as the authoritative reference. Note: ×＝，^＝**，．＝.，：＝:，＿＝_，−＝-，！＝!=．The following substitutions are used solely for plain-text compatibility in this description. No algorithmic or semantic modification has been made. ◆◆◆ ［PYTHON ILLUSTRATION CODE — note：×＝，^＝**，．＝.，：＝:，＿＝_，−＝-，！＝!=］ ◆◆◆ from collections import defaultdict def compose（f，g）： return {s： f［g［s］］ for s in g} states ＝ ［0，1，2，3］ A ＝ {0：1，1：2，2：3，3：0} B ＝ {0：1，1：0，2：2，3：3} AB ＝ compose（A，B） BA ＝ compose（B，A） witnesses ＝ ［s for s in states if AB［s］ ！＝ BA［s］］ s0 ＝ witnesses［0］ print（f"A and B commute? {len（witnesses）＝＝0}"） print（f"Witness s＝{s0}： A o B（{s0}）＝{AB［s0］}，B o A（{s0}）＝{BA［s0］}"） C ＝ D ＝ {0：1，1：2，2：3，3：0} print（f"C o D ＝＝ D o C： {compose（C，D）＝＝compose（D，C）}"） print（"All orderings collapse −− Arbitrium violated．"） ◆◆◆ ［END］ ◆◆◆ Published on July 21, 2026. DOI: 10.5281/zenodo.21471088. (shrink)

This paper addresses a structural question at the intersection of philosophy and science: why has the constitutional layer of theoretical systems remained institutionally vacant since the late 19th century, and what are the formal consequences of this vacancy? -/- We introduce a functional distinction between two layers present in any theoretical construction. The operational layer executes inference, computation, and expansion. The constitutional layer specifies what a theory is, within what scope it is valid, and what identity criteria it adopts. Historical (...) analysis from 1870 to 2000 demonstrates that the divergence of mathematical physics from philosophical methodology produced an institutional separation in which the constitutional layer was systematically left without an explicit custodian. -/- The central formal result, the Internal Closure Impossibility Theorem, proves that no operational layer can generate its own constitutive conditions as internal operations. The proof proceeds by reductio ad absurdum: assuming such internal generation exists produces an infinite regress that cannot terminate within finite computational resources. An external fixation point, termed Arbitrium, is therefore structurally necessary. -/- This result is not a reformulation of Gödel incompleteness or Tarski's truth hierarchy. Gödel operates at the syntactic level of formal systems; Tarski at the level of linguistic hierarchies. The present theorem operates at the constitutive level of theoretical construction itself. -/- The framework employed is Noology, a formal system developed within Cognitional Mechanics, which provides axiomatic foundations for the functional separation of constitutional and operational layers. Re-implementing the audit function in a form compatible with contemporary mathematical physics constitutes the logical responsibility identified by this paper. (shrink)

A critical interpretation on Zubiri is put forward as an essential element for grasping the current importance of his thought, since this puts his potential contributions to the test and proposes corrections and creative extensions within the contemporary horizon of philosophy, with regard to phenomenology, hermeneutics, neurosciences and emergentism. The book focusses mainly on the evolution of Zubiri’s philosophy, his metaphysics, noology, anthropo- logy and the relationship between science and philosophy, but also on his serious crises. It should be (...) placed in context by associating this with today’s proposals for a new realism, linked to new forms of both ontology and hermeneutics. (shrink)

