Encyclopedia/All topics/Foundation

Foundation

Articles 1,861–1,920 of 2,979. Alphabetical by title.

• Foundation Primitive Recognition Calculus Delta Real Calibration Unit Forced By

  A single measurement of how a cost function bends at its origin is enough to fix its unit, but only if that measurement is supplied from outside the discrete framework.

• Foundation Primitive Recognition Calculus Delta Real Display Real Forgetful

  A single machine-checked theorem collects the basic facts about how the framework's real numbers behave, and it proves nothing about the physical world.

• Foundation Primitive Recognition Calculus Delta Real Floor Double

  A machine-checked theorem pins down how rounding errors behave when a real number is repeatedly doubled, a small but load-bearing step in building real arithmetic from rational app

• Foundation Primitive Recognition Calculus Delta Real Lo Le Hi Cross

  A small lemma about nested rational intervals guarantees that every real number has a unique description as a shrinking chain of bounds.

• Foundation Primitive Recognition Calculus Delta Real Obs Eq Iff Value

  Two descriptions of a real number are observationally equal exactly when they pin down the same value, a bridge between what a ledger can see and what mathematics can prove.

• Foundation Primitive Recognition Calculus Delta Real Of Rat Obs Eq Iff

  When two exact rational numbers are fed into the framework's real-number construction, they are observationally equal exactly when they are the same number.

• Foundation Primitive Recognition Calculus Delta Real Value Canonical

  Every real number has a unique address in the framework's discrete ledger, and the address points back to the number.

• Foundation Primitive Recognition Calculus Delta Real Value Surjective

  A machine-checked theorem shows that the framework's discrete approximation process can name every real number, not just a convenient subset.

• Foundation Primitive Recognition Calculus Delta Real Width Real Bound

  A real number can be pinned down by nested rational intervals whose widths shrink to zero; the width bound is the rule that makes the pinning honest.

• Foundation Primitive Recognition Calculus Finite Certificate Transfer

  A machine-checked proof that finite data cannot faithfully certify the continuum, and what that limit means for the framework's ledger of recognition events.

• Foundation Primitive Recognition Calculus Finite Certificate Transfer Conservati

  A theorem about certificates shows when a statement about the continuum can be reduced to finite data, and when it cannot.

• Foundation Primitive Recognition Calculus Finite Certificate Transfer Everything

  A machine-checked proof shows that a certificate system which accepts every claim cannot actually identify anything, and what that means for the limits of finite proof.

• Foundation Primitive Recognition Calculus Finite Certificate Transfer No Sound F

  A machine-checked theorem shows that no finite system of certificates can both soundly and faithfully cover the real number line.

• Foundation Primitive Recognition Calculus Finite Certificate Transfer Sound Fait

  A machine-checked theorem shows that when finite certificates are sound and faithful, the things they certify can be counted, a result with sharp limits.

• Foundation Primitive Recognition Calculus Formal System

  A formal system is any rule-governed language that can tell two basic tokens apart; the framework proves its own minimal calculus fits inside every such system.

• Foundation Primitive Recognition Calculus Formal System Formal System Certificat

  A machine-checked proof that the primitive recognition calculus can be embedded into any formal system that can tell its two endpoints apart.

• Foundation Primitive Recognition Calculus Formal System Formal System Embedding

  A machine-checked theorem shows that any formal system able to tell two primitive tokens apart can host the recognition calculus's core structure, but it does not prove that e

• Foundation Primitive Recognition Calculus Formal System Prcembedding Into

  A formal bridge that lets a minimal recognition calculus speak inside any sufficiently expressive formal system, and the precise limit of that claim.

• Foundation Primitive Recognition Calculus Formal System Prcformal System Embeddi

  A formal system is any precise language with tokens and expressions; the theorem shows the primitive recognition calculus can be faithfully translated into any such system that can

• Foundation Primitive Recognition Calculus Formal System Prcformal System Express

  A formal system is expressive when it can tell its two starting tokens apart; Recognition Science proves its own minimal system can.

