Encyclopedia/All topics/Foundation

Foundation

Articles 1,981–2,040 of 2,979. Alphabetical by title.

• Foundation Primitive Recognition Calculus Hilbert Display Completion Born Weight

  In quantum mechanics, the Born rule turns an amplitude into a probability; in Recognition Science, bornWeight is the same operation on a finite display, and a theorem proves it mat

• Foundation Primitive Recognition Calculus Hilbert Display Completion Display

  A finite Hilbert space is a display of native F_RS[i] finite amplitudes; the bridge preserves Born weights, squared norm, and normalization.

• Foundation Primitive Recognition Calculus Hilbert Display Completion Display Nor

  In the Recognition Science framework, a formal theorem states that a quantum state's total probability is the same whether computed in its native representation or in a standa

• Foundation Primitive Recognition Calculus Hilbert Display Completion Finite Hilb

  A finite Hilbert space is a way of displaying the framework's own amplitudes, and the display provably preserves all the weights that matter.

• Foundation Primitive Recognition Calculus Hilbert Display Completion Norm Sq

  A simple mathematical tool, the squared norm, connects a framework's native amplitudes to a standard Hilbert space, preserving all comparisons.

• Foundation Primitive Recognition Calculus Inevitability

  Any formal system able to tell two distinct objects apart already contains the primitive recognition calculus, a result the framework's machine-checked library proves.

• Foundation Primitive Recognition Calculus Inevitability Admissible Foundation

  A formal system that can tell two distinct points apart already contains the seed of Recognition Science's primitive calculus.

• Foundation Primitive Recognition Calculus Inevitability Any Foundation Presuppos

  Every formal system that can tell two different symbols apart already contains the seed of Recognition Science's primitive calculus.

• Foundation Primitive Recognition Calculus Inevitability Prc Inevitability Certif

  A machine-checked theorem states that any formal system able to distinguish two distinct points already contains a primitive recognition calculus, with the external parsing work ke

• Foundation Primitive Recognition Calculus Inevitability Prcadmissible Foundation

  A formal theorem shows that any expressive formal system already contains the primitive recognition calculus, including the calculus itself.

• Foundation Primitive Recognition Calculus Inevitability Prcinevitability Target

  A formal theorem states that any foundation able to distinguish two points already contains a primitive recognition calculus, though the theorem's reach depends on how externa

• Foundation Primitive Recognition Calculus Integer Order

  A machine-checked library proves that the basic objects of Recognition Science, signed orbits, form a totally ordered line, and that this order behaves exactly like the usual order

• Foundation Primitive Recognition Calculus Integer Order Mul Recip Cancel Right A

  A formal rule about when multiplying by a reciprocal undoes itself, stated for a discrete arithmetic of signed orbits.

• Foundation Primitive Recognition Calculus Integer Order Negative Flag Mul Of Neg

  In the signed-orbit calculus, a negative times a nonnegative is negative, unless the nonnegative is zero.

• Foundation Primitive Recognition Calculus Integer Order Negative Flag Mul Of Non

  A formal theorem about a signed counting system pins down a familiar rule: a negative times a non-negative is negative, unless the non-negative is zero.

• Foundation Primitive Recognition Calculus Integer Order Num Mul Recip Num Balanc

  A small theorem about reciprocals shows how the framework's ledger keeps track of signs, and it pins down what happens when a number is zero.

• Foundation Primitive Recognition Calculus Integer Order Recip Num Balanced Negat

  A small theorem about fractions and signs shows when flipping a number upside down changes its sign, and when it cannot.

• Foundation Primitive Recognition Calculus Integer Order Recip Num Mul Num Balanc

  A precise rule about fractions in the framework's arithmetic: when the denominator is not zero, the reciprocal's numerator and the original denominator are the same size.

• Foundation Primitive Recognition Calculus Integer Order Recip Num Not Balanced N

  A small theorem about reciprocal numbers and their signs shows how the framework's discrete arithmetic keeps its order relations consistent.

