Theorems & Proofs

An index of all named results across the wiki, with direct links.

Theorems

• Theorem (Exact divisor-rectangle description of the zero class) — in Zero-Class Geometry
• Theorem (Hyperbolic baseline minus arithmetic correction) — in Zero-Class Hyperbola Gap
• Theorem (Exact first-boundary formula) — in First Boundary Model $S^{(1)}$
• Theorem (Exact total first-boundary sum) — in First Boundary Model $S^{(1)}$
• Theorem (Exact reciprocal series) — in First Boundary Model $S^{(1)}$
• Theorem (Exact weighted first-boundary series) — in First Boundary Model $S^{(1)}$
• Theorem (Zero-class degeneracy criterion) — in The basic geometry of MTMN
• Theorem (Exact divisor-rectangle description of the zero class) — in The basic geometry of MTMN
• Theorem (Exact hyperbolic decomposition of the zero-class area) — in The basic geometry of MTMN
• Theorem (Exact first-boundary formula) — in The first-boundary model
• Theorem (Exact total first-boundary sum) — in The first-boundary model
• Theorem (Cubic-order bounds for the total area) — in The first-boundary model
• Theorem (Exact evaluation of the first-boundary reciprocal series) — in The first-boundary model
• Theorem (Exact evaluation of the weighted first-boundary series) — in The first-boundary model
• Theorem (Second-boundary formula for odd $N$) — in The second-boundary model
• Theorem (Total second-boundary sum for odd $N$) — in The second-boundary model

Propositions

• Proposition (Piecewise gap formula) — in Zero-Class Hyperbola Gap
• Proposition (Complementary-residue reflection) — in Convex Hull Area Function $S(N,a)$
• Proposition (A row determines its whole boundary layer) — in The basic geometry of MTMN
• Proposition (Transpose and central symmetries) — in The basic geometry of MTMN
• Proposition (Rows and columns permute exactly at coprime indices) — in The basic geometry of MTMN
• Proposition (Coprime residue classes form permutation plots on the coprime subgrid) — in The basic geometry of MTMN
• Proposition (The zero class detects compositeness) — in The basic geometry of MTMN
• Proposition (The hull as the region between two broken lines) — in The basic geometry of MTMN
• Proposition (The divisor envelope lies above the hyperbola) — in The basic geometry of MTMN
• Proposition (Piecewise decomposition of the hyperbola gap) — in The basic geometry of MTMN
• Proposition (A closed formula for one divisor-to-divisor segment) — in The basic geometry of MTMN
• Proposition (Exact first-boundary tail formula) — in The first-boundary model
• Proposition (A rigorous enclosure for the reciprocal constant) — in The first-boundary model
• Proposition (Piecewise formula for the two-layer odd-$N$ model) — in Using the first two layers together
• Proposition (Values at $0$ and $1$) — in Residue-area polynomials
• Proposition (Reflected coefficient form) — in Residue-area polynomials
• Proposition (Weighted first moment) — in Residue-area polynomials
• Proposition (Odd moduli) — in Residue-area polynomials

Corollaries

• Corollary (Integrality) — in Convex Hull Area Function $S(N,a)$
• Corollary (Complementary-residue reflection) — in The basic geometry of MTMN
• Corollary — in The basic geometry of MTMN
• Corollary (Integral formula for the zero-class area) — in The basic geometry of MTMN
• Corollary (Trapezoidal formula for the zero-class area) — in The basic geometry of MTMN
• Corollary (Convergence of the full reciprocal series) — in The first-boundary model
• Corollary (Rigorous enclosure from finitely many exact values) — in The first-boundary model
• Corollary (Convergence of the weighted full series) — in The first-boundary model
• Corollary (Odd factorization criterion) — in Residue-area polynomials
• Corollary (Odd prime factorization corollary) — in Residue-area polynomials

Definitions

• Definition (The left-half hyperbola gap) — in Zero-Class Hyperbola Gap
• Definition (Main area functions) — in Introduction
• Definition (Proper divisors and divisor rectangles) — in The basic geometry of MTMN
• Definition (Divisor points and the lower edge of the hull) — in The basic geometry of MTMN
• Definition (The left-half hyperbola gap) — in The basic geometry of MTMN
• Definition (Residue-area polynomial) — in Residue-area polynomials

Examples

• Example (A support direction for $A_{7,2}$) — in Exact formulas using convex geometry

Remarks

• Remark (Odd moduli) — in Zero-Class Hyperbola Gap
• Remark (Two different roles often confused in notation) — in The basic geometry of MTMN
• Remark (Four geometric objects) — in The basic geometry of MTMN
• Remark (A ratio form) — in The basic geometry of MTMN
• Remark (If the midpoint is not itself a divisor point) — in The basic geometry of MTMN
• Remark (Baseline and correction) — in The basic geometry of MTMN
• Remark (A geometric encoding of a finite arithmetic problem) — in Exact formulas using convex geometry
• Remark (What formula (6.1) does and does not solve) — in Exact formulas using convex geometry
• Remark (Historical origin of the boundary program) — in The first-boundary model
• Remark (A historical numerical guess) — in The first-boundary model
• Remark (Why the coordinates $u=x+y$, $v=x-y$ are natural) — in The second-boundary model
• Remark (Even $N$) — in The second-boundary model
• Remark (Series built from the second-boundary totals) — in The second-boundary model
• Remark (Not quite a palindromic polynomial) — in Residue-area polynomials