Encyclopedia/All topics/Foundation

Foundation

Articles 2,221–2,280 of 2,979. Alphabetical by title.

• Foundation Primitive Recognition Calculus Prctype Theory Parse Tt System Expr Re

  A formal system is reflexive when every expression can be traced back to itself; the framework's machine-checked library proves this holds for the two-valued type theory it bu

• Foundation Primitive Recognition Calculus Prctype Theory Parse Tt System Express

  A machine-checked proof shows that the simplest possible two-symbol system already contains the full expressive core of Martin-Löf type theory.

• Foundation Primitive Recognition Calculus Prctype Theory Parse Tt System Not Deg

  A machine-checked proof shows the two-valued logic at the base of mathematics is rich enough to host the Recognition Science framework's primitive calculus.

• Foundation Primitive Recognition Calculus Prctype Theory Parse Type Theory Reali

  A machine-checked theorem shows that the two-element type in Martin-Löf type theory already contains the minimal structure Recognition Science needs to begin.

• Foundation Primitive Recognition Calculus Prime Axis Coherence

  Prime axis coherence is a theorem about when independent prime-number scales lock into one common power law.

• Foundation Primitive Recognition Calculus Prime Axis Coherence Character Is Rpow

  A theorem in the framework's machine-checked library shows that when independent prime factors are locked to one common scale, the resulting character is simply a power functi

• Foundation Primitive Recognition Calculus Prime Axis Coherence Log Char Log

  The natural logarithm is not just a function; in one formal account it is the unique way to assign additive weights to the prime numbers.

• Foundation Primitive Recognition Calculus Prime Axis Coherence Log Char Mul

  A simple rule about prime factors turns any assignment of numbers to primes into a function on all whole numbers that respects multiplication.

• Foundation Primitive Recognition Calculus Prime Axis Coherence Log Char Prime

  A small theorem in the framework's machine-checked library says a certain additive function reads back its own weight at every prime number.

• Foundation Primitive Recognition Calculus Prime Axis Coherence Power Law Iff Ali

  A single global power law holds exactly when the independent prime axes are locked to one common scale.

• Foundation Primitive Recognition Calculus Prime Axis Coherence Prime Axis Cohere

  Prime-axis coherence is a proved theorem about when independent prime-number scales collapse into one global power law.

• Foundation Primitive Recognition Calculus Prime Axis Coherence Weights Aligned

  WeightsAligned is a formal definition in the Recognition Science library that says when a set of prime-number weights are all proportional to a reference scale, a condition that fo

• Foundation Primitive Recognition Calculus Quantized Proof Method

  A method that turns continuous mathematical problems into finite checks, with a machine-checked library showing the reduction always works.

• Foundation Primitive Recognition Calculus Quantized Proof Method Application Stu

  Four famous unsolved problems appear in the framework's library as named placeholders, each carrying the same formal obligation but no solution.

• Foundation Primitive Recognition Calculus Quantized Proof Method Has Finite Redu

  A proof method that turns continuum problems into finite checks, with the hard Millennium problems as named targets.

• Foundation Primitive Recognition Calculus Quantized Proof Method Problem Audit

  A problem audit is a formal way of saying that checking a solution and checking a failure can both be reduced to checking a finite certificate.

• Foundation Primitive Recognition Calculus Quantized Proof Method Problem Audit F

  A machine-checked theorem shows that any continuum problem with a certificate-preserving audit reduces to finite certificates, but it does not solve any specific millennium problem

• Foundation Primitive Recognition Calculus Quantized Proof Method Quantized Proof

  A method that turns continuous problems into checkable finite cases, with its limits stated plainly.

• Foundation Primitive Recognition Calculus Quotient

  The module identifies endpoints that a recognition trace judges equivalent, and proves the identification is well-behaved enough to build on.

• Foundation Primitive Recognition Calculus Quotient Endpoint Class

  An endpoint class is a formal bucket that gathers every endpoint a recognition judgment treats as identical, and it is the smallest such bucket the framework's logic allows.

