Encyclopedia/All topics/Foundation

Foundation

Articles 2,041–2,100 of 2,979. Alphabetical by title.

• Foundation Primitive Recognition Calculus Orbit Arithmetic Succ Add Eq

  A small theorem about counting steps shows that the order of adding one step to a count does not change the total.

• Foundation Primitive Recognition Calculus Orbit Divisibility

  A machine-checked library proves that the divisibility structure of a primitive counting system exactly matches the divisibility of the natural numbers, including a native definiti

• Foundation Primitive Recognition Calculus Orbit Divisibility Nontrivial Factoriz

  A theorem in the Recognition Science library shows that a number's divisibility structure is faithfully mirrored by its ordinary integer value.

• Foundation Primitive Recognition Calculus Orbit Divisibility Not Unit Of Nat Of

  In the framework's discrete counting system, the number one is the only unit, and a machine-checked theorem proves that no other natural number behaves like it.

• Foundation Primitive Recognition Calculus Orbit Divisibility Of Nat Ne Zero Of N

  In a formal system where numbers are positions on a recognition orbit, the declaration ofNat_ne_zero_of_ne_zero proves that a nonzero natural number never maps to the zero position

• Foundation Primitive Recognition Calculus Orbit Divisibility Orbit Divisibility

  A machine-checked certificate that prime orbit positions in the recognition ledger behave exactly like prime numbers, with no hidden axioms.

• Foundation Primitive Recognition Calculus Orbit Divisibility Prime Orbit Iff To

  A theorem in the Recognition Science library shows that certain positions in a discrete counting structure are exactly the ordinary prime numbers, with no extra conditions.

• Foundation Primitive Recognition Calculus Orbit Divisibility Prime Orbit Of Unit

  A prime number is usually defined by what divides it; this theorem shows the same idea can be rebuilt from the opposite direction, using only multiplication and the number one.

• Foundation Primitive Recognition Calculus Orbit Divisibility Unit Or Eq Of Divid

  A theorem about a number system built from recognition events proves the classical prime property: a prime's only divisors are 1 and itself.

• Foundation Primitive Recognition Calculus Orbit Divisibility Unit Or Unit Of Mul

  In the framework's arithmetic of recognition events, a prime cannot be split into two non-trivial factors: one factor must always be the unit.

• Foundation Primitive Recognition Calculus Orbit Equiv Nat

  A formal bridge shows the framework's basic counting steps and ordinary whole numbers are the same thing, with proofs checked by machine.

• Foundation Primitive Recognition Calculus Orbit Euclidean

  Euclidean division, the familiar schoolbook operation of quotient and remainder, turns out to be the first arithmetic that a discrete recognition ledger can force.

• Foundation Primitive Recognition Calculus Orbit Euclidean Coprime Divides Of Div

  A small number-theory lemma about coprime numbers, proved inside the framework's machine-checked library, and what it does and does not say.

• Foundation Primitive Recognition Calculus Orbit Euclidean Divides Gcd Of Divides

  A machine-checked theorem about a discrete counting system shows that any common divisor of two numbers also divides their greatest common divisor, a property familiar from ordinar

• Foundation Primitive Recognition Calculus Orbit Euclidean Gcd Ne Zero Of Right N

  A small theorem in the framework's arithmetic library guarantees that the greatest common divisor of two counting numbers is never zero unless both are zero.

• Foundation Primitive Recognition Calculus Orbit Euclidean Normalize Ratio Den Mu

  A machine-checked theorem about reducing ratios to lowest terms shows that the framework's bookkeeping for orbits obeys the same rule every schoolchild learns for fractions.

• Foundation Primitive Recognition Calculus Orbit Euclidean Normalize Ratio Num Mu

  A machine-checked theorem shows that dividing a ratio by its greatest common divisor preserves the ratio's value, a step toward a unique reduced form.

• Foundation Primitive Recognition Calculus Orbit Euclidean Quotient Mul Divisor A

  A machine-checked proof shows that in the framework's discrete arithmetic, dividing and taking a remainder always reconstructs the original number exactly.

