1. Introduction: From Taiji to the Weave

Taiji Evolution Cosmology [1] holds that Taiji is the origin of the universe, giving rise to the two aspects, yin and yang (quantum entanglement). Countless pairs of yin and yang interweave into a network: we call each entangled pair a “knot”, and the way in which knots connect is the “weave”. The weave determines physical law—gravity, spacetime, and matter all emerge from the network. Our spacetime geometry is precisely woven by this weave—conservation structure plus quantum fluctuations give rise to spacetime geometry. The task of this paper is to turn the “weave” from metaphor into mathematics.

This paper first defines the weave and the Weaving Formula, then lists the assumptions and describes the net types of spacetime geometry, then gives the fundamental theorems and corollaries with numerical verification, and finally discusses the boundaries of the Weaving Formula and draws conclusions.

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2. Definition of the Weave

Definition 1 (Weave): A weave is a charged network: W = (V, E, T)—V is the vertex set, E the edge set, and T the vertex-tensor family. Each edge e ∈ E carries a charge a_e ∈ G (G the charge space, a finite group). For a vertex v ∈ V, let ∂v := {e ∈ E : v ∈ e} denote its incident edge set, let the valence be d_v := |∂v| (trivalent means d_v ≡ 3), and let G^{∂v} := {b : ∂v → G} be the space of incident-edge charge functions at v. The vertex tensor T_v : G^{∂v} → ℂ takes as its input the charge configuration on the edges incident to v; legs correspond one-to-one with incident edges and are not numbered.

The vertex tensor satisfies two constraints (for each v ∈ V):

1. Charge conservation: Tv ≠ 0 only if the group sum of the charges on all edges incident to v vanishes, ⊕{e∈∂v} a_e = 0;
2. Normalization: Σ_{(a_e)_{e∈∂v}} |T_v((a_e)_{e∈∂v})|² = 1.

The definition holds for an arbitrary charge space G and an arbitrary valence (different vertices may have different valences), independent of the particular group and valence chosen.

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3. The Weaving Formula

Definition 2 (Weaving Formula): The Weaving Formula defines the quantum state of the network—a superposition over all charge configurations, with amplitudes given by products of vertex tensors. A charge configuration is an assignment of charges to edges. The set of all edge-charge configurations is the function space G^E := {a : E → G} (a assigns to each edge e ∈ E a charge a_e ∈ G); |a⟩ is the quantum basis vector corresponding to the configuration a; a|∂v ∈ G^{∂v} is the restriction of the configuration a to the edges ∂v incident to v, i.e. the input of T_v:

|Ψ[W]⟩ = Σ_{a ∈ G^E} ( Π_{v∈V} T_v(a|∂v) ) |a⟩          (3.1)

Z[W] = ⟨Ψ[W]|Ψ[W]⟩ = Σ_{a ∈ G^E} | Π_{v∈V} T_v(a|∂v) |² (3.2)

The sum runs over all edge-charge configurations (charge conservation is enforced automatically by the support of the T_v); the product runs over all vertices v. Z[W] is the partition function. The expression holds for an arbitrary charge space G and an arbitrary valence—the Weaving Formula is abstract: legs are labelled by incident edges and are not numbered, the valence d_v = |∂v| is encoded in the dimension of the domain of T_v, and the specific group is merely a parameter inserted into it.

Relationship to the ground state: the Weaving-Formula state |Ψ[W]⟩ is the string-net ground state (the TV state) [2,3,4]: the allowed configurations are jointly selected by vertex charge conservation (the δ term of the tensor) and by the flat-face constraints (the Hamiltonian realization of the constraints and the numerical confirmation are given in §5 and §6; on K4 the overlap between |Ψ[W]⟩ and the ground state is 1.00000000 exactly).

Construction process: given the weave data W = (V, E, T)—

1. assign a charge a_e ∈ G to each edge, obtaining an edge-charge configuration a ∈ G^E;
2. the charge-conservation constraint (the ⊕ of the charges on the edges incident to each vertex vanishes) selects the allowed configurations;
3. the amplitude of each allowed configuration is the product of the vertex tensors Π_{v∈V} T_v(a|∂v);
4. the network quantum state is the superposition of all allowed configurations |Ψ[W]⟩ = Σ{a∈G^E} (Π{v∈V} T_v(a|∂v)) |a⟩;
5. the partition function Z[W] = ⟨Ψ[W]|Ψ[W]⟩ is the sum of squared amplitudes.

