The definition divides d by p; the theorem divides p by the discriminant – the direction of division has changed hands.
The definition of the Legendre symbol asks whether the remainder of d on division by p is a square. Yet the answer given by reciprocity is settled by the remainder of p on division by some other number. This paper measures that other number by exhaustive search, confirms it is the discriminant, and shows that the four and the eight of the two supplements share one root. NO NEW MATHEMATICAL THEOREM IS CLAIMED: quadratic reciprocity, the fact that the discriminant is the conductor, and the formulation as a Dirichlet character are all standard since Gauss. Nor does this paper prove the law, since what is offered is a measurement and not a proof.
First, the law itself was verified from the definition. Every symbol was computed from Euler’s criterion, that is without using reciprocity even once. Over all one thousand eight hundred and thirty pairs of odd primes below three hundred the two sides agreed with no failures, and both supplements held over sixty-one primes with no failures. Notice here what the definition looks at. The definition of the symbol looks at the remainder of d on division by p, and dividing p by anything appears nowhere in it, yet the right-hand side of the law is written entirely in terms of p modulo four and q modulo four.
Second, and this is the core. For each squarefree d the symbol was computed for every prime below twenty thousand, and the smallest modulus such that the symbol depends only on the remainder of p was found by searching upward from one. For all sixteen values of d tested, that smallest modulus was exactly the absolute value of the discriminant: four at d equal to minus one, eight at two, eight at minus two, twelve at three, three at minus three, five at five, twenty at minus five, twenty-four at six, twenty-eight at seven, thirteen at thirteen, seventeen at seventeen, seven at minus seven, eleven at minus eleven, twenty-one at twenty-one, twenty-nine at twenty-nine and nineteen at minus nineteen, all measured here. Nothing in the definition gives any reason for the answer to be periodic in p at all. Reciprocity is the assertion of that periodicity, and the exchange of which side does the dividing is what the word reciprocity is pointing at. That a period exists at all is the content of the law; the value of the period is only the question that comes next.
Third, the separator is d modulo four. Four times the absolute value of d is always a period but not always the smallest one. When d is congruent to one modulo four the true modulus drops to a quarter of it, giving five rather than twenty at d equal to five, thirteen rather than fifty-two at thirteen, and twenty-nine rather than one hundred and sixteen at twenty-nine, the ratio being exactly four. When d is congruent to two or three the two agree. What must be moved for the modulus to move is d modulo four, neither the size of d nor its sign, so the modulus is not a number picked for convenience. Whether the factor of four is present is settled by d modulo four and there is no room for choice, which in the language of Paper 95 places it on the side of fact rather than convention.
Fourth, the four of the first supplement and the eight of the second are one reason, not two. The first supplement is a statement about p modulo four and the second about p modulo eight, so two different numbers appear to arrive for two different reasons. But the measurements give four at d equal to minus one, eight at two and eight at minus two, all of them the absolute value of the discriminant. Furthermore, collecting the symbol of two by residue class of p modulo eight over all odd primes below twenty thousand, the value is plus one throughout for the classes one and seven and minus one throughout for three and five, with not a single class in which the values split. The two supplements are not two theorems but one statement read at d equal to minus one and at d equal to two. This corpus usually says one word, distinct roots; here it does the opposite and gives one address to two numbers that had looked separate. After Paper 31 confirmed that the pi of a solid angle and the pi of a Gaussian integral share a root, and Paper 142 that Unruh’s two pi and Hawking’s do, this is the third such confirmation.
Fifth, the law still yields no root. Setting the Jacobi symbol computed through reciprocity, which uses no exponentiation at all, beside the construction of a square root by Tonelli-Shanks at the same prime gives four steps against twenty at one million and three, seven against twenty-four at ten million and nineteen, six against twenty-seven at one hundred million and seven, and five against thirty at one billion and seven, with the root verified to satisfy the congruence in every case. And the gap is not only one of cost. Tonelli-Shanks begins by finding a non-residue, and it has no method for that other than trying two, three, five and so on in turn. A theorem gives the count, exactly half of the nonzero classes being non-residues, but it names none of them: among one to one hundred and ninety-nine there are one hundred and five non-residues at the first prime, ninety-nine at the second and ninety-seven at the third, with smallest non-residues two, two and five. Counting is a theorem; producing one is a search, and no deterministic polynomial-time method for the smallest non-residue is known. This has the shape of Paper 163, which separated a good code exists from here is a good code, and as in Paper 25 the separator is again which tools are allowed.
A census. The definition of the symbol divides d by p, d is what moves it, and its address is computation for each p. Reciprocity divides p by the absolute value of the discriminant, d modulo four is what moves it, and its address is period equals discriminant. The first supplement divides p by four and is the case d equal to minus one; the second divides p by eight and is the case d equal to two. Constructing a root is not a matter of division at all: what moves it is which tools are allowed, and it is a search rather than a theorem. The definition and the theorem look at different things, the four and the eight of the supplements are one root, the modulus is not chosen, and knowing whether a root exists stays separate, to the end, from holding one.
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.
⚠️ Scope ── what is not claimed
a census of established standard facts (Layer 2). What is not shown: any new mathematical theorem, none is claimed; a proof of reciprocity, none is given, since what is shown is a measurement. What is added: measuring the smallest period by exhaustive search and confirming that it matches the discriminant, exhibiting with a separator the cases in which four times d overshoots fourfold, and confirming that the four and the eight of the supplements share one root. What remains: the definition divides d by p and the theorem divides p by the discriminant. Whether one can still be surprised that the direction of division changes hands is what separates having read this law from having recited it, and the discovery that the four and the eight were one number all along is that surprise’s small dividend.
Reference
Yamagishi, Y. (2026). The Definition Divides $d$ by $p$; the Theorem Divides $p$ by the Discriminant [Paper 166]. Zenodo.
DOI: 10.5281/zenodo.22093052
ORCID: 0009-0006-8587-3450