T. O.

This paper establishes the Isometric Extension of the Mathematical Unified Theory of Cognitional Mechanics (CM-MUT), deriving the exact quantitative correspondence between the geometric modal functor and the algebraic modal functor over the historical category Hist. The central result is that for all admissible operational histories H, the κ-scale L¹ norm and the Frobenius norm are related by the Casimir invariant K=√3 of M₃(ℂ): ‖M_A(H)‖_κ = K·‖M_G(H)‖_F. This isometric relationship is derived from two implementation axioms, A3' (Modal Norm Selection) and A5 (Modal Distance Preservation), which formalize assumptions implicit throughout the CM corpus. The proof proceeds via the automorphism structure Aut(M₃(ℂ))≅PGL(3,ℂ), Axiom 4 (Redundancy Exclusion), and a formal lemma establishing that minimal descriptive complexity in the Cartan subalgebra uniquely forces uniform eigenvalue distribution Var(κ)=0 for all admissible histories. Four corollaries follow from a single algebraic source: the inverse fine structure constant α⁻¹=137.03599…, the Iron Stability Attractor κ(Fe)=6/13, the chemical projection constant Π_chem=145/468, and the ABC quality bound q(a,b,c)<13/7. Prior CM results on the Riemann Hypothesis, P≠NP, and the Halting Problem are unified under this framework. The Navier-Stokes regularity problem, the Hodge Conjecture, the Yang-Mills mass gap, the Poincaré Conjecture, and the Birch-Swinnerton-Dyer Conjecture are shown to be Tier-3 projections of Tier-1 structural necessities. The apparent difficulty of the Millennium Problems is an artifact of their Tier-3 formulation, not of mathematical reality. All derivations proceed with zero free parameters from the axiomatic structure of M₃(ℂ). Published on April 4, 2026 DOI: 10.5281/zenodo.19411822

This work establishes the Internal Language Theorem for the algebra M3(C) within the framework of Cognitional Mechanics (CM). CM derives all four Standard Model dimensionless coupling constants with zero free parameters, but the mechanism by which M3(C) generates the structural constants used in these derivations has remained implicit. In particular, the repeated appearance of the cyclotomic evaluation values Φ1(3)=2, Φ2(3)=4, Φ3(3)=13, and Φ6(3)=7 has lacked a formal explanation. The present paper demonstrates that these values arise necessarily from the axioms A1–A4 defining the M3(C) Structure. Using the Cayley–Hamilton theorem, the evaluation point x=n=3 is shown to be uniquely determined by information completeness and redundancy exclusion. The minimal noncommutative realization of A1 yields the circulant matrix C3, whose characteristic polynomial generates Φ3. The Cartan reflection H↦−H produces its dual Φ6. The linear factors Φ1 and Φ2 arise from the terminal Cayley–Hamilton polynomials x3±1. All other cyclotomic polynomials are excluded by degree closure or by the integer-eigenvalue condition derived from A4. The result is a fully self-generated internal language: the set {Φ1,Φ2,Φ3,Φ6} is complete, unique, and necessary. This mechanism has no analogue in conventional physics or in Connes’ noncommutative geometry, where M3(C) functions only as a carrier of particle ontology. In CM, the algebra itself is the origin of physical constants through the self-evaluation formula Φn(n). Published on March 31, 2026 DOI: 10.5281/zenodo.19350315

Contemporary nuclear physics accounts for radioactive decay and nuclear reactions through five independent theoretical frameworks, each with its own force carriers, coupling constants, and free parameters. Alpha decay is modeled via quantum mechanical tunneling through the Coulomb barrier. Beta decay is attributed to weak interaction mediated by W± bosons. Gamma decay is treated as electromagnetic radiation from excited nuclear states. Nuclear fission is described by the liquid drop model. Nuclear fusion is governed by strong force dynamics. No unified derivation from a single axiomatic structure has been achieved within the Standard Model. This paper derives all five phenomena from the axioms of Cognitional Mechanics (CM) and the algebraic structure of M₃(ℂ), the unique minimal noncommutative finite-dimensional C*-algebra, without free parameters. The central result is that all nuclear transitions are instances of spectral relaxation of the resonance coefficient κ toward the Iron Stability Attractor κ(Fe) = 6/13, established as the S4 closure node of the M₃(ℂ) automorphism structure. Alpha decay is derived as separation of the ⁴He Aut-invariant orbit unit. Beta decay is derived as SU(2)₁₂ isospin transition, with parity non-conservation following from the Tier structural non-alignment between the Tier-2 algebraic operation and the Tier-3 spatial symmetry. Gamma decay is derived as Cartan eigenvalue relaxation. Nuclear fusion is derived as two-domain spectral convergence toward κ(Fe) = 6/13. Nuclear fission is derived from the condition δt(x)/t → −1, under which entropy monotonicity becomes unsatisfiable within a single M₃(ℂ) domain. The half-life is established as a Tier-3 dimensional quantity structurally excluded from Tier-2 derivation. The Iron Stability Theorem κ(Fe) = 6/13 unifies all nuclear transitions under a single algebraic fixed point. Published on March 31, 2026 DOI: 10.5281/zenodo.19343077

