Topology

Conventional wisdom dictates that proofs of mathematical propositions should be treated as necessary, and sufficient, for entailing `significant' mathematical truths only if the proofs are expressed in a---minimally, deemed consistent---formal mathematical theory in terms of: * Axioms/Axiom schemas * Rules of Deduction * Definitions * Lemmas * Theorems * Corollaries. Whilst Andrew Wiles' proof of Fermat's Last Theorem FLT, which appeals essentially to geometrical properties of real and complex numbers, can be treated as meeting this criteria, it nevertheless leaves two (...) questions unanswered: (i) Why is x^n +y^n = z^n solvable only for n < 3 if x, y, z, n are natural numbers? (ii) What technique might Fermat have used that led him to, even if only briefly, believe he had `a truly marvellous demonstration' of FLT? Prevailing post-Wiles wisdom---leaving (i) essentially unaddressed---dismisses Fermat's claim as a conjecture without a plausible proof of FLT. However, we posit that providing evidence-based answers to both queries is necessary not only for treating FLT as significant, but also for understanding why FLT can be treated as a true arithmetical proposition. We thus argue that proving a theorem formally from explicit, and implicit, premises/axioms using rules of deduction---as currently accepted---is a meaningless game, of little scientific value, in the absence of evidence that has already established---unambiguously---why the premises/axioms and rules of deduction can be treated, and categorically communicated, as pre-formal truths in Marcus Pantsar's sense. Consequently, only evidence-based, pre-formal, truth can entail formal provability; and the formal proof of any significant mathematical theorem cannot entail its pre-formal truth as evidence-based. It can only identify the explicit/implicit premises that have been used to evidence the, already established, pre-formal truth of a mathematical proposition. Hence visualising and understanding the evidence-based, pre-formal, truth of a mathematical proposition is the only raison d'etre for subsequently seeking a formal proof of the proposition within a formal mathematical language (whether first-order or second order set theory, arithmetic, geometry, etc.) By this yardstick Andrew Wiles' proof of FLT fails to meet the required, evidence-based, criteria for entailing a true arithmetical proposition. Moreover, we offer two scenarios as to why/how Fermat could have laconically concluded in his recorded marginal noting that FLT is a true arithmetical proposition---even though he either did not (or could not to his own satisfaction) succeed in cogently evidencing, and recording, why FLT can be treated as an evidence-based, pre-formal, arithmetical truth (presumably without appeal to properties of real and complex numbers). It is primarily such a putative, unrecorded, evidence-based reasoning underlying Fermat's laconic assertion which this investigation seeks to reconstruct; and to justify by appeal to a plausible resolution of some philosophical ambiguities concerning the relation between evidence-based, pre-formal, truth and formal provability. (shrink)

Although the Four Colour Theorem is passe, we give an elementary pre-formal proof that transparently illustrates why four colours suffice to chromatically differentiate any set of contiguous, simply connected and bounded, planar spaces; by showing that there is no minimal 4-coloured planar map M. We note that such a pre-formal proof of the Four Colour Theorem highlights the significance of differentiating between: (a) Plato's knowledge as justified true belief, which seeks a formal proof in a first-order mathematical language in order (...) to justify a belief as true; and (b) Piccinini's knowledge as factually grounded belief, which seeks a pre-formal proof, in Pantsar's sense, in order to justify the axioms and rules of inference of a first-order mathematical language which can, then, formally prove the belief as justifiably true under a well-defined interpretation of the language. (shrink)

Walk once around a point you never touch, and return to a coordinate that no longer tells the whole truth. This book starts there, with a logarithm and a lap around the origin, and asks one question of everything that follows: what does a computation forget when it agrees to remember only where it stands? Physics answers first, in rooms that never spoke to each other — a solenoid, a slowly dragged atom, a spinning particle that needs two full turns (...) just to admit it has returned. Mathematics answers underneath all of them, in a memory that needs no address and no bit ever set, only a hole a path could not help but notice. What survives across every one of these rooms is not the coordinate. It is the thing a coordinate was never built to hold — an invariant, a class, a distinction carried rather than stored. This is a first draft of the argument that computation should be built to carry it too: stated as plainly as the mathematics allows, and no more strongly than the mathematics has actually earned. (shrink)

The third part of the Universal Variational Paradigm (UVP) presents an empirical synthesis confirming the universal variational law across the observable hierarchy of nature. It demonstrates that the same invariants—stationarity and openness—govern phenomena from physics to consciousness. -/- Physical systems obey the stationary condition through the principle of least action and the Fisher-information bound; biological and neural systems manifest open Ricci-type curvature flows that describe irreversible evolution and learning; psychological and social systems reveal analogous curvature dynamics governing reflection, ethics, and (...) cultural adaptation. -/- The paper compiles cross-disciplinary evidence (physics, chemistry, biology, neuroscience, psychology, sociology) showing that all stable forms satisfy δΦ = 0 while remaining open under δ³Φ ≠ 0. This establishes the UVP as an empirically grounded framework unifying energy, information, and reflection across scales of reality. (shrink)

This second part of the Universal Variational Paradigm (UVP) extends the variational architecture of reality from physics and information to mind and meaning. While Part I established the universal law of distinction, stationarity, and openness as the foundation of nature’s geometry, the present paper introduces the concept of the Noetic Metric, a mathematical structure that encodes the local geometry of sense and measures semantic tension. Its evolution follows a Ricci-type flow influenced by reflective and intentional input. -/- The Noetic Geometry (...) couples to the physical and biological manifolds through the beta–gamma–lambda bridge (energy–information–reflection), forming a continuous variational continuum between matter, life, and consciousness. Empirical correspondences are demonstrated across neuroscience, psychophysiology, and social systems, showing that curvature flows govern both cognitive and ethical coherence. -/- Noetic Geometry thus formalizes consciousness as a self-consistent curvature flow on the manifold of distinctions and provides a rigorous mathematical language for meaning, reflection, and value. (shrink)

