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To Determine an Object You Need Every Other One ── What the Yoneda Lemma Requires and What Is Lost When the Probe Is Weakened: Counted over the 156 Graphs on Six Vertices ── [Paper 244]

Yuuki Yamagishi
Aug 2026 · Zenodo (CERN European Organization for Nuclear Research)

Abstract

The Yoneda lemma says that an object is completely determined by its relations to every other object. This paper asks about that word every: where does the separating power fall when the probe is weakened? And the loss turned out not to be governed by the strength of the probe. No new mathematical theorem and no new law is claimed. Scope of this paper (scope note): no new mathematical theorem and no new law is claimed. The Yoneda lemma, the existence of cospectral non-isomorphic graphs, and the coincidence of the character tables of the dihedral group of order eight and the quaternion group are all standard. No measured value is cited; every number is obtained by exhaustive enumeration. No category theory is developed; only the statement of the lemma is used, and neither its proof nor any generalisation is treated. Nothing is said about the complexity of graph isomorphism; this is an exhaustive count in the finite case of six vertices. No representation theory is developed; the character tables are quoted as known. It is not claimed that the Yoneda lemma justifies the method of this body of work; Section 6 exists precisely to stop that claim from being made. The relation to earlier papers. Paper 152 showed that two groups can share a character table without being isomorphic; this paper counts that phenomenon as a function of the strength of the probe. Paper 153 separated three stages of forgetting and exhibited evidence that a right adjoint is absent; this paper treats the neighbouring theorem on the same shelf. Paper 102 treated the question whether one can hear the shape of a drum; Section 3 is its finite version, counted exhaustively. Paper 163 showed that there exists and here it is are different sentences; Section 6 says that determined and computable are different sentences. The setting. Take a collection of objects and a probe, a function applied to an object that returns a value. When a probe returns the same value on two objects it does not separate them. The material is the simple graphs on six vertices, of which there are 32768 labelled ones. Three probes are used: the degree sequence, the spectrum of the adjacency matrix, and the spectrum together with the triangle count. For comparison the isomorphism type itself is included. First, the 32768 labelled graphs fall to 156 isomorphism classes, counted by applying all 720 vertex permutations and taking a canonical form. Second, the degree sequence is not enough. It takes only 102 distinct values and 84 classes survive unseparated in 30 groups. Third, the spectrum is not enough either. It takes 151 values and 10 classes survive in 5 groups. Fourth, this is the core. Adding the triangle count leaves the distinct values at 151 and the unseparated classes at 10. The triangle count is the trace of the cube of the adjacency matrix divided by six, a function of the spectrum, so adding it adds no information. Whether more probes separate more depends on whether the new one can be recovered from the old. Measure more invariants and you will eventually tell them apart is false; dependent invariants advance nothing. Fifth, strength is not a total order on separating power. The smallest pair the spectrum cannot separate has degree sequences zero one one one one four and zero zero two two two two, both with four edges and no triangles, sharing the spectrum minus two, zero, zero, zero, zero, two. The degree sequence does separate that pair. A weaker probe separates where a stronger one fails. Sixth, the same happens for groups. The dihedral group of order eight and the quaternion group share a character table and are not isomorphic. Counting the elements whose n-th power is the identity, that