Encyclopedia/All topics/Foundation

Foundation

Articles 121–180 of 2,979. Alphabetical by title.

• Foundation Branch Selection Is Coupling Combiner Iff Interaction Defect Nonzero

  A single algebraic test decides whether a cost-combining rule genuinely mixes two inputs or merely adds their separate contributions.

• Foundation Branch Selection Rclcombiner Zero Separately Additive

  A small formal lemma about a two-variable polynomial is the hinge that lets one branch of a cost function survive and forces the other out.

• Foundation Branch Selection Separately Additive Iff Interaction Defect Zero

  A single number, the interaction defect, tells whether a two-input rule merely adds its parts or genuinely couples them.

• Foundation Categorical Logic Realization

  A bridge that shows the natural numbers built from pure logic are the same object category theory calls the natural-number object.

• Foundation Categorical Logic Realization Canonical Categorical Realization

  A machine-checked construction shows that the framework's arithmetic is the same no matter which formal realization you pick.

• Foundation Categorical Logic Realization Categorical Arithmetic Invariant

  A formal bridge shows that the natural numbers built inside the Recognition Science framework are the same natural numbers, no matter how the framework's internal logic is rea

• Foundation Categorical Logic Realization Category Interface Of Lawvere

  A small formal bridge lets the framework's arithmetic speak the language of category theory, without rebuilding the subject.

• Foundation Categorical Logic Realization Lawvere Nno

  A natural-number object is the categorical way to say "counting works," and this declaration pins down that structure without rebuilding category theory.

• Foundation Categorical Logic Realization Logic Nat Nno

  A natural-number object is the categorical skeleton of counting, and the framework's logic builds one from its own arithmetic.

• Foundation Categorical Logic Realization Logic Nat Nno Has Category Interface

  A machine-checked theorem shows the natural numbers built from logic alone fit a categorical template, and it proves far less than it names.

• Foundation Chemistry

  The module lays out a general framework for chemical reaction rates, but it currently establishes only abstract properties of a cost function, not chemistry itself.

• Foundation Chemistry Solv Reorg4 Cert

  A machine-checked certificate records three properties of a cost function, but says nothing about chemistry until the variables are defined.

• Foundation Circle Covering

  A circle's winding number counts how many times a loop goes around, and a machine-checked proof now shows the standard trigonometric parametrization is the right tool for defi

• Foundation Circle Covering Carrier Covering

  A single map from the real line to the unit circle, t to (cos t, sin t), is a covering map, a fact that lets topologists define winding numbers.

• Foundation Circle Covering Carrier Covering Val

  The map that sends a real number t to the point (cos t, sin t) on the unit circle is a covering map, a fact that underpins the definition of winding number.

• Foundation Circle Covering Circle Homeo Carrier

  A homeomorphism is a continuous, reversible stretching; this one proves the abstract circle and the familiar unit circle are the same topological object.

• Foundation Circle Covering Is Covering Map Trig Circle Point

  The map sending each real number t to (cos t, sin t) on the unit circle is a covering map, letting topology count loop windings.

• Foundation Circle Covering Iso E

  A single declaration in the Recognition Science library identifies the complex plane's unit circle with the real plane's unit circle, a bridge that makes the trigonometri

• Foundation Circle Covering Ulift Carrier Covering Eq Trig

  The circle's standard parametrization by cosine and sine is not just a formula: it is a covering map, a fact that underwrites the winding number.

• Foundation Circle Fundamental Simplex

  The circle's simplest loop, the path that goes around once and returns to its start, is built and verified as a formal object in the framework's machine-checked library.

• Foundation Circle Fundamental Simplex Fundamental Sphere One Singular One Simple

  A single formal declaration pins down one end of a loop around a circle; here is exactly what it proves and what it leaves alone.

• Foundation Circle H1 Computation

  A circle's one-dimensional hole is the integer line; this page shows how a machine-checked library pins that fact down.

• Foundation Circle H1 Computation Circle H1 Ziso Int Of Nonempty Homotopy Equiv O

  A machine-checked proof that the circle's one-dimensional hole is counted by the integers, and the honest limits of that result.

• Foundation Circle H1 Computation Homology One Nonempty Iso Int Of Quasi Iso At S

  A machine-checked theorem proves that any chain complex resembling a circle in one degree has the integers as its first homology group.

• Foundation Circle H1 Computation Homology One Nonempty Iso Int Of Quasi Iso Sing

  A machine-checked proof that any chain complex that looks like a circle at the level of its algebraic skeleton has the integers as its first homology group.

• Foundation Circle H1 Computation Ordinary Cellular Circle Chain Model H1 Nonempt

  Two different algebraic models of a circle have the same first homology group, a fact that is proved but not yet connected to the standard topological circle.

• Foundation Circle H1 Computation Ordinary Cellular To Reduced Comp Reduced Cellu

  A machine-checked proof shows that two ways of building a circle's skeleton are exact inverses, a small but necessary step toward a larger goal.

• Foundation Circle H1 Computation Reduced Cellular To Ordinary Comp Ordinary Cell

  Two algebraic descriptions of a circle's one-dimensional holes are shown to be interchangeable, a necessary step before either can stand in for the real geometric circle.

• Foundation Circle H1 Computation Singular Homology Functor Sphere One Int Nonemp

  A machine-checked proof that the circle's one-dimensional hole is measured by the integers, and the honest limits of what that proof covers.

• Foundation Circle Lifting

  A small formal module proves the circle can be unwound into a line, the step that lets a winding number count turns unambiguously.

