Encyclopedia/All topics/Foundation

Foundation

Articles 61–120 of 2,979. Alphabetical by title.

• Foundation Arithmetic From Logic Embed Strict Mono Of One Lt

  Counting numbers can be built from a single repeated step, and this theorem proves that the step preserves order.

• Foundation Arithmetic From Logic Log Generator Ne Zero

  Counting numbers can be built from a single repeated step, and one declaration pins down what that step is not.

• Foundation Arithmetic From Logic Lt Iff Le And Ne

  A single theorem shows that the natural numbers, built from a logic of comparison, order themselves exactly as school arithmetic expects.

• Foundation Arithmetic From Logic Pow Le Pow Iff Of One Lt

  A machine-checked theorem shows that in a specific framework, comparing powers reduces to comparing their exponents, with no hidden assumptions about bases.

• Foundation Arithmetic From Logic Pow Lt Pow Iff Of One Lt

  A machine-checked proof shows that comparing powers of a number greater than one reduces to comparing their exponents, a fact so basic it underpins the framework's constructio

• Foundation Arithmetic Of

  In Recognition Science, arithmetic is not assumed: it is forced into existence by the structure of recognition itself, and its counting numbers are the unique ones that can exist.

• Foundation Arithmetic Of Canonical Peano Surface

  A machine-checked theorem shows that the natural numbers, with zero and successor, form the unique arithmetic structure forced by the framework's logic.

• Foundation Arithmetic Of Extracted Peano Surface

  A machine-checked theorem shows that any structure satisfying the framework's basic conditions carries a full Peano arithmetic, identical in behavior to the natural numbers.

• Foundation Arithmetic Of Is Initial

  When a system provides a starting point and a way to step forward, the framework proves that only one counting structure can exist, up to relabeling.

• Foundation Arithmetic Of Logic Nat Lift Unique Fun

  A machine-checked theorem shows that the natural numbers, built from the framework's primitive logic, are the unique starting point for all counting structures.

• Foundation Arithmetic Of Peano Object

  A Peano object is the minimal structure that supports counting: a starting point, a way to move to the next thing, and nothing else.

• Foundation Arithmetic Of Realization Lift Unique Fun

  Peano arithmetic is the structure every counting system shares; a machine-checked library shows why any valid counting system maps onto it in exactly one way.

• Foundation Arrow Of Time

  Time's arrow, the stubborn one-way flow from past to future, may emerge from a purely geometric quantity called Berry phase.

• Foundation Arrow Of Time Before Asymm

  In the Recognition Science framework, time's arrow is defined by a quantity that only grows, making the relation 'before' a strict ordering.

• Foundation Arrow Of Time Before Irrefl

  A moment cannot be before itself: the simplest property of time's arrow, proved from a monotone measure of complexity.

• Foundation Arrow Of Time Before Transitive

  A formal proof shows that the framework's notion of "before" behaves like ordinary time: it is transitive, so if A is before B and B is before C, then A is before C.

• Foundation Arrow Of Time Forward Accumulates

  A machine-checked theorem shows why time has a direction: a certain measure of complexity only grows when steps run forward, and never shrinks when they run backward.

• Foundation Arrow Of Time Reverse Subtracts

  A machine-checked theorem shows that reversing a step in the framework's ledger subtracts the accumulated phase, while the measure of complexity keeps growing.

• Foundation Arrow Of Time Z Absolute Immune To Reversal

  A machine-checked theorem about absolute values underlies a proposed origin for time's arrow, but the physical bridge remains open.

• Foundation Axiom Discharge Plan

  A classical equation from 1968, once assumed as an axiom, is now proved from simpler pieces inside the Recognition Science framework.

• Foundation Axiom Discharge Plan Aczel Kannappan Via Cases

  A classical functional equation from 1966 now has a machine-checked proof that its only smooth solutions are constants, hyperbolic cosines, and ordinary cosines.

• Foundation Axiom Discharge Plan Cosh Rescaling Lemma

  A lemma that turns any smooth solution of a classical functional equation into a hyperbolic cosine, by rescaling its time variable.

• Foundation Axiom Discharge Plan Ode Cos Unit Uniqueness

  A simple differential equation pins down the cosine function exactly, and a machine-checked proof now confirms it without relying on an unexamined assumption.

• Foundation Axiom Discharge Plan Ode Cosine Case

  A machine-checked theorem pins down the only smooth function that solves a simple second-order equation with given starting values: the cosine.

• Foundation Biconditional Self Negation

  A statement that claims to be true exactly when it is false cannot exist, and the framework's library proves it.

• Foundation Biconditional Self Negation Classical Logic And Unique Minimizer Theo

  A classical logic fact, that no statement can be true exactly when it is false, is proved and applied to a recognition ledger.

• Foundation Biconditional Self Negation Complete Classical Logic And Closure

  A machine-checked theorem shows that no real-valued configuration can satisfy a statement equivalent to its own negation, and that exactly one configuration, unity, has zero defect

• Foundation Biconditional Self Negation Config Classification

  Every real-valued configuration in Recognition Science falls into exactly one of two categories: stable or outside, with no third option.

• Foundation Biconditional Self Negation Diverge Impossible

  A machine-checked proof shows no real-valued configuration can have an infinite defect, and explains why this has nothing to do with Gödel's incompleteness theorem.

• Foundation Biconditional Self Negation No General Self Negating Predicate

  A machine-checked proof shows no statement can be true exactly when it is false, a fact with a precise boundary.