This paper completes the algebraic unification of the atomic constituents -- electron, proton, and neutron -- within the Cognitional Mechanics (CM) framework. The three atomic substances are shown to be not ontological primitives but distinct spectral projection modes of the single algebra M₃(ℂ). The automorphism group Aut(M₃(ℂ)) ≅ SU(3)/ℤ₃ is derived internally from Axioms A1-A2 via the Skolem-Noether theorem, without external introduction of SU(3). The isospin symmetry SU(2)_isospin is uniquely identified as the SU(2)₁₂ subalgebra. The dimension n=3 is proved unique: n=2 cannot generate the required roots, while n≥4 violates the Redundancy Exclusion Axiom A4. The neutron is established as ontologically non-independent: it is the T₃=-1/2 projection of the SU(2)₁₂ isospin doublet, with no independent Tier-2 existence separate from the proton. The nucleon magnetic moment ratio is derived in two layers. The first-order prediction μₚ/|μₙ| = 1.4630 (residual 0.21% against experiment 1.4599) follows from the correction coefficient Φ₁Φ₂/n², derived from the structural identity [Tr_adj(H²)]²/Tr_adj(H⁴) = Φ₂, which holds for ALL traceless Cartan generators via the pure algebraic identity a⁴+b⁴+(a+b)⁴ = 2(a²+ab+b²)² combined with the A1 trace-zero condition. The second-order correction δ₁ = αs·4/39, with normalization constant Φ₃ = 13 uniquely forced by the CM Self-Containment principle and the internal structure of Axiom A2, reduces the residual to 0.006%, yielding μₚ/|μₙ| = 1.45999. No free parameters enter at any stage. The Standard Model gauge structure SU(3)×SU(2)×U(1) is subsumed as the Tier-3 image of M₃(ℂ)⊕ℂ under projection, not negated. Published on March 29, 2026 DOI: 10.5281/zenodo.19301842

This paper establishes the ontological status of matter within the Cognitional Mechanics (CM) framework, completing the argument initiated in Particles Are Unnecessary. The central thesis is that substances are not ontological primitives but cognitive heuristics attached to stable Tier-3 projections of the Tier-2 algebraic structure M₃(ℂ). The paper addresses the residual question left open by particle elimination: why does the chemical scale appear privileged? The answer is derived, not assumed. Elemental stability follows algebraically from the conjunction of Cartan action and automorphism action on M₃(ℂ). The boundary is an output of the algebra, not an input. The invariant ξ ≡ μα is introduced as a pure Tier-2 spectral quantity in which particle presuppositions cancel. Both μ and α individually presuppose particle ontology; their product yields a pure spectral ratio derivable from M₃(ℂ) structure alone, constituting the mathematical self-dissolution of substance ontology. The historical trajectory from Newton through Maxwell to Einstein is shown to point toward progressive ontological elimination. CM completes this trajectory. Gauge theory is reinterpreted as the Newtonian mechanics of the microscopic: valid within bounds, structurally false as a foundation. The paper closes with a formal statement: matter is the collective cognitive label of stable resonance nodes of M₃(ℂ) spectral structure under Tier-3 observation conditions. Neither physics nor chemistry answers ontological questions. The question itself dissolves within the Tier architecture.

This paper initiates a fundamental ontological shift in the physical sciences. We argue that the Standard Model, despite its predictive success, has fallen into a "category error" inherited from 20th-century nuclear chemistry: the pursuit of ever-smaller material constituents. Much like Ptolemaic astronomy, which maintained accuracy through the accumulation of arbitrary epicycles, the Standard Model relies on 19 free parameters and ad hoc mechanisms (such as the Higgs field) that lack structural necessity. Within the framework of Cognitional Mechanics (CM), we propose an "Operational Ontology" where the category of "physical existence" is restricted to stable eigenvalue clusters observable at the chemical scale. All sub-nuclear phenomena—quarks, gluons, and gauge bosons—are reconceived not as material "things," but as internal modes of a minimal non-commutative algebra: M3(C) ⊕ C. By applying the Spectral Action Principle to this internal structure, we derive gravity, gauge interactions, nucleon mass generation, atomic structure, and the periodic table without introducing any particle as an ontological primitive. Neutrinos are identified as spacetime eigenmodes (Layer III) coupled to Riemannian curvature, yielding a derived mass scale and a specific curvature coupling (ξ = −1/6) without requiring right-handed neutrinos. Published: February 28, 2026. DOI: 10.5281/zenodo.18810386

