Encyclopedia/All topics/Cost

Cost

Articles 61–120 of 310. Alphabetical by title.

• Cost Convexity Deriv2 Jcost One

  A single machine-checked theorem pins down the curvature of the cost function at its resting point, and it is careful about what it does not say.

• Cost Convexity Jcost As Composition

  The recognition cost function J(x) = ½(x + x⁻¹) − 1 is strictly convex on positive numbers, a shape fact that underpins its uniqueness theorem.

• Cost Convexity Jcost Strict Convex On Pos

  A short proof that the recognition cost function bends upward on positive numbers, and why that curvature matters.

• Cost Convexity Jlog Strict Convex On

  The framework's cost function has a bowl-shaped graph, and that curvature is what makes a unique solution possible.

• Cost Convexity Strict Convex On Cosh

  A machine-checked proof shows the recognition cost function is strictly convex, which guarantees it has a single lowest point.

• Cost Cosh Quadratic Lower Bound

  The hyperbolic cosine grows at least as fast as a parabola, a fact the framework's cost function inherits.

• Cost Derivative

  The cost derivative is the rate of change of the J-cost function, and its linearization is the correct first-order description of how recognition cost changes under a small scaling

• Cost Derivative Deriv Jcost Eq

  The cost function J(x) = (x + 1/x)/2 - 1 has a simple derivative, and that derivative is the key to how the framework measures harm.

• Cost Derivative Differentiable At Jcost

  The cost of recognition changes smoothly with its input, a fact that lets the framework take derivatives and linearize harm.

• Cost Derivative Lin J Eq Derivative Times X

  A small calculus identity in the Recognition Science library: the first-order change in recognition cost equals the derivative times the multiplier, and nothing more.

• Cost Derivative Lin J Matches Harm Def

  A machine-checked identity shows that a linear approximation used in harm calculations is exactly the derivative of a cost function, nothing more and nothing less.

• Cost Derivative Lin J Unit

  A small lemma about a cost function's linear behavior at its zero point, and the exact boundary of what that lemma does not say.

• Cost F Eq J On Pos Of Averaging

  Any cost function that treats reciprocal values as equally costly and averages correctly must be the function J(x) = (x + 1/x)/2 - 1.

• Cost F Eq J On Pos Of Derivation

  A single function measures the forced cost of recognition, and a machine-checked proof shows it is the only one.

• Cost Fixed Point

  The cost fixed point is the golden ratio, the unique positive number that satisfies the cost function's self-consistency equation.

• Cost Frequency Ladder

  A simple cost rule for comparing two frequencies forces a specific next note: the golden ratio.

• Cost Frequency Ladder Frequency Ratio Cost Unit

  A single theorem in a machine-checked library pins down the cost of a frequency ratio of one: it is exactly zero, and nothing else follows from it alone.

• Cost Frequency Ladder Is Self Similar Ratio

  A ratio that equals one plus its reciprocal, a property with a single positive solution: the golden ratio.

• Cost Frequency Ladder Phi Cost Fixed Point

  The golden ratio is the only positive number that equals one plus its own reciprocal, a property that makes it the cheapest nontrivial frequency ratio in a formal cost model.

• Cost Frequency Ladder Phi Is Self Similar

  The golden ratio is the one positive number that equals one plus its own reciprocal, a property that makes it the cost-minimal step up any frequency ladder.

• Cost Frequency Ladder Phi Unique Self Similar

  The golden ratio is the only positive number that equals one plus its own reciprocal, and that uniqueness is what a machine-checked proof pins down.

• Cost Functional Equation

  The cost functional equation is the unique formula for recognition cost forced by five plain conditions, established in the kernel-checked library 4.

• Cost Functional Equation Aczel

  A single functional equation, with five plain conditions, forces the unique cost function that Recognition Science uses as its starting point.

• Cost Functional Equation Aczel Law Of Logic Forces Jcost Aczel

  A simple equation for the cost of recognition has exactly one solution, and a machine-checked proof forces the result.

• Cost Functional Equation Composition Log Curvature Forces Jcost

  A single equation governs the unavoidable cost of recognition, and a machine-checked proof forces the result exactly.

• Cost Functional Equation D Alembert Continuous Of Log Curvature

  A single regularity condition, called log-curvature, turns a purely algebraic functional equation into a proof that its only solution is the familiar hyperbolic cosine.

• Cost Functional Equation D Alembert Cosh Solution Of Log Curvature

  A single functional equation, known since d'Alembert's work on vibrating strings, forces its only smooth solution to be the hyperbolic cosine.

• Cost Functional Equation D Alembert To Ode General Theorem

  A single functional equation, the d'Alembert equation, forces its smooth solutions to obey a second-order differential equation, a bridge that Recognition Science uses to prov

• Cost Functional Equation Has Log Curvature Full Filter Forces Zero

  A small technical lemma about limits does quiet but essential work: it pins down the exact meaning of curvature in the framework's cost equation.

• Cost Functional Equation Law Of Logic Forces Jcost With Regularization

  A uniqueness theorem in a machine-checked library shows that any cost function obeying five plain conditions must take one specific form, and the proof needs an extra regularity as

• Cost Functional Equation Ode Regularity Continuous Of Smooth

  A single smoothness condition turns a functional equation into a differential equation, and the framework's library proves the bridge is safe.

