# Foundational Reasoning and Geometry: Induction, Counting, Typing, and Length

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

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


- Stage 1 · required



## [Natural number](/en/mko/mko-natural-number/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261012%2F)

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 · optional



## [Proposition](/en/mko/mko-proposition/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261012%2F)

Use classical integer comparison statements and signed arithmetic witnesses. Mark unassigned free variables as open; separate equality and negation, candidate reports, actual truths, and proof terms. Do not create a universal logic language or new formal Evidence.

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


- Stage 1 · required



## [Arithmetic operations](/en/mko/mko-arithmetic-operations/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261012%2F)

Existing canonical prerequisite for this density pulse.

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


- Stage 1 · optional



## [Set membership](/en/mko/mko-set-membership/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261012%2F)

Use finite typed sets and complete characteristic tables. Outside-Fin(n) candidates are ill-typed, not blindly evaluated predicate inputs; distinguish membership from insertion/deletion and finite predicate extension. Preserve existing Set α→Prop Evidence without treating containers as all of set theory.

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


- Stage 1 · optional



## [Right triangle](/en/mko/mko-right-triangle/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261012%2F)

Qualify candidate positive integer angles and exact sum 180 before checking 90; two 90s and a zero fail. Preserve vertex/opposite-side labels and ordered acute angles; convert rational π multiples exactly, without tolerance or periodic wrapping. Candidate data are not a constructed nondegenerate triangle.

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


- Stage 1 · optional



## [Euclidean length](/en/mko/mko-euclidean-length/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261012%2F)

Use integer coordinates in one equal-unit orthogonal Euclidean L2 plane. Preserve signed displacement and squared lengths; give integer principal distances only for perfect squares. Check exact rational midpoints and both half-segment squares; separate axis path sums from direct endpoint length, retain zero distances, and do not claim all metric axioms or replay old floating-point Evidence as new.

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


- Stage 2 · optional



## [Mathematical induction](/en/mko/mko-mathematical-induction/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261012%2F)

N starts at 0; distinguish hypothesis index n from target n+1. Check explicit integer recurrence prefixes, affine-candidate base/step defects, and finite coverage only; finite samples are not induction for arbitrary n.

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


- Stage 2 · optional



## [Basic counting principles](/en/mko/mko-counting-principle/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261012%2F)

Explicitly tagged disjoint choices or full ordered products include empty branches and zero options. Later widths may depend on earlier choices; check candidate equal-width eligibility separately from total defects. Matching totals imply neither equal widths nor probabilistic independence; correct overlaps by actual set extension.

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


- Stage 2 · optional



## [Function mapping](/en/mko/mko-function-mapping/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261012%2F)

Total maps on complete finite types allow the empty domain and its unique empty table, but no total map from a nonempty source into an empty target. Distinguish graph, range, and unused codomain; outside queries are ill-typed, and compare both complete tables pointwise. Preserve existing total-function core Evidence; exclude partial, effectful, or failing programs.

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



Path source & statuspath-pulse-foundations-20261012 · candidate

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