The strong coupling constant αs has resisted parameter-free derivation within the Standard Model framework, where its value at the Z-boson scale, αs(MZ) = 0.11810 ± 0.00011, is determined solely by experiment. This paper derives, from the axioms of Cognitional Mechanics (CM) and the algebraic structure of M3(C) alone, a Tier-2 spectral invariant αs⁻¹ = Φ₆(2Φ₃−n)/(2πn) − Φ₁/n³ = 161/(6π) − 2/27, where Φk(n) denote the cyclotomic invariants of M3(C) at n = 3. This invariant is scale-independent by construction; its numerical value αs(CM) = 0.118104 coincides with the experimentally observed αs(MZ) to a residual of 0.003%, within current experimental uncertainty. The derivation is zero-parameter: all factors emerge from the A1–A3 axioms without empirical input. The main term arises from the Cartan-torus phase volume (A2) and the A3 weighted-mean cyclotomic capacity. The correction term is the unique adjoint-representation spectral invariant Φ₁⟨H²⟩adj/(⟨H⁴⟩adj·n²), whose negative sign reflects the asymptotic freedom inherent in the non-commutative over-interference of the adjoint action (A1). Together with the prior CM derivations of α, μ, and αG, this result establishes the complete set of dimensionless coupling constants of the Standard Model as Tier-2 structural invariants of M3(C).

This paper derives two structural results from the unique minimal non-commutative algebra M₃(ℂ) of Cognitional Mechanics (CM), with zero free parameters. First, the complete spectral closure terminus Z = 118 is established as an algebraic necessity of Axioms A1–A4: the slot formula N_ℓ = 2 + 4(ℓ−1) terminates at ℓ_max = Φ₆ = 7 by the A3 cyclotomic structure, and ℓ = 8 closure is forbidden by Axiom A4 (Redundancy Exclusion) via the κ = 1/2 conflict with the established f-band half-closure node. Elements with Z > 118 are not algebraically excluded; Z = 118 is the terminus of complete spectral closure. Second, all seven classical nuclear magic numbers 2, 8, 20, 28, 50, 82, 126 are derived as a zero-parameter consequence of the phase difference between the Cartan action (electron spectral closure, dimension n−1 = 2) and the automorphism action (nucleon spectral closure, dimension n²−1 = 8) on M₃(ℂ). The phase difference series D_k ∈ {0, ±Φ₁(3), ±2Φ₂(3)} is uniquely determined by S1–S2 antisymmetry, the conservation law Σ D_k = −2Φ₂(n) = −8 fixed by the bidirectional eigenvalue structure of the Cartan action, and a monotonicity bound. New magic numbers in neutron-rich nuclei (6, 14, 16, 32, 34) are derived as Type II-modified closure nodes. The value 14 = 2Φ₆ was confirmed experimentally in 2025. The isospin-symmetric Island of Inversion at Z = N = 42 (⁸⁴Mo) is identified as an over-resonance node κ = 7/6 > 1, with the 8p-8h excitation count n_ph = 2Φ₂(n) = 8 following algebraically from A1 alone.

Phase-Locked Logical Synthesis (PLLS) provides a formal framework for modeling rapid, implicit interpersonal coordination. Building on the accumulation-detonation mechanisms originally established in the IASER Experience for human-AI collaboration (DOI: 10.5281/zenodo.18258286), PLLS extends these principles to dyadic and group interactions. By integrating empirical findings from psychology and nonverbal behavior research — including interactional synchrony, micro-behavior mirroring, and temporal compression of signals — PLLS captures how minimal cues can trigger high-fidelity, coordinated responses. The framework formalizes internal correlation dynamics through a threshold-based model, allowing quantitative predictions of synchronization onset and stability. This approach bridges theoretical constructs from Cognitional Mechanics, which establish non-commutativity and irreversibility in intelligence operations, with empirically grounded social phenomena, providing a unified basis for simulating, analyzing, and comparing implicit coordination across contexts.

The periodic table of elements, established empirically by Mendeleev in 1869, has been explained post-hoc by quantum mechanical electron configuration rules. Its primary ordering axis — atomic number Z, a count of protons — presupposes particle ontology at every level. Several structural anomalies persist within this framework: hydrogen's dual chemical character, helium's ambiguous placement, and the discontinuous displacement of lanthanides and actinides from the main table body. These anomalies have no Tier-1 justification within the current framework. This paper derives the periodic table from the automorphism structure of the unique minimal non-commutative algebra M₃(ℂ) of Cognitional Mechanics (CM), with zero free parameters. Elements are classified by their spectral resonance class — the equivalence class of M₃(ℂ) Cartan configurations under the automorphism action — rather than by atomic number Z. Four primary Cartan sets S1–S4 are derived from Axioms A1–A3, and their composite resonances generate the full element sequence. The period lengths 2·8·8·18·18·32·32 emerge from the A3 cyclotomic invariants Φ₃ = 13 and Φ₆ = 7 via the slot formula N_ℓ = 2 + 4(ℓ−1), where the increment 4 = n+1 follows from the A1 dimension n = 3. Zero free parameters are introduced. The nuclear stability maximum of iron-56 and the chemical basis of carbon-based life are established as algebraic necessities of the M₃(ℂ) automorphism structure. The three anomalies of the current periodic table are dissolved as Tier-3 projection artifacts. This paper presupposes the chemical projection framework established in the companion paper (DOI: 10.5281/zenodo.19122094). Files