• Foundation Primitive Recognition Calculus Orbit Euclidean Quotient Mul Divisor T

  When one counting number divides another exactly, the framework's division operation recovers the original number when multiplied back.

• Foundation Primitive Recognition Calculus Orbit Euclidean Signed Quotient Mul Di

  When one whole number divides another exactly, the quotient times the divisor recovers the original number, a fact the framework's machine-checked library proves for its own a

• Foundation Primitive Recognition Calculus Orbit Of Nat To Nat

  A tiny formal object shows how the framework counts its own primitive steps, and what that counting does not say.

• Foundation Primitive Recognition Calculus Orbit Succ Injective

  A simple theorem about counting steps guarantees that each new step is genuinely new, and it is the first rung of a ladder that Recognition Science climbs.

• Foundation Primitive Recognition Calculus Orbit To Nat Of Nat

  A small formal lemma proves that the framework's primitive counting steps and ordinary natural numbers are the same sequence, with no hidden assumption about what counting mea

• Foundation Primitive Recognition Calculus Orbit Zero Ne Succ

  A formal proof that the first step in counting is not a repeat of the starting point, and what that proof does and does not say.

• Foundation Primitive Recognition Calculus Physical One Act Calibration

  In Recognition Science, a single measurement act forces the unit of cost to be exactly 1, and the proof is machine-checked.

• Foundation Primitive Recognition Calculus Physical One Act Calibration Canonical

  A single measurement, one unit of cost, forced to equal one: the canonical instrument is the framework's simplest calibration device.

• Foundation Primitive Recognition Calculus Physical One Act Calibration Instrumen

  A single measurement of curvature, if it reads exactly one, forces the unit of cost to be one: the framework's calibration is a theorem, not a choice.

• Foundation Primitive Recognition Calculus Physical One Act Calibration One Act I

  A one-act instrument is a formal device that fixes the unit of recognition cost to exactly 1, and the framework proves any such device must do so.

• Foundation Primitive Recognition Calculus Physical One Act Calibration Physical

  A single measurement, if it reads exactly one, forces the unit of recognition cost to be one. That is the calibration theorem.

• Foundation Primitive Recognition Calculus Prccalibration Independence

  A family of cost functions all satisfy the core laws of Recognition Science, but only one of them is the distinguished cost J, and the module proves exactly why.

• Foundation Primitive Recognition Calculus Prccalibration Independence Calibratio

  The unit of scale in a recognition cost is not forced by the cost laws; it is the one free choice the framework leaves open.

• Foundation Primitive Recognition Calculus Prccalibration Independence Cost Lambd

  A one-parameter family of cost functions all obey the same composition law, which isolates calibration as the single choice that selects the canonical cost.

• Foundation Primitive Recognition Calculus Prccalibration Target

  A single family of cost functions survives the framework's forcing, and one number, a curvature, picks out the unique member the framework needs.

• Foundation Primitive Recognition Calculus Prccalibration Target Calibration Unit

  In the Recognition Science framework, the cost function's unit of scale is not fixed by the discrete structure itself; it remains a free positive real, a gauge, until one cali

• Foundation Primitive Recognition Calculus Prccalibration Target Clog Inj

  A family of cost functions has exactly one knob left to turn, and clog_inj is the proof that turning it always changes the function.

• Foundation Primitive Recognition Calculus Prccalibration Target Cost Freedom Is

  A family of cost functions leaves exactly one free real parameter, and that parameter is a scale, not a mystery.

• Foundation Primitive Recognition Calculus Prccalibration Target Cost Lambda One

  A single equation pins down the cost of recognition at unit scale, and the proof is a matter of algebra, not physics.

• Foundation Primitive Recognition Calculus Prccalibration Target Gauge Action Tra

  The cost of recognition is fixed up to a single positive number, and this theorem says that number is the only freedom left.

• Foundation Primitive Recognition Calculus Prccalibration Target Log Curvature

  A family of cost functions leaves exactly one free real parameter, a scale the discrete structure cannot fix.

