OCME · Bilingual mathematics & derivations
Foundation Transforms: Angles, Number Lines, Equations, Factorials, and Circle Boundaries
On this page
- Points, lines, and planes
- Rational number
- Variables and algebraic expressions
- Natural number
- Arithmetic operations
- Euclidean length
- Euclidean distance in an equal-unit orthonormal frame
- Angle
- Real number
- Equation
- Factorial
- Circle boundaries and zero-radius degeneracy
- Linear equation in one variable
- Absolute value
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.
Points, lines, and planes
Existing canonical prerequisite for this density pulse.
Rational number
Existing canonical prerequisite for this density pulse.
Variables and algebraic expressions
Existing canonical prerequisite for this density pulse.
Natural number
Existing canonical prerequisite for this density pulse.
Arithmetic operations
Existing canonical prerequisite for this density pulse.
Euclidean length
Existing canonical prerequisite for this density pulse.
Euclidean distance in an equal-unit orthonormal frame
Existing canonical prerequisite for this density pulse.
Angle
Use exact nondirected degree values in [0,180]: guard splits, distinguish complements/supplements, and compute two threshold defects and three-part budgets. Sums above 180 are not angles in this entry. Infer no measurements from sketches, ray lengths, or directed rotations.
Real number
Check the rational-point embedding slice only: ordered midpoints, finite distinct points and least containing closed intervals, inclusive interval intersections/separation, and four directed offsets/distances. Touching is a point interval, not empty; the containing interval is not the point set’s topological closure. No Cauchy completeness, irrational classification, or general continuity proof.
Equation
Check original Q equation sides, finite candidates, and reversibility premises. Multiplication by zero may change solutions; with denominator x−h first exclude x=h, and cross-multiplication cannot restore it. Check both shifted sides and defects without a universal AST or general solving-completeness claim.
Factorial
Use N from 0, 0!=1, and exact BigInt products: adjacent ratios, empty-product windows, independently checked reports/base and recurrence defects, and omission of one existing factor with multiplication back. Bound canonical decimal reports; recurrence self-consistency and floating-point values are not factorial correctness. No general permutation, divisibility, or asymptotic proof.
Circle boundaries and zero-radius degeneracy
Use one equal-unit orthogonal Euclidean frame and a nonnegative integer radius. Check three squared residual/classification sets but only boundary membership, axis antipodes, radius-parameter scaling, and fixed-horizontal-offset square remainders. r=0 is only the center with collapsed endpoints; negative remainders have no real root. No general circle-API or similarity-theorem formal bridge.
Linear equation in one variable
Only a≠0 gives a one-variable linear equation over Q. Check normalized rows, parameter-dependent constant degeneracies, exact offsets around roots, and two rows’ roots/common solutions. a=0 can yield all Q or no solution but is not linear; no general linear-system or uniqueness formal Evidence.
Absolute value
Normalize rational representations before branch decisions; distinguish three-point distances from path defects, inclusive four-point radius tests, and rational-target roots/candidate audits. Negative targets have no roots and zero roots are deduplicated. Finite slices are not all real absolute-value properties, a universal triangle inequality, or general inequality solving.
Path source & status
path-pulse-foundations-20261014 · candidate