Non-Measurability Theory

1. Introduction

In physics,non-measurability theory is a theoretical framework in which all elementary particles are composed of three-dimensional objects called dark energy. Dark energy itself is non-measurable,some dark energy forms measurable matter, and some forms non-measurable matter (Table 1), The cosmological ideas of the Tao Te Ching [1] and Non-Measurability Theory are essentially the same(See Chapter 12 for a detailed discussion).We use the properties (especially the Banach-Tarski paradox) of non-measurable sets to study the frontier of physics. The expansion of the universe, the star collapses into a black hole, neutrino oscillations, and so on may be all physical manifestations of the Banach-Tarski paradox.

In ancient mathematics, some lengths that could not be measured by rational numbers caused the first mathematical crisis. In order to solve this crisis, we introduced the concept of irrational numbers. Similarly, the axiom of choice implies that there is a non-measurable set. We cannot assign a real number to the nonmeasurable set. To solve the measurement problem, we assign an interval number to each nonmeasurable set, and the property of the interval number can solve the measurement problem of the nonmeasurable set. The mathematical tool for analyzing nonmeasurable sets is interval mathematics ([7–12]). Here we need to use interval arithmetic, interval measure theory, interval probability theory, interval tensor theory and interval differential geometry. Interval arithmetic is a mature branch of mathematics, but other branches of interval mathematics are not mature enough and need us to do a lot of research.

Does nonmeasurable matter exist in our universe? Most people say no, because we can’t divide matter into complicated cases.But there is a possibility that there has always been nonmeasurable matter in the universe! At the beginning of the 20th century, the idea that energy was discontinuous revolutionized physics. Energy itself is discontinuous, that’s the law of nature.We have a similar idea that there exists matter whose mass (energy) value is a nondegenerate interval number.In other words, non-measurable matter has different masses at the same time,we call such matter nonmeasurable matter. We assume that non-measurable matter has all the properties of non-measurable sets.According to the assumptions, we can solve a large number of physics problems.(An analogy between hadronic physics and theory of nonmeasurable sets can be found in reference [14].A black hole has different masses at the same time[25] ,so a black hole is nonmeasurable ).This hypothesis is closely related to the following physics topics: (1)The mystery of the missing antimatter, (2) Quantum gravity, (3) The gravitational theory of a single particle, (4) Properties of neutrinos or quarks,(5)Dark matter and dark energy, (6) Hubble constant (7) The energy of gravitational fields, (8) Properties of black holes,(9)The accelerating expansion of the universe,(10)Cosmological constant problem,(11)Big Bang Puzzle . Evidences for the existence of non-measurable matter can be found in section 4 of the article.

We take two properties of dark energy as the principle of the theory, one is that dark energy has the property of a non-measurable set, and the other is that all matter and fields are excitation of dark energy. We do deductive reasoning on the basis of these two principles, and the first conclusion is that there is non-measurable matter, and we provide evidence for the existence of such matter. According to the second principle, there are only dark energy fields in vacuum, and no other quantum fields, which can solve the cosmological constant problem. We can see that energy is not conserved if there is non-measurable matter, which solves the black hole information paradox. We can get that the cosmological constant is not a real number, but an interval number, We can deduce that the Hubble constant of de Sitter spacetime is a non-degenerate interval constant whose value depends on the measurement method, which can explain the irreconcilable gap between the measurements obtained by different measurement methods. And then , we introduce an important principle of quantum mechanics, namely quantum intertextuality, into the theory of gravity and geometry, we get quantum gravity and quantum geometry, and we try to explain strange quantum phenomena in quantum geometry.

, we ought in fact to suppose it, if we can thereby obtain a simpler explanation of phenomena, .Riemann said in his famous paper[26].The above hypothesis can explain a large number of physical phenomena very simply, and we think that these two principles may be the truth. Experiment is the only standard to test a theory, so we need to provide more experimental evidence.

2. Preliminaries

Theorem 2.1   (Banach-Tarski Paradox, Euclidean space). (AC).If $A,B\subseteq {\mathbb{R}}^n(n\ge 3)$ are any bounded sets with nonempty interior, then A and B are equidecomposable.

Theorem 2.2   (Banach-Tarski Paradox, Sphere ). (AC).If $A,B\subseteq {\mathbb{S}}^n(n\ge 2)$ are any bounded sets with nonempty interior, then A and B are equidecomposable.

Theorem 2.3   (Banach-Tarski Paradox, Hyperbolic space). (AC).If $A,B\subseteq {\mathbb{H}}^n(n\ge 2)$ are any bounded sets with nonempty interior, then A and B are equidecomposable.

Theorem 2.4   (Banach-Tarski Paradox). (AC).Let G be a connected, finite-dimensional non-solvable Lie group acting transitively and continuously on a metric space (X, d) with no isolated points and suppose that G has negligible fixed points. Then any two subsets of X with compact closures and non-empty interiors are G-equidecomposable.

Theorem 2.5   (A continuous movement of Banach-Tarski Paradox) [54]. (AC) If $n\ge 3$, then any two bounded subsets of $R^n$ with non-empty interior are continuously equidecomposable using orientation-preserving isometries.

In particular, the solid ball in $R^3$ has a continuous paradoxical decomposition.

Remark   Some conclusions on this topic can be found in References [4,16,35,27,50].

3. Generalized Measure Theory

Definition 3.1   $\mathbb{M}=\left\{I_x^y\right|x,y\in \mathbb{R},x\le y\}$,$I_{x_1}^{y_1}\le I_{x_2}^{y_2}$ iff $x_1\le x_2$ and $y_1\le y_2$ ,$I_{x_1}^{y_1}⊑I_{x_2}^{y_2}$ iff $x_2\le x_1,y_1\le y_2$. $I_{x_1}^{y_1}+I_{x_2}^{y_2}=I_{x_1+x_2}^{y_1+y_2}$, $I_{x_1}^{y_1}·I_{x_2}^{y_2}=I_{min\{x_1{x}_2,x_1{y}_2,y_1{x}_2,y_1{y}_2\}}^{max\{x_1{x}_2,x_1{y}_2,y_1{x}_2,y_1{y}_2\}}$, ,$I_0^0=0,I_1^1=1$. Then $I_x^y$ is called an (finite) interval number. $\langle \mathbb{M},+,·,\le ,⊑,0,1\rangle$ is called ordering interval arithmetic,denoted by $\mathcal{M}$.

Remark1   $I_{x_1}^{y_1}-I_{x_2}^{y_2}{=}_{def}{I}_{x_1}^{y_1}+I_{-y_2}^{-x_2}$.$I_{x_1}^{y_1}÷I_{x_2}^{y_2}{=}_{def}{I}_{x_1}^{y_1}·I_{{\displaystyle \frac{1}{y_2}}}^{{\displaystyle \frac{1}{x_2}}}(x_2>0$ or $y_2<0)$.

Remark2   The binary operation f on this arithmetic structure can be uniformly defined in the following form: $f(I_{x_1}^{y_1},I_{x_2}^{y_2})=I_{min\{f(a,b)\mid a\in [x_1,y_1],b\in [x_2,y_2]\}}^{max\{f(a,b)\mid a\in [x_1,y_1],b\in [x_2,y_2]\}}$.

Definition 3.2   ${\mathbb{M}}^{*}=\left\{I_x^y\right|x,y\in {\mathbb{R}}^{*},x\le y\}$,$I_{x_1}^{y_1}\le I_{x_2}^{y_2}$ iff $x_1\le x_2$ and $y_1\le y_2$ ,$I_{x_1}^{y_1}⊑I_{x_2}^{y_2}$ iff $x_2\le x_1,y_1\le y_2$. $I_{x_1}^{y_1}+I_{x_2}^{y_2}=I_{x_1+x_2}^{y_1+y_2}$, $I_{x_1}^{y_1}·I_{x_2}^{y_2}=I_{min\{x_1{x}_2,x_1{y}_2,y_1{x}_2,y_1{y}_2\}}^{max\{x_1{x}_2,x_1{y}_2,y_1{x}_2,y_1{y}_2\}}$, ,$I_0^0=0,I_1^1=1$. Then $I_x^y$ is called a generalized interval number. $\langle {\mathbb{M}}^{*},+,·,\le ,⊑,0,1\rangle$ is called generalized ordering interval arithmetic,denoted by ${\mathcal{M}}^{*}$.

Remark1   $I_{x_1}^{y_1}-I_{x_2}^{y_2}{=}_{def}{I}_{x_1}^{y_1}+I_{-y_2}^{-x_2}$.$I_{x_1}^{y_1}÷I_{x_2}^{y_2}{=}_{def}{I}_{x_1}^{y_1}·I_{{\displaystyle \frac{1}{y_2}}}^{{\displaystyle \frac{1}{x_2}}}(x_2\ge 0$ or $y_2\le 0)$.$(x_2=0\Rightarrow {\displaystyle \frac{1}{x_2}}=\infty )$,$(y_2=0\Rightarrow {\displaystyle \frac{1}{y_2}}=-\infty )$

Remark2   The binary operation f on this arithmetic structure can be uniformly defined in the following form: $f(I_{x_1}^{y_1},I_{x_2}^{y_2})=I_{min\{f(a,b)\mid a\in [x_1,y_1],b\in [x_2,y_2]\}}^{max\{f(a,b)\mid a\in [x_1,y_1],b\in [x_2,y_2]\}}$.

Definition 3.3   Let X be a set,$\mathcal{A}\subseteq \mathcal{P}(X)$,$\mathcal{A}$ be a algebra ($\sigma -$algebra or Complete Boolean algebra) over X, a mapping $\nu :\mathcal{A}\to {\mathbb{M}}^{*}$ is called an interval measure on X if it satisfies the following properties (1) (2) (3) (4) .$(X,\mathcal{A},\nu )$ is called an interval measure space .A mapping $\nu :\mathcal{A}\to {\mathbb{M}}^{*}$ is called a finitely additive interval measure on X if it satisfies the following properties(1)(2)(3)($4^{\prime }$).$(X,\mathcal{A},\nu )$ is called a finitely additive interval measure space.

(1)

(Non-negativity)$\nu (A)\ge I_0^0$.

(2)

(Empty set)$\nu (\varnothing )=I_0^0$.

(3)

(Shiffman Property)If$A\bigcap B=\varnothing ,\nu (A)=I_{x_1}^{y_1},\nu (B)=I_{x_2}^{y_2}$, $min\{x_1+y_2,x_2+y_1\}=\perp ,$ $max\{x_1+y_2,x_2+y_1\}=\top$ ,then $I_{\perp }^{\top }⊑\nu (A\cup B)$.

(4)

(Countable covering property)For all countable collections ${\displaystyle {\left\{A_i\right\}}_{i=1}^{\infty }}$ of pairwise disjoint sets in $\mathcal{A}$, $\nu ({\cup }_{i=1}^{\infty }{A}_i)⊑{\Sigma }_{i=1}^{\infty }\nu (A_i)$.

(4’)

(Finite covering property)For all finite collections ${\displaystyle {\left\{A_i\right\}}_{i=1}^n}$ of pairwise disjoint sets in $\mathcal{A}$, $\nu ({\cup }_{i=1}^n{A}_i)⊑{\Sigma }_{i=1}^n\nu (A_i)$.

An interval measure on X is called an interval probability measure if (5) (Normalisation) $\nu (X)=I_1^1$.$(X,\mathcal{A},\nu )$ is called an interval probability space .

A finitely additive interval measure on X is called a finitely additive interval probability measure if (5) (Normalisation) $\nu (X)=I_1^1$.$(X,\mathcal{A},\nu )$ is called a finitely additive interval probability space .

Remark    The Shiffman Property is proved for a general measure space in [2][42].

Theorem3.1   Every interval probability measure(IPM) on a $\sigma -$ algebra is an Interval-valued Probability Measure(IVPM) on the $\sigma -$ algebra.

Proof. 

Let $(X,\mathcal{A},\nu )$ be an interval probability measure space. Thus $\nu$ satisfies:

(IPM1)

   $\nu (A)\ge I_0^0$ for all $A\in \mathcal{A}$;

(IPM2)

    $\nu (⌀)=I_0^0$;

(IPM3)

    (Shiffman property) If $A\cap B=⌀$, $\nu (A)=I_{x_1}^{y_1}$, $\nu (B)=I_{x_2}^{y_2}$, then

$$I_{min\{x_1+y_2,x_2+y_1\}}^{max\{x_1+y_2,x_2+y_1\}}⊑\nu (A\cup B);$$

(IPM4)

    (Countable covering property) For pairwise disjoint sets ${\left\{A_i\right\}}_{i=1}^{\infty }\subset \mathcal{A}$,

$$\nu \left({\displaystyle \bigcup _{i=1}^{\infty }}{A}_i\right)⊑{\displaystyle \sum _{i=1}^{\infty }}\nu (A_i);$$

(IPM5)

    $\nu (X)=I_1^1$.

We must show that $\nu$ is an IVPM, i.e.

(IVPM1)

    $\nu (⌀)=I_0^0$;

(IVPM2)

    $\nu (X)=I_1^1$;

(IVPM3)

    $\nu (A)⊑I_0^1$ for all $A\in \mathcal{A}$;

(IVPM4)

    For every partition $A={\bigcup }_{k\in K}{A}_k$, $A^c={\bigcup }_{j\in J}{B}_j$, we have

$$\nu (A)⊑I_{max\{1-{\sum }_{j\in J}{B}_j^u,{\sum }_{k\in K}{A}_k^l\}}^{min\{1-{\sum }_{j\in J}{B}_j^l,{\sum }_{k\in K}{A}_k^u\}}.$$

(IVPM1) and (IVPM2) are exactly (IPM2) and (IPM5). We first prove (IVPM3) and a useful complement relation.

