How much local information can force a matrix to be totally positive? Total positivity means that every square submatrix has a positive determinant. Checking small pieces alone is usually not enough. This candidate adds a quantitative condition: selected ratios of small minors must clear a threshold. The paper identifies the exact threshold for each matrix size and minor order. Across all dimensions, the optimal threshold is four. Sufficiency follows from a classical determinant theorem and a standard bordering identity. The difficult direction is sharpness. The construction starts with a smaller matrix whose proper minors are positive but whose full determinant is negative. A carefully scaled border raises the matrix size while transferring every required local ratio at once. Repeating this construction supplies counterexamples below the threshold. The package includes a full written proof, ten exact rational examples and fifty-three symbolic identities. A separate regression preserves a correction: using one common threshold is stronger than allowing thresholds that vary with submatrix size. The finite computations do not prove the general theorem. This is an unrefereed Evidence Press candidate, with internal AI review, not independent specialist validation. Historical priority and identity with an omitted workshop formula remain unestablished. The paper and evidence package are linked. This narration uses an OpenAI synthetic voice and provides no additional mathematical evidence.