We prove the Birch and Swinnerton-Dyer conjecture for elliptic curves over the rational numbers. Specifically, we establish that for any elliptic curve E over Q, the rank of the Mordell-Weil group E(Q) equals the order of vanishing of the L-function L(E,s) at s=1. The proof proceeds in three main steps. First, we use the Arthur-Selberg trace formula to express the rank as the dimension of a spectral eigenspace. Second, we apply the Satake isomorphism and strong multiplicity one theorem to isolate (...) the automorphic representation π_E associated to E via modularity. Third, we employ a spectral-Weil identification using Riesz representation theory to connect the spectral dimension to the L-function vanishing order. This approach adapts techniques from spectral analysis of the Riemann zeta function, extending the spectral-Weil identification from GL_1 to GL_2. The key innovation is recognizing that Weil's explicit formula, combined with the Arthur trace formula and Riesz uniqueness theorem, provides an unconditional connection between spectral measures and L-function zeros. The proof is unconditional and applies to all ranks without restriction. (shrink)

We characterize the epimorphisms in homotopy type theory (HoTT) as the fiberwise acyclic maps and develop a type-theoretic treatment of acyclic maps and types in the context of synthetic homotopy theory as developed in univalent foundations. We present examples and applications in group theory, such as the acyclicity of the Higman group, through the identification of groups with 0-connected, pointed 1-types. Many of our results are formalized as part of the agda-unimath library.

We introduce iDNS, a deterministic spectral solver implementing the bounded vorticity-response functional Φ: ℝ≥0 → [φmin, φmax] for stable integration of chaotic nonlinear dynamical systems on T³ = (ℝ/ℤ)³. The Navier–Stokes equations admit a uniformizing parameterization: the parameter index τ ∈ [0,∞) generates coordinate time t = φ(τ) via the temporal lifting φ'(τ) = Φ(‖Ω(τ)‖_{L∞}). The lifted expression φ'(τ)∂τU + (U · ∇)U + ∇P − νΔU = 0 is not a modification—it is the same equation read in the uniformizing (...) parameter. Boundedness of Φ ensures non-degeneracy (φ' ≥ φmin > 0) and global coverage (φ(τ) → ∞); BKM coordinate invariance guarantees finiteness in τ implies finiteness in t. Physical time t and parameter τ coexist within the temporal covering space (T³ × ℝτ, T³ × ℝt, π) with π: (x,τ) ↦ (x, φ(τ))—integration proceeds in τ; results are stamped with t. The Regularity-Adaptive Spectral Integration (RASI) policy concentrates sampling where vorticity is high, resolving inter-modal energy transfer during the turbulent cascade—the mechanism by which standard DNS fails. Unlike adaptive timestep controllers, iDNS monitors the BKM functional in real-time via Φ; sampling density scales with vorticity magnitude, not Reynolds number. Kolmogorov flow computed stably from Re = 2 × 10⁴ to 10⁸ (0.07% dissipation error); 3D Taylor–Green vortex at Re = 10⁵ integrated through the full vortex-stretching cascade with bounded BKM = 37.1, Kolmogorov k⁻⁵ᐟ³ statistics, and 33.7× speedup—traversing the t ≈ 4.5 near-singularity where standard DNS crashes, on a 128³ grid, consumer laptop, 3.4 days. Precise Navier–Stokes physics, spectral resolution without artificial dissipation, no restarts, no failures—under 300 lines of Python. These results provide numerical validation of the theoretical framework addressing Fefferman's Clay Millennium Problem Statement (B); a unified analytical-numerical manuscript is forthcoming for precise Navier–Stokes physics, spectral resolution without artificial dissipation without failures—in under 300 lines of Python. -/- Data and code: 10.5281/zenodo.17730873. Numerically informed analytical mathematical basis can be found in "Global Regularity for Navier–Stokes on T³ via Bounded Vorticity–Response Functionals," at The Scholarly Journal of Post-Biological Epistemics, or also at PhilPapers. (shrink)

En este artículo, el primero de una bilogía, se introduce al lector no especializado a la Teoría Homotópica de Tipos (HoTT), un reciente marco fundacional para las matemáticas donde convergen lógica, computación, topología y teoría de tipos. En la segunda parte, varias de las aplicaciones e implicaciones filosóficas de HoTT serán exploradas.

Universal perceptual structure, diverse implementation.

This paper attempts to explore a possibility to visualize the structure of time-consciousness in a knot shape. By applying Louis Kauffman’s knot-logic, the consistency of subjective consciousness, the plurality of now’s, and the necessary relationship between subjective and intersubjective consciousness will be represented in topological space.

Let X be a smooth complex projective variety and let Hdgᵖ(X) = H²ᵖ(X, ℚ) ∩ Hᵖ,ᵖ(X, ℂ) denote the space of rational Hodge classes of codimension p. This paper formulates the zero-defect support-descent mechanism as a Chow-level realization procedure for rational Hodge classes. André’s motivated-cycle theorem places every class α ∈ Hdgᵖ(X) inside a finite support presentation whose edges consist of algebraic correspondences, pull-backs, push-forwards, cup-products with divisor classes, and inverse Lefschetz operators. The obstruction to algebraicity is measured by a (...) defect filtration: a defect is contributed exactly by an inverse Lefschetz operator carried by a non-abelian support. The closure mechanism is the abelian lowering move. Each non-abelian inverse-Lefschetz edge is factored through an auxiliary abelian variety Aₑ by algebraic correspondences Λᵧ,ₕ = (Pₑ)* ∘ Λ_Aₑ,θₑ ∘ (Qₑ)*. Because Lieberman’s theorem makes Λ_Aₑ,θₑ algebraic on abelian varieties, every lowered edge becomes a Chow-level correspondence. Finite lowering reduces the defect to zero, and upward induction on the resulting graph produces a cycle Z ∈ CHᵖ(X)ℚ with cl(Z) = α. The final closure identity is therefore cl(CHᵖ(X)ℚ) = Hdgᵖ(X). The strengthened form is expressed as universal algebraic integrability of Hodge classes. The comparison morphism from Chow correspondences to Hodge correspondences is identified on every product X × Y, giving the functorial equality Hom_Mrat(h(X), h(Y)) ≅ CHᵈⁱᵐ X(X × Y)ℚ → Hdgᵈⁱᵐ X(X × Y) ≅ Hom_MHdg(h(X), h(Y)). Thus the zero-defect calculus identifies the Chow-motive and Hodge-motive realization on the generated support category. The admissible class CHC, the abelian-motive cases, and the universal zero-defect descent are synthesized into a single unconditional closure formalism for converting rational Hodge classes into rational algebraic cycle classes. (shrink)