Let $A\in \mathcal{A}$ and write

$$\nu (A)=I_{A_l}^{A_u},\nu (A^c)=I_{B_l}^{B_u}.$$

Since $A\cap A^c=⌀$ and $A\cup A^c=X$, the Shiffman property (IPM3) gives

$$I_{min\{A_l+B_u,B_l+A_u\}}^{max\{A_l+B_u,B_l+A_u\}}⊑\nu (X)=I_1^1.$$

By the definition of interval inclusion, this implies

$$min\{A_l+B_u,B_l+A_u\}\ge 1,max\{A_l+B_u,B_l+A_u\}\le 1.$$

Hence

$$min\{A_l+B_u,B_l+A_u\}=max\{A_l+B_u,B_l+A_u\}=1.$$

Therefore

$$A_l+B_u=1,B_l+A_u=1,$$

so that

$$B_u=1-A_l,B_l=1-A_u.$$

Consequently,

$$\nu (A^c)=I_{1-A_u}^{1-A_l}=I_1^1-\nu (A).$$

(1)

From (IPM1) we have $A_l\ge 0$ and $B_l\ge 0$, so $1-A_u=B_l\ge 0$, i.e. $A_u\le 1$. Thus

$$0\le A_l\le A_u\le 1,$$

which proves (IVPM3).

Now we prove (IVPM4). Let $A={\bigcup }_{k\in K}{A}_k$ and $A^c={\bigcup }_{j\in J}{B}_j$ be partitions, where all sets are pairwise disjoint. Write

$$\nu (A_k)=I_{A_k^l}^{A_k^u},\nu (B_j)=I_{B_j^l}^{B_j^u}.$$

By the countable covering property (IPM4),

$$\nu (A)⊑\sum _{k\in K}\nu (A_k)=I_{{\sum }_{k\in K}{A}_k^l}^{{\sum }_{k\in K}{A}_k^u}.$$

Hence

$$A_l\ge \sum _{k\in K}{A}_k^l,A_u\le \sum _{k\in K}{A}_k^u.$$

(2)

Similarly, for $A^c$,

$$\nu (A^c)⊑\sum _{j\in J}\nu (B_j)=I_{{\sum }_{j\in J}{B}_j^l}^{{\sum }_{j\in J}{B}_j^u},$$

so

$$B_l\ge \sum _{j\in J}{B}_j^l,B_u\le \sum _{j\in J}{B}_j^u.$$

(3)

Using the complement relation (1), namely $B_l=1-A_u$ and $B_u=1-A_l$, the inequalities in (3) become

$$1-A_u\ge \sum _{j\in J}{B}_j^l⟹A_u\le 1-\sum _{j\in J}{B}_j^l,$$

and

$$1-A_l\le \sum _{j\in J}{B}_j^u⟹A_l\ge 1-\sum _{j\in J}{B}_j^u.$$

Combining these with (2), we obtain

$$A_l\ge max\left\{\sum _{k\in K}{A}_k^l,1-\sum _{j\in J}{B}_j^u\right\},$$

and

$$A_u\le min\left\{\sum _{k\in K}{A}_k^u,1-\sum _{j\in J}{B}_j^l\right\}.$$

Therefore

$$\nu (A)=I_{A_l}^{A_u}⊑I_{max\{{\sum }_{k\in K}{A}_k^l,1-{\sum }_{j\in J}{B}_j^u\}}^{min\{{\sum }_{k\in K}{A}_k^u,1-{\sum }_{j\in J}{B}_j^l\}},$$

which is exactly (IVPM4).

Thus all the axioms of an IVPM are satisfied, and the proof is complete.    □

Remark   A formal definition of Interval-valued Probability Measure(IVPM) is proposed in [3].

Theorem3.2   [Equivalence of IVPM and IPM] Let $(X,\mathcal{A},\nu )$ be an interval-valued probability measure satisfying (IVPM1)–(IVPM4) in the sense of Theorem 3.1. Then $\nu$ is an interval probability measure. Consequently, together with Theorem 3.1, the notions of IPM and IVPM are equivalent.

Proof. 

Write $\nu (A)=I_{l(A)}^{u(A)}$, where $l(A)\le u(A)$. We verify the axioms (IPM1)–(IPM5).

First, (IPM1), (IPM2), and (IPM5) are immediate. Indeed, (IPM1) follows from (IVPM3): $\nu (A)\subseteq I_0^1$ implies $l(A)\ge 0$, hence $\nu (A)\ge I_0^0$. Moreover, (IPM2) and (IPM5) are exactly (IVPM1) and (IVPM2), respectively.

We next establish the complement identities. Applying (IVPM4) to the trivial partitions $A=A$ and $A^c=A^c$, we obtain

$$\nu (A)\subseteq I_{max\{l(A),1-u(A^c)\}}^{min\{u(A),1-l(A^c)\}}.$$

Thus

$$l(A)+u(A^c)\ge 1,u(A)+l(A^c)\le 1.$$

Interchanging A and $A^c$ gives

$$l(A^c)+u(A)\ge 1,u(A^c)+l(A)\le 1.$$

Combining these inequalities yields

$$l(A^c)=1-u(A),u(A^c)=1-l(A).$$

(1)

We now prove the Shiffman property (IPM3). Let $A\cap B=⌀$, and put $C=A\cup B$, $D=X\setminus C$. Then $A^c=B\bigsqcup D$. Applying (IVPM4) to the trivial partition of A and the partition $A^c=B\bigsqcup D$, we get

$$\nu (A)\subseteq I_{max\{l(A),1-u(B)-u(D)\}}^{min\{u(A),1-l(B)-l(D)\}}.$$

Hence

$$l(A)\ge 1-u(B)-u(D),u(A)\le 1-l(B)-l(D).$$

Using the complement identities for C, namely

$$u(D)=1-l(C),l(D)=1-u(C),$$

we obtain

$$l(C)\le l(A)+u(B),u(C)\ge u(A)+l(B).$$

Interchanging A and B gives

$$l(C)\le l(B)+u(A),u(C)\ge u(B)+l(A).$$

Therefore

$$l(C)\le min\{l(A)+u(B),l(B)+u(A)\},$$

and

$$u(C)\ge max\{u(A)+l(B),u(B)+l(A)\}.$$

Writing $\nu (A)=I_{x_1}^{y_1}$ and $\nu (B)=I_{x_2}^{y_2}$, this is exactly

$$I_{min\{x_1+y_2,x_2+y_1\}}^{max\{x_1+y_2,x_2+y_1\}}\subseteq \nu (A\cup B),$$

which is the Shiffman property.

Finally, we prove the covering property (IPM4). Let ${\left\{A_i\right\}}_{i=1}^{\infty }$ be pairwise disjoint, and put $A={\bigcup }_{i=1}^{\infty }{A}_i$. Applying (IVPM4) to the partition $A={⨆}_i{A}_i$ and the trivial partition of $A^c$, we get

$$\nu (A)\subseteq I_{max\{{\sum }_{i}l(A_i),1-u(A^c)\}}^{min\{{\sum }_{i}u(A_i),1-l(A^c)\}}.$$

By the complement identities,

$$1-u(A^c)=l(A),1-l(A^c)=u(A).$$

Hence

$$l(A)\ge \sum _{i}l(A_i),u(A)\le \sum _{i}u(A_i).$$

Thus

$$\nu (A)\subseteq I_{{\sum }_{i}l(A_i)}^{{\sum }_{i}u(A_i)}=\sum _i\nu (A_i).$$

This is (IPM4). Therefore all the axioms of an IPM are satisfied, and the proof is complete.    □

Proposition3.3   If $m^{*}$ is the Lebesgue outer measure on ${\mathbb{R}}^n$, $m_{*}$ is the Lebesgue inner measure on ${\mathbb{R}}^n$,for every subset S of ${\mathbb{R}}^n$, $\mu (S)=I_{m_{*}(S)}^{m^{*}(S)}$,then $\mu$ is an interval measure on ${\mathbb{R}}^n$ .

Proof   (1)-(2)and (4) are easy to prove .(3) is proved by Max Shiffman.

Proposition3.4   Let $\overline{d}$ be the upper asymptotic density on $\mathbb{N}$, and let $\underline{d}$ be the lower asymptotic density on $\mathbb{N}$. For every subset $A\subseteq \mathbb{N}$, define

$$\mathbb{D}(A)=I_{\underline{d}(A)}^{\overline{d}(A)}.$$

Then $\mathbb{D}$ is a finitely additive interval probability measure on $\mathbb{N}$.

Proof. 

For $A\subseteq \mathbb{N}$, set

$$\underline{d}(A)=\underset{n\to \infty }{lim\; inf}{\displaystyle \frac{|A\cap \{1,\dots ,n\left\}\right|}{n}},\overline{d}(A)=\underset{n\to \infty }{lim\; sup}{\displaystyle \frac{|A\cap \{1,\dots ,n\left\}\right|}{n}}.$$

We verify the axioms of a finitely additive interval probability measure.

(1)

Non-negativity. Since $\underline{d}(A)\ge 0$, we have

$$\mathbb{D}(A)=I_{\underline{d}(A)}^{\overline{d}(A)}\ge I_0^0.$$

(2)

Empty set. Since $\underline{d}(⌀)=\overline{d}(⌀)=0$,

$$\mathbb{D}(⌀)=I_0^0.$$

(3)

Shiffman property. Let $A,B\subseteq \mathbb{N}$ with $A\cap B=⌀$. Write

$$\mathbb{D}(A)=I_{x_1}^{y_1},\mathbb{D}(B)=I_{x_2}^{y_2},$$

so that

$$x_1=\underline{d}(A),y_1=\overline{d}(A),x_2=\underline{d}(B),y_2=\overline{d}(B).$$

For each $n\ge 1$, put

$$a_n={\displaystyle \frac{|A\cap \{1,\dots ,n\left\}\right|}{n}},b_n={\displaystyle \frac{|B\cap \{1,\dots ,n\left\}\right|}{n}}.$$

Since $A\cap B=⌀$,

$$a_n+b_n={\displaystyle \frac{|(A\cup B)\cap \{1,\dots ,n\left\}\right|}{n}}.$$

Using the standard inequalities

$$\underset{n\to \infty }{lim\; sup}(a_n+b_n)\ge \underset{n\to \infty }{lim\; sup}{a}_n+\underset{n\to \infty }{lim\; inf}{b}_n,$$

$$\underset{n\to \infty }{lim\; sup}(a_n+b_n)\ge \underset{n\to \infty }{lim\; inf}{a}_n+\underset{n\to \infty }{lim\; sup}{b}_n,$$

we obtain

$$\overline{d}(A\cup B)\ge y_1+x_2,\overline{d}(A\cup B)\ge x_1+y_2.$$

Hence

$$\overline{d}(A\cup B)\ge max\{x_1+y_2,x_2+y_1\}.$$

Similarly,

$$\underset{n\to \infty }{lim\; inf}(a_n+b_n)\le \underset{n\to \infty }{lim\; inf}{a}_n+\underset{n\to \infty }{lim\; sup}{b}_n,$$

$$\underset{n\to \infty }{lim\; inf}(a_n+b_n)\le \underset{n\to \infty }{lim\; sup}{a}_n+\underset{n\to \infty }{lim\; inf}{b}_n,$$

so

$$\underline{d}(A\cup B)\le x_1+y_2,\underline{d}(A\cup B)\le y_1+x_2.$$

Thus

$$\underline{d}(A\cup B)\le min\{x_1+y_2,x_2+y_1\}.$$

Therefore

$$I_{min\{x_1+y_2,x_2+y_1\}}^{max\{x_1+y_2,x_2+y_1\}}\subseteq I_{\underline{d}(A\cup B)}^{\overline{d}(A\cup B)}=\mathbb{D}(A\cup B).$$

This is the Shiffman property.

(4’)

Finite covering property. Let $A_1,\dots ,A_m\subseteq \mathbb{N}$ be pairwise disjoint. Then

$$\underline{d}\left({\displaystyle \bigcup _{i=1}^m}{A}_i\right)\ge {\displaystyle \sum _{i=1}^m}\underline{d}(A_i),$$

and

$$\overline{d}\left({\displaystyle \bigcup _{i=1}^m}{A}_i\right)\le {\displaystyle \sum _{i=1}^m}\overline{d}(A_i).$$

Hence

$$\mathbb{D}\left({\displaystyle \bigcup _{i=1}^m}{A}_i\right)=I_{\underline{d}({\bigcup }_{i=1}^m{A}_i)}^{\overline{d}({\bigcup }_{i=1}^m{A}_i)}\subseteq I_{{\sum }_{i=1}^m\underline{d}(A_i)}^{{\sum }_{i=1}^m\overline{d}(A_i)}={\displaystyle \sum _{i=1}^m}\mathbb{D}(A_i).$$

Thus $\mathbb{D}$ satisfies the finite covering property.

(5)

Normalization. Since

$$\underline{d}(\mathbb{N})=\overline{d}(\mathbb{N})=1,$$

we have

$$\mathbb{D}(\mathbb{N})=I_1^1.$$

Therefore $\mathbb{D}$ is a finitely additive interval probability measure on $\mathbb{N}$.    □

Theorem3.5   Let X be a set,$\mathcal{A}$ be a $\sigma$- algebra over X, $\{f_i\mid i\in K\}$ is a set of some probability measures on $(X,\mathcal{A})$, $\left\{f_i(A)\right|i\in K\}$ is an interval, ,$f_K^{-}(A)=min\left\{f_i(A)\right|i\in K\}$,$f_K^{+}(A)=max\left\{f_i(A)\right|i\in K\}$ . $\nu (A)=I_{f_K^{-}(A)}^{f_K^{+}(A)}$,then $\nu$ is an interval probability measure on $(X,\mathcal{A})$.

Proof. 

We verify the axioms of an interval probability measure.

(1) Non-negativity: Since each $f_i$ is a probability measure, $f_i(A)\ge 0$. Hence $f_K^{-}(A)\ge 0$, so $\nu (A)\ge I_0^0$.

(2) Empty set: For every $i\in K$, $f_i(⌀)=0$. Thus $f_K^{-}(⌀)=f_K^{+}(⌀)=0$, and $\nu (⌀)=I_0^0$.

