Encyclopedia/All topics/Foundation

Foundation

Articles 841–900 of 2,979. Alphabetical by title.

• Foundation Logic As Functional Equation Logic Rcl Is Unique Functional Form Of L

  A comparison operator that treats all inputs fairly must take one specific algebraic shape, a result now checked by machine.

• Foundation Logic As Functional Equation Logic Satisfies Laws Of Logic L

  A machine-checked library shows that a comparison operation on a special kind of number obeys the same structural laws as ordinary logic, and that this forces a unique functional f

• Foundation Logic As Functional Equation Logic Scale Invariant L To Real

  A property called scale invariance, defined on a special kind of number, carries over to ordinary real numbers through a bridge that preserves its meaning.

• Foundation Logic As Functional Equation Logic Transport Comparison

  A bridge that carries the laws of logic from one mathematical setting to another, and what it leaves untouched.

• Foundation Logic As Functional Equation Non Contradiction And Scale Imply Recipr

  Two basic rules about comparing quantities, non-contradiction and scale invariance, are enough to force a symmetry that makes the comparison well-posed.

• Foundation Logic As Functional Equation Rcl Is Unique Functional Form Of Logic

  Logic, treated as a comparison between quantities, forces a single algebraic form for that comparison, a result with a machine-checked proof.

• Foundation Logic As Functional Equation Route Independence Implies Multiplicativ

  A single condition on how comparisons combine forces the cost function to obey a strict multiplicative rule, a step toward the framework's unique cost.

• Foundation Logic From Cost

  Logical consistency is the minimum-cost structure of recognition configurations, and this module establishes the core theorems in a machine-checked way.

• Foundation Logic From Cost Consistent Minimum Cost

  In Recognition Science, a consistent statement is the cheapest possible state: its cost is zero exactly when its presence is balanced at one.

• Foundation Logic From Cost Contradiction Positive Cost

  In a ledger where every configuration carries a price, a contradiction is either infinitely expensive or impossible, which is how logic gets a bill.

• Foundation Logic From Cost Logical Contradiction Impossible

  A formal proof shows that within a cost-based model of propositions, a contradiction cannot exist as a stable configuration.

• Foundation Logic From Cost Mp From Cost And Logic

  A machine-checked theorem shows that in one formal model, contradictions carry positive cost, while consistent statements can be free.

• Foundation Logic From Cost Prelogical Boolean Fragment

  A theorem in a machine-checked library shows that the basic operations of logic, AND, OR, and NOT, appear as the cheapest stable states of a cost function.

• Foundation Logic From Cost Zero Cost Contradiction Forbidden

  In classical logic, a contradiction is simply impossible; in Recognition Science, the same ban appears as a fact about cost.

• Foundation Logic Real Constants Alpha Inv L Bounds

  A machine-checked theorem places the inverse fine-structure constant inside a narrow numerical window, but it does not derive the constant's value.

• Foundation Logic Real Constants Hbar L Eq Phi Inv Fifth

  A machine-checked theorem states that the reduced Planck constant equals the golden ratio to the minus fifth power, but only within a formal mirror of the real numbers.

• Foundation Logic Real Constants Kappa Einstein L

  A machine-checked library declares a framework constant for gravity's strength and proves it matches the established real-number value exactly.

• Foundation Logic Real Constants Phi L Gt One

  A single formal theorem confirms that the golden ratio, defined in a special number system, is greater than one; here is what that does and does not say.

• Foundation Logic Real Constants Phi L Gt One Point Five

  A machine-checked proof that a certain constant sits between 1.5 and 1.62, and what that bound does and does not say.

• Foundation Logic Real Constants Phi L Lt One Point Six Two

  A theorem in a machine-checked library pins the golden ratio below 1.62, confirming a bound that already held for the real-number version.

• Foundation Logic Real Constants Phi L Pos

  A machine-checked proof that the golden ratio is positive, and why that small fact matters for a framework that builds constants from logic.

• Foundation Logic Real Transcendentals

  A machine-checked library shows that the real numbers and their transcendental functions, like π and the exponential, exist inside a more primitive structure built from logic alone

• Foundation Logic Real Transcendentals Cosh L

  coshL is the hyperbolic cosine function, defined on a special number system, and it behaves exactly like the familiar one.

• Foundation Logic Real Transcendentals Cosh L Eq Exp

  A machine-checked proof that the framework's hyperbolic cosine obeys the standard exponential formula, and nothing more.

• Foundation Logic Real Transcendentals Exp L Log L

  The exponential and natural logarithm are inverse operations on positive numbers, a fact so basic that it underpins compound interest, radioactive decay, and the pH scale.

• Foundation Logic Real Transcendentals Log L Exp L

  The natural logarithm and exponential are inverse functions, and a machine-checked library confirms the same holds on its reconstructed real-number line.

• Foundation Logic Real Transcendentals Sqrt L Nonneg

  The square root of any recovered real number is never negative, a fact carried over from the standard real numbers.

• Foundation Logic Realization

  A single interface that lets different frameworks for logic plug into one forcing program, extracting arithmetic from any setting that obeys the laws.

• Foundation Logic Realization Faithful Arithmetic Interpretation

  A machine-checked proof that the arithmetic forced by the laws of logic embeds without collision into the real numbers.

• Foundation Logic Realization Has Identity Step Of Nontrivial

  A small theorem in the Recognition Science library shows that any non-trivial system of comparison must contain a distinct starting point, the seed from which its arithmetic is ext

• Foundation Logic Realization Logic Realization

  LogicRealization is a formal interface that lets different systems of logic be compared by the arithmetic they force, not by their surface details.

