OCME · Bilingual mathematics & derivations
Finite Structures: Pythagoras, Number Systems, Inference, Events, and Labeled Triangles
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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.
Right triangle
Existing canonical prerequisite for this density pulse.
Euclidean length
Existing canonical prerequisite for this density pulse.
Natural number
Existing canonical prerequisite for this density pulse.
Proposition
Existing canonical prerequisite for this density pulse.
Sample space
Existing canonical prerequisite for this density pulse.
Subset
Existing canonical prerequisite for this density pulse.
Points, lines, and planes
Existing canonical prerequisite for this density pulse.
Pythagorean theorem
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.
Integer
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.
Logical connectives
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.
Event
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.
Triangle
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.
Rational number
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.
Mathematical proof
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.
Path source & status
path-pulse-foundations-20261013 · candidate