Resumo Neste ensaio procuro analisar o movimento reversível que há entre cinema e filosofia no pensamento de Gilles Deleuze. A filosofia do cinema é um dos temas deleuzianos por excelência e Cinema 1 e Cinema 2, dois dos seus livros mais estudados. Porém, que papel teve o cinema no seu pensamento filosófico? Com o objectivo de analisar os diferentes tipos de interferências que ocorrem entre o campo estritamente filosófico e o campo não-filosófico, pretendo demonstrar de que modo o cinema não (...) foi apenas considerado como uma arte que ilustrava ideias filosóficas já elaboradas, mas como uma arte que contribuiu para a própria criação conceptual. Palavras-chave : filosofia do cinema, Gilles Deleuze, metafilosofia, noologiaMy aim with the present essay is to analyse the reversible movement that occurs in Gilles Deleuze’s thought between cinema and philosophy. Philosophy of cinema is one of the most stimulating aspects of the deleuzian thought and Cinema 1 and Cinema 2 two of his most studied books. However, which role had cinema in his philosophical thought? With the final objective of analysing the different kinds of interferences that occur between the specific philosophical field and the non-philosophical field, I want to clarify how cinema was not seen as a mere artistic illustration of some pre-existing ideas but as a philosophical contribution. Keywords : Gilles Deleuze, metaphilosophy, noology, philosophy of cinema. (shrink)

This paper establishes the structural necessity for intelligence theory to possess a meta-theoretical framework that transcends specific formal tools. Intelligence operates universally across natural understanding, formal construction, artificial creation, and social practice. If intelligence were constrained by particular tools such as calculus, set theory, or logical operators, it would become dysfunctional in domains where those tools are inapplicable. Therefore, intelligence theory must encompass all tools while remaining subordinate to none. We present Noology as the unique formal system satisfying this (...) requirement. Noology adopts consistency (Consensus) as an absolute requirement, employs mathematical language to describe consistency, yet remains independent of any mathematical axiomatic system. The system is built on three primitive notions: Ordo (non-commutative operational order), Consensus (absence of contradiction), and Arbitrium (external configuration authority). This paper demonstrates that the universal operability of intelligence structurally necessitates a meta-theoretical architecture. We introduce a three-tier framework: Tier-1 (Noology) as the regulative layer, Tier-2 (Cognitional Mechanics with M₃(ℂ) algebra) as the executive substrate, and Tier-3 (MUT/GUT) as the projective display layer for physics and mathematics. We prove that conventional tool-specific theories—logical positivism, computationalism, and physicalism—fail because they recognize only one element of the tripartite structure while excluding others. Through phenomenological classification of four intelligence domains and dominance analysis, we show that all intelligent operations are weighted combinations of Ordo, Consensus, and Arbitrium. The meta-theoretical structure generates a structural boundary for artificial systems: since AI cannot originate Arbitrium or bear irreversible responsibility, Artificial General Intelligence is definitionally incoherent. Intelligence amplification remains viable by preserving human Arbitrium as the source of configuration while employing AI as an operational instrument. This framework redirects cognitional inquiry from simulating autonomy in machines to understanding the geometry of human judgment itself. (shrink)

Modern physics is divided into two empirically successful yet theoretically incompatible frameworks: the Standard Model of quantum fields and General Relativity of spacetime geometry. Despite decades of effort, unification attempts have relied on speculative constructs — supersymmetry, extra dimensions, string theory — none of which have been experimentally confirmed. Persistent anomalies, including the muon g-2 discrepancy, electron g-2 measurement inconsistencies, fine-structure constant deviations, and the continued non-detection of dark matter or gravitons, resist explanation within either framework alone. -/- Version 1 (...) of CM-GUT identified a categorical misinterpretation at the root of this impasse: treating gravity as an internally generated interaction rather than as an external operational constraint. By formalizing operational type separation — distinguishing Type I (internal generation) from Type II (external constraint) operations within the algebra M₃(ℂ) — Version 1 provided a unified explanation of the above anomalies as natural consequences of the Type I–Type II interface, without introducing new particles, extra dimensions, or modifications to existing physics. -/- Version 2 advances the foundation on three fronts. First, the Type Separation Principle, declared as an independent axiom in Version 1, is here derived as a theorem from Axioms A1–A4 alone. The external gravitational constraint δt(x)·I lies outside the commutator closure of M₃(ℂ) by a trace identity: every element of the Type I operational closure is traceless, while δt(x)·I carries nonzero trace. The impossibility of quantum gravity follows as a structural necessity, not a phenomenological observation. Second, the axiomatic foundation is fully aligned with the CM standard axiom set A1–A4 and the Noological Tier architecture. Third, the post-Version 1 result that particle ontology is non-essential is integrated as a second independent argument for graviton non-existence: once particles are identified as Tier 3 projections rather than ontological primitives, the graviton concept becomes unavailable prior to any dynamical consideration. -/- Version 2 does not unify the Standard Model and General Relativity as independent theories. It derives both as Tier 3 projections of a single Tier 1 algebraic structure, making unification a consequence rather than a task. The falsifiable predictions of Version 1 are retained and their theoretical grounding is strengthened. -/- Published on April, 9 DOI: 10.5281/zenodo.19485015. (shrink)

