Abstract
“There are this many receptors, so this many odours can be told apart,” it is said. This paper counts that “this many” does not follow from the number of receptors. No new theorem or law is claimed.
First, counting every combination as different gives an enormous number: $2^{400}$ for $400$ receptor types, $\mathbf{120.411998}$ orders of magnitude (Section 2).
Second, fixing how many are active shrinks it sharply: exactly $5$ active gives $\mathbf{10.920220}$ orders, and $1$ to $10$ gives $19.422809$ (Section 2).
Third, reading only the strongest one gives the receptor count itself: $400$, or $2.602060$ orders (Section 2).
Fourth, and this is the core. The same $400$ yields a spread of more than $117$ orders: widest and narrowest rules differ by $\mathbf{117.809938}$ orders (Section 2).
Fifth, the effect of adding receptors also depends on the rule: doubling to $800$ adds $120.411998$ orders when every combination counts, and only $\mathbf{1.510609}$ for exactly five (Section 3).
Sixth, the separator is whether one asks about the number of receptors or about the rule for reading them: the distinguishable number is a consequence of the rule on the reading side, not of the receptors (Section 4).
Keywords: olfactory receptor, combinatorial coding, binomial coefficient, counting rule, order of magnitude, model.
Scope (what is not claimed)
No new theorem or law is claimed: that smell represents odours by combinations of many receptors, that the number of subsets is $2^{R}$ and of $k$-subsets a binomial coefficient, and that estimates of the number of distinguishable odours vary greatly with the assumptions made, are all known. No real value is claimed: $400$ is a model number placed for comparison, not a claim about any organism. No human ability is stated: how many odours people can actually tell apart is not stated. The brain is not entered: neural processing, learning and adaptation are not treated. Chemistry is not entered, which substances bind which receptors is not treated. Relation to earlier papers: Paper 323 showed that the “three” in three primaries is not a property of light. This paper stands beside it: there the number of colours came from the count on the receiving side; here the number of odours comes from the reading rule on the receiving side. Paper 182 showed that “the curse of dimensionality” is not one curse. Here too one question, “how many”, admits an answer per rule. What is added is setting four reading rules on the same $400$ and laying out their orders of magnitude, measuring a spread of $117.809938$ orders, measuring that the gain from doubling receptors also depends on the rule, and placing the separator on receptor count versus reading rule.
1. Notation and the Question
Setting: let $R$ be the number of receptor types, and let odours be told apart by which receptors they activate. Four rules are placed for what counts as “a different odour”.
$$N_{\mathrm{all}}=2^{R},\qquad N_{k}=\binom{R}{k},\qquad N_{\le 10}=\sum_{k=1}^{10}\binom{R}{k},\qquad N_{\max}=R$$The last rule reads only the most strongly activated receptor. Here $R=400$, and since the numbers are large they are compared by order of magnitude (common logarithm).
Question: once the number of receptors is fixed, is the number of distinguishable odours fixed?
2. Four Numbers From the Same $400$
Computation 2.1: at $R=400$, the orders of magnitude under the four rules were computed.
| rule | orders of magnitude |
|---|---|
| every combination different ($2^{R}$) | $120.411998$ |
| $1$ to $10$ active | $19.422809$ |
| exactly $5$ active | $10.920220$ |
| only the strongest one read | $2.602060$ |
| widest minus narrowest | $117.809938$ |
Observation 2.2 this is the core. The same $400$ yields a spread of more than $117$ orders. The receptor count is the same under all four rules, yet the answer runs from $2.602060$ to $120.411998$. What fixes the number is not the receptors but the rule for what counts as different.
Observation 2.3 the narrower the rule, the smaller the number. Constraining how many may be active cuts the choices: saying “represented by combinations” does not yet say which combinations are allowed.