• Foundation Primitive Recognition Calculus Integer Order Recip Num Not Balanced O

  A theorem about reciprocals in a discrete number system: the sign of the reciprocal's numerator is exactly the opposite of the sign of the original number's denominator.

• Foundation Primitive Recognition Calculus Integer Rational

  Before the framework can count anything, it must build the integers and rationals from scratch, out of pure distinctions.

• Foundation Primitive Recognition Calculus Integer Rational Abs Eq Zero Iff To In

  In the framework's internal arithmetic, a number is zero exactly when its absolute value is zero, a small bridge between two ways of representing signed quantities.

• Foundation Primitive Recognition Calculus Integer Rational Abs Ne Zero Of Not Ba

  A machine-checked theorem proves that in the framework's number system, a value that is not the zero element must have a positive absolute value, a small but load-bearing fact

• Foundation Primitive Recognition Calculus Integer Rational Abs Ne Zero Of To Int

  A small lemma about absolute values in a formal number system, and the precise boundary of what it proves.

• Foundation Primitive Recognition Calculus Integer Rational Negative Flag Eq True

  A small machine-checked theorem ties a bookkeeping flag for negative numbers to the ordinary integer comparison it represents.

• Foundation Primitive Recognition Calculus Integer Rational Nonneg Flag Eq True I

  A small machine-checked lemma ties a boolean flag to a mathematical property, and knowing exactly what it does not say keeps it honest.

• Foundation Primitive Recognition Calculus Integer Rational Ratio Orbit Equiv Iff

  A machine-checked theorem identifies two rational numbers exactly when their ratio orbits agree, tying the framework's discrete ledger to ordinary fractions.

• Foundation Primitive Recognition Calculus Integer Rational Signed Orbit Equiv Eq

  A signed orbit is a pair of counting numbers that records a position and a direction; the equivalence relation tells when two such records describe the same integer.

• Foundation Primitive Recognition Calculus Integer Rational Signed Orbit Equiv If

  A signed orbit is a pair of counting numbers that records a position and a direction, and the framework's library proves when two such records are the same.

• Foundation Primitive Recognition Calculus Kernel

  A machine-checked library of formal theorems certifies that the first stage of a recognition calculus has concrete logical objects, not just a paper sketch.

• Foundation Primitive Recognition Calculus Kernel Kernel First Pass Certificate

  A machine-checked certificate proves the framework's first chain of reasoning has concrete objects at every stage, without yet proving the chain's final conclusion.

• Foundation Primitive Recognition Calculus Multi Distinction Geometry

  Geometry emerges from the algebra of independent binary distinctions, not as a separate assumption.

• Foundation Primitive Recognition Calculus Multi Distinction Geometry Boundary Sq

  In the geometry of independent distinctions, the boundary of a boundary is always zero, a fact that turns simple bookkeeping into a foundation for space.

• Foundation Primitive Recognition Calculus Multi Distinction Geometry Diff Self C

  A machine-checked library records a trivial identity as a theorem, and the honest lesson is about what a formal system must not overclaim.

• Foundation Primitive Recognition Calculus Multi Distinction Geometry Face Bounda

  In the framework's discrete geometry, the boundary of a boundary is always zero, a fact that gives independent distinctions a consistent shape.

• Foundation Primitive Recognition Calculus Multi Distinction Geometry Multi Disti

  A machine-checked proof shows that the geometry of a square, and of higher-dimensional cubes, follows from the algebra of making independent binary distinctions.

• Foundation Primitive Recognition Calculus Multi Distinction Geometry Vtx

  The declaration Vtx names the four corners of a square, the simplest picture of two independent yes-or-no distinctions.

• Foundation Primitive Recognition Calculus Objecthood Registry

  A classification system that assigns every mathematical object in a theory one of seven commitment types, from forced to conventional.

• Foundation Primitive Recognition Calculus Objecthood Registry Background Object

  A machine-checked audit assigns every background object in a physical theory its proper kind of commitment, from forced to conventional.

• Foundation Primitive Recognition Calculus Objecthood Registry Classify Completio

  A formal theorem classifies the real number system as the unique completion of a countable process, and names the axiom that creates it.

