Encyclopedia/All topics/Foundation

Foundation

Articles 2,581–2,640 of 2,979. Alphabetical by title.

• Foundation Schrodinger Derivation

  The Schrödinger equation, the rule for how quantum states change in time, emerges from a discrete eight-step recognition cycle rather than being assumed as a postulate.

• Foundation Schrodinger Derivation Discrete Schrodinger Eigenmode

  The Schrödinger equation, usually assumed as a postulate, appears here as a theorem about a simple eight-step cycle.

• Foundation Schrodinger Derivation Eigenmode Evolution Scaled

  A one-line theorem about how a quantum state's shape holds under time evolution, and the precise boundary of what it proves.

• Foundation Schrodinger Derivation Omega8 Pow Eq Evolution Factor

  A single equation in a machine-checked library connects the eighth roots of unity to the time evolution of a quantum state, showing how the Schrödinger equation can emerge from a d

• Foundation Schrodinger Derivation Schrodinger Difference Eigenmode

  A machine-checked theorem shows that a single step of a discrete evolution on an eight-point space reproduces the exact phase change of a quantum state, with the familiar Schröding

• Foundation Schrodinger Derivation Schrodinger Equation Cert Inhabited

  A single machine-checked declaration certifies that the Schrödinger equation follows from a discrete eight-step recognition cycle, while leaving the continuum limit as a bounded ap

• Foundation Schrodinger Derivation Schrodinger Equation From Rs

  A machine-checked derivation shows the Schrödinger equation as the exact time-evolution of a simple eight-step cycle, not as an independent postulate.

• Foundation Schrodinger Derivation Schrodinger Remainder Bound

  A machine-checked theorem bounds the error when a discrete eight-step quantum evolution is approximated by the continuous Schrödinger equation.

• Foundation Schur Pinch

  A method for proving that certain complex functions cannot have poles, built from two classical function classes and a map between them.

• Foundation Schur Pinch Cayley Inv

  A formula that turns a bounded complex number back into one with a non-negative real part, the core of a classical mapping between two halves of the complex plane.

• Foundation Schur Pinch Cayley Norm Le One

  A simple inequality about complex numbers, proven in a machine-checked library, maps one class of functions to another and underpins a template for excluding poles.

• Foundation Schur Pinch Cayley Schur Of Herglotz

  A machine-checked theorem shows that any function with non-negative real part maps, through a specific transform, into a function bounded by one.

• Foundation Schur Pinch Is Herglotz

  A Herglotz function is a complex function whose real part never dips below zero, a positivity condition that lets analysts control wild behavior.

• Foundation Schur Pinch Phase Le Half Pi Re Nonneg

  A machine-checked theorem that looks like it bounds a complex number's phase turns out to restate a triviality; the real content lives elsewhere in the framework.

• Foundation Schur Pinch Phase Lt Half Pi Re Pos

  A small theorem about complex numbers: if a number's angle from the positive real axis stays under 90 degrees, its real part is positive.

• Foundation Seam Bridge Bridge

  A machine-checked map that carries dynamical theorems from a primitive starting point into a full physical theory.

• Foundation Seam Bridge Bridge Coverage Tag

  A three-state label that forces every part of a formal theory to say honestly where its results come from.

• Foundation Seam Bridge Bridge Op Bridge

  A bridge is a map that lets theorems about one system be carried over to another, and OpBridge is the framework's machine-checked definition of that map.

• Foundation Seam Bridge Bridge Transport Fixed Point

  A formal bridge that lets a theory's stationary states be read off a simpler underlying structure, with a proof that fixed points carry across.

• Foundation Seam Bridge Bridge Transport Iterate

  A formal bridge carries every repeated step from a basic distinction into a full physical theory, preserving the pattern exactly.

• Foundation Seesaw Mechanism Rs V3

  A back-of-envelope formula for neutrino mass, and the small set of facts a machine-checked library actually proves about it.

• Foundation Seesaw Mechanism Rs V3 Seesaw Mech Rs V3 Cert

  A formal certificate proves three general properties of a cost function, but its name refers to a physics idea it does not actually establish.

• Foundation Self Bootstrap Distinguishability

  The module proves that a formal language can tell two propositions apart, and that this fact is distinct from its own denial, without deriving any object from nothing.

• Foundation Self Bootstrap Distinguishability Bool Distinguishable

  A single theorem about the Boolean type proves that the language of mathematics already contains a distinction, before any physical theory begins.

• Foundation Self Bootstrap Distinguishability Dist Claim Self Distinguishes

  A formal statement about distinct objects turns out to be distinct from its own denial, a small but exact fact about how logic itself works.

• Foundation Self Bootstrap Distinguishability Distinguishability Forced Given Obj

  A theorem in the framework's library shows that the claim 'there exist two different things' is distinct from its own denial, a fact about logic rather than physics.

• Foundation Self Bootstrap Distinguishability Distinguishability Lifted From Bool

  A small theorem shows that if a system can tell two things apart at all, then those two things are not the same object.

• Foundation Self Bootstrap Distinguishability Meta Language Distinguishes Props

  A formal language already separates true from false; the framework's theorem certifies that distinction and names what it cannot do.

• Foundation Self Bootstrap Distinguishability Prop Ne Not

  In logic, a statement is never identical to its own denial; the Recognition Science framework records this as a formal theorem and uses it as a floor for a larger argument.

• Foundation Sibridge Closure

  A machine-checked proof that the framework's native units convert to seconds, metres, and kilograms in exactly one way, once the measured gravitational constant is supplied.

• Foundation Sibridge Closure A L Eq Of C Constraint

  In the Recognition Science framework, the length of a tick is not a free parameter: the speed of light fixes it exactly.

