Encyclopedia/All topics/Foundation

Foundation

Articles 2,641–2,700 of 2,979. Alphabetical by title.

• Foundation Singular Pair Pair Ses Degreewise Short Exact

  A machine-checked lemma in the Recognition Science library shows that an injective continuous map between spaces yields an exact sequence of homology groups at every dimension, a s

• Foundation Singular Pair Relative Homology Id Is Zero

  When a space is compared with itself, its relative homology groups vanish; here is what that theorem does and does not say.

• Foundation Singular Pair To Sset Map App Injective

  An injective continuous map between spaces forces a one-to-one correspondence at every level of the singular simplex construction.

• Foundation Singular Prism

  A machine-checked library proves that two continuously deformable shapes have identical internal structure, a key idea for how Recognition Science models change.

• Foundation Singular Prism Homotopic Maps Induce Same Homology

  Two continuous maps that can be deformed into each other produce identical algebraic measurements of a space's holes.

• Foundation Singular Prism Is Iso Homology Map Of Homotopy Equiv

  A central theorem of algebraic topology, proved in the framework's machine-checked library, shows that spaces connected by a continuous deformation have identical homology gro

• Foundation Singular Prism Prism Comp Face Bot

  A theorem about the bottom edge of a geometric prism shows the framework's library of formal theorems can certify the exact boundary behavior of a standard construction.

• Foundation Singular Prism Prism Comp Face Cancel

  A theorem about geometric building blocks shows that two different ways to build a prism from a face produce the same shape, a fact that keeps the framework's counting machine

• Foundation Singular Prism Prism Comp Face Of Le

  A machine-checked theorem in the Recognition Science library pins down how a standard geometric construction, the prism, interacts with the boundary faces of a simplex.

• Foundation Singular Prism Prism Comp Face Top

  A small identity about the edges of a geometric prism turns out to be the hinge that makes the whole framework's counting machinery consistent.

• Foundation Singular Sphere

  A single point in space, examined closely enough, carries a complete record of the space around it.

• Foundation Singular Sphere Geometry

  A machine-checked library builds the geometry of spheres from two poles and an open cover, then proves which homology groups vanish.

• Foundation Singular Sphere Geometry Abs Eq One Of Sq Eq One

  A machine-checked library of formal theorems proves that on a circle, only the two poles have a coordinate whose square is one.

• Foundation Singular Sphere Geometry H1 S1 Ne Zero

  A machine-checked proof that the circle has a one-dimensional hole, a fact classical topology has known for over a century, now lives inside the Recognition Science framework'

• Foundation Singular Sphere Geometry Sphere Dim Eq Of Homotopy Equiv

  In topology, a sphere's dimension is a matter of homotopy: the declaration sphere_dim_eq_of_homotopyEquiv proves that if a space is homotopy equivalent to an n-sphere, then it

• Foundation Singular Sphere Geometry Sphere Homology Vanish

  A theorem about spheres shows that most of their higher-dimensional holes simply do not exist, and it does so without any special assumptions.

• Foundation Singular Sphere Geometry Sphere Top Ne Zero

  The declaration proves the circle has a hole that cannot be shrunk away, a fact the framework uses to build its model of recognition.

• Foundation Singular Sphere Geometry Spheres Not Homotopy Equivalent

  Two spheres of different dimension cannot be continuously deformed into each other, a fact the framework's machine-checked library proves for its own sphere model.

• Foundation Singular Sphere Is Iso Aug H Of Path Connected

  A machine-checked theorem shows that in any path-connected space, counting the connected components is the same as counting the integers, a bridge that Recognition Science uses to

• Foundation Singular Sphere Is Iso Homology Map Aug To

  A machine-checked theorem shows that for any path-connected space, counting points by a clopen set gives an isomorphism on the zeroth homology group.

• Foundation Singular Sphere Is Zero H1

  In algebraic topology, the zero-dimensional homology of a single point is the integers; Recognition Science's machine-checked library proves all higher homology groups of a po

• Foundation Singular Sphere Is Zero H1 Of Contractible

  A machine-checked proof that a space which can shrink to a point has no one-dimensional holes, and the precise limits of that statement.

• Foundation Singular Sphere Is Zero Homology Of Contractible

  A machine-checked proof that a contractible space has no higher-dimensional holes, and a precise statement of what that does not say.

• Foundation Singular Sphere Is Zero Of Is Zero Inter

  A machine-checked proof shows that the 0th homology of the singular sphere is the integers, with higher homology vanishing.

• Foundation Singular Subdivision

  Subdivision is the act of cutting a shape into smaller pieces, a classical idea that gains new power when the pieces are kept in a discrete record.

• Foundation Singular Subdivision Abnd Comp Acone Zero

  In algebraic topology, the boundary of a cone over a simplex is the original simplex itself; a machine-checked proof now records this fact in a formal library.

• Foundation Singular Subdivision Asub Iter Support Bound

  Repeatedly subdividing a geometric object into smaller pieces leaves the object's overall shape untouched, but the process must track which pieces are which.

• Foundation Singular Subdivision Exists Asub Iter Small

  A machine-checked theorem shows that repeatedly subdividing a geometric shape into smaller pieces always produces a well-defined, finite process, a result with a precise scope.

• Foundation Singular Subdivision Exists Sd Op Iter Small

  A machine-checked theorem shows that repeatedly subdividing a space's singular chains always yields a well-defined map, and it says nothing about what those chains represent.

• Foundation Singular Subdivision T Op Chain Homotopy Succ

  A machine-checked theorem shows that repeatedly subdividing a shape's building blocks changes its boundary in a precise, controlled way.