• Foundation Primitive Recognition Calculus Objecthood Registry Classify Conventio

  In Recognition Science, the unit of cost is a free choice, like choosing inches over centimeters, and no measurement can tell the difference.

• Foundation Primitive Recognition Calculus Objecthood Registry Classify Forced Ra

  A machine-checked proof shows that every number system built on the real line must contain the rational numbers, a fact with a plain mathematical explanation.

• Foundation Primitive Recognition Calculus Objecthood Registry Classify Observabl

  In Recognition Science, a machine-checked theorem classifies observables as the probes that survive a physical quotient, and it makes no claim about which observables exist.

• Foundation Primitive Recognition Calculus Objecthood Registry Display Object Ext

  Complex numbers, Hilbert spaces, manifolds, measures, and physics display objects each carry a formal commitment tag in the Recognition Science objecthood registry.

• Foundation Primitive Recognition Calculus Omniscience

  A machine-checked library pins down exactly what it means to know everything about a sequence of yes-or-no answers, and how much of that knowledge is constructively available.

• Foundation Primitive Recognition Calculus Omniscience Llpo

  A precise boundary on what a finite observer can know by searching an infinite sequence.

• Foundation Primitive Recognition Calculus Omniscience Lpo Iff Wlpo And Markov

  A single theorem pins down exactly how much omniscience a constructive mathematician may assume, by splitting it into two weaker and independent principles.

• Foundation Primitive Recognition Calculus Omniscience Lpo Imp Llpo

  A theorem about infinite sequences of true-or-false values shows that a strong principle of knowing everything implies a weaker one, without any use of the law of excluded middle.

• Foundation Primitive Recognition Calculus Omniscience Lpo Imp Markov

  A machine-checked proof shows that one strong form of mathematical omniscience implies a weaker one, and the gap between them is exactly a known search principle.

• Foundation Primitive Recognition Calculus Omniscience Lpo Imp Wlpo

  A machine-checked proof shows that a strong form of omniscience implies a weaker one, a result that holds without the law of excluded middle.

• Foundation Primitive Recognition Calculus Omniscience Wlpo And Markov Imp Lpo

  A machine-checked proof shows that two weaker principles of omniscience, taken together, are exactly as strong as the full one.

• Foundation Primitive Recognition Calculus Orbit

  A minimal counting structure built from repeated acts of distinction, proven equivalent to the natural numbers.

• Foundation Primitive Recognition Calculus Orbit Arithmetic

  Orbit arithmetic is the arithmetic of counting repetitions in a discrete ledger, and it is exactly the arithmetic of ordinary whole numbers.

• Foundation Primitive Recognition Calculus Orbit Arithmetic Add Left Cancel

  In a formal system where counting is built from repeated acts of distinction, a theorem proves that equal sums force equal addends, a property familiar from ordinary arithmetic.

• Foundation Primitive Recognition Calculus Orbit Arithmetic Add Right Cancel

  Adding the same thing to both sides of an equation cannot hide a difference: that is what add_right_cancel proves for the framework's primitive counting positions.

• Foundation Primitive Recognition Calculus Orbit Arithmetic Add Succ Eq

  A single equation defines how counting works in the framework's primitive arithmetic.

• Foundation Primitive Recognition Calculus Orbit Arithmetic Add Zero Eq

  In a framework where counting begins from distinct marks, a single theorem states that adding nothing leaves a count unchanged, a fact so basic it is true by definition.

• Foundation Primitive Recognition Calculus Orbit Arithmetic Mul Ne Zero

  In a framework where counting starts from discrete recognition events, a machine-checked theorem proves that multiplying two nonzero counts never yields zero.

• Foundation Primitive Recognition Calculus Orbit Arithmetic Mul Succ Eq

  A single formal rule describes how multiplication behaves when one factor grows by one, and it is a theorem, not a definition.

• Foundation Primitive Recognition Calculus Orbit Arithmetic Mul Zero Eq

  A machine-checked proof that, in a discrete ledger of distinctions, counting nothing leaves you with nothing.