Encyclopedia/All topics/Algebra

Algebra

Articles 1–24 of 24. Alphabetical by title.

• Algebra Cost Algebra

  A single equation governs how the cost of recognizing two things together combines, and it forces the cost function's exact form.

• Algebra Cost Algebra Canonical Recognition Cost System Cost Inv

  A single number measures the recognition cost of any positive ratio, and the framework proves that swapping a ratio for its reciprocal leaves that cost unchanged.

• Algebra Cost Algebra Canonical Recognition Cost System Cost One

  A single algebraic rule governs how recognition costs combine, and its simplest case fixes the cost of doing nothing at zero.

• Algebra Cost Algebra Canonical Recognition Cost System Domain

  A single theorem in the framework's machine-checked library pins down where the cost function lives: all positive real numbers, no more and no less.

• Algebra Cost Algebra Continuous Bijective Preserves J Eq Id Or Inv

  A continuous, bijective map on the positive reals that preserves the cost function must be either the identity or the reciprocal map.

• Algebra Cost Algebra Cost Compose Fourfold Power Counterexample

  A simple algebraic check shows why the cost-composition operation, though natural, is not associative, and what that failure does and does not mean.

• Algebra Cost Algebra Defect Dist Le J Of Ratio Bounds

  A single machine-checked inequality says how close a composed cost stays to the simple cost of a ratio, and it does not say the bound is tight.

• Algebra Cost Algebra Defect Dist No Global Quasi Triangle

  A cost function that measures the gap between two values obeys a triangle-like bound only when the values are close, and the framework proves why the bound cannot hold globally.

• Algebra Cost Algebra Defect Dist Quasi Triangle Local

  A theorem in the framework's machine-checked library puts a precise limit on how much the cost of a ratio can grow when the two inputs stay within a bounded range.

• Algebra F2 Power

  A vector space over the two-element field is a set of binary strings where adding two strings means flipping the bits they share.

• Algebra F2 Power Axis1 Weight

  A tiny theorem about a three-bit string proves that one coordinate is on and the other two are off, a fact that anchors a larger count of story shapes.

• Algebra F2 Power Axis123 Weight

  A tiny formal theorem about a three-bit vector pins down a counting fact that narrative theory later leans on.

• Algebra F2 Power Card Weight Zero Three

  In a three-bit code, exactly one string has no 1s; a machine-checked proof pins down that elementary fact and its role in a larger counting scheme.

• Algebra F2 Power Hamming Weight Le

  A simple counting fact about binary strings, proved in a machine-checked library, that limits how many true bits any string of fixed length can carry.

• Algebra F2 Power Nonzero Card Three

  In a three-bit binary system, exactly seven of the eight possible states are nonzero; a machine-checked proof pins this down.

• Algebra F2 Power One Dim Subspace Card

  In the framework's algebra of binary strings, every nonzero vector generates a two-element subspace, and the theorem counts exactly how many such subspaces exist.

• Algebra F2 Power One Dim Subspace Closed

  In a binary vector space, every nonzero vector generates a two-element subgroup that is closed under addition, a fact that underpins a count of seven in the framework's narrat

• Algebra F2 Power Weight Zero Iff

  In the framework's algebra of on-off switches, a row of switches has zero on-positions exactly when every switch is off.

• Algebra Phi Ring Phi Equation

  The golden ratio's defining equation, phi squared equals phi plus one, is a proved theorem in a machine-checked library, not a definition.

• Algebra Phi Ring Phi Int Sq

  The golden ratio generates a number system where every quantity is an integer combination of 1 and φ, and the framework claims all its physical constants live there.

• Algebra Phi Ring Phi Psi Diff

  The golden ratio has a hidden twin, and the difference between them is the square root of five.

• Algebra Phi Ring Phi Psi Product

  The golden ratio has a quiet algebraic partner, and their product is the simplest surprise in the number system they generate.

• Algebra Phi Ring Phi Psi Sum

  The golden ratio has a twin, and the two numbers add up to exactly 1.

• Algebra Phi Ring Psi Equation

  The golden ratio's less famous sibling satisfies the same defining equation, and a machine-checked library proves it.