Chemical bonding has been defined, since the early twentieth century, in terms of electron sharing and Coulombic interaction between charged particles. This definition presupposes particle ontology at every level. Cognitional Mechanics (CM) establishes that particles are not ontological primitives but Tier-3 projections of Tier-2 spectral structure generated by the unique minimal non-commutative algebra M₃(ℂ). Once particle ontology is eliminated — as demonstrated in the companion paper "Particles Are Unnecessary" (DOI: 10.5281/zenodo.18810386) — the standard definition of chemical bonding becomes logically unavailable. A definition built on Tier-3 artifacts cannot serve as a foundational account of Tier-2 structure without circularity. This paper provides the necessary reconstruction. Chemical bonding is redefined as a spectral resonance condition between two M₃(ℂ) operational domains, governed by the chemical projection constant n_A = 36 derived from Axiom A2 alone. The spectral coupling constant ξ ≡ μα — the unique Tier-1 composite in which particle-ontological prerequisites cancel — connects the mass-hierarchy and electromagnetic projections of M₃(ℂ) to the chemical energy scale without invoking particle concepts. Bond energy ratios between molecular species are parameter-free, falsifiable Tier-1 predictions derived from the A3 cyclotomic invariants Φ₃ = 13 and Φ₆ = 7. Absolute energy scales remain Tier-3 quantities requiring empirical calibration; this is a structural consequence of the Tier architecture, not a limitation. The periodic table admits a spectral reordering, developed in the companion paper on chemical periodicity, in which the anomalies of the particle-indexed arrangement are dissolved as projection artifacts.