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4. Assumptions and the Net Types of Spacetime Geometry

4.1 Assumptions

The Weaving Formula is abstract (Definition 2: arbitrary charge space, arbitrary valence); making it concrete requires choosing the charge space and the valence, and giving the explicit form of the vertex tensor at the chosen valence. The assumptions of our universe are:

• (G1) Charge space: G = Z₂³ (an eight-element finite abelian group). This is the sole assumption-type input; replacing the group leaves the form of the formula unchanged, and only the value of lnD changes (verified by multi-weave universality scans).

Figure 1. The eight elements (eight states).

• (G2) Valence and the form of the vertex tensor: valence n = 3 (trivalent vertices). A point in parameter space, not a derived result; quartic valence and others are equally viable (verified numerically, with the tensor form changed accordingly). Under the trivalent setting the vertex tensor takes the explicit form:

T⁽ᵛ⁾_abc = δ_{a⊕b⊕c, 0} · t_ab · ω(a,b,c)            (4.1)

where the δ term is the charge-conservation constraint (T is nonzero only if a⊕b⊕c = 0); t_ab is the vertex amplitude matrix (normalized, see G3); ω is the phase data (a 3-cocycle, see G4).

• (G3) Vertex amplitude: tab (the amplitude matrix of the trivalent vertex tensor, depending on the edge charges a and b; vertex-wise normalization Σ{a,b}|t_ab|² = 1, see Constraint 2 of §2). In the principal case (TV state / string-net ground state) the amplitude is taken to be uniform, t_ab ≡ 1/8, which automatically satisfies the normalization.
• (G4) Phase: ω—a 3-cocycle satisfying the pentagon equation, with ω ∈ H³(Z₂³, U(1)) ≅ (Z₂)⁷ [5], 128 independent choices in total. The pentagon equation is the explicit form of the 3-cocycle condition δω = 1 for abelian groups (a five-term identity, for arbitrary a, b, c, d ∈ Z₂³):

ω(b,c,d) · ω(a,b⊕c,d) · ω(a,b,c⊕d) = ω(a⊕b,c,d) · ω(a,b,c) (4.2)

Substituting the assumptions G1–G4 into Definition 2, the Weaving Formula of this paper takes the following complete form. Under the trivalent setting each vertex v has exactly three incident edges; fix their numbering ∂v = {e{v,1}, e{v,2}, e{v,3}}, so that the configuration a|∂v corresponds one-to-one with the ordered triple (a{e{v,1}}, a{e{v,2}}, a{e_{v,3}})—the a, b, c in the G2 tensor element are precisely the charges of the three edges incident to that vertex. Substituting into Definition 2, the network quantum state (keeping the tensor-layer form):

|Ψ[W]⟩ = Σ_{a ∈ (Z₂³)^E} ( Π_{v ∈ V} T⁽ᵛ⁾_{a_{e_{v,1}}, a_{e_{v,2}}, a_{e_{v,3}}} ) |a⟩ (4.3)

Partition function:

Z[W] = ⟨Ψ[W]|Ψ[W]⟩ = Σ_{a ∈ (Z₂³)^E} Π_{v ∈ V} |T⁽ᵛ⁾_{a_{e_{v,1}}, a_{e_{v,2}}, a_{e_{v,3}}}|² (4.4)

Substituting further the explicit form of G2 (with a, b, c replaced by the corresponding edge charges) into the tensor layer, we obtain the fully expanded form:

|Ψ[W]⟩ = Σ_{a ∈ (Z₂³)^E} ( Π_{v ∈ V} δ_{a_{e_{v,1}} ⊕ a_{e_{v,2}} ⊕ a_{e_{v,3}}, 0} · t_{a_{e_{v,1}}, a_{e_{v,2}}} · ω(a_{e_{v,1}}, a_{e_{v,2}}, a_{e_{v,3}}) ) |a⟩ (4.5)

Fully expanded partition function (|ω| = 1, so the phase does not contribute to the weight):

Z[W] = Σ_{a ∈ (Z₂³)^E} Π_{v ∈ V} δ_{a_{e_{v,1}} ⊕ a_{e_{v,2}} ⊕ a_{e_{v,3}}, 0} · |t_{a_{e_{v,1}}, a_{e_{v,2}}}|² (4.6)

where each edge charge a_e ∈ G = Z₂³ (G1); the tensor element of a vertex is the explicit form of G2 (G2–G4):

T⁽ᵛ⁾_abc = δ_{a⊕b⊕c, 0} · t_ab · ω(a,b,c),  a, b, c ∈ Z₂³ (4.7)

The three factors each play their own role:

• δ_{a⊕b⊕c, 0} (charge-conservation projector): the tensor is nonzero only if the group sum of the three legs vanishes; the allowed edge-charge configurations are selected automatically by the δ of each vertex (Constraint 1 of §2);
• tab (vertex amplitude, G3): normalized amplitude matrix, vertex-wise normalization Σ{a,b}|t_ab|² = 1 (Constraint 2 of §2); in the principal case (TV state / string-net ground state) the amplitude is uniform, t_ab ≡ 1/8 (equal-weight superposition, the normalization being then automatic);
• ω(a,b,c) (phase data, G4): a 3-cocycle satisfying the pentagon equation; modulo coboundaries ω ∈ H³(Z₂³,U(1)) ≅ (Z₂)⁷ [5], 128 independent choices.