(3) Shiffman property: Let $A\cap B=⌀$. Write $\nu (A)=I_{x_1}^{y_1}$ and $\nu (B)=I_{x_2}^{y_2}$, i.e.

$$x_1=f_K^{-}(A),y_1=f_K^{+}(A),x_2=f_K^{-}(B),y_2=f_K^{+}(B).$$

For each $i\in K$, since $f_i$ is a probability measure,

$$f_i(A\cup B)=f_i(A)+f_i(B).$$

Let $i_0\in K$ be such that $f_{i_0}(A)=x_1$. Then

$$f_K^{-}(A\cup B)=\underset{i}{min}(f_i(A)+f_i(B))\le f_{i_0}(A)+f_{i_0}(B)\le x_1+y_2.$$

Similarly, choosing $j_0$ with $f_{j_0}(B)=x_2$,

$$f_K^{-}(A\cup B)\le f_{j_0}(A)+f_{j_0}(B)\le y_1+x_2.$$

Thus

$$f_K^{-}(A\cup B)\le min\{x_1+y_2,x_2+y_1\}.$$

On the other hand, let $i_1\in K$ satisfy $f_{i_1}(A)=y_1$. Then

$$f_K^{+}(A\cup B)=\underset{i}{max}(f_i(A)+f_i(B))\ge f_{i_1}(A)+f_{i_1}(B)\ge y_1+x_2.$$

Similarly, choosing $j_1$ with $f_{j_1}(B)=y_2$,

$$f_K^{+}(A\cup B)\ge f_{j_1}(A)+f_{j_1}(B)\ge x_1+y_2.$$

Hence

$$f_K^{+}(A\cup B)\ge max\{x_1+y_2,x_2+y_1\}.$$

Therefore

$$I_{min\{x_1+y_2,x_2+y_1\}}^{max\{x_1+y_2,x_2+y_1\}}\subseteq \nu (A\cup B),$$

which is the Shiffman property.

(4) Countable covering property: Let ${\left\{A_j\right\}}_{j=1}^{\infty }$ be pairwise disjoint sets in $\mathcal{A}$. For each $i\in K$,

$$f_i\left({\displaystyle \bigcup _{j=1}^{\infty }}{A}_j\right)={\displaystyle \sum _{j=1}^{\infty }}{f}_i(A_j).$$

Hence

$$f_K^{-}\left({\displaystyle \bigcup _{j=1}^{\infty }}{A}_j\right)=\underset{i}{min}{\displaystyle \sum _{j=1}^{\infty }}{f}_i(A_j)\ge {\displaystyle \sum _{j=1}^{\infty }}\underset{i}{min}{f}_i(A_j)={\displaystyle \sum _{j=1}^{\infty }}{f}_K^{-}(A_j).$$

Similarly,

$$f_K^{+}\left({\displaystyle \bigcup _{j=1}^{\infty }}{A}_j\right)=\underset{i}{max}{\displaystyle \sum _{j=1}^{\infty }}{f}_i(A_j)\le {\displaystyle \sum _{j=1}^{\infty }}\underset{i}{max}{f}_i(A_j)={\displaystyle \sum _{j=1}^{\infty }}{f}_K^{+}(A_j).$$

Therefore

$$\nu \left({\displaystyle \bigcup _{j=1}^{\infty }}{A}_j\right)\subseteq {\displaystyle \sum _{j=1}^{\infty }}\nu (A_j).$$

(5) Normalization: For every $i\in K$, $f_i(X)=1$. Thus $f_K^{-}(X)=f_K^{+}(X)=1$, and $\nu (X)=I_1^1$.

All the axioms are satisfied, so $\nu$ is an interval probability measure on $(X,\mathcal{A})$.    □

Definition 3.4   [Intermediate Measure Value] Let $(X,\mathcal{M},\mu )$ be a measure space, and let ${\mu }_{*}(A)$ and ${\mu }^{*}(A)$ denote the inner measure and outer measure of a set $A\subset X$, respectively.

A real number ${\mu }_m(A)$ is called an intermediate measure value of A if

$${\mu }_{*}(A)\le {\mu }_m(A)\le {\mu }^{*}(A).$$

If A is non-measurable, then

$${\mu }_{*}(A)<{\mu }^{*}(A).$$

In this case, if

$${\mu }_{*}(A)<{\mu }_m(A)<{\mu }^{*}(A),$$

then ${\mu }_m(A)$ is called a strict intermediate measure value of A.

Proposition3.6   (A Construction of an Intermediate Measure Value) Let $A\subset X$ be any set and let $\lambda \in [0,1]$. Define

$${\mu }_m(A)=(1-\lambda ){\mu }_{*}(A)+\lambda {\mu }^{*}(A).$$

Then ${\mu }_m(A)$ is an intermediate measure value of A.

In particular, if A is non-measurable and $0<\lambda <1$, then

$${\mu }_{*}(A)<{\mu }_m(A)<{\mu }^{*}(A),$$

so that ${\mu }_m(A)$ is a strict intermediate measure value.

If A is measurable, then

$${\mu }_{*}(A)={\mu }^{*}(A)=\mu (A),$$

and consequently

$${\mu }_m(A)=\mu (A).$$

Proof. 

By the fundamental property of inner and outer measures,

$${\mu }_{*}(A)\le {\mu }^{*}(A).$$

Since

$${\mu }_m(A)=(1-\lambda ){\mu }_{*}(A)+\lambda {\mu }^{*}(A),0\le \lambda \le 1,$$

${\mu }_m(A)$ is a convex combination of ${\mu }_{*}(A)$ and ${\mu }^{*}(A)$. Therefore,

$${\mu }_{*}(A)\le {\mu }_m(A)\le {\mu }^{*}(A).$$

Hence, by Definition , ${\mu }_m(A)$ is an intermediate measure value of A.

Now suppose that A is non-measurable. Then

$${\mu }_{*}(A)<{\mu }^{*}(A).$$

For $0<\lambda <1$,

$$\begin{array}{cc}\hfill {\mu }_m(A)-{\mu }_{*}(A)& =\lambda \left({\mu }^{*}(A)-{\mu }_{*}(A)\right)>0,\hfill \end{array}$$

which gives

$${\mu }_{*}(A)<{\mu }_m(A).$$

Similarly,

$$\begin{array}{cc}\hfill {\mu }^{*}(A)-{\mu }_m(A)& =(1-\lambda )\left({\mu }^{*}(A)-{\mu }_{*}(A)\right)>0,\hfill \end{array}$$

and hence

$${\mu }_m(A)<{\mu }^{*}(A).$$

Thus,

$${\mu }_{*}(A)<{\mu }_m(A)<{\mu }^{*}(A).$$

Finally, suppose that A is measurable. Then

$${\mu }_{*}(A)={\mu }^{*}(A)=\mu (A).$$

Consequently,

$$\begin{array}{cc}\hfill {\mu }_m(A)& =(1-\lambda )\mu (A)+\lambda \mu (A)\hfill \\ & =\mu (A).\hfill \end{array}$$

Therefore, the intermediate measure value coincides with the ordinary measure on measurable sets.    □

Definition 3.5   (Measurement Dependence of the Measured Value) Let A be a non-measurable set. The measured value of A is said to be measurement-dependent if the value obtained depends on the measurement method used.

For example, consider two measurement methods:

1.

Inner-measure method. The set A is measured from inside by measurable subsets $E\subseteq A$. The resulting value is

$${\mu }_{\mathrm{in}}(A)={\mu }_{*}(A).$$

2.

Outer-measure method. The set A is measured from outside by measurable supersets $F\supseteq A$. The resulting value is

$${\mu }_{\mathrm{out}}(A)={\mu }^{*}(A).$$

For a non-measurable set, it may happen that

$${\mu }_{\mathrm{in}}(A)<{\mu }_{\mathrm{out}}(A).$$

Consequently, the two measurement methods assign different values to the same set A:

$${\mu }_{\mathrm{in}}(A)\ne {\mu }_{\mathrm{out}}(A).$$

More generally, an admissible intermediate measurement method M may assign a value

$${\mu }_{*}(A)<{\mu }_M(A)<{\mu }^{*}(A).$$

Thus, the measured value of a non-measurable set can depend on the measurement method.

Remark   When a non-measurable set is measured using methods applicable to measurable sets, different measurement methods yield different values. In contrast, for a measurable set, these methods yield the same value. This provides a model for physical measurement: when non-measurable physical quantities of non-measurable matter are measured using methods applicable to measurable matter, different measurement methods may yield different values, resulting in significant measurement tension, potentially reaching a level of $5\sigma$. In contrast, for measurable matter, these measurement methods yield nearly identical values.

Definition 3.6   Let $\mu$ be an interval probability measure on $(\Omega ,\mathcal{F})$. A probability measure P is called a collapsing measurement of interval probability measure $\mu$ if and only if

$$P\mathrm{is}\mathrm{a}\mathrm{probability}\mathrm{measure}\mathrm{on}(\Omega ,\mathcal{F}),\forall A\in \mathcal{F}(P(A)\in \mu (A)).$$

Proposition3.7   [Coverage by collapsing measurements for min–max generated interval probability measures] Let $(\Omega ,\mathcal{F})$ be a measurable space, and let ${\left\{P_i\right\}}_{i\in K}$ be a nonempty family of probability measures on $(\Omega ,\mathcal{F})$. Assume that for every $A\in \mathcal{F}$, the set

$$\{P_i(A):i\in K\}$$

has a minimum and a maximum. Define

$$\nu (A)=I_{l_A}^{u_A},l_A=\underset{i\in K}{min}{P}_i(A),u_A=\underset{i\in K}{max}{P}_i(A),A\in \mathcal{F}.$$

Then $\nu$ is an interval probability measure on $(\Omega ,\mathcal{F})$.

Let

$$C(\nu )=\left\{P:\begin{array}{c}P\mathrm{is}\mathrm{a}\mathrm{probability}\mathrm{measure}\mathrm{on}(\Omega ,\mathcal{F}),\hfill \\ P(A)\in \nu (A)\forall A\in \mathcal{F}\hfill \end{array}\right\}.$$

Then, for every $A\in \mathcal{F}$,

$$\{P(A):P\in C(\nu )\}=\nu (A)=I_{l_A}^{u_A}.$$

Equivalently, for every $A\in \mathcal{F}$ and every $r\in \nu (A)$, there exists a collapsing measurement $P\in C(\nu )$ such that

$$P(A)=r.$$

Proof. 

By Theorem 3.5, $\nu$ is an interval probability measure. It remains to prove the coverage property.

Fix $A\in \mathcal{F}$. Since the minimum and maximum are attained, choose $P_{-},P_{+}\in {\left\{P_i\right\}}_{i\in K}$ such that

$$P_{-}(A)=l_A,P_{+}(A)=u_A.$$

Let $r\in [l_A,u_A]=\nu (A)$. If $l_A=u_A$, then $r=l_A=u_A$, and we may take $P=P_{-}$. Otherwise, set

$$\lambda ={\displaystyle \frac{u_A-r}{u_A-l_A}}\in [0,1],$$

and define

$$P=\lambda P_{-}+(1-\lambda )P_{+}.$$

Since $P_{-}$ and $P_{+}$ are probability measures, P is also a probability measure on $(\Omega ,\mathcal{F})$. Moreover,

$$P(A)=\lambda P_{-}(A)+(1-\lambda )P_{+}(A)=\lambda l_A+(1-\lambda )u_A=r.$$

For every $B\in \mathcal{F}$, we have

$$l_B\le P_{-}(B)\le u_B,l_B\le P_{+}(B)\le u_B.$$

Hence

$$l_B\le P(B)=\lambda P_{-}(B)+(1-\lambda )P_{+}(B)\le u_B,$$

so

$$P(B)\in I_{l_B}^{u_B}=\nu (B).$$

Therefore $P\in C(\nu )$. Thus every $r\in \nu (A)$ is attained by some $P\in C(\nu )$. Consequently,

$$\nu (A)\subseteq \{P(A):P\in C(\nu )\}.$$

The reverse inclusion follows immediately from the definition of $C(\nu )$. Hence

$$\{P(A):P\in C(\nu )\}=\nu (A)=I_{l_A}^{u_A}.$$

This completes the proof.    □

4. The Foundation of Non-Measurability Theory

Lemma 4.1    Properties of non-measurable sets( Refer to the book[33]):

(1)

Non-measurable sets have no definite measure,the measure of a non-measurable set depends on the measurement method.

(2)

If the intersection of a non-measurable set and a measurable set is an empty set, then the union of the non-measurable set and the measurable set is non-measurable.

(3)

A non-measurable set is neither a continuous (connected) nor a discrete set.

(4)

The complement of a non-measurable set is non-measurable.

(5)

The cardinality of a non-measurable set is uncountable.

(6)

(The Shiffman Property)If$A\bigcap B=\varnothing ,\nu (A)=I_{x_1}^{y_1},\nu (B)=I_{x_2}^{y_2}$, $min\{x_1+y_2,x_2+y_1\}=\perp ,$ $max\{x_1+y_2,x_2+y_1\}=\top$ ,then $I_{\perp }^{\top }⊑\nu (A\cup B)$.

(7)

(Finite covering property) For all finite collections ${\displaystyle {\left\{A_i\right\}}_{i=1}^n}$ of pairwise disjoint sets in $\mathcal{A}$, $\nu ({\cup }_{i=1}^n{A}_i)⊑{\Sigma }_{i=1}^n\nu (A_i)$.

(8)

(Separation invariance) If $d(A,B)>0$, then $\nu (A\cup B)=\nu (A)+\nu (B)$.

(9)

Various versions of Banach-Tarski Paradox[27].

Definition 4.1   If the mass of a matter is a real number ,then the matter is called measurable matter. If the mass of a matter is a non-degenerate interval number ,then the matter is called a non-measurable matter.If the lower bound of the interval mass of a non-measurable matter is 0, the matter is called a completely non-measurable matter; otherwise, it is called a partially non-measurable matter.If A and B have different masses , A and B are equidecomposable ,then we call this phenomenon the physical manifestation of the Banach-Tarski Paradox.

