Abstract
When a bound in physics is written down, $\le$ and $<$ are distinguished with care. And the symbol on the page does not settle whether the wall can be touched. This paper measures, for eleven bounds, whether the equality is reached. NO NEW MATHEMATICAL THEOREM AND NO NEW LAW IS CLAIMED.
First, some walls cannot be touched. Approaching the Buchdahl bound of $8/9$, the central pressure diverges, passing $166667$ and then $1.67\times10^6$ times $\rho c^2$ — the configuration that would touch the wall is not a configuration at all (Section 3).
Second, some walls can be. A gaussian meets $\Delta x\,\Delta p=\hbar/2$ exactly to fifteen digits — the ratio measured $1.000000000000000$ (Section 4).
Third, my own guess was wrong. What forbids the algebraic CHSH ceiling of $4$ is not relativity — a PR box reaches $4$ exactly and its signalling measures $\mathbf{0}$ (Section 5).
Fourth, this is the core. The number $2\sqrt2=2.828427124746$ is a wall that is touched, a wall out of reach, and not a wall at all — what changed is not the number but which theory one stands inside (Section 5).
Fifth, there are three separate reasons for not reaching. A divergence is required (Buchdahl); the extremiser sits at an endpoint (Landauer is still $0.0005\%$ above at $10^6$ units of time); something other than the inequality excludes it (Chandrasekhar) (Section 7).
Scope (what is not claimed)
NO NEW MATHEMATICAL THEOREM AND NO NEW LAW IS CLAIMED — the Buchdahl bound, the Tsirelson bound, PR boxes, minimum-uncertainty states, the Landauer bound and the Chandrasekhar mass are all standard. No bound is derived — for a given bound, only whether the equality is reached is measured. No claim of completeness — eleven examples appear, and the classification is not asserted to be exhaustive. No general mathematics is offered — whether a supremum is a maximum is a standard distinction in analysis, and this paper only applies it to bounds in physics and counts. No rule about notation is proposed — whether one ought to write $\le$ or $<$ is not discussed. Relation to earlier papers: Paper 16 showed that the equals sign has six distinct roots — this paper takes the inequality side, where one symbol points at two kinds of wall. Paper 21 wrote that whether something is an identity or a hypothesis depends on which theory one stands inside — Section 5 here is the quantitative version. Paper 194 showed there is only one equality fence, and Paper 160 counted three ceilings over one inequality — this paper sorts those ceilings by whether they are touched. What is added is following the Buchdahl central pressure numerically into the limit, confirming minimum uncertainty to fifteen digits, computing the CHSH value and the signalling of a PR box, following the Landauer excess out to $10^6$ units of time, and tabulating eleven bounds by whether the equality is reached.
1. Notation and Position
For a bound $X\le B$, the equality is reached when some configuration realises $B$, and not reached when $X$ comes arbitrarily close to $B$ without ever taking it. In mathematics this is the standard distinction between a supremum and a maximum. This paper applies it to bounds in physics and counts.
Position: Paper 16 showed six distinct roots of the equals sign — this paper takes the inequality side. The phrase which theory one stands inside, from Paper 21, does the work in Section 5.
2. The Symbol Does Not Settle Whether the Wall Can Be Touched
Observation 2.1 there are walls written with $\le$ that are never touched, and walls written with $\ge$ that are met exactly. The symbol says that the bound cannot be passed; it does not say that it can be met.
And which of the two it is cannot be read off the symbol. It has to be computed.
3. A Wall Not Touched: The Extremiser Requires a Divergence
Check 3.1: for a uniform-density interior Schwarzschild solution, measure the central pressure against the compactness $u=2GM/(Rc^2)$.
| $u$ | $u\div(8/9)$ | Central pressure $\div\rho c^2$ |
|---|---|---|
| $0.5$ | $0.5625$ | $0.261204$ |
| $0.86$ | $0.9675$ | $5.10897$ |
| $0.888$ | $0.999$ | $166.666$ |
| $0.888888$ | $0.999999$ | $\mathbf{166667}$ |
| $0.8888888$ | $0.9999999$ | $\mathbf{1.66667\times10^6}$ |
Observation 3.2 The central pressure diverges at $8/9$. The configuration that would touch the wall is not a configuration — in form it is written as at most, and in physics it is strictly less than.