Cognitional Mechanics (CM) establishes an axiomatic framework that formalizes the structural mechanisms of intelligence as a self-contained operational system, abstracting intelligence through non-commutative operations, convergence of semantic states, and structurally inaccessible domains, without invoking physical observables or psychological primitives. While CM exhibits a clear structural correspondence with Quantum Mechanics (QM)---most notably in its non-commutative operator structures and bounded transitions---it departs fundamentally in its treatment of discreteness. Within the standard Hilbert-space formalism, non-commutativity and spectral discreteness are mathematically independent properties: position and (...) momentum operators, for instance, are non-commuting yet possess continuous spectra. In QM, however, the physical framework couples non-commutativity to empirical constants and measurement postulates, producing discreteness as a structural feature of the theory. CM identifies this coupling as a projection condition of the physical layer rather than a structural necessity of the algebra, and demonstrates that non-commutativity is sustainable in a continuous setting without discretization. This analysis is situated within an explicit three-tier architecture. Tier-1 (Noology) constitutes the regulative source. Tier-2 (CM) constitutes the executive operational domain, in which M₃(ℂ) is the unique minimal non-commutative substrate. Tier-3 (MUT/GUT) constitutes the projective display layer, in which physical theories including QM appear as projections of Tier-2 dynamics. CM and QM share a common Tier-1 origin without being in direct correspondence: their shared non-commutativity descends from Tier-1, while QM's discreteness is introduced at Tier-3 by projection conditions absent from Tier-2. The operational limit constant c regulates internal operability and convergence as a logical boundary condition, distinguishing it from Planck's constant ℏ, which fixes physical quantization through empirical calibration. CM thus clarifies that its resemblance to QM is structural rather than physical, and provides a vantage point from which the projection origin of QM's characteristic features is examinable. (shrink)

What is logotherapy based on? How does logotherapy differ from psychotherapies or other traditional forms of therapy? What disorders does logotherapy help with? These are the questions to which the 'Handbook for logotherapists' gives a clear and consistent answer. The Handbook starts from two logotherapeutic premisses, according to which there is a Noological dimension in human consciousness, which differs from the psychic dimension of consciousness and human has a will to purpose. These premisses are basic assumptions set by Viktor Frankl (...) himself. ‘Handbook for Logotherapists’ aims to verify these premisses by starting with studying the Noological dimension of human consciousness in relation to the psychic dimension. Based on research, the Handbook draws e.g. the following conclusions: a) The Noological dimension differs in its function and content from the psychic dimension, b) the human will to purpose is a characteristic of the Noological dimension that cannot be found in the psychic dimension, and c) there can be disorders in the Noological dimension that originate from the experienced conflict between the functioning of (the agents of) the Noological dimension (self-distancing and self-transcendence) and the situational reality. Purpose cannot be invented, but is to be found outside of a person, in the situational reality. Searching, finding, realizing and implementing purposes always involve the functioning of both self-transcendence and self-distancing. That is why self-distancing and self-transcendence are the primary agents of the Noological dimension. However, the process of searching, finding and implementing a purpose can become corrupted, as purpose may be missing, it may be lost or gone, it may be distorted, or the situation may have distorted the purpose, even if the functioning of the agents of the Noological dimension is healthy. Although these distortions with experiencing a purpose often manifest themselves unfavorably also in a person's somatic condition and in the functioning of the psychic dimension in many different ways, they are - nevertheless - distortions that are a person's healthy reaction to a sick state or situation, and their basis is in the Noological dimension of consciousness. With the help of logotherapy, it is possible to separate disorders of the Noological dimension from disorders of the psychic dimension, to detect what kind of distortion with experiencing a purpose is involved, and to treat a particular distortion with logotherapy's own treatment methods. 'Handbook for logotherapists' shows why logotherapy as a specific form of therapy is the most appropriate aid for the treatment of purpose-centered distortions and demonstrates, how these distortions can be detected and treated with logotherapeutic tools. -/- . (shrink)