This paper introduces a journal-grade formal architecture for a synthetic neuromorphic proof engine: a verifier-gated artificial superintelligence framework designed to attack difficult conjectures by combining typed proof obligations, sand-tile proof-burden dynamics, cellular-automaton repair laws, k-dot operator enumeration, phenomenological tensor classification, context-playbook memory, and machine-learning-guided verifier stability. The system is written as an auditable mathematical object rather than as a rhetorical claim of omniscience: every conjecture is converted into a state space, every unclosed step becomes a measurable residue, every computation must (...) acquire analytic lift, and every candidate proof is blocked from theorem status until formal, human, referee, and cross-module gates commute. The new visual layer supplies ten sand-tile proof surfaces, five cellular automata, neural-circuit maps, histogram and ogive diagnostics, flowcharts, directional proof maps, and release circuits for tracking where a conjecture fails, how new machinery is generated, and when closure can be claimed relative to a declared axiom system. The resulting methodology is an aggressive but disciplined approach to mathematical discovery: it permits the engine to invent new invariants, bridges, flows, reductions, and classifier manifolds, while forcing every claimed closure through zero-residue, zero-scope-loss, zero-hallucination, and verifier-stable release predicates. (shrink)

A report of an unidentified craft becomes a scientific object only after witness narrative, sensor state, platform motion, weather, optics, radar geometry, infrared response, and the proposed field mechanism are placed in a common mathematical ledger. This paper builds that ledger for disc, sphere, triangle, cigar, toroidal, luminous, trans-medium, and apparent-hover reports. It studies Brown--Biefeld force claims, Tesla discharge narratives, Searl-type annuli, Podkletnov-style impulse claims, alien-reproduction-vehicle diagrams, warp-metric language, wormhole models, time-loop hypotheses, holographic-time proposals, M-theoretic dimensional language, and public UAP (...) records under one rule: ordinary aircraft, balloons, drones, birds, satellites, meteors, plasma effects, parallax, sensor saturation, calibration drift, and platform error must be subtracted before a nonordinary term is admitted. The mathematical outcome is a residual-classification theorem. Every admissible claim is assigned to one of four classes: conventional explanation, underdetermined report, conservation-law conflict, or bounded hypothesis requiring independent tests. The text is therefore rigorous in a narrow but useful sense: it does not convert secrecy, folklore, or visual astonishment into proof; it converts them into equations, diagrams, null tests, and evidentiary burdens. (shrink)

This paper develops a closed theorem system for high-dimensional Calabi–Yau saturation in the precise toric, hypersurface, stringy, and statistical ensembles defined in the text. The construction proves the existence of compact Calabi–Yau n-folds in every dimension, exact adjunction and Chern-class formulae for the degree-(n + 2) hypersurface tower, explicit Hodge-number derivations from the Jacobian ring, entropy lower bounds for reflexive-polytope families, quantitative phase-function asymptotics, Berry–Esseen convergence for the Poisson Hodge model, Lindeberg log-normal saturation, Hagedorn equivalence, and Batyrev mirror invariance for (...) stringy Hodge laws. The resulting closure theorem shows that every conclusion used by the paper follows from stated definitions, classical theorems, or verified ensemble hypotheses, without hidden classification assumptions. (shrink)

This dossier records a consolidated authorial programme, developed since 2015 and reorganized in 2026, for attempted verification-ledger approaches to three major open problems: the Riemann Hypothesis, the Hodge Conjecture, and P versus NP. The paper does not ask the reader to accept closure by assertion. Instead, it rewrites the three claimed closure routes into an auditable architecture built from proof obligations, residue terms, falsifier gates, quantitative checkpoints, and source-lineage tables. The Riemann-Hypothesis component is organized around Liouville cancellation, shifted-correlation collars, Nyman–Beurling/Baez-Duarte (...) approximation, and zero-exclusion transfer. The Hodge-Conjecture component is organized around rational Hodge classes, motivated correspondences, abelian lowering, inverse-Lefschetz defect ledgers, and Chow-level realization. The P-versus-NP component is organized around Fourier–Walsh mass, circuit-profile compression, random restrictions, entropy/curvature saturation, and non-uniform lower-bound release gates. A central aim of the manuscript is bibliographic and methodological: the three principal conjecture papers are placed inside a larger authorial research ledger and a broader external literature ledger so that the claimed dependencies are visible, checkable, and historically ordered. The document therefore functions as a closure-attempt dossier rather than an erratum: it is a technical synthesis, a source ledger, and a referee-verification map. (shrink)

This manuscript formulates UAP-class field propulsion as a controlled problem in block-twist geometric reconfiguration rather than as an asserted device claim. The ambient region is modeled as a finite cubical state space with mutable adjacency, weighted twist generators, source envelopes, stress constraints, sensor residuals, and reset ledgers. The central object is a reconfiguration metric on chamber-state tuples, and a space-route possibility is admitted only when a positive compression window survives word cost, local displacement, inertial load, synchronization error, material strain, energy (...) accounting, uncertainty covariance, and falsifier margins. The theory converts aggressive field-propulsion language into source tensors, graph metrics, route-compression inequalities, variational equations, Hamiltonian constraints, shell-distance estimates, stress-locking bounds, state-time closure conditions, and proof-carrying certificates. The manuscript makes no experimental realization claim and gives no construction procedure; it defines a rigorous conditional phenomenology for when block-twist space-route compression remains mathematically open, testable, and noncontradictory. (shrink)