• Foundation Primitive Recognition Calculus Quotient Endpoint Class Eq Of Same

  When a recognition ledger judges two endpoints equivalent, the framework's formal library proves they occupy the same class, a step that makes counting by sameness possible.

• Foundation Primitive Recognition Calculus Quotient Endpoint Class Lift

  When two endpoints of a trace are judged equivalent, any function that respects that equivalence can be lifted to the class itself.

• Foundation Primitive Recognition Calculus Quotient Endpoint Class Of

  A formal construction groups endpoints that a trace judges the same; it does not say which endpoints are physically identical.

• Foundation Primitive Recognition Calculus Quotient Examples

  A quotient collapses states that no observable can tell apart; the examples show when the collapse is total, trivial, or exactly the definition.

• Foundation Primitive Recognition Calculus Quotient Examples Empty Observable Pha

  When no measurement can tell two states apart, the physical quotient fuses them into one: a toy example of how recognition forces equivalence.

• Foundation Primitive Recognition Calculus Quotient Examples Projective State Dis

  In the Recognition Science framework, two states are physically identical exactly when no admitted observable can tell them apart.

• Foundation Primitive Recognition Calculus Quotient Examples Quotient Examples He

  A machine-checked theorem bundles three examples showing how physical states collapse or separate when observables are admitted or withheld.

• Foundation Primitive Recognition Calculus Quotient Examples Separating Gauge Fam

  When every possible measurement is allowed, no two distinct states can ever look the same.

• Foundation Primitive Recognition Calculus Quotient Selection

  When two states look identical to every possible measurement, the framework's calculus treats them as one physical state, and it proves this collapse is forced, not chosen.

• Foundation Primitive Recognition Calculus Quotient Selection Forced Iff

  When two states look identical to every available measurement, the framework's mathematics identifies them, and this identification is not a choice but a logical consequence.

• Foundation Primitive Recognition Calculus Quotient Selection Gauge From Indistin

  When two states look identical to every available measurement, a forced quotient identifies them, and the identification is exact.

• Foundation Primitive Recognition Calculus Quotient Selection Identified Of Obs E

  When two states look identical to every available measurement, the theory treats them as one state, and this theorem makes that collapse precise.

• Foundation Primitive Recognition Calculus Quotient Selection Obs Equiv Refl

  Observational equivalence, the relation that collapses states no experiment can tell apart, is reflexive: every state is indistinguishable from itself.

• Foundation Primitive Recognition Calculus Quotient Selection Obs Equiv Symm

  When two states look identical through every available measurement, the relation is symmetric: if x is indistinguishable from y, then y is indistinguishable from x.

• Foundation Primitive Recognition Calculus Quotient Selection Observable Descends

  When two states look identical to every measurement you can make, the framework's mathematics says you may treat them as one physical state without losing any information.

• Foundation Primitive Recognition Calculus Quotient Selection Proj Injective Of S

  A machine-checked theorem about when collapsing indistinguishable states changes nothing, and the exact boundary of what it proves.

• Foundation Primitive Recognition Calculus Rational Field

  A rational number is a ratio of two whole numbers; Recognition Science rebuilds this familiar object from a discrete record of recognition events.

• Foundation Primitive Recognition Calculus Rational Field Div Mul Cancel

  In a number system built from recognition events, division cancels cleanly: dividing by a nonzero number and then multiplying by it returns the original value.

• Foundation Primitive Recognition Calculus Rational Field Inv Mul Cancel

  A machine-checked theorem confirms that in the framework's arithmetic, multiplying a nonzero number by its reciprocal always yields one, the same rule that governs ordinary fr

• Foundation Primitive Recognition Calculus Rational Field On Prcrat Normalized Re

  The cost of recognition in the framework's rational number system does not depend on which equivalent fraction you use to compute it.

• Foundation Primitive Recognition Calculus Rational Field Positive Ne Zero

  A positive rational number in this framework is one that is greater than zero, and the theorem positive_ne_zero proves that such a number cannot be zero.

