OCME · Bilingual mathematics & derivations
Foundational Reasoning and Geometry: Induction, Counting, Typing, and Length
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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.
Natural number
Existing canonical prerequisite for this density pulse.
Proposition
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.
Arithmetic operations
Existing canonical prerequisite for this density pulse.
Set membership
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.
Right triangle
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.
Euclidean length
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.
Mathematical induction
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.
Basic counting principles
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.
Function mapping
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.
Path source & status
path-pulse-foundations-20261012 · candidate