In the fourteenth plateau of A Thousand Plateaus, Deleuze and Guattari develop a dichotomy between two kinds of space – the smooth and the striated. What I want to focus on in this chapter is the status of these two conceptions of space. As Deleuze and Guattari note, these two forms of space are only discovered in a mixed form, yet are capable of being analysed de jure through their separation. In this sense, the plateau on the smooth and striated (...) can be seen as something like a transcendental deduction of their ontology of spaces. I will explore what Deleuze and Guattari mean when they say that they want to construct a theo-noology of smooth and striated spaces. I want to look at the ethical implications of this distinction, before looking at some alternative approaches to the issue of space. It should be noted that the question of the striation of space is one that is shared by Bergson, Heidegger, Merleau-Ponty and Sartre among others. The novelty of Deleuze and Guattari’s account is in their formulation of the notion of smooth space as a response. I will begin by looking at the notion of striated space itself, and in particular will explore the degree to which we should see it as a structure or as a method, and the interrelation between these two characterisations. (shrink)

RESUMO Neste ensaio analiso as diferentes modalidades de pensamento que ocorrem entre a filosofia e as imagens em movimento partindo da distinção metafilosófica elaborada por Gilles Deleuze entre “pensar” e “filosofar”. Esta é uma distinção fundamental para a possível elaboração de uma filosofia do cinema, ou, pelo menos, para afirmar que “o cinema filosofa”, uma tese atualmente imersa num certo equívoco. Neste sentido, como possível resolução para tal mal-entendido, sugiro uma adequada designação deleuziana de “pensar com conceitos” e “pensar com (...) imagens”, num processo recíproco fundamental entre o campo filosófico e o não filosófico das artes. Começando com uma introdução à noologia em Deleuze e uma descrição dessas ideias e do seu valor estético, prossigo com uma análise mais detalhada sobre imagens em movimento, metáforas e adaptação cinematográfica de modo a, dentro de uma abordagem pós-continental e pós-analítica, questionar se o cinema filosofa. ABSTRACT In this essay I analyse the different modalities of thought that occur between philosophy and moving images, beginning with Gilles Deleuze’s metaphilosophical distinction between “thinking” and “philosophizing”. This is an essential distinction for the possible elaboration of a film philosophy, or at least one which claims that “film philosophizes,” a thesis that is nowadays immerged in a certain misconstruction. In this sense, I suggest, as a conceivable resolution to this misunderstanding, a more proper Deleuzian designation of “thinking with concepts” and “thinking with images,” in a fundamental reciprocal process between the philosophical and non-philosophical fields of the arts. Starting with an introduction to Deleuze’s noology and a description of these ideas and their aesthetic value, I proceed with a closer analysis of moving images, metaphors, and film adaptation in order to question, within a post-continental-analytic approach, whether film philosophizes. (shrink)