Remark1   A set with inner measure zero and positive outer measure is called a completely nonmeasurable set; otherwise, it is called a partially nonmeasurable set.

Remark2   Photon, electron , moun and tau are measurable particles.Higgs,w boson, quarks,neutrinos , gravitons , dark matter,black holes , the universe ,dark energy , gluons,colored particles, magnetic monopoles ,Bose-Einstein condensate are most likely non-measurable. The expansion of the universe, the star collapses into a black hole, neutrino oscillations, parity nonconservation, and so on are all physical manifestations of the Banach-Tarski paradox.

Property 4.1    The non-measurable matter has the properties of three-dimensional nonmeasurable sets.Such as

(1) 

The mass of a non-measurable matter is a non-degenerate interval number,the value of the mass is dependent on the measurement,the mass lower bound of completely non-measurable matter is zero.A non-measurable set can yield different measurement values when measured by different methods, while applying those same methods to a measurable set always produces the same value. This provides a mathematical model for measuring the physical properties of non-measurable matter, implying that the measured mass of non-measurable matter depends on the measurement method.

(2) 

The non-measurable particles have the property of oscillation( the reactants depend entirely on the relative positions of the nonmeasurable particles).For example, neutrinos have a property called neutrino oscillation.In other words,the nonmeasurable matter has the physical manifestation of the Banach-Tarski Paradox.

(3) 

Non-measurable particles and non-measurable antiparticles behave differently, such as neutrinos and antineutrinos[28],quarks and anti-quarks[29].Although non-measurable particles and non-measurable antiparticles have the same mass and shape, they have different spins,which causes them to behave differently.This is also a physical manifestation of the Banach-Tarski Paradox.

(4) 

The completely non-measurable matters cannot be measured directly, they are detected by indirect measurements. For example, neutrinos, Higgs, etc. are indirectly proved to exist through the products of interaction. its shape is complex and full of wormholes.

(5) 

The completely non-measurable matter has a strong ability to penetrate. For example,neutrinos pass through matter easily.When helium-4 is cooled below the lambda point (2.17 K) and enters the superfluid phase (He II), a remarkable phenomenon occurs: liquid helium can flow through extremely narrow gaps, capillaries, or porous media (such as sintered metal powders and clay plugs) with virtually no resistance. These passages can be so narrow that ordinary liquids, and even gas molecules, can barely pass through

(6) 

( Unilateral Gravity) completely non-measurable matter can generate gravity, but gravity cannot act on it.

Remark    The non-measurable theory is based on two principles, one of which is that dark energy is non-measurable, and the other is that all matter is composed of dark energy.All the conclusions of this paper can be proved on the two principles. the two principles are supported by experimental data, as are the conclusions derived from them.

If you want to talk about the properties of dark energy, then you have to make sure that dark energy exists[5-6]. It is an accepted view in modern physics that fields exist.We believe that dark energy is the fundamental field in the universe,the dark energy field is non-measurable,other fields are first excitation of the dark energy field , and all particles are second excitation of the dark energy field.

Fundamental 4.1( Non-measurability )   Dark energy is non-measurable.

(Dark energy is a field, it is non-measurable.)

The equation-of-state parameter governs the rate at which the dark energy density evolves. For a perfect, unchanging vacuum energy, we have $w=-1$ . Current experimental bounds tell us that $w=-1$ is the central preferred value. Riess team got this statement in the paper[15]: , This[20] resulted in $H_0=73.8\pm 2.4km{s}^{-1}Mp{c}^{-1}$ , which coupled with the 7-year WMAP results [21] yielded $w=-1.08\pm 0.10$, or an estimate of the effective number of relativistic species of $Neff=4.2\pm 0.7$, . Dark energy seems to work in a different way than anything previously observed. The density of anything is the amount of matter in a given volume, and dark energy is an unusual phenomenon because even as the volume of the universe increases with expansion, the density stays the same. As the universe expands, new dark energy is being created all the time, meaning its density remains the same. You can think of it this way, when you get more space, you actually get more dark energy, it’s like getting something for nothing. You can explain this if you assume that dark energy is nonmeasurable.

The Banach-Tarski paradox was first proposed by Banach and Tarski in 1924. This theorem states that if the axiom of choice holds, a three-dimensional solid sphere can be divided into finite (non-measurable) parts and then formed into two complete spheres of the same radius simply by rotating and moving them elsewhere.The process can be represented by the following formula, $S\to S+S$. A stronger version of this paradox is that any two subsets in a three-dimensional Euclidean space with a non-empty interior are equidecomposable .Dark energy is constantly recombining and decaying, so that more and more dark energy can be obtained.

Fundamental 4.2 ( Universality )   All matter and fields are excitation of dark energy.

(Dark energy is the ground state, and it is precisely under the physical action of the Banach-CTarski paradox that dark energy undergoes quantum excitation, transitioning from the ground state to various quantum states and matter.)

Heisenberg worked on a unified field theory of elementary particles[30][31] [43]., All the elementary particles are made of the same substance ,which we may call energy or universal matter ;they are just different forms in which matter can appear., Heisenberg wrote in his book[44].Heisenberg argued that it was unnecessary to introduce a separate operator for each type of elementary particle (such as electrons, protons, and mesons). He proposed that there is only one fundamental elementary entity, a , Fundamental Field,, and that all observed elementary particles and excited states are eigenstates of this unified field under different symmetry-breaking patterns and quantum states., If the different parts of the universe were not as the organs of the same body, they would not react one upon the other; they would mutually ignore each other, and we in particular should only know one part. We need not, therefore, ask if Nature is one, but how she is one., Poincare wrote in his book[45]. Einstein attempted to construct a unified field theory in which electromagnetism and gravity would emerge as different aspects of a single fundamental field.

According to QFT,there are many quantum fields in the universe, particles are excitation of their respective fields.A key open question in fundamental physics is the cosmological constant problem.Experimental measurements show that the cosmological constant is extremely small, in contrast with the theoretical prediction, which is some 120 orders of magnitude larger.We can solve this problem by assuming that the existence of more fundamental fields,the vacuum state has only a dark energy field and no other fields or particles.If dark energy is the fundamental field, then the dark energy field would unify the four interactions, and the matter and field as we know it today are just different manifestations of dark energy.

References [22–24] provide strong evidence for this hypothesis. Matter is converted into vacuum energy ( dark energy), as the matter falls through the event horizon.

Quantum fluctuations also provide strong evidence for this hypothesis.Our universe is a quantum fluctuation of vacuum [36,37].

The team[38] determined that matter makes up about 31.5 percent of the total amount of matter and energy in the universe, and dark energy makes up 68.5 percent. About 20 percent of total matter is made up of regular matter, which includes stars, galaxies, atoms and life, while about 80 percent is made up of dark matter, whose mysterious nature is unknown but which may consist of subatomic particles that have yet to be discovered. Everything in the universe is made up of dark energy, and dark energy itself is nonmeasurable, so from the point of view of probability, most matter should be nonmeasurable, and only a few can form measurable matter, which can also explain that most things in the universe are unknown, and almost everything in the universe is dark matter and dark energy.

“ we ought in fact to suppose it, if we can thereby obtain a simpler explanation of phenomena, .Riemann said in his famous paper[26].

Remark    Dark energy is the most elementary object in the universe.Dark energy itself is nonmeasurable,dark energy can produce more dark energy,some dark energy forms measurable matter, and some forms nonmeasurable matter .

Corollary 4.3   There is non-measurable matter in the universe.

Remark    Much of the matter in the universe may be non-measurable, and we provide some examples for Corollary 4.1 below.

non-measurable matter	measurable matter
quark	tau
neutrino	neutron
dark energy	proton
gluon	atom
superfluid	molecule
dark matter	planet
black hole	electron
Bose-Einstein condensate	neutron star
fractionated electrons	comet
non-Abelian anyon	satellite
Higgs	photon,
W boson	moun
⋮	⋮

non-measurable matter	measurable matter
quark	tau
neutrino	neutron
dark energy	proton
gluon	atom
superfluid	molecule
dark matter	planet
black hole	electron
Bose-Einstein condensate	neutron star
fractionated electrons	comet
non-Abelian anyon	satellite
Higgs	photon,
W boson	moun
⋮	⋮

Remark    Table 1 describes the possible non-measurable matter in the universe.

Tentative view 4.1   Quarks are non-measurable particles.

Evidence   (1)The mass of a quark depends on how you measure it,a quark has different masses at the same time.Because quarks can never be free, the mass of quarks can only be represented by it in some kind of interaction. So the mass of quarks is not unique, but depends on the way they are confined. Any quark mass should be footmarked to indicate the method by which the value of that mass was obtained.

(2)Quarks and anti-quarks behave differently.[28]

Remark    B. W. Augenstein [14] draws an extensive analogy between quarks and non-measurable sets.

Tentative view 4.2   Neutrinos are non-measurable particles.

Evidence   (1), Oscillations, [17-19] refer to the way neutrinos change flavor as they travel. A neutrino born as one flavor (electron, muon, or tau neutrino) will eventually morph into the other varieties, and the probability of appearing as a different flavor depends on how far it has gone.Particles are objects that can move, rotate and boost. The reactant depends on where you messasure it,Neutrinos constantly change their position relative to the detector as they travel.Neutrinos do not change as they travel, which is the physical manifestation of the Banach-Tarski Paradox.The three types of neutrinos are different manifestations of the same non-measurable particle.

(2)Neutrinos and antineutrinos behave differently[29] .

(3)The extremely weak interaction and strong penetrating ability of neutrinos can be regarded as manifestations of their , non-measurability, in physical experiments.

(4)If neutrino oscillations are merely manifestations of the same non-measurable particle, then the neutrino may have different masses under different measurement methods. In this sense, even the zero mass assigned to the neutrino in the Standard Model may represent the mass revealed by a particular measurement method.

Tentative view 4.3   The energy of the gravitational field is non-measurable.[51]

Evidence   (1)There is no universal definition of local gravitational energy density.

(2)Gravitational energy pseudotensors are coordinate- and gauge-dependent.

(3)Different coordinate systems or frames can yield different values of gravitational energy density.

(4)The local expression for gravitational energy is not an ordinary tensor and therefore is not a generally covariant local observable in the usual sense.

(5)The Brown-CYork quasi-local energy depends on the observer and boost gauge.

(6)Different observers can obtain different Brown-CYork quasi-local energies.

Tentative view 4.4   Dark matter is non-measurable .

Evidence   (1) Dark matter particles have not yet been unambiguously detected through non-gravitational interactions. Dark matter is currently established primarily through its gravitational effects.

(2) Dark matter has a very high penetrating ability due to its extremely weak non-gravitational interactions with ordinary matter.

(3)Dark-matter halos do not undergo sustained dissipative collapse like ordinary baryonic matter.

(4)Observations of colliding galaxy clusters indicate that the dark-matter component undergoes little or no significant deceleration during the collision [56].

Tentative view 4.5   Black holes are non-measurable.

Evidence    (1)A Black hole has different masses at the same time (the ref [25] provides strong evidence for this phenomenon).A star with a specific mass collapses into a black hole with many different masses at the same time. In other words, the star was measurable matter, the star collapsed into a black hole and became non-measurable matter. The matter collapses and splits into a lot of non-measurable matter, which recombines to form black hole.Black holes are likely to be dark energy stars[22],the inside of a black hole has a different cosmological constant than the outside.So,the formation of a black hole is the physical manifestation of the Banach-Tarski Paradox.

(2)If mass growth of black holes only occurred through accretion or merger, then the masses of these black holes would not be expected to change much at all.But the gravitating mass of a black hole can increa se with the expansion of the universe independently of accretion or mergers.So, black holes are most likely dark energy stars, and the non-measurability of dark energy could explain the mystery of black hole mass growth. [32]and [33] provide the first observational evidence linking black holes to dark energy.This paper provided a theoretical explanation for mass growth of black holes. A black hole is dark energy, dark energy is constantly decaying and recombining to produce more dark energy, which is the physical manifestation of the Banach-Tarski Paradox.

Tentative view 4.6   The universe is non-measurable.

Evidence    Is the expansion of the Universe accelerating? All signs point to yes[34]. The expansion of the universe was propelled by a repulsive gravitational force generated by an exotic form of matter.Here, the expansion of the universe is nothing more than a physical manifestation of the Banach - Tarski paradox . Viewing the universe as a non-measurable set, the decomposition and reorganization of its non-measurable subsets could drive cosmic expansion. Galaxies within it may be regarded as measurable components, where the distance between two galaxies changes over time. When the rate of such replication increases, it could explain the accelerated expansion of the universe.

Tentative view 4.7   W boson is non-measurable.

Evidence    In 2022, the CDF II Collaboration at the Fermilab Tevatron performed a high-precision measurement of the W-boson mass using the full Run II dataset, obtaining$M_W=80433.5\pm 9.4$ MeV/$c^2$. This result showed a deviation of approximately $7\sigma$ from the Standard Model electroweak global-fit prediction, thereby triggering widespread discussion about possible physics beyond the Standard Model. The result was published in Science[39]. However, the CMS Collaboration at the Large Hadron Collider (LHC) subsequently performed an independent high-precision measurement using 13 TeV proton-Cproton collision data. Published in Nature[40] in 2026, the CMS measurement yielded$M_W=80360.2\pm 9.9$ MeV, in close agreement with the Standard Model prediction.The same particle yielding different measurements in different instruments may be regarded as a manifestation of the particle’s non-measurability.

Compared with the W-boson mass, the experimental measurements of the Z-boson mass exhibit very little tension among different detectors. At the Large Electron-CPositron Collider (LEP), the four independent experiments ALEPH, DELPHI, L3, and OPAL performed precision measurements of the Z-boson resonance using their respective detector systems. Their measurements of the Z-boson mass and other electroweak parameters were subsequently combined by the LEP Electroweak Working Group. The combined result was $(M_Z=91.1875\pm 0.0021,\mathrm{GeV})$, demonstrating the remarkable consistency of the independent LEP measurements. Thus, in terms of the Z-boson mass, no significant experimental tension between the four LEP detectors was observed. [61]

Tentative view 4.8   H boson is non-measurable.