• Foundation Primitive Recognition Calculus Prccategory Theory Parse

  Category theory's basic building blocks, truth values and subobjects, turn out to contain the minimal core that Recognition Science needs to get started.

• Foundation Primitive Recognition Calculus Prccategory Theory Parse Category Theo

  Category theory's basic building blocks already contain the minimal structure Recognition Science needs to begin its work.

• Foundation Primitive Recognition Calculus Prccategory Theory Parse Top Ne Bot

  A theorem that truth and falsity are distinct is the smallest possible guarantee that a system of logic is not empty.

• Foundation Primitive Recognition Calculus Prccategory Theory Parse Topos System

  A machine-checked theorem shows that the mathematical universe used by Recognition Science has at least two distinct truth values, so it is not a trivial one-point system.

• Foundation Primitive Recognition Calculus Prcchain Bridge

  A machine-checked bridge shows that the framework's first physical output, the golden ratio, lives in a countable field, never needing the full continuum.

• Foundation Primitive Recognition Calculus Prcchain Bridge Delta Cost Feeds Rs Ch

  A single machine-checked theorem connects the framework's cost of recognition to its first physical outputs, and shows those outputs never need the full continuum of real numb

• Foundation Primitive Recognition Calculus Prcchain Bridge Jcost Log Curvature On

  A single number, the curvature of a cost curve at its resting point, pins down the exact form of a universal cost function.

• Foundation Primitive Recognition Calculus Prcchain Bridge Jcost Log Eq Clog One

  A single formula reveals the hidden geometry of the framework's foundational cost function, and shows what that geometry does and does not force.

• Foundation Primitive Recognition Calculus Prcchain Bridge Phi In Minimal Field

  The golden ratio, long known as a geometric proportion, turns out to live inside a small, countable number system that the Recognition Science framework builds from its first princ

• Foundation Primitive Recognition Calculus Prccompleteness Independence

  A machine-checked proof shows that the real numbers' completeness is not forced by the cost laws, but is a separate, uncountable commitment.

• Foundation Primitive Recognition Calculus Prccompleteness Independence Countable

  The real numbers are complete, but no countable subfield can be: a theorem shows why the continuum is exactly what completeness buys.

• Foundation Primitive Recognition Calculus Prccompleteness Independence Jcost Is

  A single machine-checked theorem confirms the canonical cost function obeys its own defining laws, and proves that completeness is an extra commitment, not a consequence.

• Foundation Primitive Recognition Calculus Prccompleteness Independence Real Has

  The real numbers are the unique number system where every bounded collection has a least upper bound; a machine-checked proof shows this property is an independent commitment, not

• Foundation Primitive Recognition Calculus Prccompleteness Independence T Not Com

  A machine-checked theorem shows that the real numbers' defining completeness property cannot be derived from the Recognition Science cost axioms.

• Foundation Primitive Recognition Calculus Prccost On Field

  A single countable field contains every constant physics needs, and the cost function never leaves it.

• Foundation Primitive Recognition Calculus Prccost On Field Cost And Constants Sh

  A single countable field of real numbers holds the cost function, its iterates, and the constants π, φ, e, and α⁻¹, so the framework's arithmetic never needs the full continuu

• Foundation Primitive Recognition Calculus Prccost On Field Jcost Alpha Inv Mem T

  The inverse fine-structure constant, like pi and the golden ratio, lives inside a countable field of real numbers that the cost function never leaves.

• Foundation Primitive Recognition Calculus Prccost On Field Jcost Iterate Mem T

  A cost function that maps a countable field into itself, and what that closure means for the constants of physics.

• Foundation Primitive Recognition Calculus Prccost On Field Jcost Mem T

  A machine-checked theorem shows a single countable field of real numbers holds both the recognition cost function and the constants built from it.

• Foundation Primitive Recognition Calculus Prccost On Field Jcost Phi Mem T

  The golden ratio's recognition cost stays inside a countable field, a small set that never needs the full continuum.

• Foundation Primitive Recognition Calculus Prccost On Field Jcost Pi Mem T

  The recognition cost of pi is a number that belongs to the same countable field as pi itself, a fact the framework proves without any special assumptions.