In A Thousand Plateaus, Deleuze and Guattari carve out an image of thought and a path for philosophy that is connected to the figure of the cosmic artisan. This article situates the artisan in relation to both past and future, comparing this figure to that of the artist in the work of another great philosopher who desired to bring forth a new beginning for philosophy, Martin Heidegger. After discussing multiple ways that Deleuze and Guattari's thought is world-destroying in terms of (...) past ontological and epistemological commitments, the article considers the cosmic artisan/philosopher's diagrammatic role in light of the technological-scientific advancements within the field of quantum physics. Through these elaborations, it becomes clear that Deleuze and Guattari are articulating the need for a new sensibility – to the imperceptible, the virtual, the compossible and the molecular. Deleuze and Guattari's noological nomadism opens a way forward, where world-destroying becomes linked to creation, and the figure of modernity becomes the possibility of a new relation to the earth – a cosmic earth. (shrink)

This article proffers a critical reading of multiliteracy pedagogy and a materialism of the multimodal and machinic. A critical stance is taken against the mesmerising modes of representation that run rampant across our ocular territories. The article assesses the dangers of fetishizing technologies. To this end, Multiple Literacies Theory (MLT) is read through a Guattarian theoretical prism to emphasise four chief points: (1) the role of the unconscious, (2) the role of affect (affectus in the Spinozian sense; contrary to feeling (...) which immobilises, affect moves us), (3) the apparent becoming cyborg of the human race and (4) the desire to desire repression. The author contends that while qualitative multiplicities play a central role for MLT, it is neither inconsistent nor contradictory to consider the role of the machinic in greater detail. It is argued that more research must be directed into the cartographic, ethological and noological mechanisms at work in plastic universes of reference (new technologies and the arts) which transform the virtual and transfigure the dynamic of affect. (shrink)

It is well-known that Gilles Deleuze (and Félix Guattari) are critical of the notion of ideology. However, it is not very clear why they seem to be so dismissive of it. In what follows, I will begin my discussion by showing what Deleuze means by ideology and reconstructing why Deleuze thinks that ideology is a misused concept and that the misuse of the concept warrants its dismissal. In Anti-Oedipus, the insufficiency or inadequacy of the concept of ideology can be understood (...) in light of the opposition between production and representation. Next, I will show that—in A Thousand Plateaus—this very opposition between production and representation is relaxed in favor of the notion of assemblage. By looking at Althusser’s theory of ideology, I will argue that ideology is not merely a cognitive representation in the sense that it can be considered as having its own material existence. If so, the concept of ideology does not have to be too narrowly conceived. Rather, I propose that Deleuze and Guattari offer a different kind of ideology critique through noology and the critical analysis of microfascism. Finally, I will conclude by arguing that Deleuze is still enacting a novel form of ideology critique despite his dismissal of the very terminology. (shrink)

This paper dissents from the canonical interpretation of Zubiri’s philosophy, which tends to conceive the way from On essence (1962) to Sentient Intelligence (1980 - 1983) as a project of phenomenological refinement of philosophy as noology. On the contrary, the continuity of both works is shown in the analysis of notions of “fact” and “essence”. This is the basis to conclude that it makes sense to conceive Zubiri’s philosophy as phenomenology, as long as it’s not characterized as an absolute (...) and unfounded knowledge. (shrink)

This article orientates Deleuze & Guattari’s pragmatic semiotics towards a semiotics of law. This pragmatic semiotics is explored, and directly related to the theory of emergence and complexity that is also a key feature of Deleuze & Guattari’s work. It is suggested that the development of these aspects of Deleuze & Guattari’s thought in relation to law allows the contours of a noological legal theory to be sketched out. Noology is the study of images of thought, their emergence, their (...) genealogy, and their creation. A first exploration of this noological legal theory is then carried out by the conceptualisation of nome law as the first emergence of law as theorised by Deleuze & Guattari in the plateau “1837: Of the Refrain” from “A Thousand Plateaus”. This is a conceptualisation of law’s emergence in a far-from-equilibrium palaeolithic hunter-gatherer pack, and contrasts to accounts of law’s origin in a founding violence or mythical contract. It is the ‘big bang’ of legality, and the opening up of a first image of legality, problematic of social organisation, and anthropomorphic knowledge space. (shrink)

This article reflects on the social reality in the work of the Spanish philosopher Xavier Zubiri. For this, a detailed investigation of the most important milestones is carried out where social reality is worked in accordance with the work of the philosopher. And, in addition, a possible “noology of society” is suspected that would be fundamental to understand the human today. And for this the work Sentient Intelligence becomes fundamental.