• Foundation Logic Realization Positive Ratio Faithful

  A machine-checked proof shows that the arithmetic forced by the framework's laws embeds without collision into the positive real numbers.

• Foundation Logic Realization Positive Ratio Has Identity Step

  A machine-checked proof shows that any continuous, positive-ratio comparison system obeying the Laws of Logic has a nontrivial identity step, the seed from which arithmetic is extr

• Foundation Logic Realization Positive Ratio Interpret Injective

  A machine-checked theorem shows that the arithmetic a recognition process forces internally cannot collapse into the real numbers, a fact with a precise scope.

• Foundation Magnitude Of Mismatch

  A comparison that gives one answer for a pair of things must give the same answer regardless of order; the framework proves this is the only consistent reading.

• Foundation Magnitude Of Mismatch Asymmetric Not Single Valued

  A comparison that gives different answers when you swap the two things being compared cannot be a single, well-defined function on the pair.

• Foundation Magnitude Of Mismatch Equality Cost Single Valued

  A comparison that gives one answer regardless of order is the same thing as a symmetric comparison, and the equality-induced cost is one such comparison.

• Foundation Magnitude Of Mismatch Forces

  A single comparison function, applied to a pair without ordering, must treat both orders alike: the framework proves symmetry is forced, not chosen.

• Foundation Magnitude Of Mismatch Magnitude Of Mismatch Forced

  A comparison that gives one answer must give the same answer either way around; the framework's library proves this equivalence and names what it does not.

• Foundation Magnitude Of Mismatch Single Valued Implies Symmetric

  If a comparison between two things yields one value regardless of order, then that comparison is symmetric: the theorem is a plain fact about functions, proved in a machine-checked

• Foundation Magnitude Of Mismatch Single Valued On Unordered Pair

  A comparison that gives one answer regardless of the order of its inputs is, by definition, symmetric; a machine-checked theorem makes this equivalence precise.

• Foundation Magnitude Of Mismatch Symmetric Implies Factors Through

  A comparison that ignores the order of its two inputs is exactly the same thing as a comparison made on an unordered pair.

• Foundation Many Worlds From Jcost

  A branch of reality becomes observable only when its recognition cost crosses a fixed threshold set by the golden ratio.

• Foundation Many Worlds From Jcost Many Worlds3 Cert

  A machine-checked certificate for three basic facts about a cost function, and a warning about what it does not prove.

• Foundation Mass Weak Bases

  Quarks mix because the framework's cube assigns them to different axes depending on whether you ask about mass or about the weak force.

• Foundation Mass Weak Bases Cabibbo Largest Angle

  A machine-checked theorem ranks the quark mixing angles by a simple numerical gap, but it does not compute the angles themselves.

• Foundation Mass Weak Bases Ckm Hierarchy From Torsion Gaps

  A machine-checked theorem derives the observed ordering of quark mixing strengths from a single structural number: the gap between two torsion values.

• Foundation Mass Weak Bases Edge Dressed Prefers Axis0

  In the framework's model of particle generations, the middle generation's preferred axis is fixed by a simple count of bit flips, not by any fitted parameter.

• Foundation Mass Weak Bases Even Flip Involution

  A small symmetry in a three-generation model: flipping two of three axes twice brings every state back to itself.

• Foundation Mass Weak Bases Weak Complement Is Identity

  A small formal lemma about how three generations of quarks label the axes of an eight-dimensional space, and why that labeling matters for the CKM matrix.

• Foundation Mathlib Cohomology Bridge Circle H1 Mathlib Computation Iff Iso Int

  A formal bridge contract states when a machine-checked computation of the circle's first homology group is equivalent to a specific algebraic fact, and what that equivalence d

• Foundation Mathlib Cohomology Bridge Circle H1 Znonzero Of Mathlib Circle Linkin

  A machine-checked theorem proves the circle's first cohomology is nontrivial, a fact the framework needs to force three spatial dimensions.

• Foundation Mathlib Cohomology Bridge Mathlib Circle Linking Backend Nonempty Iff

  A formal bridge shows that the existence of a linking structure in the framework's library is exactly equivalent to a non-trivial cohomology group of the circle.

• Foundation Mathlib Cohomology Bridge Mathlib Circle Linking Backend Of Circle H1

  A formal bridge connects a machine-checked computation of the circle's first cohomology group to the framework's proof that linking forces three spatial dimensions.

• Foundation Maximal Forcing Admissible Realization

  A framework for deriving reality's laws uses a simple rule: never give up a degree of freedom without a fight.

• Foundation Maximal Forcing Admissible Realization Admissibility Class

  A formal container for what a physical theory is allowed to be, and the rule for narrowing it without fiat.

• Foundation Maximal Forcing Admissible Realization Forced After Tightening

  A claim forced on a wider class of allowed worlds remains forced when the class is narrowed, a monotonicity fact with a precise boundary.

• Foundation Maximal Forcing Admissible Realization Forced Of Forced Under Tighten

  A simple logical guarantee: if a claim is already forced, adding more rules cannot un-force it.

• Foundation Maximal Forcing Admissible Realization Legitimate Tightening

  In Recognition Science, a tightening is a rule that narrows which worlds count as possible; a legitimate one must prove it is not just a free choice.

• Foundation Maximal Forcing Admissible Realization Tightening Does Work

  A machine-checked proof shows that when a claim becomes forced only after adding a deeper law, some previously possible world must have been excluded.