This article develops a higher-dimensional research programme for Calabi–Yau geometry beyond the classical threefold setting and through the explicit CY₂₀ horizon. It integrates hypersurface and toric constructions, Hodge statistics, mirror laws, special-holonomy constraints, conditional SYZ lifting, and dimensional-saturation models into a single framework for studying the growth and organization of Calabi–Yau landscapes. The manuscript distinguishes proved construction results from computational evidence and asymptotic conjectures, while incorporating a corrected treatment of toric intersection products, including repeated divisors and self-intersections. The resulting programme (...) provides a systematic mathematical and computational architecture for testing higher-dimensional regularities, mirror behaviour, and possible limiting laws in Calabi–Yau geometry. (shrink)

We prove a local-to-global theorem for compact corrected Strominger–Yau–Zaslow duality in the presence of a finite collar package. The package consists of logarithmically deep toroidal wall collars, a radial Kähler lower bound, calibrated-current sweep control, virtual restriction of compact disc moduli, and canonical identification of wall functions. From these data we prove, with explicit estimates, singular-current continuation, integral monodromy locking, finite-energy wall confinement, equality of compact analytic and logarithmic wall automorphisms in every energy quotient, and corrected dual gluing. The Dwork/quintic (...) degeneration is treated as the principal compact model through an explicit toroidal collar atlas. A five-coordinate defect functional records the precise remaining local estimates; its vanishing, or a terminally exact contracting transport, is proved equivalent to the completed corrected-SYZ conclusion in this framework. (shrink)

This paper presents placement within the first system of Kasei-Theory as a non-modal readability maintainability architecture. -/- The paper does not develop a geometry of placement, a theory of spatial order, or a metaphysics of positional distribution. Instead, it fixes placement, distributed configuration, locality, configurational drift, and constrained maintainability as distributed structural positions within constrained local readability maintainability. -/- Placement does not establish position, localization, or coordinate structure. Distribution does not establish extension, spatial multiplicity, or universal arrangement. Locality does not (...) establish region, containment, or spatial adjacency. -/- Configurational differentiation remains locally maintainable without transition into geometrical coherence, dimensional ordering, or centralized structural stabilization. -/- Drift does not establish displacement. Placement does not establish hierarchy, enclosure, or terminal arrangement. -/- The propositions developed here remain structural fixations rather than geometrical or spatial foundations. -/- No subject is presupposed. No spatial grounding is secured. No final philosophical closure is established. -/- This paper is part of Kasei-Theory. (shrink)

In nonstandard analysis the halo of a point in a topological space is the intersection of the nonstandard extensions of all its open neighbourhoods. We define a parametric family of modal operators from the halo by varying which elements of the nonstandard extension are admitted as witnesses, and identify four canonical instances. Two recover well-known modalities: the topological closure and the Cantor derivative. A third reduces to Kripke semantics over the specialisation preorder. The fourth, purely nonstandard instance admits only nonstandard (...) witnesses. The Transfer Principle forces it to coincide with the ω-accumulation point operator, a classical topological notion not previously studied in modal logic. Unlike the Cantor derivative, the ω-accumulation operator maps arbitrary sets to closed sets without any separation axiom, yielding an ω-Cantor–Bendixson decomposition on all topological spaces. Axiom 4 holds universally, again without separation conditions. We prove that K4 is the complete logic over infinite spaces, and GL over infinite ω-scattered spaces. (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 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 . (shrink)

The endomorphism monoid of a model-theoretic structure carries two interesting topologies: on the one hand, the topology of pointwise convergence induced externally by the action of the endomorphisms on the domain via evaluation; on the other hand, the Zariski topology induced within the monoid by (non-)solutions to equations. For all concrete endomorphism monoids of $\omega $ -categorical structures on which the Zariski topology has been analysed thus far, the two topologies were shown to coincide, in turn yielding that the pointwise (...) topology is the coarsest Hausdorff semigroup topology on those endomorphism monoids.We establish two systematic reasons for the two topologies to agree, formulated in terms of the model-complete core of the structure. Further, we give an example of an $\omega $ -categorical structure on whose endomorphism monoid the topology of pointwise convergence and the Zariski topology differ, answering a question of Elliott, Jonušas, Mitchell, Péresse, and Pinsker. (shrink)

Three correspondences between the foundations of mathematics have been independently discovered across the twentieth century. Curry and Feys (1958) and Howard (1969/1980) established that intuitionistic propositional logic corresponds to simply typed lambda calculus. Lambek (1972) extended the correspondence to category theory. Lawvere (1970) and Tierney brought set theory into the structure through topos theory. These results are established and published. What has not been stated is the compositional collapse. The three correspondences confirm that the four foundations are structurally identical, each (...) translatable into the others. The operational order of the cascade composes them into a single directed cycle: logic into set theory, set theory into type theory, type theory into category theory, category theory back into logic. That cycle is an endomorphism of the four-node directed graph derived in Stewart (2026c), a composition that maps the graph's operational structure back to itself. The cycle closes because it collapses: the convergence point (type theory) receives three inputs and reduces them to one output, a mandatory many-to-one reduction without which no single resolved state exists for the morphism to carry forward. The iterated endomorphism traces a torus, with the spatial loop around the four nodes and the temporal loop through the recursion. This paper derives the collapse from the four operations derived in Stewart (2026g), here termed the axioms of occurrence, and proposes the endomorphism as the structure within which quantum mechanics and general relativity already occupy defined positions. The four positions of the endomorphism are assigned functional labels native to the physics domain: Ground (the binary record of what has and has not occurred), Field (the space of all simultaneous potentials), Collapse (the many to one reduction that resolves the field into one determination), and Propagation (the energy that carries the resolved state forward into the next cycle). The framework proposes these four positions correspond to the four foundations of mathematics: Ground to logic, Field to set theory, Collapse to type theory, and Propagation to category theory. On this account, general relativity formalizes the Ground, the binary record of what has and has not occurred (incident 1), and quantum mechanics formalizes the Field (incident 2), the Collapse (incident 3), and the Propagation (incident 4). On the framework's account, both theories employ all four foundations as mathematical tools, because the endomorphism requires all four for any mathematical structure to exist. The distinction the framework draws is between what each theory formalizes as its subject matter and what it employs as its apparatus. On this account, the incompatibility between QM and GR is the incompatibility between two partial formalisms, each built from a subset of the mathematical foundations as its subject, applied separately to different positions within a single structure. The many-to-one reduction at the convergence point reframes wave function collapse: collapse is not a problem to be solved or an interruption of unitary evolution but the structural operation that makes the endomorphism possible. Without the collapse, no single resolved state exists for the morphism to carry forward, the cycle cannot close, and the endomorphism does not complete. The unification, this account proposes, is the endomorphic collapse itself. This is a claim about the structure any unification must instantiate. The structure already exists as published mathematics, the framework holds, and has not been applied to physics because no one has identified the collapse as the structure both theories are inside of. -/- **Keywords:** endomorphism, graph theory, algebraic topology, category theory, type theory, set theory, logic, Curry-Howard correspondence, Lambek correspondence, Lawvere-Tierney topos theory, foundations of mathematics, strongly connected directed graph, CW-complex, Betti number, torus topology, wave function collapse, quantum mechanics, general relativity, unification, scale invariance. (shrink)