• Foundation Primitive Recognition Calculus Rational Field Positive Normalize

  A machine-checked theorem ensures that the framework's ratio objects keep their sign when simplified, a small but load-bearing step in building its number system.

• Foundation Primitive Recognition Calculus Rational Field Rational Field Certific

  A machine-checked certificate that the framework's rational numbers form a genuine field, with division and positivity behaving exactly as in ordinary arithmetic.

• Foundation Primitive Recognition Calculus Real Boundedness Modulus

  A small fixed threshold in a recognition ledger's cost function guarantees that nearby entries stay close, a step toward building real numbers from discrete records.

• Foundation Primitive Recognition Calculus Real Boundedness Modulus Prc Real Boun

  A small step in a formal proof system that guarantees Cauchy sequences of rational numbers stay within bounds, a prerequisite for defining real numbers.

• Foundation Primitive Recognition Calculus Real Boundedness Modulus Prcboundednes

  A tiny rational number, one eighth, is the threshold that keeps the framework's recognition ledger from growing without bound.

• Foundation Primitive Recognition Calculus Real Boundedness Modulus Prccauchy Seq

  A machine-checked proof shows that sequences of rational numbers that converge under the framework's cost function stay within a fixed interval, a key step toward defining rea

• Foundation Primitive Recognition Calculus Real Cauchy

  A Cauchy sequence is the classical way to build real numbers from rationals; in Recognition Science it becomes a ledger of recognition costs that closes in on a limit.

• Foundation Primitive Recognition Calculus Real Cauchy Lt Iff To Rat Lt

  A machine-checked theorem ties a new way of ordering rational numbers to the familiar one, without claiming to define the real numbers themselves.

• Foundation Primitive Recognition Calculus Real Cauchy Prccauchy Seq

  A Cauchy sequence is a standard way to build real numbers from rationals; the framework's PRCCauchySeq is its version, built on a cost-based notion of closeness.

• Foundation Primitive Recognition Calculus Real Cauchy Prcjcost Distance Self Zer

  In a framework where recognition has a forced cost, the cost of recognizing a thing as itself is exactly zero, a fact that anchors how the framework builds real numbers.

• Foundation Primitive Recognition Calculus Real Cauchy Prcjcost Distance Symmetri

  A machine-checked theorem shows that a certain way of measuring the gap between two numbers treats them identically, no matter which is named first.

• Foundation Primitive Recognition Calculus Real Cauchy Prcreal Cauchy Certificate

  A machine-checked proof that the framework's real numbers exist as a completed structure, not just as an unfinished process.

• Foundation Primitive Recognition Calculus Real Cauchy Prcsquare Gap To Rat

  A machine-checked theorem shows how the Recognition Science framework measures distance between rational numbers, and what that measurement does not say.

• Foundation Primitive Recognition Calculus Real Cauchy Real Cauchy Certificate

  A machine-checked library of formal theorems proves that the framework's rational arithmetic can build a complete number system, a step toward treating real numbers as a recog

• Foundation Primitive Recognition Calculus Real Cauchy Zero Lt Of Positive

  A small theorem in a machine-checked library shows that a number's positivity is enough to place it after zero, a bridge that lets a theory of recognition build the real numbe

• Foundation Primitive Recognition Calculus Real Complete Ordered Field

  A real number is a completed orbit of a ledger, built from rational bookkeeping entries that settle ever closer together.

• Foundation Primitive Recognition Calculus Real Complete Ordered Field Prc Real C

  A formal certificate that lists the exact conditions under which a recognition-based number system would become the real numbers.

• Foundation Primitive Recognition Calculus Real Complete Ordered Field Prcjcost D

  A distance between two numbers that does not change when you shift both by the same amount is a familiar geometric idea, and a machine-checked proof now forces the result for a spe

• Foundation Primitive Recognition Calculus Real Complete Ordered Field Prcreal Ad

  In building real numbers from a discrete recognition ledger, addition is the first operation proven safe to use.