3. The Effect of Adding Receptors
Computation 3.1: receptors were doubled from $400$ to $800$.
| rule | $R=400$ | $R=800$ |
|---|---|---|
| every combination different | $120.411998$ | $240.823997$ |
| exactly $5$ active | $10.920220$ | $12.430830$ |
Observation 3.2 the gain from doubling also depends on the rule. Under the first rule the orders rise by $120.411998$; under exactly five, by only $1.510609$: the first grows in orders proportionally to the count, the second only as its fifth power.
Observation 3.3 so “more receptors, more odours” holds to a degree set by the rule. The same doubling doubles the orders under one rule and adds fewer than two under another: even the sensitivity to the receptor count is decided on the reading side.
4. The Separator
Observation 4.1 the separator is whether one asks about the number of receptors or about the rule for reading them. With the first fixed, the second moves the answer by $117.809938$ orders: the distinguishable number is a consequence not of the count on the receiving side but of how what is received is counted.
Observation 4.2 placed beside Paper 323. 323 showed that the “three” in three primaries is not a property of light. There the number of colours came from a count on the receiving side; here, from the same count, the number changes utterly with the reading. One step past the count lies the rule. Paper 182 showed that “the curse of dimensionality” is not one curse. Here too one question admits an answer per rule.
Observation 4.3 what is not said. Real receptor counts, human discrimination, neural processing and odour chemistry are not treated. What is shown is that “how many” becomes one number only once a rule is fixed.
5. Census and Closing
- $400$ receptor types assumed (Section 1): a model number.
- every combination: $\mathbf{120.411998}$ (Computation 2.1): the widest.
- $1$ to $10$ active: $19.422809$ (Computation 2.1): constrained, smaller.
- exactly $5$: $\mathbf{10.920220}$ (Computation 2.1): smaller still.
- strongest only: $2.602060$ (Computation 2.1): the count itself.
- spread $\mathbf{117.809938}$ orders (Computation 2.1): the core.
- doubling adds $120.411998$ (Computation 3.1): wide rule.
- doubling adds $\mathbf{1.510609}$ (Computation 3.1): narrow rule.
- sensitivity to the count depends on the rule (Observation 3.3): the reading side.
- receptor count, or reading rule (Observation 4.1): the separator; Papers 323, 182.
Closing: How many odours can be told apart is set not by the number of receptors but by the rule for reading them. With $400$ receptor types, counting every combination as different gives $120.411998$ orders, exactly five active gives $10.920220$, and reading only the strongest gives $2.602060$, a spread of $117.809938$ orders: the receptor count is the same throughout; only the rule for what counts as different changed. Doubling to $800$ adds $120.411998$ orders under the widest rule and only $1.510609$ under exactly five: the first grows in orders with the count, the second only as its fifth power. So how far “more receptors, more odours” holds is decided on the reading side. Placed among the earlier papers: beside Paper 323’s “the number comes from the count on the receiving side”, as the stage of rules one step past the count, and counting with Paper 182’s pattern one question with an answer per rule. To be honest: $400$ is a model number, and nothing is said about how many odours people can actually tell apart.
References
- Buck, L., & Axel, R. (1991). A novel multigene family may encode odorant receptors: a molecular basis for odor recognition. Cell.
- Yamagishi, Y. (2026). “The Curse of Dimensionality” Is Not One Curse [Paper 182]. YOI: https://www.yamagishiyuuki.com/yoi/182; DOI: 10.5281/zenodo.22104170
- Yamagishi, Y. (2026). The “Three” in Three Primaries Is Not a Property of Light [Paper 323]. YOI: https://www.yamagishiyuuki.com/yoi/323; DOI: 10.5281/zenodo.22209319
Reference
Yamagishi, Y. (2026). How Many Odours Can Be Told Apart Is Set Not by the Number of Receptors but by the Rule for Reading Them [Paper 851]. YOI: https://www.yamagishiyuuki.com/yoi/851
ORCID: 0009-0006-8587-3450
PDF (English) · PDF (Japanese) · Date proof of the English PDF (OpenTimestamps .ots) · SHA256SUMS
The record page for this paper (in Japanese, with the PDFs in each language and the date proof): YOI 851. This paper is a preprint and has not been peer reviewed.