Evidence    The mass of the H boson is dependent on the measurement method. , As shown in Table V, the measured values of the Higgs boson mass for the $H\to \gamma \gamma$ and $H\to Z{Z}^{*}\to 4\iota$channels are $125.98\pm 0.42(stat)\pm 0.28(syst)GeV$ and $124.51\pm 0.52(stat)\pm 0.06(syst)$ GeV, respectively. The compatibility between the mass measurements from the two individual channels is at the level of $2.0\sigma$ corresponding to a probability of $4.8\%$., [41].

Tentative view 4.9   Superfluids are non-measurable.

Evidence   (1) Zero-Viscosity Flow: A superfluid experiences no frictional resistance when it flows. If a circulating flow is initiated in a closed ring-shaped container, it can persist indefinitely because there is no resistance to stop it.We regard the Cooper pairs in a superconductor as a superfluid. The charged Cooper pairs pass through the crystal lattice without impediment, flowing free of scattering and resistance.This shows the extremely strong penetrating ability of non-measurable matter.

(2)Climbing-film effect (Rollin film): A superfluid such as superfluid helium can form a thin film along the inner walls of a container and climb upward and potentially flow over the rim, driven by surface interactions and superfluid properties, until the levels equilibrate or, in the classic demonstration, liquid can drain from the container.This shows the unilateral gravity of non-measurable matter.This is discussed in Section 9.

Tentative view 4.10   Fractionated electrons is non-measurable.

Evidence   The evidence that electrons can recombine appears in the fractional quantum Hall effect. The fractional quantum Hall effect allows for the ballistic transport of fractional electrons, which is not allowed in known theories, but in our theory, this phenomenon has the same interpretation as the superconductivity mechanism: Electrons are composed of non-measurable particles, which can be recombined into a new particle at high magnetic and low temperatures, so that fractional electrons can appear.

“Recently, however, interest in fractionated electrons has exploded, because it turns out that they have a kind of collective memory. To put this more concretely: After you move them around one another, their subsequent behavior reliably reflects how you treated them. Because of this , memory,, fractional electrons-known as anyons-are promising ingredients for building up and storing quantum information, and ultimately for making quantum computer, .Frank Wilczek mentioned in an article[47]. The phenomenon is most likely the physical manifestation of the Banach-Tarski Paradox.If fractionated electrons are non-measurable, then the phenomenon can be explained by Banach-Tarski Paradox.

Tentative view 4.11   Non-Abelian anyons are non-measurable.

Evidence   “Indistinguishability of particles is a fundamental principle of quantum mechanics1. For all elementary and quasiparticles observed to date, including fermions, bosons and Abelian anyons, this principle guarantees that the braiding of identical particles leaves the system unchanged. However, in two spatial dimensions, an intriguing possibility exists: braiding of non-Abelian anyons causes rotations in a space of topologically degenerate wavefunctions. Hence, it can change the observables of the system without violating the principle of indistinguishability"[48].The phenomenon is most likely the physical manifestation of Banach-Tarski Paradox.If anyons are non-measurable, then the phenomenon can be explained by Banach-Tarski Paradox.

Corollary 4.4   Energy is not conserved.

If there is non-measurable matter, then energy is not conserved, because the total energy of the non-measurable matter depends on how the parts of it are put together. We now extend the energy value from real values to interval values, the interval energy value of an object depends on the way all the parts of the object are put together, so the energy is no longer conserved on interval values.The law of energy change is described by the axiom of interval value measure,which is also a generalization of the law of energy conservation.According to the axiom of interval measures, if there is no decomposition-recombination involved, it appears that energy is conserved, but if there is decomposition-recombination involved, Banach-Tarski paradox phenomenon occurs, and energy is not conserved.

A framework for problem solving 1 (The mystery of the missing antimatter:Banach-Tarski mechanism ) The following Banach-Tarski mechanism may provide an answer to the mystery of the disappearance of antimatter. Let us consider a non-measurable particle of this type, whose particle and antiparticle are the same particle. According to the Banach-Tarski paradox, different combinations of non-measurable particles may produce different products. In experiments[28], the researchers observed that measurements of the oscillations of experimental neutrinos and antineutrinos showed that neutrinos have a higher probability of oscillating than antineutrinos, which means that more matter is produced.This means that the answer to the puzzle is hidden in a non-measurable particle, which is the same particle as the antiparticle.Dark matter may have such particles in abundance that matter and antimatter disappear, leaving a surplus of matter.

A framework for problem solving2(Big Bang Puzzle ): Instead of introducing an inflation field, we can use the properties of dark energy to solve this problem. Dark energy has the properties of a non-measurable set (including the Banach-Tarski paradox). At the beginning of the universe, there was only dark energy in the universe, which had a very high temperature and pressure, and violent interactions among various subsets of dark energy. Dark energy was decomposed and recombined at a very high rate, and a lot of dark energy was copied, resulting in a rapid increase in the density of the universe. In order to ensure that the density of dark energy remained constant at interval values, the universe needed exponential expansion. Since replication is preserving the original properties, this explains the high flatness and temperature uniformity of the universe.

A framework for problem solving3(Black hole information paradox ): If energy is not conserved, then the step , energy carried away by radiation = decrease in black hole mass, does not hold. A black hole could continue to radiate without losing mass (or its mass change could be independent of the radiated energy), and the outcome of , complete evaporation leaving only thermal radiation, would not occur. The classical form of the information paradox would therefore not arise.

5. On the Hypotheses Which Lie at the Bases of Geometry

“Still more complicated relations may exist if we no longer suppose the linear element expressible as the square root of a quadric differential. Now it seems that the empirical notions on which the metrical determinations of space are founded, the notion of a solid body and of a ray of light, cease to be valid for the infinitely small. We are therefore quite at liberty to suppose that the metric relations of space in the infinitely small do not conform to the hypotheses of geometry; and we ought in fact to suppose it, if we can thereby obtain a simpler explanation of phenomena".Riemann said in his famous paper[26].

“In fact we go even further and claim that even for an entangled pair of particles, in a quantum theory of gravity there must be a Planckian bridge between them, albeit a very quantummechanical bridge which probably cannot be described by classical geometry"[46].In a very interesting attempt to connect the geometry of space-time and fundamental properties of quantum mechanics Maldacena and Susskind proposed the so-called ER=EPR conjecture.It is highly likely that the new geometry we are discussing may give an explanation to ER=EPR.

By adding quantum intertextuality to gravitational theory and geometry, we can reach quantum gravity and quantum geometry.We know that Riemannian geometry is a mathematical tool for general relativity, but it cannot explain the phenomena at the Planck scale. We need to generalize Riemannian geometry, but the generalization must explain quantum mechanical phenomena. Quantum contextuality is a principle of quantum mechanics, which states that the values of some physical quantities are uncertain and depend on the measurement methods. To unify general relativity and quantum mechanics, introducing quantum contextuality into general relativity is a worthwhile idea.We assume that there is a quantum matter that has many mass values, depending on how it is measured. This matter we call non-measurable matter,so we have introduced quantum contextuality into the gravity theory, making general relativity and quantum mechanics organically merged together. This hypothesis also explains many strange phenomena in particle physics and has received some experimental support. But we can also deduce the existence of non-measurable matter through the properties of two dark energies. We take these two assumptions as the principles of this theory, which naturally leads to the conclusion that space-time is non-measurable, and physical quantities about space-time also have quantum contextuality, such as distance, volume, angle and other geometric concepts that depend on measurement methods.This is a possible way to unify general relativity and quantum mechanics.

The following are deductive conclusions of the two fundamentals( non-measurability and universality of dark energy), and these conclusions are supported by cosmological data and modern quantum mechanical experiments.

Definition 5.1   Let$I_i\subseteq R$.$\{f(x,a_1,\dots ,a_n)=0|a_1\in I_1,\dots ,a_n\in I_n\}$ is called an interval equation on R. $\{x\in R\mid \exists a_1\in I_1\dots \exists a_n\in I_n(f(x,a_1,\dots ,a_n)=0)\}$ is called the solution set of the interval equation.

Remark    The following equations can be regarded as interval equations.

Corollary 5.1   The cosmological constant $\Lambda$ is a non-degenerate interval constant.

Proof. 

Dark energy is non-measurable, so its energy density is a non-degenerate interval

$${\rho }_{\Lambda }=[{\rho }_{min},{\rho }_{max}],{\rho }_{min}<{\rho }_{max}.$$

The energy density and the cosmological constant satisfy

$${\rho }_{\Lambda }={\displaystyle \frac{c^4}{8\pi G}}\Lambda .$$

Since c and G are real constants, the factor $8\pi G/c^4$ is a real constant. Solving for $\Lambda$ gives

$$\Lambda ={\displaystyle \frac{8\pi G}{c^4}}{\rho }_{\Lambda }.$$

Because $c,G>0$, this factor is positive. Multiplying a non-degenerate interval by a positive real constant again yields a non-degenerate interval. Hence

$$\Lambda =\left[{\displaystyle \frac{8\pi G}{c^4}}{\rho }_{min},{\displaystyle \frac{8\pi G}{c^4}}{\rho }_{max}\right],$$

with lower bound strictly smaller than upper bound. Therefore $\Lambda$ is a non-degenerate interval constant.    □

Remark The conclusion is the starting point for quantum gravity.

In order to establish a rigorous foundation for the theory of gravity, we need to generalize Riemann geometry,we use interval tensor theory to establish interval Riemannian geometry.So,the theory of gravity moves from scalar theory to tensor theory and finally to interval tensor theory.

The set of some probability measures forms a mathematical structure ( Theorem 3.4).Similarly,the set of some Riemannian manifoles forms a mathematical object, and the set of Riemannian metric on these manifoles forms an interval tensor. We call such a set of structures an interval manifold, and the set of Riemannian measures an interval Riemannian metric.

Definition 5.2   Let A be a cluster of order-n tensors ,$a_{ij}$ be the component of a($a\in \mathbf{A}$). If$\left\{a_{ij}\right|a\in \mathbf{A}\}$ is an interval number, then A is called a generalized interval tensor.If A has only one element, it is called a degenerate generalized interval tensor, otherwise it is called a nondegenerate generalized interval tensor.

Remak Here the concept of generalized interval tensor is a generalization of the concept of interval tensor in ref [7]and [8] .Sometimes, generalized interval tensors are simply written as interval tensors,but the meaning of the interval tensor here is different from the reference [7]and [8].

Definition 5.3   Let T be a cluster of Riemannian manifolds on the smooth manifold M, G be the cluster of Riemannian metrics in T . If the components of the Riemannian metrics g of T form a interval tensor at each point, then we call $\{\langle M,g\rangle |g\in G\}$ an interval Riemannian manifold.

If the components of the Riemannian metrics of T form a degenerate interval tensor at each point of M, and we call $\{\langle M,g\rangle |g\in G\}$ a measurable manifold(Riemannian manifold).Otherwise,we call $\{\langle M,g\rangle |g\in G\}$ a nonmeasurable manifold.

Using this interval metric tensor, we can calculate the length of the curve, the distance between two points, and curvature. In the degenerate case, the interval metric tensor is exactly the Riemannian metric. In the non-degenerate case, the value of a geometric quantity defined by $ds$ may be a nondegenerate interval number.

The above concepts can be extended to pseudo-Riemannian manifolds.

According to Einstein’s vacuum field equation $R_{\mu \nu }=\Lambda g_{\mu \nu }$ and Corollary 5.1, we naturally come to the following conclusion:

Corollary 5.2   The de Sitter metric $g_{\mu \nu }$ is a non-degenerate interval metric.

Proof. 

Let

$$\Lambda \in [{\Lambda }_1,{\Lambda }_2],0\le {\Lambda }_1<{\Lambda }_2.$$

In inflationary coordinates, the de Sitter metric is

$$d{s}^2=-d{t}^2+e^{2Ht}\left(d{x}^2+d{y}^2+d{z}^2\right),H=\sqrt{{\displaystyle \frac{\Lambda }{3}}}.$$

Thus

$$g_{tt}=-1,g_{xx}=g_{yy}=g_{zz}=e^{2Ht}=e^{2t\sqrt{\Lambda /3}},$$

and

$$g_{\mu \nu }=0(\mu \ne \nu ).$$

Fix $t>0$. For the spatial components,

$${\displaystyle \frac{d}{d\Lambda }}{e}^{2t\sqrt{\Lambda /3}}=e^{2t\sqrt{\Lambda /3}}·2t·{\displaystyle \frac{1}{2\sqrt{3\Lambda }}}={\displaystyle \frac{t}{\sqrt{3\Lambda }}}{e}^{2t\sqrt{\Lambda /3}}>0.$$

Hence $g_{xx}$ is strictly increasing in $\Lambda$. Therefore

$$g_{xx}\in \left[e^{2t\sqrt{{\Lambda }_1/3}},e^{2t\sqrt{{\Lambda }_2/3}}\right].$$

Since ${\Lambda }_1<{\Lambda }_2$, this is a non-degenerate interval. The same conclusion holds for $g_{yy}$ and $g_{zz}$.