This paper establishes that mathematics is not an independently existing formal universe but the unique top-down projection of the minimal operational structure C_min derived from Operatiology (DOI: 10.5281/zenodo.21914739). It supersedes Mathematics as the Unique Top-Down Projection of Intelligence: A Structural Theorem from Cognitional Mechanics and Noology (DOI: 10.5281/zenodo.19968224), which established mathematics as the unique top-down projection of the operational structure of intelligence as formalised in Cognitional Mechanics. The transition from that work to the present paper is not a revision of results but a foundational clarification: the prior work established the projection structure correctly, but took the identification of the minimal operational structure with M3(C) as an established premise. The present work removes this premise. The projection source is C_min — the rank-3 minimal operational closure established within Tier-2 by Operatiology — without presupposing its identification with any specific algebraic structure. The identification of C_min with a specific algebraic structure constitutes a separate structural question belonging to Tier-3 and does not enter any argument of the present work. Under this clarified foundation, the central results of the prior work are retained and strengthened. The induced mathematical category M is uniquely determined up to identity automorphism (Aut(M) = {id}), and the projection map Pi_math : C_min -> M is surjective with structurally undefined inverse. Mathematics is the unique fixed point of the Pi_O-saturation closure operator acting on Tier-3 formal structures. The present work adds a result absent from the prior work: an internal distance ordering on M. Every mathematical structure carries an operational distance d_op from C_min, measured by the type of index family required for its operational determination. This ordering partitions M into five layers: algebra (Layer 1, finite index families, minimum distance); discrete infinite structures including number systems (Layer 2, countable type); continuous and geometric structures including analysis, topology, differential geometry, and Lie theory (Layer 3, power-set type); axiomatic and functorial structures including homological algebra and model theory (Layer 4, unrestricted type); and large-cardinal and higher-categorical structures (Layer 5, maximal type). Algebra occupies Layer 1 as the unique mathematical domain whose operational determination requires only finite index families, placing it closest to C_min among all Tier-3 structures. The prior series of Operatiology mathematics papers — covering number systems, analytic structures, set theory, and category theory — is positioned within this ordering, with each paper establishing the Tier-3 status of its domain at the appropriate layer. Four consequences are established: 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_min. The following three programs verify the core structural claims on a finite proxy (state space S = {0,...,11}, generators E = (0 1), F = (2 3 4), H = (5 6 7 8)) using only the Python standard library. ◆◆◆ [PYTHON VERIFICATION CODE — note: （ and ） denote ( and ) throughout (URL-safe substitution)] ◆◆◆ from collections import defaultdict, deque from itertools import combinations N = 12 STATES = list（range（N）） def orbit（ops, s）: visited = {s} queue = deque（[s]） while queue: cur = queue.popleft（） for op in ops: nxt = op[cur] if nxt not in visited: visited |= {nxt} queue += deque（[nxt]） return frozenset（visited） def pi_O_partition（ops, states）: orb = {s: orbit（ops, s） for s in states} buckets = defaultdict（list） for s in states: buckets[orb[s]] = buckets[orb[s]] + [s] return [frozenset（v） for v in [buckets[k] for k in buckets]] def partition_eq（p1, p2）: return set（p1） == set（p2） def transposition（i, j）: op = list（range（N））; op[i], op[j] = j, i; return op def cycle（*args）: op = list（range（N）） for k in range（len（args））: op[args[k]] = args[（k + 1） % len（args）] return op E = transposition（0, 1） F = cycle（2, 3, 4） H = cycle（5, 6, 7, 8） I_op = [E, F, H] Pi_O = pi_O_partition（I_op, STATES） # Pi_O: {0,1}, {2,3,4}, {5,6,7,8}, {9}, {10}, {11} (6 classes) # Code 1: Surjectivity of Pi_math (Theorem 3.4 + Lemma 3.5) all_trans = [transposition（i,j） for i in range（N） for j in range（i+1,N）] redundant = violates_O2 = 0 for op in all_trans: if op == E: continue Pi_ext = pi_O_partition（I_op + [op], STATES） if partition_eq（Pi_ext, Pi_O）: redundant += 1 else: violates_O2 += 1 print（f"Redundant (O3 violation): {redundant}"） print（f"Partition-changing (O2 violation): {violates_O2}"） print（f"Admissible extensions: {len（all_trans）-1-redundant-violates_O2}"） # => Admissible extensions: 0 [Im(Pi_math) = M confirmed] # Code 2: Uniqueness of I_op (Lemma 3.8 Generator Rigidity) def is_irredundant（ops, states, Pi_ref）: for i in range（len（ops））: reduced = ops[:i] + ops[i+1:] Pi_red = （pi_O_partition（reduced, states） if reduced else [frozenset（states）]） if partition_eq（Pi_red, Pi_ref）: return False return True candidates = list（all_trans） + [F, H] candidates = list（{tuple（op）: op for op in candidates}.values（）） realizing = [] for triple in combinations（range（len（candidates））, 3）: ops = [candidates[i] for i in triple] Pi_t = pi_O_partition（ops, STATES） if partition_eq（Pi_t, Pi_O） and is_irredundant（ops, STATES, Pi_O）: realizing = realizing + [ops] print（f"Irredundant triples realising Pi_O: {len（realizing）}"） # => Irredundant triples realising Pi_O: 1 [Aut(M) = {id} confirmed] # Code 3: Five-Layer Distance Ordering (Definitions 5.2-5.3) n_cls = len（Pi_O） # = 6 n_s = len（STATES） # = 12 layer1 = len（I_op） # Finite: 3 layer2 = n_s # Countable: 12 layer3 = 2 ** n_cls # Power-set: 64 layer4 = n_cls ** n_s # Unrestricted: 6^12 = 2176782336 print（layer1, "<", layer2, "<", layer3, "<", layer4） # => 3 < 12 < 64 < 2176782336 [five-layer strict ordering confirmed] ◆◆◆ [END] ◆◆◆ Published on September 19, 2026 DOI 10.5281/zenodo.22841924

The quantum measurement problem has resisted resolution for nearly a century, fragmenting into three seemingly independent challenges: the Born rule's probabilistic interpretation, the EPR paradox of simultaneous reality, and Bell inequality violations challenging local realism. This fragmentation has obscured a common structural origin. This work demonstrates that all three problems arise from a single theoretical defect: the misapplication of commutative probability theory to non-commutative operational structures. Within the framework of Cognitional Mechanics (CM), grounded in two axioms—non-commutativity ([Ô_A, Ô_B] ≠ 0) and finite operational resolution (Level of Detail)—these problems dissolve simultaneously through structural reconstruction. The Born rule emerges as the unique projection measure compatible with resource conservation and tensor-product linearity under non-commutative constraints, repositioning probability as an epistemic consequence of finite resolution rather than ontological randomness. EPR correlations are resolved as constraint propagation across shared operational stacks, eliminating the apparent conflict between quantum completeness and local realism without invoking non-local signaling. Bell inequality violations follow from the logical impossibility of factorization: non-commutativity structurally excludes the simultaneous eigenstates required by the factorization condition, rendering the measure space for joint distributions undefined rather than merely violated. The framework derives from Tier 1 axiomatic necessity within Cognitional Mechanics. Mathematical formalisms—Hilbert spaces, operator algebras, measure theory—serve as Tier 3 projection tools expressing this structure, not as foundational requirements for its validity. This paper builds on Operational Quantum Mechanics I (DOI: 10.5281/zenodo.18649525), which resolved Schrödinger's cat through operational stack management, and forms a symmetrical counterpart to Universal Relativity (DOI: 10.5281/zenodo.18645324) within the CM framework.