• Foundation Circle Lifting Is Covering Map Trig

  The real number line winds around a circle like thread on a spool: the declaration isCoveringMap_trig makes that picture precise enough for a machine to check.

• Foundation Circle Lifting Std Simplex Contractible Space

  The standard simplex, the building block of topological shapes, is contractible: it can be shrunk to a single point without tearing.

• Foundation Circle Lifting Std Simplex Simply Connected Space

  A standard simplex, the building block of shapes in topology, has no holes: every loop drawn on it can shrink to a point.

• Foundation Circle Lifting Trig Circle Point Eq Iff

  A single theorem identifies when two real numbers land on the same point of a circle, a fact that underpins the winding number.

• Foundation Circle Lifting Trig Circle Point Eq Iff Exp

  Two real numbers land on the same point of a circle exactly when they differ by a whole number of turns, a fact that underwrites the winding number.

• Foundation Circle Param Constant Sphere One Singular One Simplex Face One

  A formal proof that a constant path on a circle has both ends at the same point, and why that humble fact anchors a larger project.

• Foundation Circle Param Constant Sphere One Singular One Simplex Face Zero

  A machine-checked proof that the two ends of a constant path on a circle are the same point, and what that does not say about the circle's fundamental loop.

• Foundation Circle Param Constant Sphere One Singular One Simplex Faces Eq

  A circle's simplest building block, a constant path at a basepoint, has two ends that coincide; a machine-checked theorem records this trivial fact.

• Foundation Circle Param Continuous Trig Circle Vector

  The unit circle's standard parametrization, (cos t, sin t), is continuous: a small change in the angle produces a small change in the point.

• Foundation Circle Param Sphere One Base Vector Mem Sphere

  A unit circle needs a starting point; this theorem proves the obvious candidate actually lies on the circle.

• Foundation Circle Param Trig Circle Point Two Pi

  The unit circle's standard trigonometric parametrization, (cos t, sin t), returns to its starting point after one full turn of 2π.

• Foundation Circle Winding

  A number that counts how many times a path wraps around a circle, and the machine-checked proof that this count is a stable, well-defined invariant.

• Foundation Circle Winding Chain

  A machine-checked proof that the number of times a loop winds around a circle is a genuine topological invariant, not just a geometric accident.

• Foundation Circle Winding Chain Closed Singular One Chain List Spans Cycles Of F

  A machine-checked proof shows that counting how many times a loop winds around a circle gives a complete classification of all closed loops on the circle.

• Foundation Circle Winding Chain Closed Singular One Cycle Boundary Generate Of Z

  A winding number counts how many times a loop wraps around a circle; a machine-checked proof shows this count is consistent across different ways of drawing the loop.

• Foundation Circle Winding Chain Closed Singular One Cycle List Boundary Generate

  A winding number measures how many times a loop wraps around a circle; a formal proof shows this invariant respects the basic rules of adding and subtracting paths.

• Foundation Circle Winding Chain Cycle Winding Integral Of Free Boundary Kernel D

  A circle's loops carry a number that counts how many times they wrap around, and a machine-checked proof shows this count behaves like a boundary detector.

• Foundation Circle Winding Chain Oriented Cyclic Families Explicit Raw Prism Gene

  A winding number is a count of how many times a path loops around a circle; a machine-checked library proves the counting respects boundaries, but the full classification of loops

• Foundation Circle Winding Path Displacement Loop Int Mul

  The winding number counts how many times a loop wraps around a circle; a machine-checked proof pins down exactly when that count is an integer.

• Foundation Circle Winding Path Homotopic Rel Const Of Loop Winding Zero

  On a circle, a loop that winds around zero times can be shrunk to a point, and the framework's machine-checked library proves it.

• Foundation Circle Winding Path Lift Endpoint Eq Of Winding Zero

  A winding number of zero means a loop on a circle can be untangled to a point, and the framework proves a precise version of that fact.

• Foundation Circle Winding Path Lift Shifted Exists Norm Bound

  A path on a circle can be unwound into a line, and the unwinding is always confined within a finite band.

• Foundation Circle Winding Path Winding Fundamental Loop

  The winding number counts how many times a path wraps around a circle; one full loop has winding number 1.

• Foundation Circle Winding Trig Circle Point Add Int Mul Period

  A single formal theorem pins down the exact period of the circle's defining map, and it is the keystone for measuring how far a path winds around the circle.

• Foundation Ckmhierarchy From Phi Ladder

  The six quark masses, spanning five orders of magnitude, are placed on a geometric ladder where each step multiplies by the golden ratio.

• Foundation Ckmhierarchy From Phi Ladder Ckm Hierarchy One Statement

  The Standard Model's six quarks span five orders of magnitude in mass; this theorem places them on a geometric ladder with a fixed ratio between steps.

• Foundation Ckmhierarchy From Phi Ladder Mass Geometric

  A machine-checked theorem says that in one framework, quark masses must sit on a ladder where each step multiplies by the golden ratio.

• Foundation Ckmhierarchy From Phi Ladder Mass Ratio Top Up Above 30000

  The heaviest quark is more than 30,000 times heavier than the lightest, a gap the framework derives from a single scaling number.

• Foundation Ckmhierarchy From Phi Ladder Mass Ratio Top Up Pos Band

  A machine-checked theorem pins the ratio of the heaviest to the lightest quark mass to a specific positive band, but it stops far short of matching experiment.

• Foundation Ckmhierarchy From Phi Ladder Quark Rungs Strict Ordering

  A machine-checked theorem orders the six quark masses by placing each on a rung of a golden-ratio ladder, but it does not itself predict any measured mass.