Catalan's Constant Is Irrational. Master Determinant Manuscript: Rigidity Gates, Coupled Ledger, and Locked Cascade.

Catalan's Constant Is Irrational. Master Determinant Manuscript: Rigidity Gates, Coupled Ledger, and Locked Cascade. Version 9.2 Creator Lance Thomas Davidson ORCID: 0009-0006-1245-1644 Publication date 7 September 2026 Version 9.2. This record replaces every earlier draft of the reconciled cascade. Resource type Preprint Language English Related work Built from Zhi-Wei Sun, Catalan's Constant Is Irrational, posted on arXiv as 2609.04176, version 1, 3 September 2026, subject class General Mathematics. This paper argues that Catalan's constant is not a ratio of two integers. Catalan's constant is the alternating sum of the reciprocals of the odd squares: one minus one-ninth plus one-twenty-fifth minus one-forty-ninth, and so on. The argument was first assembled by Zhi-Wei Sun. He built a matrix from the leftover pieces of that series (the tails), completed it by a Newton interpolation, split the matrix into a Pascal block, a diagonal of tails, and a Cauchy block, divided the prime-power cost into small, middle, and large ranges, and locked a width ratio of one-twentieth so that the surviving quadratic coefficient is strictly negative. Every object that carries the proof in these pages comes from that construction. This manuscript does not invent a second architecture. It treats Sun's product as one number, gives each named factor a single real logarithm and a single prime factorization, and applies one operator (called rigidity) that returns either an identity those maps already force or a named leftover. The work proceeds in a fixed order. First a polynomial in one variable is built from the tails and the residual matrix and is shown not to vanish when that variable equals Catalan's constant. No assumption that the constant is rational is used at this stage. Only after nonvanishing is proved does the paper assume, for contradiction, that Catalan's constant is a fraction. Clearing denominators already controlled by the construction produces a nonzero integer. The height of that integer (how large it is on the logarithmic scale of the square of the cutoff parameter) is then bounded. After matching rows cancel and integer factors are paired, three quantities remain: oscillation of an Archimedean placement functional, a slack integral split across the three prime ranges, and a two-adic gain coming from the factorial prefactor. Their combination is a positive margin of about zero point nine six seven. For large cutoff that forces the integer to lie strictly between zero and one, which is impossible. The constant therefore cannot be a fraction. Several choices generate more than one factor at once (the width ratio one-twentieth, the interpolation degree twice the cutoff, the falling-factorial basis, and the Vandermonde normalization). Those choices are locks. Changing one of them alone produces certificates that no longer describe the same product. Rigidity does not use how fast the tails tend to zero as an input at the quadratic scale. What the residual and rank lemmas use is an exact two-term recurrence: each tail plus the next tail equals the reciprocal of an odd square. The size of the tail is a separate property. On a block of selected indices that size contributes only a lower-order logarithm, so it occupies no quadratic slot. That zero is a score, not a missing estimate. A selected-index measure and an Archimedean functional appear in the middle of the paper. They bound one oscillation term. They are not a second proof. The same printed decimals cannot be pasted onto the polynomial evaluated at one-half: the identity that turns a tail into an explicit rational uses the hypothesis that Catalan's constant equals the chosen fraction, and that identity fails at one-half. A rebuild that redefines the tail as a formal remainder at an arbitrary point is a different product. This file does not claim a height bound for that product. A proposed global sign rule (all Cauchy-Binet terms like-signed away from Catalan's constant) is false, because the early partial sums already straddle one-half, and is not used. Labels Q1, Q2, and Q3 are names of derivation chains written while the ledger was assembled. They are identifiers, not open questions. The line that closes the paper is the clipped-height bound on the integer described above. Catalan's constant is shown not to be a ratio of integers by reconstructing Sun's determinant cascade as one product. A polynomial built from series tails is proved nonzero at the constant before rationality is assumed. Rationality would produce a nonzero integer whose logarithmic height is strictly negative at the scale of the square of the cutoff, with explicit positive margin about zero point nine six seven. Each named factor is scored once. Tail decay is not a quadratic input. Evaluation at a foreign point is a different product and is not this claim.