• Cost Functional Equation Ode Regularity Differentiable Of Smooth

  A smoothness assumption that lets a functional equation become a differential equation, and the exact limit of what it proves.

• Cost Functional Equation Strict

  A sharper version of the cost equation needs only two conditions, not five, and it was once vacuous until a fix gave it real content.

• Cost Functional Equation Strict Composition Log Curvature Forces Jcost Unconditi

  A single equation pins down the cost of recognition from just two conditions, with no hidden assumptions.

• Cost Functional Equation Strict Law Of Logic Forces Jcost Of Log Calibration

  A single equation pins down the only possible cost of recognition, and a stricter version of the proof needs just two assumptions.

• Cost Gauge Orbit Classification

  A machine-checked proof shows that every well-behaved cost function in the framework is either a simple sign check or a signed power, with no other options.

• Cost Gauge Orbit Classification Charges At Two Iff Not Sign Gauge

  A single number, the cost at ratio two, decides whether a recognition ledger is a pure sign detector or something richer.

• Cost Gauge Orbit Classification Gauge Orbit Is Signed Power Family Of Six Expone

  A machine-checked proof shows that under one extra assumption, every cost function in the framework belongs to one of two simple families.

• Cost Gauge Orbit Classification Nontrivial Is Signed Power

  A machine-checked theorem pins down every non-degenerate recognition cost as a simple signed power, and says precisely where the proof stops.

• Cost Gauge Orbit Classification Sign Gauge Sees Orientation Only

  A single theorem in a machine-checked library pins down the simplest possible cost rule: it reads only whether a ratio is positive or negative, nothing else.

• Cost Gauge Orbit Classification Strict Somewhere Iff Charges At Two

  A single condition on the cost at one number, 2, decides whether a recognition cost function is trivial or structured, and the proof is machine-checked.

• Cost Gauge Orbit Classification Vanishes At Two Iff Trace Two

  A single number, the trace of the ratio 2, decides whether a cost function collapses to a trivial sign gauge or carries real information.

• Cost Gauge Orbit From Real Character

  A machine-checked proof shows that under the framework's structural conditions, every cost function is either a simple sign gauge or a signed power, and nothing else.

• Cost Gauge Orbit From Real Character Gauge Orbit Is Sign Or Odd Power Family Ref

  A proposed tidy classification of cost functions fails: the framework's own axioms admit a family of exceptions, so the classification is false.

• Cost Gauge Orbit From Real Character Real Character Factorization Hypotheses Of

  A machine-checked theorem shows that any cost function satisfying the structural ledger conditions automatically has the real-character factorization form, and it is silent on whic

• Cost Gauge Orbit From Real Character Sign Gauge Native Cost Character Exponent Z

  A simple three-valued cost function, which only records the sign of a ratio, turns out to be a genuine recognition cost with a character exponent of zero, a result with sharp limit

• Cost Gauge Orbit From Real Character Sign Gauge Native Cost Character Not Odd Po

  A cost function that reads only the sign of a number turns out to be irreducible: it cannot be written as any odd power of that number, a fact the framework's machine-checked

• Cost Gauge Orbit From Real Character Sign Gauge Native Cost Not Odd Power Genera

  A simple three-valued cost function proves it cannot be reproduced by any odd-power rule, a result that sharpens the classification of recognition costs.

• Cost Gauge Orbit From Real Character Sign Gauge Native Cost Real Character Candi

  A simple rule that assigns a cost based only on the sign of a ratio turns out to be a structural solution, and its character is exactly the sign function itself.

• Cost Gauge Orbit From Real Character Signed Power Native Cost One Not Sign Gauge

  A machine-checked theorem shows that two different rules for assigning recognition costs cannot be the same rule, no matter how they are compared.

• Cost Gauge Orbit From Real Character Structural Sans Anchor Real Character Facto

  A machine-checked theorem proves that every structural cost function admits a real-character factorization, but it does not identify which factorization.

• Cost Geometric Root

  In Recognition Science, the cost of telling two states apart has a hidden geometric shape, and that shape forces the golden ratio.

• Cost Geometric Root Cosh Sub One Eq Two Sinh Sq Half

  A single hyperbolic identity that reframes the cost of recognition as a squared distance, and the exact limits of what that reframing proves.

• Cost Geometric Root Jcost Chain Excess Identity

  A single equation governs how the cost of recognizing two events in sequence exceeds the cost of recognizing them separately.

• Cost Geometric Root Jcost Eq Cosh Log Sub One

  A single formula ties the cost of recognizing a change to the hyperbolic cosine of its logarithmic size, and the formula's proof is checked by machine.

• Cost Geometric Root Jcost Subdivision Trivializes

  A proved theorem about splitting a cost into ever finer steps shows that an infinitely refinable ledger collapses, which forces discreteness as a structural necessity.

• Cost Geometric Root Jcost Superadd Strict Same Sign

  In a ledger that prices distinctions, combining two changes in the same direction always costs more than the sum of their separate prices.

• Cost Geometric Root No Cost Floor Under Refinement

  A proved theorem shows that splitting a distinction into ever finer steps drives its recognition cost to zero, which forces any ledger with a positive cost floor to stop refining.

• Cost Jcost Core

  Jcost core is the compatibility module that re-exports the canonical J-cost definitions and supplies the structural instances older Intelligence modules relied on.

• Cost Jcost Logic

  A single formula, forced by five plain conditions, prices every act of recognition in this framework.