This paper shows the relevance of Zubirian noology for metaphysics in the 21st century, which is marked by the rise of neurosciences and artificial intelligence. Zubiri’s noological analysis, which is closely linked to neurobiology, contributes to offering a new conception of human reality and intelligence.

The subject of ethics has already been studied by scholars who have focused on the work of the philosopher Xavier Zubiri. Nevertheless, within the framework of what we might call ‘Zubiri’s ethics’ (Aranguren, Ellacuria, D. Gracia, etc.), not all the presuppositions of a postmetaphysical Zubirian ethics have yet been analyzed. By Zubirian ‘post-metaphysical ethics’ we understand that which derives from noology or the theory of sentient intelligence. As this theory affects both the affecting feeling and the tending will according (...) to what Zubiri called ‘intellectionism’, the aim of the present work is the study of ethics from the perspective of the third vector, the tending will, following what Zubiri called ‘possessional intelligence’. The result is that the description of the moral fact allows us to approach the concepts of ‘power’ and ‘duty’ from new viewpoints, highly fruitful in present-day ethics. (shrink)

This paper deals with all types of articles, their use and their “no-use“, as one of the traditional parts of speech, as “one of the odious instruments of our grammar" (Alexis François, La grammaire du purisme et l'Académie française au 18ème siècle, Paris, Société Nouvelle de Librairie et d’Édition, p. 70). It creates a real dividing line for the language phenomenon, because it divides languages into two distinct categories: languages with article and languages without article. Beyond the grammatical technicality created (...) by the article system and the absence of article within the nominal group, this presentation allows us to rethink the noological and phenomenological necessity of this grammatical category. It is therefore necessary to wonder about the absence of articles in languages ​​with articles (French, German, English, Italian, Macedonian, etc.) and on the absence of articles in languages ​​without articles (Croatian, Serbian, Russian, Turkish, etc.). In languages ​​devoid of any article system, a determinant of numeral origin is related to the article. It appropriates an indefinite adjective value: “(a) certain”. Both possibilities or both ways to express articles in those languages ​make it possible to enrich the binary tensor in psychomechanics of language. It happens as soon as we add the median extensities, not reducible to the traditional scale known as “definite article” – “indefinite article” – “zero article”. (shrink)

Nos seus tratados de filosofia prática, Kant propõe uma bússola para a vida política: a constituição de uma sociedade cosmopolita, que culmine o processo de republicanização dos Estados, evite a guerra e permita desenvolver as potencialidades humanas. Dois caminhos se oferecem para isso: a Sociedade das Nações e a República Universal. O artigo explora as possibilidades de um e de outro, e trata de mostrar de que modo a filosofia kantiana é uma filosofia da paz, tanto numa perspectiva noológica como (...) numa perspectiva politica, quais são as virtualidades de cada um dos caminhos propostos, e como seria preciso reconstruir hoje a filosofia política kantiana para orientar a actual construção da Ordem Internacional e da Paz Mundial. /// In his political treatises, Kant proposes a principle of orientation for the political life: the constitution of a cosmopolitan society that brings to completion the process of republicanization of the states, avoids war and allows the growth of the human potentialities. In order to achieve these goals Kant considers two ways: the Society of Nations and the Universal Republic. The present article explores the possibilities inherent to the one and to the other, and aims at showing how the kantian philosophy is indeed a philosophy of peace, and this both in the noologic sense as well as in a political perspective, shows the virtualities of each one of the proposed ways, and also demonstrates in what measure it would be necessary today to reconstruct Kant's political philosophy in order to bring forward the construction of the International Order and of the World Peace. (shrink)