Finite matrices and dot products — Exact Proof Explorer

Ten independently checked constructive results establish total finite matrix entries, unique natural dot products, commutativity, and exact signed two-by-two determinant components.

10 theorem bodies · 13 proof edges · 263 tactic lines · 2 layers

Alpha v34 checked-use · first admitted v20 · independently kernel and Lean verified; not Stable

Current library: Alpha v34, 4,223 checked-use theorems; Stable remains 432. Historical first admissions, original proof editions, and non-admitted aliases are preserved. Exact original first-admission records.

10 theorems
01
MD0001 · beta_matrix_cell_exists

Every requested finite matrix coordinate has a witnessed beta-decoded entry.

layer 0 · 9 lines · Alpha v34 independently verified · alpha_closed; checked-use authorized; not Stable
MD0002 · beta_matrix_cell_functional

The decoded natural value of every flattened matrix cell is unique.

layer 0 · 17 lines · Alpha v34 independently verified · alpha_closed; checked-use authorized; not Stable
MD0003 · beta_matrix_cell_exists_unique

Every finite beta-coded matrix entry has exactly one actual value.

layer 1 · 26 lines · Alpha v34 independently verified · alpha_closed; checked-use authorized; not Stable
MD0004 · beta_dot_product_exists

Two arbitrary finite coded vectors have an exactly witnessed dot product.

layer 0 · 22 lines · Alpha v34 independently verified · alpha_closed; checked-use authorized; not Stable
MD0005 · beta_dot_product_functional

The exact finite dot-product value is independent of its coding witness.

layer 0 · 82 lines · Alpha v34 independently verified · alpha_closed; checked-use authorized; not Stable
MD0006 · beta_dot_product_exists_unique

Every pair of coded finite vectors has exactly one natural dot product.

layer 1 · 26 lines · Alpha v34 independently verified · alpha_closed; checked-use authorized; not Stable
MD0007 · beta_dot_product_empty

The exact dot product of two empty finite vectors is zero.

layer 0 · 14 lines · Alpha v34 independently verified · alpha_closed; checked-use authorized; not Stable
MD0008 · beta_dot_product_commutative

Finite natural dot products are constructively symmetric.

layer 0 · 37 lines · Alpha v34 independently verified · alpha_closed; checked-use authorized; not Stable

Exactly 10 displayed theorems have independently verified Alpha checked-use authority; none is admitted to Stable. Body-only enrollment never grants checked theorem use.

Separate complete second-wave branches: Full T13 proof · Alpha v27.