This formula is the common starting point of all the fundamental theorems of §5, and the object of the item-by-item numerical evaluation in §6.

4.2 The Net Types of Spacetime Geometry

The diamond net in the quartic realization of the Weaving Formula can describe our three-dimensional spacetime geometry. The local rule of the diamond net is unique, while the overall stacking is not; the stacking order of the layers gives different net types. The local rule is: each vertex emits four legs, all of equal length, with pairwise angle 109.4712° (a regular tetrahedron), and the shortest cycle is a six-membered ring. Viewed along the stacking axis, it is built by stacking triangular lattice layers: the nearest-neighbour distance within a layer is a = √(8/3) ≈ 1.633, larger than the leg length 1, so there are no legs within a layer; two adjacent layers are related in only two ways—shifted within the layer by a/√3 ≈ 0.943 with an axial separation of 1/3, or aligned at the same position with an axial separation of 1. The two relations alternate, the former giving three slanted legs and the latter one straight leg, which together make exactly four legs per vertex.

Figure 2. The local structure of the cubic diamond (3C) net: one vertex emits four legs, the bond angle is 109.4712°, and the nearest-neighbour distance within a layer is a = √(8/3) ≈ 1.633.

The net type is selected by two requirements: the four legs must be equivalent and of equal length, and their directions must be symmetric—the unique symmetric solution for the bond angle is the tetrahedral coordination, otherwise the net would not be isotropic; and the net as a whole must be macroscopically isotropic, i.e. the bond length, bond density and elastic response are the same along every direction. Together, the two requirements give the cubic diamond (3C) net. Nets with the same local rule but a different stacking order (for example hexagonal diamond [6]) are only statistically isotropic and do not satisfy macroscopic isotropy. This paper therefore takes the cubic diamond (3C) net to describe three-dimensional spacetime geometry. Observed from different angles, the cubic diamond (3C) net presents different three-dimensional patterns; we choose one angle and show it below. The thickened part is the structure of a six-membered ring in three-dimensional space; the six-membered ring is not coplanar and undulates in the chair form.

Figure 3. One view of the cubic diamond (3C) net; the thickened part is one six-membered ring.

To study the area law we need to project the three-dimensional diamond net onto a two-dimensional plane: an orthogonal projection along the stacking axis (the body diagonal ⟨111⟩ of cubic diamond, four equivalent directions) gives the two-dimensional honeycomb net, i.e. the trivalent realization of the Weaving Formula, as shown below.

Figure 4. The two-dimensional honeycomb net obtained by the orthogonal projection of the cubic diamond (3C) net along the stacking axis, i.e. the trivalent realization of the Weaving Formula.

4.3 The Elementary Unit of Spacetime Geometry

In real physics, we take the elementary unit of spacetime geometry to be the three-bit quantum entanglement relation of binary digits superposed in eight states, denoted a_P. Philosophically, we call it a knot. In the geometric model, the elementary unit of spacetime geometry is each edge (bond) of the network, whose physical scale is likewise denoted a_P.

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5. Fundamental Theorems

The theorems of this section are all derived directly from the Weaving Formula (Definition 2).

5.1 Theorem 1 (Area law, f = L−1)

Theorem: the bipartite entanglement entropy of the Turaev–Viro (TV) state [3] (flat, equal-weight superposition—“ground state” refers to the ground state of the Levin–Wen string-net Hamiltonian [2] H(W) = −Σ_v A_v − Σ_p B_p, where A_v is the vertex charge-conservation projector and B_p the flat-face projector; numerical confirmation in §6) under a simple cut (the boundary of the region is a single closed curve) is:

S(A) = (L−1)·lnD                                        (5.1)

where L is the number of cut edges and D = |G| = 8 is the quantum dimension of an edge (bond) [7,8].