• Foundation Biconditional Self Negation No Self Negating Config

  Classical logic itself forbids any configuration from satisfying the statement 'this configuration is not stable', a fact Recognition Science isolates and names.

• Foundation Biconditional Self Negation Self Negation Implies False

  A machine-checked proof shows no configuration can satisfy the statement "I am false," a classical-logic fact with a specific boundary against Gödel's incompleteness

• Foundation Bitkernel Families3

  A module that proves three general facts about a cost function, and honestly records that it proves nothing about its named subject yet.

• Foundation Bitkernel Families3 Bitkernel3 Cert

  A small machine-checked certificate shows that a certain cost function has three basic properties, but says nothing about dark energy or any specific physical system.

• Foundation Bitkernel4 Deep From Jcost

  A machine-checked file proves three general facts about a cost function, but its subject-specific meaning depends on a definition it does not contain.

• Foundation Bitkernel4 Deep From Jcost Bitkernel4 Deep Cert

  A formal certificate that packages three basic properties of a cost function, and the honest note that it proves nothing about the physics it was named for.

• Foundation Bool From Logic

  Boolean truth and falsity arise from the simplest possible act: making a distinction with two sides and nothing else.

• Foundation Bool From Logic Decoy Constant Bool Map Rejected

  A machine-checked proof shows that a map sending both sides of a distinction to the same value cannot be a faithful encoding, a small step in building Boolean logic from a single a

• Foundation Bool From Logic Decoy Swapped Bool Map Rejected

  A small formal proof shows why a deliberately wrong way of mapping a two-sided distinction to ordinary true and false cannot behave like a logical operation.

• Foundation Bool From Logic Eq Iff To Bool Eq

  A distinction has exactly two sides, and those two sides are exactly the two Boolean values, no more and no less.

• Foundation Bool From Logic From Bool To Bool

  A two-sided distinction and the machine's true and false are the same structure, and a round trip between them changes nothing.

• Foundation Bool From Logic To Bool And

  A two-sided distinction, with no numbers attached, is enough to recover the familiar Boolean operations of logic.

• Foundation Bool From Logic To Bool From Bool

  A small formal bridge shows that the two sides of a logical distinction translate exactly into the two Boolean values, with nothing lost and nothing added.

• Foundation Bool From Logic To Bool Not

  A machine-checked proof that the logical operation of negation, when translated into ordinary Boolean values, is exactly the standard NOT operation.

• Foundation Boolean Projection From Mark

  A set with at least two elements can be collapsed to true and false in many ways; choosing one distinguished point makes the collapse canonical.

• Foundation Boolean Projection From Mark Bool Projection Canonical Given Mark

  A two-valued shadow of a larger space becomes canonical exactly when someone first names two distinct points to keep apart.

• Foundation Boolean Projection From Mark Bool Projection Not Canonical Without Ma

  A Boolean value is a two-way choice, and a set with more than two elements cannot make that choice on its own.

• Foundation Boolean Projection From Mark Boolean Projection From Mark Cert

  A two-valued shadow of a set is canonical only after someone names a distinguished point, and the framework's certificate records exactly that boundary.

• Foundation Born Rule Forcing

  In quantum mechanics, the Born rule says the probability of an outcome is the squared amplitude. Recognition Science claims its own framework forces that rule, not from experiment

• Foundation Born Rule Forcing Contextual Measure Hybrid Witness Zero

  A machine-checked proof shows that a measurement rule which depends on the situation can differ from the standard quantum rule, and pins down exactly where they agree.

• Foundation Born Rule Forcing Contextual Measure Phase Invariant

  A theorem in the Recognition Science framework shows that the standard quantum probability rule, the Born rule, is the only possible choice once a few plain conditions are fixed.

• Foundation Born Rule Forcing Fourth Power Sum Pos Of Normalized

  A tiny lemma about eight numbers forces a key part of the Born rule in the Recognition Science framework.

• Foundation Born Rule Forcing Norm Complex Cos Of Real Of Nonneg

  A small lemma about complex numbers, the cosine bridge, is the hinge that lets a forced probability rule reach its final form.

• Foundation Born Rule Forcing Norm Complex Sin Of Real Of Nonneg

  A small formal lemma about the sine function, and the role it plays in a larger proof about measurement.

• Foundation Born Rule Forcing Occupied Modes Two Branch Card Le Two

  A small lemma about two-branch states bounds how many modes a state can occupy, and that bound is a step toward a larger uniqueness result.

• Foundation Born Rule Forcing Sector Measure Hybrid Witness Zero

  A machine-checked proof shows that the standard quantum probability rule is the only one that survives four plain conditions, and that a proposed alternative fails its own test.

• Foundation Branch Selection

  A structural requirement on how costs combine forces the unique form of a fundamental function, ruling out a competing alternative.

• Foundation Branch Selection Additive Branch Not Coupling

  A structural condition on how costs combine forces one of two possible cost functions, and rules out the other.

• Foundation Branch Selection Interaction Defect Eq Zero Of Separately Additive

  A simple formula detects whether two inputs to a combining rule merely add or genuinely interact, and that distinction decides which branch of cost functions the framework allows.

• Foundation Branch Selection Interaction Defect Rclcombiner

  A single formula detects whether a cost function treats its inputs as independent or as genuinely coupled, and that distinction settles a fork in the framework's derivation.