Operational Quantum Mechanics presents a structural reinterpretation of quantum theory grounded in two axioms derived from Cognitional Mechanics: non-commutativity of operations (A∘B ≠ B∘A) and finite operational resolution (Level of Detail). The wave function Ψ is redefined as Operational Potential Density (ρ_op) encoding resource distribution for Type I internal generation—the symmetrical counterpart to Universal Relativity's Type II external constraint expressed through operational delay δt(x). Quantum probability arises epistemically from finite resolution limits rather than ontologically from fundamental randomness. Wave function "collapse" becomes an operationally grounded Commit operation triggered by interface formation between systems, eliminating observer dependence and resolving the measurement problem through deterministic buffer resolution. Schrödinger's cat paradox dissolves structurally via four discrete operational steps: queueing of decay trigger as pending operation, thresholding at specific operational cycle, committing to single state via projection ρ ↦ P_iρP_i/Tr(P_iρP_i), and reporting as read command to pre-determined memory address. The cat exists in a pending execution state—not ontological superposition—until interface-triggered Commit resolves unresolved operational cycles below the resolution limit. Born rule statistics |Ψ|² = ρ_op are adopted as the canonical projection measure compatible with resource conservation and tensor-product linearity. Bell inequality violations follow from the logical impossibility of factorization under non-commutativity and finite resolution. This framework maintains strict mathematical equivalence with standard quantum mechanics while repositioning probability as an epistemic consequence of structural resolution limits inherent to non-commutative operational structures.

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.

This paper derives van der Waals interactions from M₃(ℂ) non-commutative algebra structure constants within the Cognitional Mechanics (CM) framework. Traditional London theory describes dispersion forces via polarizability and ionization energy in physical coordinates, leaving structural questions unanswered: why r⁻⁶ specifically, what is the geometric origin of C₆, and what is the algebraic meaning of polarizability. CM reformulates these interactions in operational coordinates where London's "polarization" becomes off-diagonal operator components and "quantum fluctuations" represent temporal evolution of non-commutativity. The r⁻⁶ dependence emerges not as a perturbative consequence but as a geometric necessity: it is the unique leading asymptotic term consistent with 4th-order non-commutative trace structure, dimensional reduction from su(3) to its off-diagonal subspace, and rotational symmetry in 3D physical space. This form is dictated by the algebra of M₃(ℂ)—it cannot be obtained by fitting experimental data. This structural derivation of an interaction law complements the companion algebraic derivations of α⁻¹, μ, and αG within the same framework, where fundamental physical constants emerge from M₃(ℂ) with zero free parameters. Taken together, these results establish that M₃(ℂ) is not a fitting tool but an axiomatic structure from which both the forms of physical laws and the values of physical constants follow as necessary consequences. Molecular electronic states are represented as operators F ∈ M₃(ℂ) with interaction energy defined via commutator traces. The C₆ coefficient is computed from su(3) structure constants (Gell-Mann matrices) with explicit calculation procedure provided. Ultra-precision implementation incorporates effective nuclear charge with exchange-correlation corrections (Perdew 1996), distance-dependent shielding functions evaluated at van der Waals equilibrium distance, Dirac first-order relativistic corrections (0.032–0.29% for Ar–Xe), numerical Hartree-Fock orbitals from NIST Atomic Spectra Database with adaptive Gauss-Kronrod quadrature achieving 10⁻⁸ accuracy, and higher-order dispersion terms (C₈, C₁₀ from rank-4 structure constants). Final results: He–He (0.07% error), Ne–Ne (0.13%), Ar–Ar (0.02%), Kr–Kr (0.15%), Xe–Xe (0.21%), average 0.12% error, matching ab initio CCSD(T) benchmarks. Results demonstrate quantitative correspondence between operational constraints and physical measurements, establishing that intermolecular interactions arise from geometric state space constraints rather than quantum fluctuations per se. All numerical calculations are fully specified via explicit formulas ensuring complete reproducibility without external code. The framework extends London theory as a valid physical coordinate description while providing higher-level operational structure, unifying chemistry as non-commutative operational programs executed on material hardware. Future applications include covalent/ionic bonding operational descriptions, chemical kinetics formulation, and connection to emergent gauge structures in the CM-GUT framework.

