Catalan's Constant Is Irrational. Master Determinant Manuscript: Rigidity Gates, Coupled Ledger, and Locked Cascade.
Year: 2026
Type: preprint
Abstract: 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: Catalan's constant is the alternating sum of the reciprocals of the odd squares. Whether that number is a ratio of integers is the question treated below. A determinant construction addressing the same question was posted by Zhi-Wei Sun on 3 September 2026 as arXiv:2609.04176v1, under General Mathematics. The citation is given once, for provenance of certain assembled factors. It is not a licence to read the two documents as drafts of a single proof. This paper IS NOT a revision of Sun’s architecture, nor a defence of his closer. It is an independent, internally consistent framework. What it takes from his construction is scaffolding: assembled factors, not the argument that follows from them. Sun wrote down weighted tails, a residual matrix, a Newton completion of a selected minor, a Pascal–Cauchy splitting of that minor, a division of prime-power cost into three ranges, and a width ratio of one-twentieth at which a particular quadratic comparison becomes negative. Those constructions are used here as already-named factors. They are not used here as his closing argument, his coefficient table, or his rule for what a logarithm is allowed to count. The bibliographic debt stops there. Later sections do not lean on his last inequality, do not adopt his Newton columns, and do not treat his raw quadratic as a slot of the integer formed below. A reader who came for a reconstruction of that preprint will be reading the wrong file. Note to the reader: This manuscript is independent of Sun’s paper in the only sense that matters for a proof: it defines a different product, scores that product by a different rule, and closes on a different coefficient attached to a different integer. Shared vocabulary — tail, residual, Cauchy–Binet, saturation — does not make the arguments the same. Shared vocabulary is what makes the confusion likely. The rest of this note exists to prevent that confusion. Expect a change of register after the scalar has been built. The opening sections produce a polynomial in one variable and prove that the polynomial is nonzero when the variable is Catalan's constant. Rationality is not in force. It is not needed. Only after that nonvanishing is secured does the text assume, temporarily, that the constant is a fraction. Denominators already controlled by the construction are then cleared. What remains is a nonzero integer. The rest of the work is an account of the height of that one integer. The operator that governs the account is rigidity. It acts on factors the cascade has already multiplied in. It returns either an identity those maps already force or a leftover that has been given a name. Each named factor is allowed one real logarithm and one prime factorization. Those two readings are the same object. Scoring a factor twice, under two names, is forbidden. Leaving a factor unscored is forbidden. Renormalizing one place of a co-dependent pair without the complementary place is forbidden. Division by a rational already present in the cascade acts on every place at once, because that is how division acts on a factorization. Sun’s text does not introduce this operator and does not police this census. Importing his estimates into a ledger that runs by these rules will manufacture contradictions that this file does not contain. The Newton completion is the first concrete fork. Falling-factorial columns are required here because their pivots are the factorials that later define the prefactor and the two-adic count of the same product. Binomial difference columns have unit pivots. They do not produce those factorials. Sun’s stage describes binomial Newton columns. If those columns are kept, the later two-adic normalization written below is not an identity of the matrix that was completed. If the falling-factorial columns are kept, his binomial description is not a description of this matrix. The two completions are not interchangeable by a change of notation. A second fork is the Pascal remainder after the node-alternant has been divided by its Vandermonde. That remainder is named here as an integer cofactor. It is present at every prime. Its real logarithm is shown to be of lower order than the scale at which the final coefficient is decided, so it cannot hide a quadratic term and cannot occupy a sixth finite slot. Sun’s expansion contains the same remainder and discards it from valuations because its exponents are nonnegative. That discard is legitimate for a lower bound. It does not settle the size of the real logarithm. This file settles that size. The two treatments are not the same lemma. A third fork concerns two distinct powers of two. The Cauchy block contributes an explicit two-power whose logarithm belongs to that block. Separately, the factorial prefactor and the fixed Vandermonde differ by a two-adic valuation that must be removed on every place at once when the scalar is normalized. Those are not one two-power written twice. Confusing them is how a two-adic mass can vanish from a denominator ledger while remaining in the real size of the unnormalized determinant. This manuscript keeps them apart by definition. A fourth fork is the count of leading logarithms. After the named factorials, selected weights, Cauchy denominators, and Vandermondes are assembled, the internal residual block has one leading coefficient. The complete scalar, which also carries the Newton prefactor and the product of all weights in the denominator, has another. Removing the fixed Vandermonde produces a third. The prime-power denominator bound computed for that last scalar has the opposite of the third. All three numbers are correct for their own objects. None may be moved onto another object. In particular the residual-block logarithm is not the logarithm of the complete real scalar. Sun’s narrative proceeds as if a single cancelled leading row remained after the singular layer, and as if one finite budget could then be read off. That collapse is not available here. What survives at the finite quadratic scale, once singular rows have cancelled and each integer factor has been paired with its occupancy, is a short list. The list contains oscillation of an Archimedean placement functional when the selected indices move, one slack obtained by splitting the same local form across three disjoint prime-power supports, and the two-adic logarithm forced by the falling-factorial prefactor after the Vandermonde is removed. Their combination is a positive margin of about zero point nine six seven. That combination is the coefficient of the clipped-height bound. It is the only coefficient that closes the paper. Sun’s last comparison is not on that list. After the singular layer cancels, his text sets the three range integrals against a raw quadratic fixed by the width ratio and treats the difference as the height coefficient. This file records that raw quadratic as an arithmetic check of the locked ratio. It then states that the check is not the Stirling image of any named factor on the witness set, not a tail contribution, and not a scale-split cell. Two later draft positives were formed from that check: the difference between slack and the raw quadratic, and that difference plus the two-adic term. Both appear in this manuscript. Both are superseded. Neither is an alternative closing line. A reader searching for Sun’s margin as the engine of the contradiction will not find it in the QED paragraph. Index sets are not free to migrate from one side of the ledger to the other. Saturation of every odd prime-power layer is proved on the initial segment of indices. No other segment is shown to saturate every such layer at once. The real estimate and the denominator estimate are therefore required to occupy that same segment. An interval block near the middle of the range is used only to test the continuum formula for the Archimedean functional. That block is not the witness. Adding a maximum on the block to a minimum on the initial segment treats two extrema as if they were one Cauchy–Binet term. Sun’s text does not forbid that addition. This text does. Unique factorization is used as an identification, not as decoration. The real logarithm of a named integer and the occupancy of that integer are one logarithm. The Archimedean functional on the witness measure and the Stirling list of the named factorials on that measure are therefore one number. Writing both on the same line of the height bound double-counts those factorials. A ledger that keeps a real estimate and an odd-prime occupancy estimate as independent controls can perform that double-count without noticing. This ledger cannot. The three prime ranges are not three separate profits. Every monomial in the absolute-value identity of a Cauchy–Binet summand belongs to exactly one range, or to a remainder family already of lower order. The corresponding integrals occupy three disjoint continuum supports. Their signed combination is a single slack. The unsigned sum of the same three quantities is not a slot. Adding a fourth copy of the local form, under a new name, is excluded by the support partition. Sun computes the three integrals. This paper adds the rule that they may not be spent twice. The tail is two properties, and only one of them enters the residual. Consecutive tails sum to the reciprocal of an odd square. A phase version of the same identity has vanishing quadratic transport defect. Those identities are what the residual expansion and the rank theorem consume. How large a tail is, and how fast the series remainder tends to zero, is a different property. On a block of selected indices th
Author: Lance Thomas Davidson
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7210264249
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