# Finite Structures: Pythagoras, Number Systems, Inference, Events, and Labeled Triangles

Deepen cross-domain question families on seven existing knowledge objects and their prerequisites, keeping mathematical sources unchanged and checking boundaries in exact finite slices.

[← Return to results](/en/paths/)

This is a suggested learning sequence, not a universal logical dependency or a formal proof.


- Stage 1 · required



## [Right triangle](/en/mko/mko-right-triangle/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261013%2F%3Ffrom%3D%252Fen%252Fpaths%252F)

Existing canonical prerequisite for this density pulse.

[Related questions](/en/explore/?mko=mko-right-triangle) [Concept guides](/en/derivations/?q=mko-right-triangle)


- Stage 1 · required



## [Euclidean length](/en/mko/mko-euclidean-length/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261013%2F%3Ffrom%3D%252Fen%252Fpaths%252F)

Existing canonical prerequisite for this density pulse.

[Related questions](/en/explore/?mko=mko-euclidean-length) [Concept guides](/en/derivations/?q=mko-euclidean-length)


- Stage 1 · required



## [Natural number](/en/mko/mko-natural-number/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261013%2F%3Ffrom%3D%252Fen%252Fpaths%252F)

Existing canonical prerequisite for this density pulse.

[Related questions](/en/explore/?mko=mko-natural-number) [Concept guides](/en/derivations/?q=mko-natural-number)


- Stage 1 · required



## [Proposition](/en/mko/mko-proposition/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261013%2F%3Ffrom%3D%252Fen%252Fpaths%252F)

Existing canonical prerequisite for this density pulse.

[Related questions](/en/explore/?mko=mko-proposition) [Concept guides](/en/derivations/?q=mko-proposition)


- Stage 1 · required



## [Sample space](/en/mko/mko-sample-space/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261013%2F%3Ffrom%3D%252Fen%252Fpaths%252F)

Existing canonical prerequisite for this density pulse.

[Related questions](/en/explore/?mko=mko-sample-space) [Concept guides](/en/derivations/?q=mko-sample-space)


- Stage 1 · required



## [Subset](/en/mko/mko-subset/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261013%2F%3Ffrom%3D%252Fen%252Fpaths%252F)

Existing canonical prerequisite for this density pulse.

[Related questions](/en/explore/?mko=mko-subset) [Concept guides](/en/derivations/?q=mko-subset)


- Stage 1 · required



## [Points, lines, and planes](/en/mko/mko-point-line-plane/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261013%2F%3Ffrom%3D%252Fen%252Fpaths%252F)

Existing canonical prerequisite for this density pulse.

[Related questions](/en/explore/?mko=mko-point-line-plane) [Concept guides](/en/derivations/?q=mko-point-line-plane)


- Stage 2 · optional



## [Pythagorean theorem](/en/mko/mko-euclid-pythagorean-theorem/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261013%2F%3Ffrom%3D%252Fen%252Fpaths%252F)

Use positive integer legs in an explicit Euclidean right-side protocol. Keep square fields ordered; only perfect squares yield positive integer hypotenuses or missing legs. A negative candidate c fails nondegenerate-length eligibility despite zero square residual. Preserve prior FELRA, vector-core, and declared side-model Evidence; no arbitrary affine bridge is completed.

[Related questions](/en/explore/?mko=mko-euclid-pythagorean-theorem) [Concept guides](/en/derivations/?q=mko-euclid-pythagorean-theorem)


- Stage 2 · optional



## [Integer](/en/mko/mko-integer/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261013%2F%3Ffrom%3D%252Fen%252Fpaths%252F)

Use explicit Z and rational candidates. Euclidean division requires b>0 and 0≤r<b, not truncated negative remainders. Separate net displacement from step magnitude, retain ordered opposites and directional differences, and classify zero/negative integers versus noninteger fractions; finite slices are not ring axioms.

[Related questions](/en/explore/?mko=mko-integer) [Concept guides](/en/derivations/?q=mko-integer)


- Stage 2 · optional



## [Logical connectives](/en/mko/mko-logical-connectives/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261013%2F%3Ffrom%3D%252Fen%252Fpaths%252F)

Evaluate two complete eight-row 0/1 classical tables for conjunction/disjunction, material implication/biconditional, both De Morgan routes, and inclusive/exclusive disjunction. Material implication is not causation; rowwise truths do not prove all inference rules or establish a general logic language.

[Related questions](/en/explore/?mko=mko-logical-connectives) [Concept guides](/en/derivations/?q=mko-logical-connectives)


- Stage 2 · optional



## [Event](/en/mko/mko-event/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261013%2F%3Ffrom%3D%252Fen%252Fpaths%252F)

Declare a nonempty finite S and distinguish F=P(S) from partition-union families. Compute complement/union/intersection, partial-block witnesses and least containing events, typed observations, and first occurrence. Finite S does not automatically make every subset measurable, and probabilities/independence are not supplied.

[Related questions](/en/explore/?mko=mko-event) [Concept guides](/en/derivations/?q=mko-event)


- Stage 2 · optional



## [Triangle](/en/mko/mko-triangle/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261013%2F%3Ffrom%3D%252Fen%252Fpaths%252F)

Use integer-labeled Euclidean A/B/C vertex data: distinctness/noncollinearity witnesses, qualified opposite-side squares, cyclic/swapped coordinate determinants, opposite midpoints, and three squared segments. Degenerate candidates are not triangles and finite coordinates are not general construction proofs.

[Related questions](/en/explore/?mko=mko-triangle) [Concept guides](/en/derivations/?q=mko-triangle)


- Stage 3 · optional



## [Rational number](/en/mko/mko-rational-number/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261013%2F%3Ffrom%3D%252Fen%252Fpaths%252F)

Represent rationals by integer numerators and nonzero denominators, including negative input denominators; outputs are exact reduced positive-denominator fractions. Check sum/difference, order and signed difference, guarded division/multiplication back, and missing-addend substitution. No floating-point approximation or complete Rat/field construction is claimed.

[Related questions](/en/explore/?mko=mko-rational-number) [Concept guides](/en/derivations/?q=mko-rational-number)


- Stage 3 · optional



## [Mathematical proof](/en/mko/mko-proof/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261013%2F%3Ffrom%3D%252Fen%252Fpaths%252F)

Use finite proposition labels, given facts, and single-premise MP edges. Check shortest depths, typing, candidate traces row by row, and rule-removal effects. Unseeded cycles cannot prove themselves and invalid rows never become premises. This toy protocol is neither general logical completeness nor human review or formal Evidence.

[Related questions](/en/explore/?mko=mko-proof) [Concept guides](/en/derivations/?q=mko-proof)



Path source & statuspath-pulse-foundations-20261013 · candidate

[Original paths JSON](/data/architecture/learning-paths.json)