Therefore the metric components form a generalized interval tensor at each point. Moreover,

$$g_{xx}({\Lambda }_1)=e^{2t\sqrt{{\Lambda }_1/3}}\ne e^{2t\sqrt{{\Lambda }_2/3}}=g_{xx}({\Lambda }_2),$$

so the metric tensor cluster

$$\{g_{\mu \nu }(\Lambda )\mid \Lambda \in [{\Lambda }_1,{\Lambda }_2]\}$$

contains more than one element. By Definition 5.2, it is a non-degenerate generalized interval tensor. Hence, by Definition 5.3, the de Sitter metric $g_{\mu \nu }$ is a non-degenerate interval metric.    □

Theorem 5.3   Let the cosmological constant $\Lambda$ of de Sitter space take values in a non-degenerate closed interval

$$\Lambda \in [{\Lambda }_1,{\Lambda }_2],0\le {\Lambda }_1<{\Lambda }_2.$$

Use flat slicing coordinates for de Sitter space, so that the metric is

$$d{s}^2=-d{t}^2+e^{2Ht}\left(d{x}^2+d{y}^2+d{z}^2\right),H=\sqrt{{\displaystyle \frac{\Lambda }{3}}}.$$

Fix $t>0$. Then the following geometric quantities are continuous functions of $\Lambda$, and their ranges are non-degenerate closed intervals; hence they are non-degenerate interval-valued:

$$L(\Lambda )=\sqrt{{\displaystyle \frac{3}{\Lambda }}},H(\Lambda )=\sqrt{{\displaystyle \frac{\Lambda }{3}}},R(\Lambda )=4\Lambda ,$$

$$D(t)=D_0{e}^{Ht},dV=e^{3Ht}dxdydz.$$

Moreover, the Ricci tensor

$$R_{\mu \nu }=\Lambda g_{\mu \nu }$$

forms a generalized interval tensor. When ${\Lambda }_1<{\Lambda }_2$, the whole Ricci tensor cluster is a non-degenerate generalized interval tensor.

Proof. 

We first recall a basic fact.

Basic fact. If $f(\Lambda )$ is continuous on the closed interval $[{\Lambda }_1,{\Lambda }_2]$, then by the extreme value theorem and the intermediate value theorem its range is a closed interval

$$\{f(\Lambda )\mid \Lambda \in [{\Lambda }_1,{\Lambda }_2]\}=[f_{min},f_{max}].$$

If $f_{min}<f_{max}$, this interval is non-degenerate.

Now we compute each quantity.

1. 

Curvature radius $L(\Lambda )$

$$L(\Lambda )=\sqrt{{\displaystyle \frac{3}{\Lambda }}}=\sqrt{3}{\Lambda }^{-1/2}.$$

Differentiate:

$${\displaystyle \frac{dL}{d\Lambda }}=\sqrt{3}\left(-{\displaystyle \frac{1}{2}}\right){\Lambda }^{-3/2}=-{\displaystyle \frac{\sqrt{3}}{2{\Lambda }^{3/2}}}.$$

Since $\Lambda >0$, we have

$${\displaystyle \frac{dL}{d\Lambda }}<0.$$

Thus $L(\Lambda )$ is strictly decreasing on $[{\Lambda }_1,{\Lambda }_2]$. Hence

$$L_{min}=L({\Lambda }_2)=\sqrt{{\displaystyle \frac{3}{{\Lambda }_2}}},L_{max}=L({\Lambda }_1)=\sqrt{{\displaystyle \frac{3}{{\Lambda }_1}}}.$$

Therefore

$$L(\Lambda )\in \left[\sqrt{{\displaystyle \frac{3}{{\Lambda }_2}}},\sqrt{{\displaystyle \frac{3}{{\Lambda }_1}}}\right].$$

Since ${\Lambda }_1<{\Lambda }_2$, this interval is non-degenerate.

2. 

Hubble parameter $H(\Lambda )$

$$H(\Lambda )=\sqrt{{\displaystyle \frac{\Lambda }{3}}}={\displaystyle \frac{1}{\sqrt{3}}}{\Lambda }^{1/2}.$$

Differentiate:

$${\displaystyle \frac{dH}{d\Lambda }}={\displaystyle \frac{1}{\sqrt{3}}}·{\displaystyle \frac{1}{2}}{\Lambda }^{-1/2}={\displaystyle \frac{1}{2\sqrt{3\Lambda }}}.$$

Since $\Lambda >0$,

$${\displaystyle \frac{dH}{d\Lambda }}>0.$$

Thus $H(\Lambda )$ is strictly increasing on $[{\Lambda }_1,{\Lambda }_2]$. Hence

$$H_{min}=H({\Lambda }_1)=\sqrt{{\displaystyle \frac{{\Lambda }_1}{3}}},H_{max}=H({\Lambda }_2)=\sqrt{{\displaystyle \frac{{\Lambda }_2}{3}}}.$$

Therefore

$$H(\Lambda )\in \left[\sqrt{{\displaystyle \frac{{\Lambda }_1}{3}}},\sqrt{{\displaystyle \frac{{\Lambda }_2}{3}}}\right],$$

which is non-degenerate.

3. 

Scalar curvature $R(\Lambda )$

$$R(\Lambda )=4\Lambda .$$

Differentiate:

$${\displaystyle \frac{dR}{d\Lambda }}=4>0.$$

Thus $R(\Lambda )$ is strictly increasing. Hence

$$R(\Lambda )\in [4{\Lambda }_1,4{\Lambda }_2],$$

and this interval is non-degenerate because ${\Lambda }_1<{\Lambda }_2$.

4. 

Ricci tensor

In flat slicing coordinates the metric matrix is

$$g_{\mu \nu }=diag\left(-1,e^{2Ht},e^{2Ht},e^{2Ht}\right),H=\sqrt{{\displaystyle \frac{\Lambda }{3}}}.$$

The Ricci tensor satisfies

$$R_{\mu \nu }=\Lambda g_{\mu \nu }.$$

4.1

Diagonal components

For the time-time component:

$$R_{tt}=\Lambda g_{tt}=\Lambda (-1)=-\Lambda .$$

Differentiate:

$${\displaystyle \frac{d{R}_{tt}}{d\Lambda }}=-1<0.$$

So $R_{tt}$ is strictly decreasing, and its range is

$$R_{tt}\in [-{\Lambda }_2,-{\Lambda }_1].$$

Since ${\Lambda }_1<{\Lambda }_2$, we have $-{\Lambda }_2<-{\Lambda }_1$, so this is a non-degenerate interval.

For a spatial diagonal component, e.g.

$$R_{xx}=\Lambda g_{xx}=\Lambda e^{2Ht}=\Lambda e^{2t\sqrt{\Lambda /3}}.$$

Let

$$f(\Lambda )=\Lambda e^{2t\sqrt{\Lambda /3}}.$$

Differentiate:

$${\displaystyle \frac{df}{d\Lambda }}=e^{2Ht}+\Lambda e^{2Ht}·2t{\displaystyle \frac{dH}{d\Lambda }}.$$

Using

$${\displaystyle \frac{dH}{d\Lambda }}={\displaystyle \frac{1}{2\sqrt{3\Lambda }}},$$

we get

$${\displaystyle \frac{df}{d\Lambda }}=e^{2Ht}+\Lambda e^{2Ht}·2t·{\displaystyle \frac{1}{2\sqrt{3\Lambda }}}=e^{2Ht}\left(1+{\displaystyle \frac{\Lambda t}{\sqrt{3\Lambda }}}\right)=e^{2Ht}\left(1+t\sqrt{{\displaystyle \frac{\Lambda }{3}}}\right)=e^{2Ht}(1+tH).$$

Since $t>0$ and $H>0$, we have

$${\displaystyle \frac{df}{d\Lambda }}>0.$$

Thus $R_{xx}$ is strictly increasing on $[{\Lambda }_1,{\Lambda }_2]$, and its range is

$$R_{xx}\in \left[{\Lambda }_1{e}^{2t\sqrt{{\Lambda }_1/3}},{\Lambda }_2{e}^{2t\sqrt{{\Lambda }_2/3}}\right].$$

The same holds for $R_{yy}$ and $R_{zz}$.

4.2

Off-diagonal components

For $\mu \ne \nu$,

$$R_{\mu \nu }=\Lambda g_{\mu \nu }=0.$$

Thus the corresponding interval is

$$[0,0],$$

which is degenerate, not non-degenerate.

4.3

The whole Ricci tensor cluster

Consider the tensor cluster

$$\{R_{\mu \nu }(\Lambda )\mid \Lambda \in [{\Lambda }_1,{\Lambda }_2]\}.$$

Since

$$R_{tt}(\Lambda )=-\Lambda ,$$

we have

$$R_{tt}({\Lambda }_1)=-{\Lambda }_1\ne -{\Lambda }_2=R_{tt}({\Lambda }_2)$$

because ${\Lambda }_1<{\Lambda }_2$. Hence the cluster contains at least two distinct elements. By Definition 5.2, it is a non-degenerate generalized interval tensor.

However, the non-degeneracy refers to the whole Ricci tensor cluster, not to every individual component. The off-diagonal components $R_{\mu \nu }=0$$(\mu \ne \nu )$ correspond to the degenerate interval $[0,0]$.

5. 

Geodesic distance $D(t)$

Consider two comoving points with coordinate differences $(\Delta x,\Delta y,\Delta z)$. Let

$$D_0=\sqrt{{(\Delta x)}^2+{(\Delta y)}^2+{(\Delta z)}^2}>0.$$

In flat slicing, the physical distance is

$$D(t)=e^{Ht}{D}_0=D_0{e}^{t\sqrt{\Lambda /3}}.$$

Differentiate with respect to $\Lambda$:

$${\displaystyle \frac{dD}{d\Lambda }}=D_0{e}^{Ht}·t{\displaystyle \frac{dH}{d\Lambda }}.$$

From Section 2,

$${\displaystyle \frac{dH}{d\Lambda }}={\displaystyle \frac{1}{2\sqrt{3\Lambda }}}>0.$$

Therefore, for $t>0$,

$${\displaystyle \frac{dD}{d\Lambda }}>0.$$

So $D(t)$ is strictly increasing in $\Lambda$. Its range is

$$D(t)\in \left[D_0{e}^{t\sqrt{{\Lambda }_1/3}},D_0{e}^{t\sqrt{{\Lambda }_2/3}}\right].$$

Since ${\Lambda }_1<{\Lambda }_2$, this interval is non-degenerate.

6. 

Volume element $dV$

The spatial volume element in flat slicing is

$$dV=e^{3Ht}dxdydz=e^{3t\sqrt{\Lambda /3}}dxdydz.$$

For a fixed non-zero coordinate volume $\Delta V_0=\Delta x\Delta y\Delta z>0$, the volume is

$$\Delta V(\Lambda )=e^{3t\sqrt{\Lambda /3}}\Delta V_0.$$

Let

$$F(\Lambda )=e^{3Ht}.$$

Differentiate:

$${\displaystyle \frac{dF}{d\Lambda }}=e^{3Ht}·3t{\displaystyle \frac{dH}{d\Lambda }}=e^{3Ht}·3t·{\displaystyle \frac{1}{2\sqrt{3\Lambda }}}={\displaystyle \frac{3t}{2\sqrt{3\Lambda }}}{e}^{3Ht}.$$

For $t>0$,

$${\displaystyle \frac{dF}{d\Lambda }}>0.$$

Thus the volume element is strictly increasing in $\Lambda$. Its range is

$$dV\in \left[e^{3t\sqrt{{\Lambda }_1/3}}dxdydz,e^{3t\sqrt{{\Lambda }_2/3}}dxdydz\right].$$

Equivalently,

$$\Delta V\in \left[e^{3t\sqrt{{\Lambda }_1/3}}\Delta V_0,e^{3t\sqrt{{\Lambda }_2/3}}\Delta V_0\right].$$

This interval is non-degenerate because ${\Lambda }_1<{\Lambda }_2$.

Note: The original text states that $dV(\Lambda )$ is strictly decreasing. This is incorrect; it is strictly increasing.

7. 

Conclusion

For $\Lambda \in [{\Lambda }_1,{\Lambda }_2]$ with $0\le {\Lambda }_1<{\Lambda }_2$ and fixed $t>0$:

1.

The curvature radius L, Hubble parameter H, scalar curvature R, comoving distance $D(t)$, and volume element $dV$ are continuous functions of $\Lambda$.

2.

Their ranges are non-degenerate closed intervals; hence they are non-degenerate interval-valued.

3.

The Ricci tensor $R_{\mu \nu }=\Lambda g_{\mu \nu }$ has every component as an interval number.

4.

The whole Ricci tensor cluster $\left\{R_{\mu \nu }(\Lambda )\right\}$ is a non-degenerate generalized interval tensor.

5.

However, the off-diagonal components of the Ricci tensor correspond to the degenerate interval $[0,0]$. Therefore, one cannot claim that every component is a non-degenerate interval number.

Thus, the geometric quantities of de Sitter space are indeed interval-valued. The scalar-type quantities, distance, and volume are non-degenerate interval-valued. The Ricci tensor as a whole is a non-degenerate generalized interval tensor, while individual components may be degenerate.    □

Remak   This result can be generalized to space-time with matter.

Corollary 5.4   The Hubble constant is non-degenerate interval valued, and different methods may obtain different values.

Experimental evidence of Corollary 5.4    Hubble constant values for several teams can be found in the references of the paper[15] by Riess team. It is not that the measurements are not accurate enough, but that the constant themselves is a non-degenerate interval number , and the constant itself have different values under different measurements.Recently, Riess team[15], find a $5\sigma$ difference with the prediction of $H_0$ from Planck CMB observations under , CDM, with no indication that the discrepancy arises”.

6. Interpreting Quantum Phenomena in Non-Measurable Space-Time

In quantum mechanics, we cannot talk about the probability of a particle appearing in a region of space, only about the probability of a particle being detected in a region of space, because there is no definite probability of a particle before it is measured. Now, we can use interval probability to talk about the probability of a particle appearing in space, and the specific probability value is dependent on the measurement.

Proposition 6.1   Let $(X,\Sigma ,\mu )$ be a $\sigma$-finite measure space, and let $A\in \Sigma$ satisfy

$$\mu (A)>0\mathrm{and}\mu (X\setminus A)>0.$$

For every normalized pure state $\psi \in L^2(X,\mu )$, i.e.

$${\parallel \psi \parallel }^2={\int }_X{\left|\psi (x)\right|}^{2}d\mu (x)=1,$$

define the probability of detecting the particle in the region A by

$$p_A(\psi )={\int }_A{\left|\psi (x)\right|}^{2}d\mu (x).$$

Then the set of all such probabilities over all pure states is exactly

$$\{p_A(\psi ):\psi \in L^2(X,\mu ),\parallel \psi \parallel =1\}=[0,1].$$

Proof. 