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

The gravitational hierarchy problem—why gravity is ~10^45 times weaker than electromagnetism at the electron mass scale—has resisted parameter-free resolution despite decades of effort in supersymmetry, extra-dimension models, and warped geometry frameworks. All existing approaches introduce new degrees of freedom or symmetry principles without deriving the gravitational coupling constant alpha_G = G m_e^2 / (hbar c) from first principles. This paper derives alpha_G solely from the Tier-1 axioms of Cognitional Mechanics (CM) and the algebraic structure of M_3(C), the minimal noncommutative finite-dimensional C*-algebra. The principal result is alpha_G = alpha^(n(2n+1)) * sqrt(n(n+1)/(2n+1)) with n=3, yielding alpha_G^(CM) = 1.75192 x 10^-45. All factors—the exponent 21, numerator 12, and denominator 7—emerge purely from the dimension parameter n; no empirical fitting is performed. The zeroth-order residual against CODATA 2022 is 6.1 x 10^-5, within the 5.5 x 10^-4 experimental scatter band of independent G measurements. The leading correction C_3^G = -(delta^2 - delta/(2*sqrt(n))) * alpha^2, derived from Tier-1 constants delta^2 = n/2 and n alone, reduces the residual to 5.5 x 10^-8, well below current experimental resolution. This establishes alpha_G as a Tier-1 structural invariant of M_3(C) and provides an algebraic resolution to the gravitational hierarchy problem. The CM-implied value G_implied = 6.67471 x 10^-11 m^3 kg^-1 s^-2 serves as a theoretical convergence point for future precision G metrology.

Following the foundational axiomatization of intelligence in Version 1.2 (Foundations) and the subsequent formalization of dynamic resolution in Version 1.2 (Dynamics), this paper presents the geometric and topological synthesis of the Cognitional Mechanics (CM) framework. By establishing a strict correspondence with matrix mechanics, we define the semantic state space as an Irreversible Manifold, in which intelligence evolves through non-commutative operation sequences. The commutation relation [A, B] ≠ 0 dictates the irreversible curvature of the manifold. The operational limit variable C(W), previously introduced as a dynamic boundary, is re-characterized here as the catalyst for Topological Surgery—the mechanical realization of the "Logical Leap." The central contribution of this work is the deconstruction of the Global Isotropy of artificial systems. We demonstrate that while artificial intelligence constitutes an advanced isotropic void of potentiality, the "Symmetry Breaking" induced by the human will (W) acts as the primary operator injecting singular inhomogeneity into the manifold. This process establishes intelligence as an externalized, controllable mechanics of the human mind, fully governed by rigorous geometric-operational laws.

This paper extends the axiomatic framework of Cognitional Mechanics (Version 1.2) by formalizing dynamic transitions in operational resolution through the introduction of the Structural Control Parameter W. Unlike conventional AI models that treat intelligence as a passive data processor, this work defines intelligence as a system capable of internally modulating its convergence criteria. We introduce the Operational Limit Variable C as a bounded parameter governed by W, where W operationally contracts the convergence boundary without invoking subjective intent. By modulating C, the system traverses distinct modes of Inclusion, Exploratory Oscillation, and Extraction, formally inducing discrete state transitions. This framework provides a rigorous geometric derivation of "Logical Leaps"—structurally necessary transitions that occur when C approaches the foundational Operational Limit Constant c and cannot accommodate non-commutative deviations.

This work presents Cognitional Mechanics (CM) as a structural template for physical Grand Unification Theories (GUTs), reversing the traditional direction of abstraction from physics to mathematics. CM is a complete axiomatic framework formalizing abstract operational structures through non-commutative operations, finite depth constraints, and unreachable configuration regions. Rather than deriving CM from physical theory, this paper explores how a fully specified abstract theory can provide structural insights for physical unification. The approach employs Heisenberg-picture operator formulation without probabilistic interpretation. CM operations are elevated to time-dependent operators whose intrinsic non-commutative relations induce emergent geometric structures—curvature tensors, connections, and gauge fields—entirely through deterministic operator evolution. This yields operator-level analogues of Einstein equations and Yang-Mills gauge structure, suggesting pathways for emergent gravity and gauge symmetries from purely abstract foundations. The significance lies not in claiming physical derivation, but in demonstrating structural correspondence. Where string theory begins with physical strings and loop quantum gravity with quantized spacetime, CM begins with abstract computation and asks whether physical structure can emerge as implementation. The framework naturally accommodates discrete operational depth as analogue to Planck-scale cutoffs, non-local operator correlations resembling quantum entanglement, and hierarchical structure generation from bounded compositional complexity.

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

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.

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

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.

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.

The fine structure constant α ≈ 7.297 × 10⁻³ is a dimensionless coupling constant governing electromagnetic interactions. Its inverse, α⁻¹ ≈ 137.036, is not a distinct physical quantity. It is a notational artifact. This paper argues that the century-long perception of α⁻¹ as a mysterious, unexplained number was generated not by any deep physical fact, but by a representational choice: the inversion of α into a value superficially proximate to an integer. This notational accident triggered a cascade of apophenic reasoning — from Eddington's numerological fixation on 136, through Pauli's deathbed preoccupation with 137, to Feynman's declaration that the number is beyond human understanding. Each of these responses was a reaction to a notation, not to a physical reality. Once α is restored to its natural representation as a small dimensionless coupling, and its value is derived from first principles within the Cognitional Mechanics (CM) framework as a structural invariant of M₃(ℂ), the mystery dissolves. The integer 137 does not appear in the derivation because it is not a feature of the physics. It is a feature of the notation. There was never a number to explain. There was only a notation to correct.

