What fixes the denominator is not the determinant but the largest invariant factor – this paper applies a brake to the corpus itself.
Paper 39 wrote that the exponents of the Planck system are half-integers because the dimension-exponent matrix of the reduced Planck constant, the gravitational constant and the speed of light has determinant of absolute value two. Paper 03 wrote that the Kolmogorov scales come in quarters because the matrix of the kinematic viscosity and the dissipation rate has determinant of absolute value four. And both called this one and the same theorem. This paper shows that the theorem is one of upper bounds and not of equalities, and writes the right invariant. NO NEW MATHEMATICAL THEOREM IS CLAIMED. The Smith normal form, the equality of a lattice index with a determinant, and the functoriality of pullback and pushforward are all standard. The conclusions of Papers 03 and 39 are not withdrawn; what is corrected is the way the reason is written.
First, transporting a quantity along a map is not one operation. A differential form pulls back along any smooth map: under the squaring map the pullback of the differential of y is twice x times the differential of x, defined at every point. A vector field, by contrast, cannot be transported. Pushing the field in the x direction forward along the same map, the point y equal to one receives plus two from x equal to plus one and minus two from x equal to minus one: one point receiving two different vectors. Forms are contravariant and pull back along any map; vector fields are covariant and need a diffeomorphism. This is not even a question of orientation but of injectivity. Measures, on the other hand, push forward. Pushing the uniform measure on the unit interval forward by the squaring map gives the reciprocal of twice the square root of y, and with four million samples the agreement was within one part in a thousand (computed here). What a density picks up is the reciprocal of the absolute value of the derivative, not of the derivative.
Second, what separates a form from a density is orientation. The reflection that exchanges the coordinates has determinant minus one. The area form changes sign under it and the area measure does not: the first carries the determinant and the second its absolute value. The same holds for a general linear map: the matrix with rows two and one, and one and three, has determinant five, and measuring the area of the image of the unit square on a four thousand by four thousand grid gives five point zero zero zero two three nine (computed here). One symbol carries two transport rules, and orientation separates them. A measure is a positive quantity and has no orientation.
Third, the address of the corpus’s absolute determinant is a volume in exponent space. A dimension-exponent matrix is a map of lattices, and the absolute value of its determinant is the index of the sublattice, that is, the volume of a fundamental domain. Counting how many points of a one hundred and twenty-one by one hundred and twenty-one window lie in the sublattice spanned by Paper 03’s matrix gives three thousand six hundred and ninety-one out of fourteen thousand six hundred and forty-one, a ratio of zero point two five two one, matching the reciprocal of the absolute determinant, zero point two five (computed here, boundary effects included). So the absolute value is there because a volume has no orientation, the same root as the absolute value on a density in the second section. This is a rare occasion on which the corpus confirms a shared root rather than separating one. But the address must be written to the end: this volume is not a physical volume but a volume in the lattice of dimension exponents, the space made by the logarithms of units. Without writing that, the quantity begins to look like some physical volume.
Fourth, and this is the core. The absolute determinant is an upper bound on the denominator, not the denominator. The inverse of an integer matrix is its adjugate divided by its determinant, and the adjugate is integral, so the denominators of the entries divide the absolute determinant. That is an upper bound and not an equality. Here is a counterexample: the diagonal matrix with entries two and two has absolute determinant four, yet the entries of its inverse are halves only and no quarter appears anywhere. And the counterexample is not contrived: take an area and a squared time as the basis and this is exactly the matrix. Building any quantity from them needs square roots at worst and a fourth root never appears. The denominator that actually occurs is the largest invariant factor, and it equals the absolute determinant exactly when the cokernel is cyclic.
Fifth, both of the corpus’s instances turned out to be cyclic. Computing the invariant factors here, Paper 03’s matrix gives one and four, Paper 39’s gives one, one and two, and Paper 38’s basis gives one, one and one. So Paper 03’s cokernel is cyclic of order four and quarters really do occur, and Paper 39’s is cyclic of order two and half-integers really do occur. Neither conclusion is withdrawn; what is corrected is the reason. The form in which Paper 03 wrote it, that an absolute determinant of two gives square roots and one of four gives fourth roots, reads an upper bound as an equality. The correct statement is that a largest invariant factor of two gives square roots and one of four gives fourth roots, the absolute determinant being merely a multiple of it. The two agreed in both instances because both cokernels happened to be cyclic. This is the familiar shape in which a brake laid down by a first edition had not been applied to the first edition itself.
The census comes out as follows. A function pulls back along any map. A differential form pulls back too, and carries the determinant. A vector field does neither and needs a diffeomorphism. A measure or density pushes forward and carries the absolute determinant. Cohomology is contravariant and pulls back; homology is covariant and pushes forward. Transporting a quantity along a map is therefore not one operation, and the direction is not a matter of taste: it is forced by what the object is. Paper 147’s separator was the coefficients, Paper 148’s the test object, and this paper’s is orientation, all three having the shape that a thing can be written in a common language without sharing a root.
IMPORTANT (scope): a census of established standard material, together with a correction to the corpus itself (Layer 2). What is not claimed: any new mathematical theorem (there is none) / the withdrawal of the conclusions of Papers 03 and 39 (they are not withdrawn). What is added: giving the corpus’s absolute determinant the address of a volume in exponent space, writing that it shares a root with the absolute value on a density, stopping the claim that the absolute determinant fixes the denominator by means of a counterexample, and recomputing the corpus’s two instances to confirm that their conclusions survive. What remains: the corpus wrote three times that the determinant is where a root of a given order comes from. Every one of the numbers was right. But the reason that produced the right numbers was one invariant too coarse. Reading an upper bound as an equality cannot be caught while the two happen to agree; to catch it one has to build, oneself, an example in which they do not.
On the making of this work: The ideas and content of this work stem from the author’s own considerations. Assistance from an AI (a large language model) was used for structuring, English translation, and checking the algebra. Any remaining errors or misinterpretations are solely the author’s. Feedback and corrections are sincerely appreciated.
Reference
Yamagishi, Y. (2026). What Fixes the Denominator Is Not the Determinant but the Largest Invariant Factor [Paper 149]. Zenodo.
DOI: 10.5281/zenodo.22083590
ORCID: 0009-0006-8587-3450