is the homomorphisms from a cyclic group, gives six and two at n equal to two, which separates them. The character table failed and counting maps from a single object succeeded. Yet at n equal to four the counts are eight and eight and do not separate. The same shape of probe changes its power when the test object changes, and which object works cannot be known in advance. That is why Yoneda demands every object. Seventh, this is a fence. The lemma reads as saying that a thing is determined by its relations, and the method of this body of work, writing what a thing separates rather than what it is, has a similar shape. Similarity is not justification. The Yoneda lemma is a theorem inside a category, about Hom sets and natural transformations, and a methodology is not such an object, so the lemma says nothing about it. Separate Yoneda as a theorem, where the Hom functor is fully faithful and the statement is proved, from Yoneda as a metaphor, where things are determined by relations and nothing is proved. Speaking the second with the authority of the first is the accident this body of work has spent its time avoiding. And determined does not mean computable: Yoneda says the object is determined and gives no way to find it. Closing. The Yoneda lemma demands every object, and replacing that word by a finite list loses something. What is lost is not governed by the strength of the probe: the degree sequence separated a pair the spectrum missed, and adding triangles gained nothing at all. The separator is whether the new probe can be recovered from the existing ones. Measure more and you will know is correct only when what is measured is independent. 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. ----- 米田の補題は、対象は他のすべての対象との関係で完全に決まると言う。本稿が問うのは、その「すべて」である——テストする相手を減らすと、どこで分離能が落ちるか。しかも落ち方は、プローブの強さでは決まらなかった。新しい数学定理も新しい法則も主張しない。 本稿の射程(射程注記):新しい数学定理も新しい法則も主張しない。米田の補題、同スペクトル非同型グラフの存在、二面体群と四元数群が同じ指標表を持つことは、いずれも標準的である。測定値を引かない——本稿の数はすべて全数え上げで得たものである。圏論を導入しない——米田の補題の主張だけを使い、証明も一般化も扱わない。グラフ同型判定の計算量を論じない——6頂点という有限の場合の全数え上げである。表現論を論じない——指標表を既知として引くだけである。米田の補題が本体系の方法を正当化するとは主張しない——むしろ本稿の第6節は、そう言いたくなることを止めるために書かれている。 既刊との関係。論文152 は「同じ指標表を持ちながら、同型でない」を示した——本稿はその現象を、プローブの強さの関数として数える。論文153 は忘却関手の三段を分け、右随伴が無いことの証拠を挙げた——本稿は同じ圏論の棚の、隣の定理を扱う。論文102 は「太鼓の形は聴き分けられるか」を扱った——本稿の第3節はその有限版の全数え上げである。論文163 は「在る」と「これだ」が別の文だと示した——第6節は「決まる」と「求まる」が別の文だと言う。 設定。対象の集まりと、そこから情報を引き出すプローブを考える。プローブとは、対象に当てて値を返す関数である。プローブが二つの対象に同じ値を返すとき、そのプローブは二つを分離できない。題材は 6 頂点の単純グラフで、ラベル付きで 32768 個ある。プローブは三つ用意する。次数列、隣接行列のスペクトル、そしてスペクトルと三角形の個数を組にしたもの。比較のために、同型類そのものを置く。 第一に、32768 個は同型類 156 に落ちる。720 通りの頂点の並べ替えをすべて当てて正規形を取り、数えた。 第二に、次数列では足りない。相異なる値は 102 しかなく、84 類が 30 の組の中で分離されずに残る。 第三に、スペクトルでも足りない。相異なる値は 151 で、10 類が 5 組で残る。 第四に、これが本稿の芯である。三角形の個数を足しても、相異なる値は 151 のまま、分離できない類は 10 のままであった。三角形の個数は隣接行列の三乗のトレースを 6 で割ったものであり、スペクトルの関数である。したがって足しても情報が増えない。プローブを増やしたときに分離能が上がるかどうかは、増やしたものが既存のプローブから復元できるかで決まる。「もっと多くの不変量を測れば、いつかは分かる」は正しくない——独立でない不変量をいくら足しても、一歩も進まない。 第五に、強さは分離能の全順序を与えない。スペクトルで分離できない最小の組を取り出すと、次数列が 0,1,1,1,1,4 のものと 0,0,2,2,2,2 のものであり、どちらも辺が 4 本、三角形が0 個で、共有しているスペクトルはマイナス 2、0、0、0、0、2 である。この二つは次数列では分離される。弱いプローブが分けて、強いプローブが分けない。 第六に、群でも同じことが起きる。二面体群 D4 と四元数群 Q8 は同じ指標表を持ち、同型でない。ところが n 乗して単位元になる元の個数、すなわち巡回群からの準同型の個数を数えると、n が 2 のとき 6 と 2 になり、二つを分ける。指標表は分けられなかったのに、たった一つの相手からの写像を数えるだけで分かれた。ところが n が 4 のときは 8 と 8 で、分けない。同じ形のプローブでも、相手を取り替えると分離能が変わる。どの相手が効くかは、あらかじめ分からない。だから米田は「すべての相手」を要求する。 第七に、これは柵である。米田の補題は「対象は、他との関係で決まる」と読める。本体系の方法——ものが何かではなく、何と何を分けるかで書く——と形が似ている。しかし似ていることは正当化ではない。米田の補題は圏の内部の定理であり、Hom 集合と自然変換という具体的な対象についての主張である。方法論はその対象ではないので、補題は方法論について何も言っていない。定理としての米田と、比喩としての米田を分ける。後者を前者の権威で語ることが、本体系がずっと避けてきた事故である。そして「決まる」は「求まる」を意味しない——米田は対象が決まると言うだけで、求め方を与えない。 結び。米田の補題が要求しているのは「すべての相手」である。その「すべて」を有限で置き換えると、落ちるものがある。しかも落ちるかどうかは、プローブの強さでは決まらない——次数列が分ける組をスペクトルが分けず、三角形を足しても一つも増えなかった。分離子は、そのプローブが既存のプローブから復元できるかである。「もっと測れば分かる」は、独立なものを測るときだけ正しい。 作成にあたって:本稿の着想と内容は、著者自身の考察に基づくものです。文章の構成整理や英訳、数式の確認には AI(大規模言語モデル)の助力を得ました。最終的な内容の解釈や誤りがあれば、それらはすべて著者の責に帰します。お気づきの点があれば、ご教示いただければ幸いです。

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