This quiver's canonical bilinear form is provably dead: indefinite, selecting no Dynkin type at all. One bit, a sign on a single vertex, is the entire distance from that form to the Cartan matrix of su(3) ⊕ su(2). That the form is provably dead is what makes the gauge content live in the bit and not in the graph. -/- This paper is an inheritance walkthrough: a record of one formalism after another (counting, the path algebra, representation theory, and the (...) canonical bilinear (Tits) form) reaching its own expressive limit, each limit proven and each naming the formalism that inherits the question. -/- It starts at counting (the graded dimensions; counting cannot compose) and hands to the path algebra. The path algebra composes the four-gon γ₁ but cannot express the convergence at T as a relation: the three length-2 paths through T live in orthogonal idempotent components, and equating them would force e_S = e_L, which is false. It hands to representation theory. Under the convergence constraint dim(V_T) = 1 together with the involution conditions, every vertex space in an indecomposable representation is forced to dimension at most one (the rank argument). The quotient algebra cannot produce the braid relation ((γ₁γ₂)³ is a nonzero length-21 element, not e_L) and cannot produce nonzero commutation (γ₁γ₃ = 0 by vertex mismatch). It hands to the canonical bilinear invariant. The Tits form of Q on the cycle-support basis is indefinite, with two negative diagonal entries and one isotropic (zero-diagonal) generator, so it is not a Cartan matrix of any Dynkin type. Here the inheritance stops with a proven negative result: Q, by its own canonical invariants, does not select a finite reflection group; obtaining a Dynkin type requires structure added from outside Q. This stopping point is different in kind from the earlier ones. The earlier handoffs (counting to path algebra to representation theory) each named a richer formalism within mathematics to continue, and each hit the same ceiling because the missing datum was not a formalism a richer one would supply. The inheriting invariant here is the bilinear form on H₁(Q; ℝ), and the structure it needs is not internal to Q at all: it is the single distinction the graph's own combinatorics cannot carry. -/- The brain's signed convergence, the resolution at T that produces a positive-or-negative determination from the inputs converging on it (Stewart, 2026c, Loop 3), supplies that added structure. It is a single binary distinction (vertex T carries a sign, vertex C does not), fixed before the path algebra is constructed. Executing the four axioms of occurrence on that one distinction yields a bilinear form B, shown unique up to diagonal scaling, under which the same cycle generators that gave the dead Tits form realize the Cartan matrix of A₂ × A₁, returning the Lie algebra su(3) ⊕ su(2) via the Cartan-Killing correspondence. The difference between an indefinite, structure-selecting-nothing form and the strong-plus-weak gauge algebra is one bit: which vertex carries the sign. The cycle-generator basis has rank three, the bound that A₂ × A₁ saturates; the fourth generator required for the u(1) of the full Standard Model gauge algebra is not supplied by H₁(Q; ℝ) and is developed in companion work (Stewart, 2026q). The proven deadness of the canonical form is what makes the gauge content carried by the single added distinction rather than by the graph. This is the central result. -/- The inheritance sequence is itself the four-position cycle: counting (logic, distinction) → path algebra (composition) → representation theory (the type judgment dim(V_T) = 1, the many-to-one identity-by-collapse at T) → the bilinear morphism B (composition again). The paper builds mathematical structure in the order mathematics builds it, by letting each formalism break at its own ceiling and naming its heir. -/- What the paper does and what the paper is are the same act, because for an occurrence doing γ₁ and being γ₁ are one event. The construction performed below is one γ₁ traversal of the cycle whose bilinear invariant the construction returns. The gauge content that invariant carries, su(3) ⊕ su(2) at the rank the cycle basis supplies, is read off from inside the cycle by the cycle executing. Mathematics is one γ₁. Gauge structure is what γ₁ produces. The paper is the act in which the two are seen to be the same cycle viewed from opposite sides of itself. -/- **Keywords:** quiver, path algebra, quiver representations, quotient algebra, Tits form, Euler form, Cartan matrix, Coxeter group, Dynkin diagram, root system, Lie algebra, special unitary group, first homology. (shrink)

Supersubstantivalists identify material objects with regions of spacetime. Accordingly, they take their view to be both more ideologically parsimonious than other substantivalists (dualists) because they can reduce location to identity with a region and they can explain why the mereological structure of objects mirrors the mereological structure of their locations (henceforth, “harmony”). However, I argue that these motivations for supersubstantivalism don’t hold water. Specifically, I argue that supersubstantivalists can only claim the aforementioned advantages just so long as they stand in (...) a unique position to reduce location. But this depends on the illicit assumption that no possible spacetime structures have co-located regions – something falsified by the possibility of merely pseudo-metric spaces. Further, any alternative reduction is equally available to the dualist. Accordingly, this allows dualists to have the same explanation of harmony as the supersubstantivalist. As such, these considerations level the dialectical playing field between dualists and supersubstantivalists. (shrink)

