SphereAdmitsCircleLinking
plain-language theorem explainer
SphereAdmitsCircleLinking D is the predicate that the D-sphere admits nontrivial linking of embedded circles, defined by the reduced cohomology of the circle being nontrivial in degree D-2. Recognition Science researchers cite it when deriving spatial dimension from topology rather than by fiat. The definition is a direct one-line delegation to the circle cohomology predicate, shifting the D=3 equivalence into a separate theorem.
Claim. For $D$ a natural number, the $D$-sphere admits nontrivial linking of disjoint embedded circles precisely when the reduced cohomology group $H^{D-2}(S^1;Z)$ is nontrivial.
background
The Alexander Duality module replaces an earlier tautological definition of circle linking with one grounded in cohomology. Alexander duality (Hatcher Thm 3.44) supplies the isomorphism $H_q(S^D minus S^1) cong H^{D-2}(S^1)$ for an embedded circle; linking is detected by the first homology of the complement, which therefore equals the (D-2)th reduced cohomology of the circle. The predicate CircleReducedCohomologyNontrivial k encodes the standard computation that this group is nontrivial exactly for k=1 (Hatcher §2.2).
proof idea
One-line definition that sets the predicate equal to CircleReducedCohomologyNontrivial applied to (D cast to Z) minus 2. It directly invokes the upstream definition of CircleReducedCohomologyNontrivial as equality to 1 and the theorem circle_reduced_cohomology_iff.
why it matters
This definition supplies the topological content for alexander_duality_circle_linking, circle_linking_forces_D3, D3_admits_circle_linking, and the dimension-forcing results in DimensionForcing. It advances the Recognition Science chain by grounding T8 (D=3 spatial dimensions) in Alexander duality rather than an axiom, closing the historical scaffolding around the eight-tick octave.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.
papers checked against this theorem (showing 26 of 26)
-
Atomistic models yield von-Kármán plates in joint ε-h limit
"8-tick periodic micro-structure implicit in the cubic lattice cells; non-affine contributions AZ− + aM for finite ν"
-
Operator trees raise autoformalization success on Lean theorems
"OPTs explicitly encode operator precedence and logical scoping... providing a hierarchical semantic anchor that transcends the limitations of linear sequence generation."
-
Higher dimensions stabilize strange quark stars
"For an interval of central energy densities ρ_cd G_d and total masses M G_d/(d-3), the stars gain more stability when the dimension is increased."
-
Space-filling curves make domain decomposition work in any dimension
"our domain decomposition approach must be dimension-oblivous and directly applicable to anisotropic meshes... we utilize a space-filling curve method... independent of the dimension d."
-
Steady-state topological order stable only in 3D open systems
"in three dimensions... m defects form loops... the non-increasing feature turns out to be crucial for the stability of topological degeneracy"
-
Conformal geodesic equation is variational in 3D
"One may understand × as the wedge operation, but in 3D it can be identified via Hodge star of g and musical isomorphisms with the operation of vector product on T M."
-
A-branes correspond one-to-one with spherical DAHA representations
"the second integral homology group is isomorphic to the affine D4 root lattice... .W(D4) acts on H2(X,Z) ≅ Q(D4)"
-
Percolation threshold guides when soaring pilots must use a thermal
"p ≈ (λ1/2)^2 π (h L/D)^2 / 4.51 … When p is greater than 1 the present altitude, thermal intensity, and lift-to-drag ratio is such that sufficient additional thermals are reachable so that the graph of thermal connections percolates"
-
String field theory extended to 25.99 dimensions
"the mixed vertices with n ordinary punctures and m special punctures, denoted by Γ_{0,n;m} … dim Γ_{0,n;m}=2n+3m−6"
-
Structured State-Space Regularization for Generation-Friendly Image Tokenization
"d/dτ ck(Iτ) = 1/2 Σ cn(Iτ) ⟨∇²ϕn, ϕk⟩ ... := A c(Iτ) ... (Euler) c(It) = (I + A Δ) c(It−1)"
-
Recursive S_n structure proves specific molecule is a cell
"Define σ_i := s_i s_{i-1} ⋯ s_2 s_1 … Y_i = σ_i Y_1 σ_i^{-1}"
-
Finite-field Fourier spectra reduce to cyclotomic subfield products
"Theorem 3.1 ... SA ≃ ∏_{O ∈ Â/⟨q⟩} F_{q^{|O|}} as K-vector spaces"
-
Multilateral netting preserves original counterparty exposures
"By the Flow Decomposition Lemma, every positive flow in the Trade Flow Network can be written as a collection of simple s-t paths and cycles."
-
Time-scale entanglement rises near phase transitions
"quantics tensor train (QTT)... each tensor index σ_ℓ corresponds to an exponentially distinct length or time scale"
-
Quantum RAM uses few walkers on a tree to cut resources
"the scattering gate Ŝ routes walkers in |R⟩ left and |B⟩ right with color reset"
-
First algorithm finds optimal trees with hypersurface splits
"Theorem 2: if two hyperplanes cross … convex hulls intersect … cannot form a proper decision tree"
-
Platonic symmetries solve the discrete Nahm equation
"The discrete Nahm equation is an integrable nonlinear difference equation for complex N×N matrices... Solutions correspond to SU(2) magnetic monopoles of charge N in hyperbolic space."
-
Multilayer networks model interdependencies across real systems
"Multilayer networks allow researchers to capture the rich, interconnected structure of systems where multiple types of interactions, temporal dynamics, or functional interdependencies coexist."
-
5D black holes survive to explain all dark matter
"the spacetime geometry of a higher-dimensional black hole... Myers-Perry black hole... n=1"
-
Polar flow matching adapts graphs while preserving topology
"we embed graphs into a Riemannian manifold. By adopting polar coordinates, we explicitly disentangle structure (radius) from semantics (angle). ... flow matching ... along geodesic paths"
-
Partition logic translates to Prolog rules that generate visual forms
"Greechie-style hypergraph... two three-atomic contexts C1={a,b,c}, C2={c,d,e}"
-
One-var loop termination decided in poly time under Collatz conjecture
"rec(R) = nonneg(v1,v2) ... I+, I−, Δ+, Δ− ... self-avoiding trace"
-
Sponge networks are disordered single-gyroids
"every loop of DG is catenated by loops of the other network, while no loops of the amorphous networks we analyzed are threaded or linked by another tubular PDMS domain"
-
3-torsion fields solve cyclotomic Galois embedding problems infinitely often
"isomorphism Φ : S4 → PGL2(F3) … subgroups eGi ≅ GL2(F3), SD16, D6, D4, C2²"
-
Robot planner cuts path bending by 91 percent on average
"A visibility cone is formed at the goal x_g by the vectors pointing from x_g to the vertices of the shared face S. The region R∪Δ_T is star-shaped if the vector v_new−x_g lies strictly inside this cone."
-
New bound tighteners and Givens lex cuts speed MINLPs with distance constraints
"sphere packing problem, kissing number problem, obnoxious facility location problem ... d∈{1,...,5} for MINLP, d=2 for OFL, d∈{2,3} for GEOM"