• Foundation Primitive Recognition Calculus Frscarrier

  A machine-checked library proves that all recognition calculations stay within a countable set of numbers, never touching the full continuum.

• Foundation Primitive Recognition Calculus Frscarrier Alpha Inv Is Term

  A machine-checked theorem confirms that the inverse fine-structure constant is a valid symbol in a finite language of numbers, not a claim about its value.

• Foundation Primitive Recognition Calculus Frscarrier Carrier Values Countable

  A machine-checked proof shows the framework's basic arithmetic values form a countable set, not the full continuum of real numbers.

• Foundation Primitive Recognition Calculus Frscarrier Carrier Values Proper

  A machine-checked proof shows that the set of numbers the Recognition Science framework can compute with is countable, and therefore cannot be the whole real number line.

• Foundation Primitive Recognition Calculus Frscarrier Carrier Values Subset

  A machine-checked proof shows that every value the Recognition Science framework can compute with belongs to a specific countable field, not the full continuum of real numbers.

• Foundation Primitive Recognition Calculus Frscarrier Has Protocol Display

  A protocol display is the framework's guarantee that any number its syntax can write down can also be produced by one of its basic processes.

• Foundation Primitive Recognition Calculus Frscarrier Rat Is Term

  A machine-checked theorem confirms that every rational number is a valid expression in the framework's finite language of constants.

• Foundation Primitive Recognition Calculus Frscomplex Amplitude

  Quantum amplitudes normally live in the complex numbers, but Recognition Science shows a finite description can carry them.

• Foundation Primitive Recognition Calculus Frscomplex Amplitude Born Weight Nonne

  In quantum mechanics, the Born rule turns a complex amplitude into a probability; a machine-checked theorem shows the framework's own version is always nonnegative.

• Foundation Primitive Recognition Calculus Frscomplex Amplitude Display Born Weig

  A machine-checked theorem shows that a finite, exactly described complex number gives the same probability weight whether computed inside its own system or in the familiar complex

• Foundation Primitive Recognition Calculus Frscomplex Amplitude Eval Im Mem

  A machine-checked theorem pins down where the imaginary part of a complex amplitude lives, and what that location means for the framework's description of reality.

• Foundation Primitive Recognition Calculus Frscomplex Amplitude Frsi Amplitude He

  Quantum states can be written with exact, finite descriptions; the complex numbers are only the display screen.

• Foundation Primitive Recognition Calculus Frscomplex Amplitude Normalized Iff Di

  A machine-checked theorem shows that a quantum state written in a finite, exact notation is normalized in that notation exactly when its display in ordinary complex numbers is norm

• Foundation Primitive Recognition Calculus Generable Real

  Even with a countable list of starting constants, most real numbers can never be written down by finite arithmetic.

• Foundation Primitive Recognition Calculus Generable Real Const Mem

  A machine-checked theorem states that every named constant in a countable family belongs to the field of generable reals, the smallest field closed under arithmetic and containing

• Foundation Primitive Recognition Calculus Generable Real Display Exceeds Generat

  A machine-checked theorem shows that some real numbers can be exhibited but never built from a finite recipe, drawing a hard line between what analysis can display and what a discr

• Foundation Primitive Recognition Calculus Generable Real Gen Field Countable

  A machine-checked theorem shows that only countably many real numbers can be built from any countable list of starting constants, while uncountably many others remain forever out o

• Foundation Primitive Recognition Calculus Generable Real Gen Field Is Operationa

  A small, countable field of real numbers can carry every operation a recognition system needs, even though it misses most of the continuum.

• Foundation Primitive Recognition Calculus Generable Real Gen Field Proper

  In Recognition Science, the set of real numbers that can be finitely generated from any countable list of constants is always countable, and therefore never the whole real line.

• Foundation Primitive Recognition Calculus Generable Real Rat Mem

  Every rational number can be built from scratch by field operations alone, no matter which constants a system names.