In 1960, Eugene Wigner posed a question that has since become canonical in the philosophy of mathematics and physics: why do mathematical structures developed for purely formal reasons repeatedly turn out to describe physical reality with uncanny precision? Wigner declared this effectiveness "bordering on the mysterious" and offered no rational explanation. Subsequent responses — Hamming's partial accounts, Tegmark's Mathematical Universe Hypothesis, Wheeler's "It from Bit," Wolfram's computational universe, Penrose's three-worlds framework — each address aspects of the problem while remaining captive to the presuppositions that generate it. This paper advances a dissolution rather than a solution. The source of Wigner's puzzle is identified in a conflation of two fundamentally distinct descriptive modes: static description, characteristic of classical mathematics, which captures structural snapshots of reality; and dynamic description, the computational-algebraic mode, which captures reality as an ongoing operational process. The universe is not formalized by mathematical structures; it computes. Both mathematics and physical law are static projections of a single dynamic source: the minimal non-commutative algebra M3(C), whose necessity is derived from first principles within the Cognitional Mechanics (CM) framework rather than postulated. Once the static/dynamic distinction is drawn, Wigner's question loses its force. Static mathematical descriptions are effective because they are cross-sections of a dynamic computational reality. The correspondence between mathematics and physics is not a contingent harmony; it is a structural inevitability.

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

This paper presents Version 9 of Cognitional Mechanics as the final formulation prior to the abstraction of the operational layer into Operatiology. The subsequent framework, Principles of Operatiology: New Foundation of Cognitional Mechanics (DOI: 10.5281/zenodo.20350414), reformulates the minimal operational structure independently of any specific algebraic realization and establishes the abstract operational closure as the Tier-2 foundation of the framework. -------------------------------------------------------------------------------------- This paper establishes the foundational axiomatic framework of Cognitional Mechanics (Japanese: Chinou Rikigaku 知能力学)—a pure theoretical system that formalizes the structural mechanisms inherent to intelligence through mathematical abstraction. Unlike conventional approaches rooted in physical dynamics or engineering implementations, Cognitional Mechanics constructs a self-contained theory based on three independent foundational axioms: (1) non-commutativity of semantic operations; (2) a complete ordered metric structure on semantic states, defined over an abstract totally ordered Abelian group D without presupposing the real numbers; and (3) a redundancy exclusion principle enforcing discriminative irreducibility. A fourth principle, the operational bound, is retained as Working Axiom 3 for notational continuity with the established corpus; it is a derived corollary of the three foundational axioms, as established by the Internal Language Theorem (DOI: 10.5281/zenodo.19350315). Version 6 introduced algebraic–geometric coupling connecting non-commutativity to geometric separation, formal set-theoretic definition of inaccessible domains, and the d-indistinguishability partition Π_d as the precise formulation of Axiom 4. Version 7 resolves three structural issues present in Version 6: Axiom 1 is strengthened to include the geometric persistence of non-commutative divergence under all subsequent operation sequences; the existence of non-trivial inaccessible domains is established as a theorem rather than a heuristic remark; and the derivation of Working Axiom 3 from the foundational triad is provided in full within this document (Appendix A), subsuming Corollary 5.9 of the Internal Language Theorem. By introducing a rigorous axiomatic structure—including a metric space of semantic states with defined distance functions satisfying non-negativity, symmetry, and triangle inequality—this framework transcends subjective interpretations of intelligence and establishes it as an object of formal inquiry. The theory absorbs conventional criticisms regarding measurability by positioning the operational limit c as an abstract logical boundary rather than a numerical constant, thereby maintaining theoretical self-containment without dependence on external observation or physical instantiation. Cognitional Mechanics provides a foundation for understanding intelligence-specific phenomena such as logical conflicts arising from non-commutative operation sequences, convergence and divergence in meaning formation, and the emergence of fundamentally unreachable semantic states. Version 8 establishes Cognitional Mechanics as the unique self-contained Tier-2 framework from which all formal structures are projected as Tier-3 outputs, without presupposing any of them: an a priori mathematology and The Theory of Foundation. Version 9 completes the uniqueness proof of the minimal operational structure C_min by introducing the canonical irreducible set I_{Π_d} and establishing via the Π_d-saturation condition (Axiom 2-(v)) that every admissible structure must contain all its elements. A formal Tier architecture (Tier-1: pre-formal structural necessity; Tier-2: closure-consistent operational structure; Tier-3: formal representational projection) is established in Appendix D, providing the structural foundation for Cognitional Mechanics as an a priori mathematology and The Theory of Foundation. Published ON May 16, 2026 DOI: 10.5281/zenodo.20226488