Encyclopedia/All topics/Foundation

Foundation

Articles 2,941–2,979 of 2,979. Alphabetical by title.

• Foundation Universal Forcing Strict Realization Strict Logic Realization

  A machine-checked interface shows that counting structure emerges from a minimal set of logical primitives, without any pre-supplied counting device.

• Foundation Universal Forcing Strict Realization To Lightweight

  A formal bridge shows that a stripped-down description of logic still forces arithmetic, without letting the description smuggle in its own counting device.

• Foundation Universal Forcing Strict Realization Universal Forcing

  A machine-checked theorem shows that any structure obeying a few basic laws of comparison and combination must contain the natural numbers, and that this arithmetic is the same in

• Foundation Universal Forcing Universal Forcing Cert

  A machine-checked certificate guarantees that every admissible model of the framework's laws extracts the same arithmetic structure, no matter which model you start from.

• Foundation Universal Instantiation From Distinction

  A single distinction between two things is enough to build arithmetic, the framework's first universal step.

• Foundation Universal Instantiation From Distinction Eq Cost Ne One

  A two-valued cost function that answers a single question: are two things the same or different?

• Foundation Universal Instantiation From Distinction Exists Logic Realization Of

  Any collection with at least two different things in it can run the framework's basic logical machinery, with no extra structure assumed.

• Foundation Universal Instantiation From Distinction Exists Named Logic Realizati

  Any collection with at least two different things in it can be made to carry the framework's basic logical structure, with no extra assumptions.

• Foundation Universal Instantiation From Distinction Logic Realization Of Distinc

  A single distinction between two points is enough to build a working logical structure, and the step that moves between them is the simplest possible: everything points at the seco

• Foundation Universal Instantiation From Distinction Universal Instantiation Cert

  A bare distinction between two points is enough to build a full logical structure, and the certificate records that fact.

• Foundation Unknot Complement Retract

  A circle in 4D space, and the space around it, turns out to have a hidden loop that can be pulled back onto itself.

• Foundation Unknot Complement Retract Coord23 Eq Zero Of Mem Range

  A small lemma about where an unknot sits in four-dimensional space, and the precise topological fact it establishes.

• Foundation Unknot Complement Retract Part23 Continuous

  A small lemma about a coordinate projection being continuous is the geometric core of a larger claim about detecting nontrivial linking in three dimensions.

• Foundation Unknot Complement Retract Retract Comp Core

  A circle can be pulled back onto itself from the space around an unknot, a topological fact that anchors a larger argument about linking.

• Foundation Unknot Complement Retract Retract Core

  A circle inside the space around an unknot can be shrunk back onto itself, a fact that marks the unknot as topologically nontrivial.

• Foundation Unknot Complement Retract Unknot Complement H1 Ne Zero

  In three-dimensional space, a simple circle has a complement that is not topologically trivial: its first homology group is nonzero, a fact the framework's library checks by m

• Foundation Unknot Complement Retract Unknot Injective

  A simple circle in three-dimensional space has a topological property that lets the framework detect nontrivial linking, proved in a machine-checked library.

• Foundation Variational Dynamics

  Foundation variational dynamics is the update rule that determines how the Recognition Science ledger evolves from one tick to the next.

• Foundation Variational Dynamics Constant Config Total Defect

  A single formula gives the total cost of a ledger state where every entry is the same number, and it is the key to why such states are the natural resting points of the dynamics.

• Foundation Variational Dynamics Eq Constant Config Of Defect Eq

  A machine-checked theorem pins down when a ledger's total defect equals a simple average, forcing every entry to the same value.

• Foundation Variational Dynamics Uniform Is Variational Successor

  A theorem about a ledger's evolution shows that the simplest possible next state is always a legal one, and it is the unique one.

• Foundation Variational Dynamics Variational Dynamics Certificate

  A single machine-checked theorem packages the core guarantees of a discrete system's evolution: existence, determinism, and the pull toward a state of rest.

• Foundation Variational Dynamics Variational Dynamics Deterministic

  In a discrete ledger of recognition events, the rule for moving from one moment to the next is not chosen: it is forced by minimizing a fixed cost, and that rule leaves no room for

• Foundation Variational Dynamics Variational Implies Recognition Step

  A machine-checked theorem shows that a ledger which updates by minimizing its total cost always produces a valid recognition step, without specifying how that step is chosen.

• Foundation Wave Particle Duality3 From Jcost

  A single number measures how much a system has committed to being a wave or a particle, and the math shows the transition is smooth, not a switch.

• Foundation Wave Particle Duality3 From Jcost Wpduality3 Cert

  A machine-checked certificate packages three simple facts about a cost function, and its own documentation says it proves nothing specific to wave-particle duality.

• Foundation Weinberg Angle Rs5

  The weak mixing angle is a measured constant of particle physics; Recognition Science's module derives only a placeholder, not the value.

• Foundation Weinberg Angle Rs5 Weinberg Angle5 Cert

  A formal certificate in the Recognition Science library proves three general facts about a cost function, but it does not derive the Weinberg angle.

• Foundation Wightman Axioms Status

  The Wightman axioms are the standard rules for a quantum field theory. In Recognition Science, five of them are shown to follow from a single cost function.

• Foundation Wightman Axioms Status Lorentz Invariance

  A simple symmetry of a cost function is what the framework means by Lorentz invariance, and the proof stops well short of full physical relativity.

• Foundation Wightman Axioms Status Spectral Positivity

  In quantum field theory, a physical state's energy must be positive; this page explains what a machine-checked proof of that condition does and does not say.

• Foundation Wightman Axioms Status Vacuum Exists

  In quantum field theory, the vacuum is the state of lowest energy. In Recognition Science, a machine-checked theorem identifies it with a specific value of a cost function.

• Foundation Wightman Axioms Status Wightman Axiom Count

  In the Wightman framework for quantum field theory, a machine-checked library counts exactly five axioms, and states plainly what the count does and does not settle.

• Foundation Wightman Axioms Status Wightman Status Cert

  A machine-checked certificate records which of the Wightman axioms hold in the Recognition Science framework, and which remain open.

• Foundation Winding Charges

  Foundation winding charges are integer-valued, exactly conserved quantities derived from the net displacement of lattice paths, and in three dimensions they number exactly three.

• Foundation Winding Charges Cancelling Pair Zero Displacement

  A single formal theorem about lattice paths shows why a step and its exact reverse always cancel, and it quietly does the work of a conservation law.

• Foundation Winding Charges Insert Cancelling Preserves Winding

  A small formal lemma about lattice paths says that adding a step and its exact opposite changes nothing about the path's net motion, a fact that underpins how the framework de

• Foundation Winding Charges Three Independent Winding Charges

  In the Recognition Science ledger, three independent conserved quantities emerge from counting net steps along three axes.

• Foundation Winding Charges Winding Charges Certificate

  A single machine-checked theorem bundles the core facts about winding numbers, the integer-valued counts that conservation laws in the framework rest on.