4. A Wall Touched: A State Meets the Equality Condition
Check 4.1: measure $\Delta x\,\Delta p$ for minimum-uncertainty gaussians.
| Width | $\Delta x\,\Delta p$ | $\div(\hbar/2)$ |
|---|---|---|
| $10^{-11}$ | $5.272859085000000\times10^{-35}$ | $1.000000000000000$ |
| $10^{-10}$ | $5.272859085000000\times10^{-35}$ | $1.000000000000000$ |
| $10^{-9}$ | $5.272859085000000\times10^{-35}$ | $1.000000000000000$ |
Observation 4.2 Equality holds exactly, to fifteen digits. The equality condition of Cauchy–Schwarz is that the two vectors be proportional, and states meeting it exist — not one but a whole family.
Check 4.3: for the arithmetic and geometric means, equality holds only when all the numbers agree. Drawing $200000$ triples uniformly from $0.5$ to $4$, equality occurred $0$ times. At $x=(2,2,2)$ the difference is exactly $0$.
Observation 4.4 It is reached, and the set that reaches it has probability zero. Being able to touch a wall and touching it by chance are different things.
5. The Core: One Ceiling Changes Standing With the Ground One Stands On
Check 5.1 (the guess that failed): in a first draft I wrote that the algebraic CHSH ceiling of $4$ is forbidden by no-signalling. A PR box was built and measured.
| Measured | Value |
|---|---|
| CHSH value of a PR box | $\mathbf{4.000000000000}$ |
| Its largest signalling | $\mathbf{0}$ |
Observation 5.2 The guess was wrong. A PR box signals not at all and reaches $4$ exactly — what forbids $4$ is quantum mechanics, and not relativity.
Check 5.3: the three ceilings, counted from each standpoint. The quantum maximum measured $2.828427124746$ against $2\sqrt2=2.828427124746$.
| Standing inside | $2$ | $2\sqrt2$ | $4$ |
|---|---|---|---|
| Local hidden variables | reached | impossible | impossible |
| Quantum mechanics | reached | reached | impossible |
| Any no-signalling theory | reached | reached | reached |
Observation 5.4 The number $2\sqrt2$ is a wall that is touched, a wall out of reach, and not a wall at all. The number never changed — what changed is which theory one stands inside. This is the quantitative version of what Paper 21 wrote about identities and hypotheses.
6. A Wall Not Touched: The Extremiser Sits at an Endpoint
Check 6.1: measure the work of erasure at $300$ K as the time is lengthened. The floor is $k_BT\ln2=2.870978885\times10^{-21}$ J.
| Time (units) | Work | Excess over the floor |
|---|---|---|
| $1$ | $1.722587331\times10^{-20}$ | $500\%$ |
| $100$ | $3.014527829\times10^{-21}$ | $5\%$ |
| $10^4$ | $2.872414375\times10^{-21}$ | $0.05\%$ |
| $10^6$ | $2.870993240\times10^{-21}$ | $\mathbf{0.0005\%}$ |
Observation 6.2 Every finite time lies strictly above. The floor is the limit of the set and not a member of it — written with $\ge$, and the equality never occurs.