• Foundation Primitive Recognition Calculus Grow Delta Forced No Enumeration

  A machine-checked proof shows that no forced process can list the continuum, a result that anchors what Recognition Science can and cannot derive.

• Foundation Primitive Recognition Calculus Grow Delta Forced No Enumeration No En

  A machine-checked theorem shows that no infinite list can capture every infinite binary sequence, a result with a 19th-century pedigree.

• Foundation Primitive Recognition Calculus Grow Eta Completion M0a

  A construction that builds the real numbers from a ledger of rational ratios, without ever writing a decimal point.

• Foundation Primitive Recognition Calculus Grow Eta Completion M0a Cross Diff Of

  A small lemma about rational numbers that anchors a larger construction, and what it deliberately leaves unproved.

• Foundation Primitive Recognition Calculus Grow Eta Completion M0a Cross Diff Sel

  A single line in a machine-checked library proves that subtracting a rational number from itself always yields zero, a small but load-bearing step in building real numbers from rec

• Foundation Primitive Recognition Calculus Grow Eta Completion M0a Cross Diff Swa

  A small algebraic fact about rational differences that behaves like signed subtraction, and what it does not say.

• Foundation Primitive Recognition Calculus Grow Eta Completion M0a Cross Diff Tri

  A simple algebraic identity about fractions that underpins the framework's construction of real numbers from recognition sequences.

• Foundation Primitive Recognition Calculus Grow Eta Completion M0a Cross Eq Of Eq

  A machine-checked proof shows that the framework's construction of real numbers from recognition sequences loses no information: distinct rationals stay distinct.

• Foundation Primitive Recognition Calculus Grow Eta Completion M0a Equiv Equivale

  A formal proof that two sequences of ratios are interchangeable when they eventually agree, and the precise limits of that claim.

• Foundation Primitive Recognition Calculus Grow Eta Completion M0a Eta Respects C

  A small lemma in a machine-checked library proves that two rational numbers that agree in the cross-difference sense also agree as limits of constant sequences.

• Foundation Primitive Recognition Calculus Grow Eta Completion M0a Mk Eq Mk Of Eq

  A theorem about when two sequences of ratios count as the same real number, and the precise sense in which the framework's real numbers are built from them.

• Foundation Primitive Recognition Calculus Grow Forced Trichotomy

  A discrete ordering that never needs to guess: every two positions compare themselves by pure structure, not by omniscience.

• Foundation Primitive Recognition Calculus Grow Forced Trichotomy Forced Order De

  In the framework's discrete ledger, comparing two positions is a finite computation, not an act of omniscience.

• Foundation Primitive Recognition Calculus Grow Forced Trichotomy Leq Antisymm St

  A formal proof that a forced ordering on discrete positions is antisymmetric, and what that proof deliberately leaves out.

• Foundation Primitive Recognition Calculus Grow Forced Trichotomy Leq Total Bool

  In the framework's discrete recognition ledger, every two positions can be compared by a finite computation, with no appeal to classical logic.

• Foundation Primitive Recognition Calculus Grow Forced Trichotomy Leq Trichotomy

  A structural ordering on the framework's primitive objects is total, decidable, and needs no classical omniscience.

• Foundation Primitive Recognition Calculus Grow Integer Divisibility

  A machine-checked library proves that divisibility, the workhorse of elementary number theory, survives intact inside a universe built from discrete recognition events.

• Foundation Primitive Recognition Calculus Grow Integer Divisibility Balanced Of

  A small theorem in the framework's machine-checked library says two orbits carry the same integer exactly when they are balanced, a fact that underpins a divisibility relation

• Foundation Primitive Recognition Calculus Grow Integer Divisibility Balanced To

  In the Recognition Science framework, a small theorem called balanced_toInt_eq says that two objects with the same balance also have the same integer value, a bridge between a stru

• Foundation Primitive Recognition Calculus Grow Integer Divisibility Dvd Z

  A formal definition of divisibility for signed orbits, proven to behave like ordinary integer divisibility.