OCME · Bilingual mathematics & derivations
Foundation Gaps: Natural Numbers, Expressions, Sets, and Finite Limit Models
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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
Fix N including 0; list successor windows, two partial predecessors, closed-interval sizes, and rational candidates for N. Zero has no natural predecessor in this protocol; do not conflate it with truncated pred, negative values, or fractions.
Set
Use finite sets of an explicit integer type: deduplication/sorting, equality with both differences, candidate membership partitions, and complete powersets of at most four elements. The empty set’s powerset contains the empty set; no unrestricted comprehension or new axiom is asserted.
Set membership
Existing canonical prerequisite for this density pulse.
Function mapping
Existing canonical prerequisite for this density pulse.
Arithmetic operations
Use rational arithmetic, or natural inputs with Q results: ordered four operations, parenthesized pairs, inverse candidates with signed defects, and natural eligibility of differences/quotients. Division by zero is undefined, negative differences are not truncated to zero, and quotients are not integer division.
Subset
Compute all A-outside-B obstructions, non-strict/strict inclusion with both witnesses, three chain-obstruction sets, and typed candidate masks. Empty-set inclusion and equality are not strict inclusion, and finite checks are not universal set proofs.
Sample space
Use explicitly declared possible-outcome protocols: ordered products, recording images and original fibers, missing/outside candidate outcomes, and separate outcome/event masks. Assign no probabilities and infer neither uniformity, independence, nor completeness of an unknown experiment.
Tends-to relation
Both filters are explicit finite principal filters, with a total map on complete finite types. Check image generators, f(S)-outside-T obstructions, typed candidates, and two composition-image routes; empty generators and the bottom filter are allowed. This finite model does not interpret a bare arrow as a general sequence/point limit or create new formal Evidence.
Variables and algebraic expressions
Specify four small expressions over Q, not a universal AST: two environments, simultaneous variable exchange, lexical shadowing, and the original domain of a/(x−b). A zero numerator does not remove an excluded point; expressions are not equations.
Path source & status
path-pulse-foundations-20261011 · candidate