• Foundation Singular Subdivision T Op Chain Homotopy Zero

  A formal theorem in the Recognition Science library shows that a certain subdivision operator agrees with the identity at the lowest dimension, up to a controlled error term.

• Foundation Singular Subdivision T Op Iter Chain Homotopy Succ

  In algebraic topology, a chain homotopy is a formal way to say two ways of cutting a space into pieces give the same answer about holes; here a machine-checked proof shows one such

• Foundation Singular Subdivision T Op Iter Chain Homotopy Zero

  Subdividing a topological space into smaller pieces is a standard tool in algebraic topology; one framework theorem shows exactly how the first step of that process behaves.

• Foundation Smgauge Algebra

  The Standard Model's force carriers number exactly twelve, and a machine-checked proof derives that count from the symmetries of a cube.

• Foundation Smgauge Algebra Factor Count

  A machine-checked theorem counts the three gauge factors of the Standard Model, and the count is 3.

• Foundation Smgauge Algebra Hyper Gen Count

  In the Standard Model of particle physics, the weak hypercharge force has exactly one kind of force carrier, a fact the Recognition Science framework derives from its cube-automorp

• Foundation Smgauge Algebra Sm Total Gen Count

  The Standard Model's gauge forces are carried by exactly 12 force-mediating fields; a machine-checked library now derives that count from a cube's symmetries.

• Foundation Smgauge Algebra Smgauge Algebra Cert

  A machine-checked certificate records that the Standard Model's three gauge groups have exactly 8, 3, and 1 generators, adding to 12.

• Foundation Smgauge Algebra Smgauge Factor

  A small formal object names the three forces of the Standard Model and counts their force carriers, tying a cube's symmetry to the number 12.

• Foundation Smgauge Algebra Weak Gen Count

  The weak nuclear force, which drives radioactive decay, is carried by exactly three force particles, a count the Recognition Science framework derives from the geometry of a cube.

• Foundation Smhypercharge From Cube

  A machine-checked library shows the Standard Model's hypercharge assignments fit exactly into the cube-completion's 1/6 unit, with all anomaly sums vanishing.

• Foundation Smhypercharge From Cube Generation Weyl State Count Eq 16

  One generation of Standard Model matter contains exactly sixteen distinct particle states, a count that a machine-checked proof verifies from a cube-based counting scheme.

• Foundation Smhypercharge From Cube Su2 Squared U1 Anomaly6 Eq Zero

  A machine-checked library proves the Standard Model's weak hypercharge assignments cancel exactly, a consistency condition that must hold for the theory to be mathematically s

• Foundation Smhypercharge From Cube Su3 Squared U1 Anomaly6 Eq Zero

  The Standard Model's hypercharge assignments pass a consistency check: their quantum anomalies cancel exactly, a fact a machine-checked proof verifies.

• Foundation Smhypercharge From Cube Three Generation Weyl State Count Eq 48

  A machine-checked theorem shows that three generations of Standard Model particles contain exactly 48 Weyl states, matching the size of a cube's signed permutation group.

• Foundation Sociology

  A framework for measuring social distance, where the cost of a gap between groups follows a single forced mathematical curve.

• Foundation Spatial Topology Forcing

  A compact, featureless substrate with no preferred direction must wrap into a 3-torus, giving space exactly three dimensions.

• Foundation Spatial Topology Forcing Isotropy Forces B1 Eq 3

  A machine-checked proof shows that a space with no preferred direction must have exactly three independent directions, if it is flat, compact, and orientable.

• Foundation Spatial Topology Forcing Self Similarity Forces Flat

  A machine-checked theorem shows that if the universe's basic recognition process is self-similar, space cannot be curved.

• Foundation Spatial Topology Forcing Spatial Dimension Eq 3

  A compact, flat, featureless 3-manifold must be a 3-torus, and its three independent directions are the three spatial dimensions.

• Foundation Spatial Topology Forcing Spatial Topology Forcing

  A machine-checked theorem derives three spatial dimensions from symmetry constraints, and it stops well short of claiming the universe is a 3-torus.

• Foundation Spatial Topology Forcing Spatial Topology Forcing Cert

  A machine-checked certificate bundles the proof that the recognition substrate's symmetry forces three-dimensional space, without claiming to derive the physical bridge.

• Foundation Spatial Topology Forcing Spatial Topology Forcing Cert Inhabited

  A machine-checked certificate packs the argument that the universe's spatial substrate must be a 3-torus, yielding three dimensions.

• Foundation Spatial Topology Forcing Substrate Symmetry Properties

  A compact, flat, featureless 3-torus is the only spatial shape that satisfies five symmetry conditions at once.

• Foundation Spin Statistics

  Foundation spin statistics derives the spin-statistics connection from the eight-tick recognition cycle: spin-1/2 states anticommute, spin-1 states commute, and Pauli exclusion fol

• Foundation Spin Statistics Boson Rotation Phase Pos One

  A machine-checked theorem shows that in the Recognition Science framework, an integer-spin particle returns to its original quantum state after a full rotation, a fact with deep co

• Foundation Spin Statistics Exchange Sign Fermion

  When two identical particles swap places, the laws of quantum mechanics can flip the sign of their shared wavefunction; this is the exchange sign, and it decides whether matter can

• Foundation Spin Statistics Fermion Rotation Phase Neg One

  A spin-1/2 particle returns to its quantum state only after two full rotations, and the minus sign that marks the first turn is the same sign that keeps two electrons apart.

• Foundation Spin Statistics Pauli Exclusion Simple

  A simple algebraic fact about complex numbers underlies the Pauli exclusion principle in the Recognition Science framework.

• Foundation Spin Statistics Spin Statistics Certificate

  The spin-statistics theorem links a particle's spin to its behavior when two identical particles are swapped; this page explains what a machine-checked proof of that link does