For any normalized pure state $\psi$,

$$0\le p_A(\psi )={\int }_A{\left|\psi (x)\right|}^{2}d\mu (x)\le {\int }_X{\left|\psi (x)\right|}^{2}d\mu (x)=1.$$

Hence

$$\{p_A(\psi ):\parallel \psi \parallel =1\}\subseteq [0,1].$$

Since $\mu (A)>0$, by $\sigma$-finiteness there exists a measurable set $B\subseteq A$ such that

$$0<\mu (B)<\infty .$$

Define

$$u(x)=\left\{\begin{array}{cc}{\displaystyle \frac{1}{\sqrt{\mu (B)}}},\hfill & x\in B,\hfill \\ 0,\hfill & x\notin B.\hfill \end{array}\right.$$

Then

$${\parallel u\parallel }^2={\int }_X{\left|u(x)\right|}^{2}d\mu (x)=1,$$

so u is a normalized pure state. Since u is supported inside A,

$$p_A(u)={\int }_A{\left|u(x)\right|}^{2}d\mu (x)=1.$$

Thus the probability 1 is attained.

Similarly, since $\mu (X\setminus A)>0$, there exists a measurable set $C\subseteq X\setminus A$ such that

$$0<\mu (C)<\infty .$$

Define

$$v(x)=\left\{\begin{array}{cc}{\displaystyle \frac{1}{\sqrt{\mu (C)}}},\hfill & x\in C,\hfill \\ 0,\hfill & x\notin C.\hfill \end{array}\right.$$

Then

$${\parallel v\parallel }^2={\int }_X{\left|v(x)\right|}^{2}d\mu (x)=1,$$

so v is a normalized pure state. Since v is supported outside A,

$$p_A(v)={\int }_A{\left|v(x)\right|}^{2}d\mu (x)=0.$$

Thus the probability 0 is attained.

Now let $t\in [0,1]$ be arbitrary. Define

$${\psi }_t=\sqrt{t}u+\sqrt{1-t}v.$$

Because u and v have disjoint supports,

$${\int }_{X}u{(x)}^{*}v(x)d\mu (x)=0.$$

Therefore

$$\parallel {\psi }_t{\parallel }^2={t\parallel u\parallel }^2+(1-t){\parallel v\parallel }^2=t+(1-t)=1,$$

so ${\psi }_t$ is a normalized pure state.

Now compute its detection probability in A. Since $v=0$ on A,

$$p_A({\psi }_t)={\int }_A{\left|\sqrt{t}u(x)+\sqrt{1-t}v(x)\right|}^{2}d\mu (x)=t{\int }_A|u(x){|}^{2}d\mu (x)+(1-t){\int }_A{\left|v(x)\right|}^{2}d\mu (x).$$

But

$${\int }_A{\left|u(x)\right|}^{2}d\mu (x)=1,{\int }_A{\left|v(x)\right|}^{2}d\mu (x)=0.$$

Thus

$$p_A({\psi }_t)=t·1+(1-t)·0=t.$$

So every $t\in [0,1]$ is attained by some pure state.

Hence

$$[0,1]\subseteq \{p_A(\psi ):\parallel \psi \parallel =1\}.$$

Combining both inclusions gives

$$\{p_A(\psi ):\parallel \psi \parallel =1\}=[0,1].$$

   □

Proposition 6.2   [Interval probability measure generated by pure states] Let $(X,\mathcal{F})$ be a measurable space, and let

$$\left\{\right|{\Psi }_i\rangle \mid i\in K\}$$

be a nonempty family of normalized pure states. For each $i\in K$ and $A\in \mathcal{F}$, define

$$P_i(A)=\langle {\Psi }_i|P_A|{\Psi }_i\rangle ,$$

where $P_A$ is the projection operator corresponding to A. Assume that for every $A\in \mathcal{F}$, the set

$$\{P_i(A)\mid i\in K\}$$

is an interval and that

$$P_K^{-}(A)=\underset{i\in K}{min}{P}_i(A),P_K^{+}(A)=\underset{i\in K}{max}{P}_i(A)$$

exist. Define

$$\nu (A)=I_{P_K^{-}(A)}^{P_K^{+}(A)},A\in \mathcal{F}.$$

Then $\nu$ is an interval probability measure on $(X,\mathcal{F})$.

Moreover, every pure-state probability measure $P_i$ is a collapsing measurement of $\nu$, i.e.

$$P_i(A)\in \nu (A),\forall A\in \mathcal{F}.$$

Proof. 

Each $P_i$ is an ordinary probability measure on $(X,\mathcal{F})$. By hypothesis, for every $A\in \mathcal{F}$, the set $\{P_i(A)\mid i\in K\}$ is an interval with extrema $P_K^{-}(A)$ and $P_K^{+}(A)$. Hence the family $\{P_i\mid i\in K\}$ satisfies the hypotheses of Theorem 3.5. Applying Theorem 3.5 directly,

$$\nu (A)=I_{P_K^{-}(A)}^{P_K^{+}(A)}$$

is an interval probability measure on $(X,\mathcal{F})$.

For every $i\in K$ and $A\in \mathcal{F}$,

$$P_K^{-}(A)\le P_i(A)\le P_K^{+}(A),$$

so

$$P_i(A)\in I_{P_K^{-}(A)}^{P_K^{+}(A)}=\nu (A).$$

Thus $P_i$ is a collapsing measurement of $\nu$.    □

The physical interpretation of collective state of all pure quantum states    Before measurement, the quantum system is in a collective state of all pure quantum states. The physical interpretation of this collective state of all pure quantum states is as follows: the probability of a particle appearing in a nontrivial region is the interval$[0,1]$. The fact that the probability of a particle appearing in this region is an interval probability means that this region is essentially completely non-measurable; we merely regard it as measurable. After measurement, one pure quantum state is selected, and the physical meaning of this quantum state is the Born statistical interpretation.

Geometric interpretation of quantum phenomena.    

We assume that the underlying structure of physical space is a three-dimensional completely non-measurable set containing a large number of holes. These holes enable nonlocal connections between different regions of space, thereby providing a possible explanation for the phenomenon of quantum entanglement between particles.

Similarly, we assume that space is non-measurable, which can explain the double-slit interference experiment.The wave-particle duality of particles may, in essence, be a manifestation of the non-measurability of spacetime.We define a probability space for the particle double-slit interference experiment. Let event A denote that the particle passes through hole a, where hole a is a non-measurable region. Similarly, let event B denote that the particle passes through hole b, where hole b is also a non-measurable region. The sample space is the set$\{A,B\}$.In this case, event A and event B do not have uniquely determined probabilities. Instead, we assign interval-valued probabilities to these events, while the probability of the sample space itself is exactly 1.We know that the probability distribution of particles on the screen is related to whether we detect which hole the particle passes through. If we do not detect which hole the particle passes through, the particle passes through the holes with interval probabilities. If we detect which hole the particle passes through, the particle passes through the hole with ordinary probabilities. These two procedures produce different probability distributions on the screen.

7. Quantum Contextuality of Gravity Equations

Corollary 7.1( Quantum Contextuality )   Gravity equations are quantum contextuality.

Since dark energy is non-measurable, some physical parameters are non-degenerate interval numbers. So gravity equations are no longer a tesor equation, but a cluster of tensor equations.

Matter is converted into vacuum energy ( dark energy), as the matter falls through the event horizon[22-24].The interior of a black hole is a vacuum with a different cosmological constant, and the universe before the Big Bang was a vacuum with a different cosmological constant. The deeper principle behind this principle may be the interconversion between matter and dark energy,dark energy particles are probably the most fundamental particles in the universe.Using this assumption ,we can solve the problem that general relativity breaks down at the singularity.The star that formed the black hole collapsed into dark energy and nothing else, so the energy tensor of matter is zero,the vacuum field equation can describe the interior of a black hole. Similarly, a vacuum field equation with a different cosmological constant could describe the universe before the Big Bang.

The following equations are in accordance with the non-measurability and universality of dark energy.

Corollary 7.2 (Interval tensor equation for quantum gravity in the vacuum )

$\{R_{\mu \nu }=\Lambda g_{\mu \nu }|\Lambda \in I\}$

Remark1 The universe within a black hole or prior to the Big Bang contained only dark energy, devoid of matter, but with a distinct cosmological constant. Therefore, this equation can describe the state of the universe in black holes or during the pre-Big Bang era.

Corollary 7.3 ( Interval tensor equation for quantum gravity )

$\{R_{\mu \nu }-{\displaystyle \frac{1}{2}}{g}_{\mu \nu }R=\kappa (T_{\mu \nu }-{\displaystyle \frac{\Lambda }{\kappa }}{g}_{\mu \nu })|\Lambda \in I,T_{\mu \nu }\in \mathbf{T}\}$.

Where I is an interval constant,$\mathbf{T}$ is an interval tensor.

Remark1 This is a cluster of Einstein field equations, and this equation is an interval equation, and studying the solution set is the only way to understand the physical meaning of this equation.If the solution set is a nondegenerate generalized interval tensor, then the space-time is nonmeasurable.The basic concept of interval equations can be found in Reference [13].

Remark2 The interval tensor equation for quantum gravity has generalized covariance. Because of physical considerations, Einstein insisted that the right-hand side ofthe field equations can be written as the sum of quantities representing the energy-momentum of matter and the energy- momentum of the gravitational field.However, such an equation does not have generalized covariance.Using interval tensors to describe gravitational field can solve this problem.

Remark3 This theory can explain not only the macroscopic phenomena of general relativity but also the microscopic phenomena of quantum mechanics.There is no doubt that this equation can explain the macroscopic phenomena of general relativity.The remaining problem is to derive the Schrodinger ( Dirac ) equation from the equation of quantum gravity.

Remark4 The pseudo energy-momentum tensor of the gravitational field depends on the coordinate system, which means that the energy of the gravitational field depends on the measurement. That is, the gravitational field energy is nonmeasurable.

Remark5 In a quantum state, the particle has a definite position and velocity under a measurement, and therefore a definite energy-momentum tensor under the measurement(Although the particle doesn’t have a definite momentum when a particle has a definite position, the particle will have a momentum under the measurement).The set of all energy-momentum tensor of the quantum state is described in terms of an interval tensor.So we’ve solved the gravity problem of a single particle.

8. Background Independent Quantum Theory on De Sitter Space

In Feynman’s path integral, if one compares a set of neighboring paths, the phases or positions in the cycle will differ greatly. This means that the waves associated with these paths will almost exactly cancel each other out. However, for some sets of neighboring paths the phase will not vary much between paths. The waves for these paths will not cancel out such paths correspond to Bohr, s allowed orbits.In general, these orbits that cannot be cancelled are called allowed orbits. The principle of least action says that the actual path taken by the system is an extremum of S, and a reasonable interpretation of the Feynman path integral can only be that particles are coming from allowed orbits, which are geodesics in some nonmeasurable spacetime.

Here’s an analogy that will allow us to build a background independent quantum theory.

Allowed orbits in Minkowski space-time ≃ Geodesics in non-measurable space-time.

We try to build a background independent quantum theory, making quantum mechanics dependent on gravity.There is a specific cosmological constant under each measurement, so we get a de Sitter space, in which there may be a unique geodesic between two points. If there is a geodesic between two points in a non-measurable de Sitter space, there are many different geodesics at the same time. We construct a background independent quantum theory by analogy with Feynman’s path integral theory, the geodesics of non-measurable spacetime are similar to the allowed orbits of Minkowski spacetime.

Definition 8.1 (Propagator on the non-measurable de Sitter space). Let the cosmological constant take values in a non-degenerate interval $[{\Lambda }_1,{\Lambda }_2]$, $0\le {\Lambda }_1<{\Lambda }_2$, and let $\rho (\Lambda )$ be a probability density on this interval satisfying

$$\rho (\Lambda )\ge 0,{\int }_{{\Lambda }_1}^{{\Lambda }_2}\rho (\Lambda )d\Lambda =1.$$

For each $\Lambda \in [{\Lambda }_1,{\Lambda }_2]$, set

$$H(\Lambda )=\sqrt{{\displaystyle \frac{\Lambda }{3}}}.$$

Use Poincar, coordinates and choose the time coordinate t as the parameter along the worldline, so that the spatial coordinates are functions of t:

$$x=x(t),y=y(t),z=z(t),$$

with

$$\dot{x}={\displaystyle \frac{dx}{dt}},\dot{y}={\displaystyle \frac{dy}{dt}},\dot{z}={\displaystyle \frac{dz}{dt}}.$$

For this $\Lambda$, the de Sitter metric is

$$d{s}_{\Lambda }^2=-d{t}^2+e^{2H(\Lambda )t}\left(d{x}^2+d{y}^2+d{z}^2\right).$$

The geodesic Lagrangian is therefore

$$L_{\Lambda }={\displaystyle \frac{1}{2}}m\left[-1+e^{2H(\Lambda )t}\left({\dot{x}}^2+{\dot{y}}^2+{\dot{z}}^2\right)\right].$$

Let $x_{\Lambda }^{\mu }(t)=(t,x(t),y(t),z(t))$ be a worldline from $(x_0,t_0)$ to $(x^{\prime },t^{\prime })$ under the value $\Lambda$, and let

$$S_{\Lambda }\left[x_{\Lambda }\right]={\int }_{t_0}^t{L}_{\Lambda }dt$$

be its action. Let $x_{\Lambda ,\mathrm{geo}}(t)$ denote a geodesic between these two points for this $\Lambda$. Then the propagator on the non-measurable de Sitter space is defined by

$$K(x^{\prime },t^{\prime };x_0,t_0)=A{\int }_{{\Lambda }_1}^{{\Lambda }_2}\rho (\Lambda )exp\left({\displaystyle \frac{i}{\hslash }}{S}_{\Lambda }\left[x_{\Lambda ,\mathrm{geo}}\right]\right)d\Lambda .$$

where A is the normalization constant. The corresponding probability is

$$P(x^{\prime },t^{\prime };x_0,t_0)={\left|K(x^{\prime },t^{\prime };x_0,t_0)\right|}^2.$$

$P(x^{\prime },t^{\prime };x_0,t_0)$ is called the probability of going from $(x_0,t_0)$ to $(x^{\prime },t^{\prime })$.

