Purple Mathematics: The Complete Theory (Third Edition) — From the Bridge between Zero and Infinity to the Scale Law of the Balance

The Third Edition gathers the thirteen published works of the Purple Mathematics series and adds a fourteenth chapter, The Wavy Line, which appears first in this edition and was not separately published. Its subject is what the framework's structures are worth when the dividing line they are drawn on is not flat. The Invariance Ledger of Chapter 10 classified every purple observable by its behaviour under deformation — arithmetic invariants, affine covariants, metric covariants — and has stood unchallenged since. Chapter 14 shows that the classification settles less than it appears to, because the tiers are entangled: the determination residue, an arithmetic invariant, is recoverable exactly from the gap lengths, which are metric covariants, with 100.00% agreement over 455,052,507 primes and not one disagreement. Hence, Ledger classifies objects, not readings — deformation cannot destroy determination, but it can destroy access to it. Three consequences follow. The direction of deformation decides everything: out-of-plane displacement is silent, confirming the gauge lemma; longitudinal strain destroys the geometric channel outright. The apex-to-apex line, drawn from the first descriptive account and never given a work to do, turns out to be a gauge: a heave moves every height and no position, so its corners stay above the primes and a reading taken through them survives a deformation of four times the apex height — the real limit being an admissibility ceiling on the wall itself, A_crit = h / (2 max |sin(pi g / L)|), governed by the tightest triangles. The tolerance of the weaker reading is scale-free at about 0.7 units from 10^6 to 10^10, so the relative tolerance decays like 1/ln x and the geometric channel thins asymptotically. The new chapter's computations were carried to 10^10 on a purpose-built resumable engine whose output validates against the program's own verified S(10^10) = 6.43e12; the Determination Law's 1/ln x decay is confirmed to 1.4% over three decades. Two predictions of the author's were refuted by those computations, and one method error was caught before publication; all three are recorded rather than repaired quietly. The Ledger of corrections now carries a third public amendment alongside the two the Second Edition already reported.