• Foundation Sibridge Closure A M A T Eq Of C Hbar

  The framework's native units connect to everyday seconds and kilograms through a single measured constant, with the conversion algebra proved in a machine-checked library.

• Foundation Sibridge Closure A T A M Eq Of C G

  A machine-checked theorem fixes the ratio between two of the framework's units using only the speed of light and Newton's constant, without fitting.

• Foundation Sibridge Closure Hbar Rs Mul G Rs

  A simple identity inside the framework's own units: the product of its reduced Planck constant and gravitational constant equals one over pi, a fact that anchors the bridge to

• Foundation Sibridge Closure Si Bridge Closed Under Three Constraints

  A machine-checked proof shows that once three physical constants are matched, the framework's own units convert to seconds, metres, and kilograms in exactly one way.

• Foundation Sibridge Closure Si Bridge Closure Cert Inhabited

  The declaration proves that a consistent set of conversion factors exists to translate the framework's native units into SI units, conditional on measured values.

• Foundation Sibridge Closure Tau0 Eq Sqrt Pi Planck Time

  A machine-checked proof shows that the framework's native unit of time equals the square root of pi times the Planck time, a conversion that depends on one measured constant.

• Foundation Sibridge Closure Tau0 Predicted Seconds Pos

  A machine-checked theorem fixes the framework's fundamental unit of time as the square root of pi times the Planck time, a specific number of seconds.

• Foundation Simplicial Ledger

  The ledger that records recognition events is built from tetrahedra, not cubes, and a theorem proves its smallest self-consistent cycle needs exactly eight steps.

• Foundation Simplicial Ledger Eight Tick Uniqueness

  A formal proof shows any self-consistent recognition loop needs at least eight steps, a lower bound with a surprisingly simple engine.

• Foundation Simplicial Ledger H Local Global Unification

  A machine-checked library states a link between the cost of a whole system and the cost of its smallest pieces, and carefully marks what remains a hypothesis.

• Foundation Simplicial Ledger Is Recognition Loop

  A recognition loop is a closed chain of tetrahedra that must visit every possible three-bit pattern, forcing any such cycle to have at least eight steps.

• Foundation Simplicial Ledger Local Global Unification

  A theorem in the framework's machine-checked library says that when a whole system is at its lowest cost, every part of it is too, but the proof leans on a hypothesis that rem

• Foundation Simplicial Ledger Recognition Loop Has Surjection

  A recognition loop in the simplicial ledger must visit every one of the eight possible 3-bit local patterns at least once.

• Foundation Simplicial Ledger Simplex3

  A tetrahedron is the atom of volume in a ledger that records recognition events, and it forces a minimum of eight ticks per closed loop.

• Foundation Simplicial Ledger Simplicial Ledger

  A ledger of recognition events, usually pictured as a cubic grid, can instead be built from tetrahedra; the framework defines that shape and proves one property about the loops it

• Foundation Simplicial Ledger Simplicial Sheaf

  A sheaf is a way to stitch local data into a global picture; this one assigns a recognition potential to each tetrahedron in a simplicial ledger.

• Foundation Singular Mayer Vietoris

  A machine-checked proof that a space's shape can be assembled exactly from the shapes of two overlapping pieces, with a precise account of what happens at the seam.

• Foundation Singular Mayer Vietoris Epi Homology Map Of Elementwise

  A machine-checked theorem shows that a certain map between homology groups is surjective under a simple condition, and the proof rests on a classical topological tool.

• Foundation Singular Mayer Vietoris Exists Sd Op Iter Mem Small Span

  In algebraic topology, the singular chain complex of a space can be built from simplices that stay inside one of two open sets; a machine-checked proof shows this subcomplex is clo

• Foundation Singular Mayer Vietoris Is Iso Homology Map Chain Succ

  A machine-checked theorem shows that when two open sets cover a space, the homology of the whole can be rebuilt from the homology of the pieces.

• Foundation Singular Mayer Vietoris Is Iso Homology Map Of Elementwise

  A machine-checked proof shows that when two open sets cover a space, their individual homology groups assemble into the homology of the whole, with no hidden gaps.

• Foundation Singular Mayer Vietoris Mv Ses Short Exact

  A machine-checked proof shows that the small singular chain groups of two open sets fit into an exact sequence, the algebraic backbone of the Mayer-Vietoris theorem.

• Foundation Singular Mayer Vietoris Mv Sum Epi Of Left Univ

  A theorem about a special case in the Mayer-Vietoris sequence, where one of the two open sets is the whole space.

• Foundation Singular Mayer Vietoris Mv Sum Epi Zero

  A machine-checked theorem shows that when two open sets cover a space, every zero-dimensional cycle can be built from cycles living inside one of the two sets.

• Foundation Singular Mayer Vietoris Sub Sd Op Iter Eq Bnd Of Boundary

  In algebraic topology, a standard tool says the boundary of a boundary is zero; a machine-checked library has now verified a version of this for a framework built on discrete recog

• Foundation Singular Pair

  A singular pair is the basic setup of algebraic topology: a space, a subspace inside it, and the homology of the leftover part.

• Foundation Singular Pair Chain Map Comp Gen Retract

  A machine-checked lemma about topological spaces shows that an injective map between spaces induces an injective map on their singular chain complexes, with a precise algebraic ret

• Foundation Singular Pair Gen Comp Gen Retract

  A single lemma in a machine-checked library shows that an injective map between spaces cannot lose information when passed through the framework's discrete ledger.

• Foundation Singular Pair Pair Homology Map Comp Zero

  A machine-checked lemma about singular homology says that a certain two-step map always lands on zero, a fact that anchors a longer exactness proof.