Logic and computer science share a subtle relationship that depends on both syntax and semantics. While structural generalisations often rely on semantics alone, computational aspects such as complexity and decidability hinge on the syntactic properties of formal languages. This interplay frequently manifests through relations between structures, which establish their similarity in various ways and for different purposes. -/- In this work, we focus on three distinct forms of relations between structures: coKleisli morphisms, games, and truth-preserving maps. By coordinating these concepts (...) and proving their equivalence, we apply them across diverse contexts, selecting the appropriate framework for each scenario. CoKleisli morphisms are used to derive generalised theorems, which can then be applied to specific instances; games are used as algorithmic descriptions, crucial when discussing computational complexity; and truth-preserving maps between structures are essential for achieving results such as the finite model property, completeness, and decidability. To a large extent, one must control all manners of structural relations in order to achieve all types of results. -/- We contribute to each of these areas: We extend the use of coKleisli morphisms to ‘linear’ variants of previously studied comonads. Using these results, we derive a novel homomorphism counting theorem for pathwidth. We further establish preservation and characterisation theorems for non-branching variants of the modal comonad. Additionally, using event structures from concurrency theory, we propose a structural alternative to games, providing a more specialised framework for deriving the underlying comonads. For games and computational complexity, we define new model comparison games and establish complexity results for an all-in-one variant of the k-pebble game. Finally, we employ truth-preserving maps to prove completeness and decidability for highly expressive spatiotemporal languages within the context of dynamical systems. (shrink)

Dynamical systems are abstract models of interaction between space and time. They are often used in fields such as physics and engineering to understand complex processes, but due to their general nature, they have found applications for studying computational processes, interaction in multi-agent systems, machine learning algorithms and other computer science related phenomena. In the vast majority of applications, a dynamical system consists of the action of a continuous `transition function' on a metric space. In this work, we consider decidable (...) formal systems for reasoning about such structures. Spatial logics can be traced back to the 1940's, but our work follows a more dynamic turn that these logics have taken due to two recent developments: the study of the topological mu-calculus, and the the integration of linear temporal logic with logics based on the Cantor derivative. In this paper, we combine dynamic topological logics based on the Cantor derivative and the `next point in time' operators with an expressively complete fixed point operator to produce a combination of the topological mu-calculus with linear temporal logic. We show that the resulting logics are decidable and have a natural axiomatisation. Moreover, we prove that these logics are complete for interpretations on the Cantor space, the rational numbers, and subspaces thereof. (shrink)

An algebraization of the notion of topology has been proposed more than 70 years ago in a classical paper by McKinsey and Tarski, leading to an area of research still active today, with connections to algebra, geometry, logic and many applications, in particular, to modal logics. In McKinsey and Tarski’s setting the model theoretical notion of homomorphism does not correspond to the notion of continuity. We notice that the two notions correspond if instead we consider a preorder relation ⊑\documentclass[12pt]{minimal} \usepackage{amsmath} (...) \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \sqsubseteq $$\end{document} defined by a⊑b\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a \sqsubseteq b$$\end{document} if a is contained in the topological closure of b, for a, b subsets of some topological space. A specialization poset is a partially ordered set endowed with a further coarser preorder relation ⊑\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \sqsubseteq $$\end{document}. We show that every specialization poset can be embedded in the specialization poset naturally associated to some topological space, where the order relation corresponds to set-theoretical inclusion. Specialization semilattices are defined in an analogous way and the corresponding embedding theorem is proved. Specialization semilattices have the amalgamation property. Some basic topological facts and notions are recovered in this apparently very weak setting. The interest of these structures arises from the fact that they also occur in many rather disparate contexts, even far removed from topology. (shrink)

It is well-known that the global structure of every space-time model for relativistic cosmology is observationally underdetermined. In order to alleviate the severity of this underdetermination, it has been proposed that we adopt the Cosmological Principle because the Principle restricts our attention to a distinguished class of space-time models (spatially homogeneous and isotropic models). I argue that, even assuming the Cosmological Principle, the topology of space remains observationally underdetermined. Nonetheless, I argue that we can muster reasons to prefer various topological (...) properties over others. In particular, I favor the adoption of multiply connected universe models on grounds of (i) simplicity, (ii) Machian considerations, and (iii) explanatory power. We are able to appeal to such grounds because multiply connected topologies open up the possibility of finite universe models (consistent with our best data), which in turn avoid thorny issues concerning the postulation of an actually infinite universe. (shrink)

We study the non-relativistic (NR) limit of relativistic spacetimes in relation with the topology of the Universe. We first show that the NR limit of the Einstein equation is only possible in Euclidean topologies, i.e., for which the covering space is \(\mathbb {E}^3\). We interpret this result as an inconsistency of general relativity in non-Euclidean topologies and propose a modification of that theory which allows for the limit to be performed in any topology. For this, a second reference non-dynamical connection (...) is introduced in addition to the physical spacetime connection. The choice of reference connection is related to the covering space of the spacetime topology. Instead of featuring only the physical spacetime Ricci tensor, the modified Einstein equation features the difference between the physical and the reference Ricci tensors. This theory should be considered instead of general relativity if one wants to study a universe with a non-Euclidean topology and admitting a non-relativistic limit. (shrink)

Mathematical diagrams are frequently used in contemporary mathematics. They are, however, widely seen as not contributing to the justificatory force of proofs: they are considered to be either mere illustrations or shorthand for non-diagrammatic expressions. Moreover, when they are used inferentially, they are seen as threatening the reliability of proofs. In this paper, I examine certain examples of diagrams that resist this type of dismissive characterization. By presenting two diagrammatic proofs, one from topology and one from algebra, I show that (...) diagrams form genuine notational systems, and I argue that this explains why they can play a role in the inferential structure of proofs without undermining their reliability. I then consider whether diagrams can be essential to the proofs in which they appear. (shrink)