Remak1 Similarly, we can establish quantum mechanics on other non-measurable Spaces.The theory established here can change our quantum theory, we can get a background independent quantum theory.

Remark2 There is a specific cosmological constant under each measurement, so this quantum theory depends on the cosmological constant.

Remark3 In general relativity, particles travel along geodesics. In quantum mechanics, particles come from every possible path at the same time. This is an obvious contradiction. In non-measurability theory, there are many different geodesics between two points at the same time, which resolves the contradiction between quantum mechanics and general relativity, the particles come along the geodesic at the same time.

9. The Unilateral Gravitational Effect of a Completely Non-Measurable Matter

9.1. Concept and Fundamental Principle

In this chapter, the non-measurable matter under discussion are completely non-measurable matter. Within the framework of the non-measurability theory, the gravitational interaction between matter and spacetime is no longer constrained by reciprocity. When certain forms of matter enter a non-measurable state, their coupling to measurable spacetime undergoes a fundamental transformation, exhibiting a property known as unilateral gravity.

Unilateral gravity refers to a state in which completely non-measurable matter continues to generate gravitational fields, curve spacetime, and influence the motion of measurable matter, yet does not respond to external gravitational fields. In other words, such matter acts as a one-way gravitational source: it exerts gravity but does not feel it.

The Weak Equivalence Principle (WEP) is a fundamental principle of gravitational physics. It states that, in a given external gravitational field, the free-fall acceleration of a body is independent of its composition, mass, and internal structure. Equivalently, the ratio between gravitational mass and inertial mass is the same for all bodies, conventionally expressed as ${\displaystyle \frac{m_g}{m_i}}=1$.Therefore, if two bodies are placed at the same location in the same gravitational field, they should undergo the same free-fall acceleration. Whether this principle also applies to completely non-measurable matter remains an open question. Castello et al. [53] proposed a model-independent cosmological method to test whether dark matter obeys the weak equivalence principle, emphasizing that its validity for dark matter has not yet been directly established. Similarly, de Matos [52] investigated whether vortices in rotating superfluids could violate the weak equivalence principle and proposed experimental tests of this possibility. These studies provide independent theoretical motivations for examining whether completely non-measurable matter may have a gravitational response different from that of ordinary measurable matter. Within the framework of non-measurability theory, this possibility is extended further: completelynon-measurable matter may not only violate the weak equivalence principle, but may exhibit unilateral gravity, in which it generates gravitational fields and influences measurable matter while exhibiting a strongly suppressed or uncertain response to external gravitational fields.

9.2. Physical Implications and Cosmological Significance

The existence of unilateral gravity has profound implications. It provides a natural explanation for the non-collapsing nature of dark matter:completely non-measurable matter can shape spacetime curvature and account for observed galactic rotation curves and gravitational lensing effects, yet it will not collapse under gravity into compact structures. Moreover, gravitational asymmetry implies that measurable matter will be attracted by non-measurable matter, but the reverse interaction does not occur — a property that could fundamentally reshape the evolution of cosmic structures.

From a theoretical standpoint, unilateral gravity suggests that the generation of gravity and the response to gravity are fundamentally distinct processes.

9.3. Experimental and Observational Evidence

While direct detection of completely non-measurable matter remains beyond current technological capabilities, several lines of indirect evidence align with the unilateral gravity framework:

• Neutrino: “when an external gravitational field is considered in the weak-field approximation, we have observed that the gravitational mass does not undergo the same redefinition as the inertial one, and hence a violation of WEP arises., [60]
  SN1987A[56] Supernova Neutrino Event Neutrinos from Supernova SN1987A were detected approximately three hours earlier than the corresponding optical signals. Within the framework of general relativity, such a time lead would require an extremely fine-tuned propagation model to be consistent, suggesting a possible deviation from symmetric gravitational lensing behavior.
  TXS 0506+056 Neutrino Flare Event (IceCube, [55]) In the TXS 0506+056 blazar event, the reconstructed arrival direction of the high-energy neutrino IceCube-170922A was approximately 0.1, from the position of TXS 0506+056, indicating a small angular offset between the neutrino arrival direction and the , -ray source position. This misalignment is considered unlikely to be a mere statistical fluctuation, and may instead indicate an intrinsic asymmetry in gravitational lensing effects experienced by neutrinos compared to photons.
• Dark matter halo morphology: Observations indicate that dark matter forms extended, non-collapsing halos around galaxies, consistent with the prediction of Unilateral gravity.
• Constraints on dark-matter spikes: Stellar-orbit observations of $S_2$ have placed stringent dynamical constraints on dark-matter spikes around Sgr A, excluding sufficiently extended spikes and, for steep initial halo profiles, disfavoring the Gondolo-CSilk adiabatic-growth prediction at the $95\%$ confidence level.[57]
• Creep effect: The , unilateral gravity-like, phenomenon of superfluid – the creep effect: One of the most astonishing properties of superfluid helium is its ability to defy gravity, climbing up the walls of a container until it spills out. The traditional explanation relies on the combined effects of its zero viscosity and the surface film. However, within the framework of non-measurability theory, this can be interpreted as a kind of , ignoring" of the directionality of gravity at the microscopic scale.
• The lack of gravitational lensing effects for neutrinos and gravitational waves: Although general relativity predicts that neutrinos and gravitational waves should exhibit gravitational lensing when propagating through gravitational fields, no unambiguous direct observational evidence for either neutrino gravitational lensing or gravitational-wave gravitational lensing has yet been established. Within the framework of unilateral gravity, one may further hypothesize that gravitational interactions possess a directional character: a gravitational field can be generated by matter or energy distributions, including non-measurable matter, while the response of certain objects to external gravitational fields may be restricted. In this case, neutrinos and gravitational waves might not exhibit the gravitational lensing effects predicted by standard general relativity, or there might be no observable gravitational lensing phenomenon at all. Therefore, systematic searches for gravitational deflection, focusing, multiple imaging, and waveform modulation of neutrinos and gravitational waves could provide an important observational avenue for testing unilateral gravity and distinguishing it from standard theories of gravity.
  , Surely neutrinos, in common with other forms of matter and energy, experience gravitational interactions . Where is the observational evidence to support this assertion? No analogue of the classic Einstein , Eddington demonstration of the deflection of starlight by the Sun is in prospect., [58], Overall, we find no compelling evidence for lensing in the observed gravitational-wave signals from any of these analyses., [59]

10. A Model of Dark Energy:Dark Energy Is the Gravitational Field

In general relativity, the gravitational interaction between objects is understood as a manifestation of the curvature of space-time; in cosmology, the apparent repulsive effect between galaxies is attributed to dark energy. A possible deeper picture is that these two phenomena are fundamentally unified: dark energy itself may be the gravitational field.We assume that the vacuum state contains only one field, namely dark energy, which permeates the entire universe. Since we know that the gravitational field exists continuously throughout the universe, we therefore conclude that dark energy is the gravitational field.The accelerated expansion of the universe is merely the physical manifestation of the Banach-CTarski paradox acting on dark energy.Since dark energy can undergo expansion, this implies that dark energy decomposes and then recombines, thereby generating a larger volume of space.

If dark energy is the gravitational field, then the measured value of dark energy may depend on the measurement method: in a local freely falling reference frame, the effects of the gravitational field can be eliminated, so the corresponding local gravitational energy density can be taken to be zero; whereas inferring it from the history of cosmic expansion yields a nonzero effective cosmological mean density.This indicates that dark energy is completely non-measurable matter.For measurable matter, physical quantities are represented by single-valued tensors; for the non-measurable gravitational field, the measurement results cannot be determined by a single tensor, but are instead represented by an interval-valued tensor.For the interval-valued tensor of the gravitational field, the energy component is an interval whose lower bound is zero.

11. Falsifiable Theoretical Predictions

Falsifiable theoretical predictions11.1 The mass of non-measurable particle is interval valued, and the specific real value depends on the measurement method.With improved measurement precision, the tension between the two measurements could reach the 5, level.

Falsifiable theoretical predictions11.2 Non-measurable particles have an effect similar to the oscillation of neutrinos.

Falsifiable theoretical predictions11.3 Unilateral gravitational effects of completely non-measurable matter, including violations of the Weak Equivalence Principle (WEP).

12. Tao Te Ching and Non-Measurability Theory

The cosmological ideas of the Tao Te Ching [1] and Non-Measurability Theory are essentially the same.The Tao in the Tao Te Ching is also dark energy, and the properties of the Tao are the same as those of dark energy. The Tao that can be spoken of is not the eternal Tao(Chapter 1), the Tao in the Tao Te Ching cannot be seen, heard or touched(Chapter 14), the Tao is hidden and nameless(Chapter 41), which corresponds precisely to the first principle of Non-Measurability Theory: dark energy is non-measurable. , The Tao gives birth to One; One gives birth to Two; Two gives birth to Three; Three gives birth to all things,, (Chapter 42)which means that the Tao gives birth to all things. This corresponds precisely to the second principle: all matter and fields are excitations of dark energy.

Chapter 25 of the Tao Te Ching describes the nature of the Tao: , There is something formed from chaos, existing before Heaven and Earth. Silent and vast, it stands alone and does not change; it moves throughout all things without ceasing and may be regarded as the Mother of the world. I do not know its name; I call it the Tao. For lack of a better name, I call it Great. Great means that it moves onward; moving onward means going far; going far means returning., These are also descriptions of the properties of dark energy. , Great means that it moves onward; moving onward means going far; going far means returning., This passage describes cosmic expansion and contraction: , going far means returning, suggests that, after expanding to a certain extent, the universe will eventually contract. This hypothesis must be tested through observations over time.

Chapter 40 of the Tao Te Ching states: , Reversal is the movement of the Dao; weakness is the function of the Dao. All things under heaven are born from being, and being is born from non-being., This expresses the idea that all things arise from the Dao and eventually return to the Dao. This can be understood as corresponding to the excitation of dark energy into matter and fields, while matter and quantum fields may, through decay, annihilation, or other physical processes, be transformed back into dark energy.

13. Summary

“The grand aim of all science is to cover the greatest number of empirical facts by logical deduction from the smallest number of hypotheses or axioms., Einstein said in the article[49]. The foundation of Non-measurability theory is made up of two basic principles. The first principle is that dark energy is non-measurable, and the second is that All matter and fields are excitation of dark energy. From the point of view of the field, the dark energy field is non-measurable, and all matter is excited by the dark energy field.

These two principles can be used to solve fundamental problems in physics, such as: quantum gravity, unified field theory, cosmological constants, Hubble constants, neutrinos, high-temperature superconductivity mechanisms, fractional charges, energy problems of gravitational fields, gravitational problems of single particles, the mystery of black hole mass growth, the mystery of the disappearance of antimatter, the properties of dark matter and dark energy, and so on.

The concept of non-measurable sets comes from the theory of real variable functions, the most famous theorem about this concept is the Banach-Tarski paradox, , the mathematics is very beautiful, unfortunately ,mathematics and physics here have diverged ways, people generally believe that it is impossible to get non-measurable matter in the real world, which requires extremely complex segmentation to get non-measurable matter, but there is also a case: the most basic matter in the universe is non-measurable, and other matter is made up of this most basic matter. This coincides with our current physical discovery that most of the matter in the universe is unknown.

Dark energy has a strange property: as the universe expands, the density of dark energy remains constant, it’s very much like something came out of nothing. As long as it is assumed that dark energy is non-measurable, it has the property of non-measurable sets, that the dark energy is excited into various matters, and the matter disappears into a vacuum, that is to say, the dark energy is constantly reorganizing and decomposing, according to the Banach-Tarski paradox, this situation is possible, the space is getting bigger and bigger, and the dark energy is also increasing, and the opposite is also possible, which can also be explained by the nature of the non-measurable set.

For the second principle, all matter in the universe is made up of dark energy, Heisenberg once proposed that all matter is made up of the same entity, and unfortunately, there was no concept of dark energy in his time. This principle asserts that this fundamental substance is dark energy. In the case of black holes, this is the end of all matter, where all matter becomes dark energy. There are several articles that try to prove that black holes are made up of dark energy. In turn, black holes produce all matter. In quantum field theory, the elementary particle is the excitation of its corresponding field, and this excitation comes from the vacuum, but the quantum field theory believes that the vacuum is the ground state of the quantum field, and this assertion believes that these fields are the most basic, and that an electron field is needed in order to generate electrons, and an electromagnetic field is needed in order to generate photons, and so on, quantum field theory believes that these fields are ubiquitous, and there are as many fields as there are elementary particles. To use an analogy, quantum field theory seems to say that in order to make a steamed bun, there needs to be flour everywhere in the universe. In fact, in order to make a steamed bun, you only need to have flour in your little place, and this flour is also made.This metaphor is used to say that the quantum field discussed in quantum field theory is not fundamental, but is made up of a more fundamental field, which we think of as dark energy.

The first conclusion is that there is non-measurable matter in the universe. Neutrinos, quarks, Gluons, superfluids , W bosons, Z bosons, magnetic monopoles, Higgs , black holes, Cooper pairs, fractional electrons, dark matter are most likely non-measurable.This conclusion can explain a wide range of physical phenomena.

According to the first principle, the cosmological constant is not a real value, but an interval value. It can be deduced that the concepts of space-time and geometry are quantum intertextuality. We incorporate quantum intertextuality into the theory of gravity and geometry, so we get a quantum theory of space-time and geometry.

We can deduce that all geometric quantities in de Sitter space are interval-valued.

The equations of quantum gravity can explain not only gravitational phenomena but also quantum phenomena. In addition, if dark energy is the most fundamental field, then the field equation of dark energy is sufficient to unify all fields, which is also the direction of our research.

We put forward several falsifiable theoretical predictions, in particular the unilateral gravitational effect of non-measurable matter, which encompasses the violation of the weak equivalence principle for that matter.