hbar is not on the mathematical side of the inequality – the uncertainty principle is not one theorem.
One phrase, the uncertainty principle, points at several theorems that measure different things. This paper calls the roll and writes each address. NO NEW MATHEMATICAL THEOREM AND NO NEW PHYSICAL LAW IS CLAIMED. Kennard (1927), Weyl (1928), Robertson (1929), Hardy (1933), Hirschman (1957) and Beckner (1975), Amrein-Berthier (1977) and Benedicks (1985), Donoho-Stark (1989), Tao (2005) and Mandelshtam-Tamm (1945) are all standard.
First, and this is the core. hbar is not on the mathematical side of the inequality. The inequality for the product of a time width and an angular-frequency width, whose lower bound is one half, is a theorem about square-integrable functions and their Fourier transforms, and it contains no physical constant whatsoever. This paper confirms symbolically that a Gaussian attains equality and that the coefficient describing its width cancels, leaving one half. The position-momentum relation acquires half of hbar as its bound because one physical identification has been passed through exactly once: the de Broglie-Planck relation, that momentum is hbar times wavenumber. The one half comes from mathematics; the hbar comes from physics. The only quantum step is that single substitution.
Second, even that one half carries a convention. Writing frequency per turn rather than per radian turns the same theorem’s bound into one over four pi. The four pi appearing there is two times two pi: the two is the extremal one half, and the two pi converts turns to radians. And the four pi in Planck’s-constant form, h over four pi, is the same two times two pi, since hbar is h divided by two pi. So the two four pi are not distinct roots; they are one root. And neither is a solid angle. The counterexample is dimension-independence: the bound holds componentwise in the same form whatever the spatial dimension, a particle on a line obeys the same bound, and there is no sphere there. Were the constant a solid angle the bound would move with dimension. It does not move. This is the shape Paper 46 laid down: change the Fourier convention and the number moves. Geometry must not be read into a number that a convention can move.
Third, the variance form says nothing when the variance diverges. The Cauchy density is a probability density whose second moment diverges (verified here). A rectangular pulse has finite support in time, yet the square of its transform decays only like the inverse square of frequency, so its frequency variance diverges: widening the window fourfold, this paper obtains estimates of 1137, 4541 and 18172, rising by roughly a factor of four each time. The sentence that one cannot be localised in time and frequency at once is therefore not, as it stands, a theorem. It becomes one only after choosing the ruler by which localisation is measured: variance gives Kennard, measure of support gives Benedicks, rate of decay gives Hardy. These are different theorems. This has the shape of the brake Paper 106 places on stationarity and fat tails. A tool built on variance works only in a world that has one.
Fourth, the discrete side has an arithmetic root. On a finite cyclic group the product of the numbers of non-zero entries is at least the size of the group (Donoho-Stark). The additive form, that the sum of those numbers is at least the size of the group plus one, holds only when the size is prime (Tao). Building evenly spaced picket fences, this paper shows that at sizes 9, 12 and 16 the product attains exactly the size of the group while the sum is 6, 7 and 8 against the required 10, 13 and 17. So for composite sizes the additive form is false. At sizes 7, 11 and 13, by contrast, four thousand random trials with randomly chosen support sizes give minimum sums of 8, 12 and 14, exactly the size plus one. What supplies the improvement is not analysis. A picket fence can be built precisely because the size has a proper divisor, and primality forbids it. An arithmetic condition sits inside a theorem of analysis. Paper 86 listed four roots of necessity – topology, symmetry, algebra, dimension – and what appears here is none of them. It is the factorisability of an integer.
Fifth, lay eight of them side by side and every one measures something different: variance (Kennard, Weyl), the expectation of a commutator (Robertson), rate of decay (Hardy), measure of the support (Amrein-Berthier, Benedicks), entropy (Hirschman-Beckner), count of non-zero entries (Donoho-Stark), primality (Tao), and a lifetime (Mandelshtam-Tamm). None implies the others. The variance can be infinite while the support is finite (the rectangular pulse), and the support can be infinite while the variance is finite (a Gaussian). The two rulers hold counterexamples against each other. As for the entropic form, this paper computes the sum of the differential entropies of the two probability densities for three Gaussians of different widths and obtains 0.306853 in every case, that is the logarithm of e over two. A different functional with a different extremum is a different theorem.
Sixth, the energy-time relation is not the same theorem as the position-momentum one. Robertson’s inequality is a statement about two self-adjoint operators, and time is not such an operator (Papers 54, 127). What the Mandelshtam-Tamm relation gives is a timescale over which some observable changes, not a variance of time. The same symbol is serving as the name of a different quantity, the same shape as the argument about logical types that Paper 118 makes for its three clocks.
Finally, what then was Fourier saying? A statement about linear waves. Sound, water waves and price series alike: a short pulse occupies a wide band. This is a fact engineering uses daily and it has no bearing on quantum mechanics. Quantum mechanics added two things – that a particle is described by a wave, and that momentum is hbar times wavenumber. Whatever makes uncertainty feel like a principle was brought by those two, not by the inequality. What nevertheless remains surprising is that hbar is finite. Let hbar tend to zero and the bound vanishes. The bound exists because of mathematics, but it bites because hbar is finite. Mathematics supplies the shape; physics supplies the scale.
IMPORTANT (scope): a census of established standard material (Layer 2). Every theorem cited is standard. What is not claimed: any new mathematical theorem (there is none) / any new physical law (there is none) / that the uncertainty relations are in error (they are not) / that quantum mechanics is dispensable (it is not; what is separated is where the mathematics ends and the physics begins). What is added: separating the mathematical one half from the physical hbar, judging the two four pi to share one root, producing two witnesses for the vacuity of the variance form, entering primality into the census as a root, and separating energy-time as a different logical type. What remains: mathematics supplies the shape and physics supplies the scale. Until the ruler is named, nothing has yet been said.
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). $\hbar$ Is Not on the Mathematical Side of the Inequality [Paper 129]. Zenodo.
DOI: 10.5281/zenodo.22079681
ORCID: 0009-0006-8587-3450