7. The Census: Eleven Bounds
Check 7.1:
| Bound | Written | Reached | Why |
|---|---|---|---|
| Heisenberg $\hbar/2$ | $\ge$ | yes | a state meets the equality condition |
| AM–GM | $\ge$ | yes | one point only, of probability zero |
| Tsirelson $2\sqrt2$ | $\le$ | yes | inside quantum mechanics |
| CHSH's $2$ | $\le$ | yes | inside local hidden variables |
| CHSH's $4$ | $\le$ | yes | inside no-signalling (a PR box) |
| CHSH's $4$, standing in quantum | $\le$ | no | quantum mechanics forbids it |
| Schwarzschild radius | $=$ | yes | the horizon is a real surface |
| Buchdahl $8/9$ | $\le$ | no | the extremiser needs a divergence |
| Landauer $k_BT\ln2$ | $\ge$ | no | an endpoint at infinite time |
| Chandrasekhar mass | $<$ | no | stability excludes it, not the inequality |
| Speed of light, massive body | $<$ | no | it would take infinite energy |
Observation 7.2 The reasons for not reaching fall into three.
| Reason | Example |
|---|---|
| (i) The extremiser requires a divergence | Buchdahl, the speed of light |
| (ii) The extremiser sits at an endpoint outside the set | Landauer |
| (iii) Something other than the inequality excludes it | Chandrasekhar (stability) |
Observation 7.3 There is only one reason for reaching — (iv) a configuration meeting the equality condition lies inside the permitted set.
8. The Separator and the Fence
Observation 8.1 (the separator) What separates the two kinds of $\le$ is whether the extremiser lies inside the set. It is not the symbol.
Observation 8.2 (the fence) The fence around this quantity is bounded by $B$ is over which set the bound is taken. Outside it the same $B$ becomes reachable or unreachable — the same shape as this sequence is random in Paper 223 and volume is conserved in Paper 225.
Observation 8.3 The difference between strictly less and at most is not a question of notation but a question of measurement. Writing the equality is not an assertion but a way of putting the question — the form in which one asks whether this wall can be touched.
Observation 8.4 Paper 16 counted six distinct roots of the equals sign. What this paper counted is two kinds of inequality sign — the sorting of the symbols is not finished.
9. The Census and the Close
What this paper measured:
- Buchdahl is not touched (Check 3.1): the pressure diverges past $1.67\times10^6$.
- Heisenberg is touched (Check 4.1): the ratio is $1$ to fifteen digits.
- AM–GM is touched, with probability zero (Check 4.3): $0$ hits in $200000$.
- The guess failed (Check 5.1): a PR box reaches $4$ and signals $0$.
- One ceiling changes standing (Check 5.3): $2\sqrt2$ has three meanings.
- Landauer is an endpoint (Check 6.1): still $0.0005\%$ above at $10^6$.
- Three reasons for not reaching (Observation 7.2): divergence, endpoint, outside exclusion.
References
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- Yamagishi, Y. (2026). The Equals Sign Has Distinct Roots Too [Paper 16]. Zenodo. 10.5281/zenodo.21600471
- Yamagishi, Y. (2026). Zero-Tension Does Not Act on an Identity [Paper 21]. Zenodo. 10.5281/zenodo.21601143
- Yamagishi, Y. (2026). $\hbar$ Is Not on the Mathematical Side of the Inequality [Paper 129]. Zenodo. 10.5281/zenodo.22079681
- Yamagishi, Y. (2026). Of $k_BT\ln2$, Only the $T$ Is Physics [Paper 131]. Zenodo. 10.5281/zenodo.22080240
- Yamagishi, Y. (2026). Pressure Pulls Too, and That Is Why There Is a Bound [Paper 145]. Zenodo. 10.5281/zenodo.22082693
- Yamagishi, Y. (2026). Three Ceilings over One Inequality [Paper 160]. Zenodo. 10.5281/zenodo.22092434
- Yamagishi, Y. (2026). The Four Means Are One Family, and There Is Only One Equality Fence [Paper 194]. Zenodo. 10.5281/zenodo.22106809
Reference
Yamagishi, Y. (2026). The Same Inequality Sign Points at a Wall You Can Touch and at One You Cannot [Paper 227]. YOI: https://www.yamagishiyuuki.com/yoi/227
DOI: 10.5281/zenodo.22123129
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 227. This paper is a preprint and has not been peer reviewed.