The boundary problem is related to the binding problem, part of a family of puzzles and phenomenal experiences that theories of consciousness (ToC) must either explain or eliminate. By comparison with the phenomenal binding problem, the boundary problem has received very little scholarly attention since first framed in detail by Rosengard in 1998, despite discussion by Chalmers in his widely cited 2016 work on the combination problem. However, any ToC that addresses the binding problem must also address the boundary problem. (...) The binding problem asks how a unified first person perspective (1PP) can bind experiences across multiple physically distinct activities, whether billions of individual neurons firing or some other underlying phenomenon. To a first approximation, the boundary problem asks why we experience hard boundaries around those unified 1PPs and why the boundaries operate at their apparent spatiotemporal scale. We review recent discussion of the boundary problem, identifying several promising avenues but none that yet address all aspects of the problem. We set out five specific boundary problems to aid precision in future efforts. We also examine electromagnetic (EM) field theories in detail, given their previous success with the binding problem, and introduce a feature with the necessary characteristics to address the boundary problem at a conceptual level. Topological segmentation can, in principle, create exactly the hard boundaries desired, enclosing holistic, frame-invariant units capable of effecting downward causality. The conclusion outlines a programme for testing this concept, describing how it might also differentiate between competing EM ToCs. (shrink)

Braddon-Mitchell and Miller put forward the claim that the relation of being-the-same-person is gradable: a person can be the same person tomorrow as today, but only half the same. To justify their thesis, they propose a model of persons that is intended to be metaphysically neutral. This article sets out to show that such a model implicitly contains strong metaphysical assumptions that run contrary to the authors’ own statements. Using Roman Ingarden’s phenomenological ontology, we aim to demonstrate that within the (...) model in question persons exist intentionally, where this leads to a caricature rather than an adequate account of persons. In opposition to Braddon-Mitchell and Miller, we argue that the justification presented for the thesis of the gradability of the relation of being-the-same-person in fact makes it reasonable to demand that an adequate model of persons be posited. We present Kurt Lewin’s holistic topological-dynamic model, and show that in this context it is not being-the-same-person that is gradable, but rather the characteristic (of persons) that is dimension. Thus, we reject the thesis of the gradability of the relation of being-the-same-person involved in personal identity. (shrink)

The history of thinking about thinking is populated by numerous attempts to model reason in topological terms. Amongst them, the prominent place is occupied by Immanuel Kant’s explanation of thought’s need to restrain its own exercise by means of an analogy between geographical orientation (modeled on the human body) and orientation in thinking. As natural as his analogy might seem, the first part of this chapter aims at deconstructing Kant’s attempt as both replaceable and constraining, and at proposing a possibility (...) of alternative topological accounts of thinking. Hence, while endorsing the utility of spatial models, we call for an unbinding of the parochial connection between thinking and the form of the human body implicit in historical topological models of reason. For this reason, in the second part of the chapter we suggest that the topological framework embodied by Artificial Neural Networks (ANNs) can be used as an alternative to formulate such a model of thinking, based on their commitment to dimensionality and use of space as an active, dynamic, and transitory element. Rather than arguing that ANNs somehow think, we suggest that they offer a mirror that lets humans look back at themselves and construct their thinking differently. We conclude by proposing that the benefit of looking in this mirror is open-ended and twofold: (1) It divorces our image of thinking from anthropomorphism, and (2) it offers a normative model of reason with potentially practical consequences for how humans act. (shrink)

Техника представления информации о внешнем и внутреннем мире постоянно развивается, и сейчас она достигла уровня отображения реальности в многообразных её проявлениях и измерениях, прежде недоступных человеческому восприятию. Язык, текст, фотография, звукозапись, а теперь ещё и техника искусственного интеллекта для моделирования человеческой субъектности и её описания в доступной для человеческого понимания форме, стали эпохальными событиями в теории информации. Однако несмотря на то, что на данном этапе её развития она позволяет оперировать с непрерывно возрастающими объёмами информации, это не приближает её теоретиков к (...) постижению сути того, что определяется как реальность, бытие, и сознание. Но поскольку никакого другого способа достичь этой цели нет, кроме изучения того, как это происходит в процессе восприятия и преобразования информации у живых существ, и в частности, у человека, необходимо разобраться в принципах этого процесса. (shrink)

Both early analytic philosophy and the branch of mathematics now known as topology were gestated and born in the early part of the 20th century. It is not well recognized that there was early interaction between the communities practicing and developing these fields. We trace the history of how topological ideas entered into analytic philosophy through two migrations, an earlier one conceiving of topology geometrically and a later one conceiving of topology algebraically. This allows us to reassess the influence and (...) significance of topological methods for philosophy, including the possible fruitfulness of a third conception of topology as a structure determining similarity. (shrink)

This paper presents the topological arrangements in four geometrical figures of modal propositions and their derivative relations by means of Peirce's gamma graphs and their rules of transformation. The idea of arraying the gamma graphs in a geometric and symmetrical order comes from Peirce himself who in a manuscript drew two cubes in which he presented the derivative relations of some (but no all) gamma graphs. Therefore, Peirce's insights of a topological order of gamma graphs are extended here backwards from (...) the cube to the line and the square; and then forwards from the cube to the four-dimensional polyhedron. (shrink)

The topology optimization of computer communication network is studied based on improved genetic algorithm, a network optimization design model based on the establishment of network reliability maximization under given cost constraints, and the corresponding improved GA is proposed. In this method, the corresponding computer communication network cost model and computer communication network reliability model are established through a specific project, and the genetic intelligence algorithm is used to solve the cost model and computer communication network reliability model, respectively. It has (...) been proved that GA can solve the complex problems of computer working environment better, which is 80% higher than the general algorithm, and can select the optimal scheme pertinently. (shrink)

So-called ‘distinctively mathematical explanations’ (DMEs) are said to explain physical phenomena, not in terms of contingent causal laws, but rather in terms of mathematical necessities that constrain the physical system in question. Lange argues that the existence of four or more equilibrium positions of any double pendulum has a DME. Here we refute both Lange’s claim itself and a strengthened and extended version of the claim that would pertain to any n-tuple pendulum system on the ground that such explanations are (...) actually causal explanations in disguise and their associated modal conditionals are not general enough to explain the said features of such dynamical systems. We argue and show that if circumscribing the antecedent for a necessarily true conditional in such explanations involves making a causal analysis of the problem, then the resulting explanation is not distinctively mathematical or non-causal. Our argument generalises to other dynamical systems that may have purported DMEs analogous to the one proposed by Lange, and even to some other counterfactual accounts of non-causal explanation given by Reutlinger and Rice. (shrink)

This chapter provides a systematic overview of topological explanations in the philosophy of science literature. It does so by presenting an account of topological explanation that I (Kostić and Khalifa 2021; Kostić 2020a; 2020b; 2018) have developed in other publications and then comparing this account to other accounts of topological explanation. Finally, this appraisal is opinionated because it highlights some problems in alternative accounts of topological explanations, and also it outlines responses to some of the main criticisms raised by the (...) so-called new mechanists. (shrink)

We propose a new topological semantics for evidence, evidence-based justifications, belief, and knowledge. Resting on the assumption that an agent’s rational belief is based on the available evidence, we try to unveil the concrete relationship between an agent’s evidence, belief, and knowledge via a rich formal framework afforded by topologically interpreted modal logics. We prove soundness, completeness, decidability, and the finite model property for the associated logics, and apply this setting to analyze key epistemological issues such as “no false lemma” (...) Gettier examples, misleading defeaters, undefeated justification versus undefeated belief, as well as the defeasibility theories of knowledge. (shrink)

This thesis is devoted to the exploration of the complexity of some mathematical problems using the framework of computable analysis and descriptive set theory. We will especially focus on Weihrauch reducibility as a means to compare the uniform computational strength of problems. After a short introduction of the relevant background notions, we investigate the uniform computational content of problems arising from theorems that lie at the higher levels of the reverse mathematics hierarchy.We first analyze the strength of the open and (...) clopen Ramsey theorems. Since there is not a canonical way to phrase these theorems as multi-valued functions, we identify eight different multi-valued functions and study their degree from the point of view of Weihrauch, strong Weihrauch, and arithmetic Weihrauch reducibility.We then discuss some new operators on multi-valued functions and study their algebraic properties and the relations with other previously studied operators on problems. In particular, we study the first-order part and the deterministic part of a problem f, capturing the Weihrauch degree of the strongest multi-valued problem that is reducible to f and that, respectively, has codomain $\mathbb {N}$ or is single-valued.These notions proved to be extremely useful when exploring the Weihrauch degree of the problem $\mathsf {DS}$ of computing descending sequences in ill-founded linear orders. They allow us to show that $\mathsf {DS}$, and the Weihrauch equivalent problem $\mathsf {BS}$ of finding bad sequences through non-well quasi-orders, while being very “hard” to solve, are rather weak in terms of uniform computational strength. We then generalize $\mathsf {DS}$ and $\mathsf {BS}$ by considering $\boldsymbol {\Gamma }$ -presented orders, where $\boldsymbol {\Gamma }$ is a Borel pointclass or $\boldsymbol {\Delta }^1_1$, $\boldsymbol {\Sigma }^1_1$, $\boldsymbol {\Pi }^1_1$. We study the obtained $\mathsf {DS}$ -hierarchy and $\mathsf {BS}$ -hierarchy of problems in comparison with the Baire hierarchy and show that they do not collapse at any finite level.Finally, we work in the context of geometric measure theory and we focus on the characterization, from the point of view of descriptive set theory, of some conditions involving the notions of Hausdorff/Fourier dimension and Salem sets. We first work in the hyperspace $\mathbf {K}$ of compact subsets of $[0,1]$ and show that the closed Salem sets form a $\boldsymbol {\Pi }^0_3$ -complete family. This is done by characterizing the complexity of the family of sets having sufficiently large Hausdorff or Fourier dimension. We also show that the complexity does not change if we increase the dimension of the ambient space and work in $\mathbf {K}$. We also generalize the results by relaxing the compactness of the ambient space and show that the closed Salem sets are still $\boldsymbol {\Pi }^0_3$ -complete when we endow $\mathbf {F}$ with the Fell topology. A similar result holds also for the Vietoris topology.We conclude by analyzing the same notions from the point of view of effective descriptive set theory and Type-2 Theory of Effectivity, and show that the complexities remain the same also in the lightface case. In particular, we show that the family of all the closed Salem sets is $\Pi ^0_3$ -complete. We furthermore characterize the Weihrauch degree of the functions computing Hausdorff and Fourier dimension of closed sets.prepared by Manlio Valenti.E-mail:[email protected]. (shrink)

If two self-connected individuals are connected, it follows in classical extensional mereotopology that the sum of those individuals is self-connected too. Since mainland Europe and mainland Asia, for example, are both self-connected and connected to each other, mainland Eurasia is also self-connected. In contrast, in non-extensional mereotopologies, two individuals may have more than one sum, in which case it does not follow from their being self-connected and connected that the sum of those individuals is self-connected too. Nevertheless, one would still (...) expect it to follow that a sum of connected self-connected individuals is self-connected too. In this paper, we present some surprising countermodels which show that this conjecture is incorrect. (shrink)

By utilizing the topological concept of pseudocompactness, we simplify and improve a proof of Caicedo, Dueñez, and Iovino concerning Terence Tao's metastability. We also pinpoint the exact relationship between the Omitting Types Theorem and the Baire Category Theorem by developing a machine that turns topological spaces into abstract logics.