Proof (four steps):

1. The boundary of region A consists of L cut edges, each carrying a charge ∈ Z₂³;
2. Total-flux conservation: summing the flat-face constraints of all faces inside region A, the interior edge charges cancel pairwise, leaving the boundary charges subject to 1 global constraint Σ⊕(boundary charges) = 0 (guaranteed by the spherical topology and the single closed boundary curve), so the number of independent boundary charges is L−1;
3. Property of the TV state: given the independent boundary charges, the interior of the region is determined uniquely by the face constraints—the Schmidt vectors are classified by the boundary charges and are orthogonal;
4. Hence the Schmidt rank is 8^(L−1) (8 choices per boundary charge) and the entropy is S = ln(rank) = (L−1)·ln8. ∎

Numerical verification:

Graph	Cut	L	f measured
K4	single vertex {0}	3	2
triangular prism	upper triangle {0,1,2}	3	2
cube	half {0,1,2,3}	4	3

5.2 Theorem 2 (Region-intrinsic entropy, f = E_B − F_B)

Theorem: for an arbitrary cut, the number of degrees of freedom of the bipartite entropy of the TV state is:

f = E_B − F_B = V_B − c₀(B)                             (5.2)

where B is the complementary region of the cut: E_B = number of edges interior to B, F_B = number of faces interior to B, V_B = number of vertices of B, and c₀(B) = number of connected components of the induced subgraph on B.

Sketch of proof:

1. The TV state is encoded on the B side by a Schmidt decomposition; the number of nonzero eigenvalues of ρ_A equals the number of reachable B-side configurations;
2. The independent degrees of freedom on the B side are the interior edge charges of B subject to the face constraints inside B (one independent constraint per face);
3. The number of independent degrees of freedom is E_B − F_B (for connected B the face constraints are independent; the Euler characteristic χ = 1 guarantees no redundancy);
4. The Euler formula V_B − E_B + F_B = c₀(B) gives f = E_B − F_B = V_B − c₀(B). ∎

Numerical verification: all 18 data points (every cut of K4 / triangular prism / cube) match; an independent cross-check by linear algebra (rank computation over GF(2)) is fully green. f = L−1 is the special case (for the cut-complementary region E_B = L, F_B = 1).

5.3 Theorem 3 (Thermodynamic-limit saturation, c_∞ = 1)

The area-law coefficient c(L) = (L−1)/L: K4 (L=3 → c=2/3), triangular prism (3 → 2/3), cube (4 → 3/4) → thermodynamic limit:

c_∞ = 1                                                 (5.3)

Twofold confirmation: (1) extrapolation of f = L−1; (2) the SVD entropy of the TRG [9] fixed-point tensor is exactly ln8.

Significance of the theorem: the area-law entropy S = c(L)·L·lnD (c(L) = (L−1)/L is the area-law coefficient, see above) divided by the horizon area A = L·a_P² gives the entropy density η = S/A = c(L)·lnD/aP²; in the thermodynamic limit c∞ = 1 and the entropy density saturates at η_∞ = lnD/a_P² (a_P is the physical scale of an edge (bond), see Corollary 3).

5.4 Theorem 4 (Universal mass-gap law)

Theorem: the mass gap of the TRG [9] fixed point of the group-constrained tensor network (domain of validity |G| ≥ 3):

m* = −ln((|G|−2)/(|G|−1))                              (5.4)

depends only on the group order |G|, independent of the group structure (abelian/nonabelian). Domain of validity |G| ≥ 3: for |G| = 2 the argument (|G|−2)/(|G|−1) = 0, so m* = −ln 0 → +∞ diverges, outside the domain of this law (Z₂/toric-code [10]-type string nets do have a gap, but it is not described by this formula); the principal group Z₂³ of this paper, with |G| = 8, is unaffected.

Numerical verification:

Group	Order	m* measured	m* predicted	Deviation
Z₃	3	0.693147	0.693147	0.00%
Z₅	5	0.287682	0.287682	0.00%
Z₆	6	0.223144	0.223144	0.00%
Z₇	7	0.182322	0.182322	0.00%
Z₈	8	0.154151	0.154151	0.00%
S₃ (nonabelian)	6	0.223144	0.223144	0.00%

The nonabelian group S₃ gives exactly the same spectrum as the abelian group of the same order—the mass gap depends only on |G|, independent of the group type.

The corollaries below are all direct consequences of the theorems above: Corollaries 1 and 2 follow from Theorem 1 (horizon and discrete applications of the area law), and Corollary 3 combines Corollary 1 with Theorem 3 (saturation coefficient).

5.5 Corollary 1 (Logarithmic correction to the black hole entropy)

Corollary 1: taking the horizon to be a simple closed surface (L edges (bonds)), Theorem 1 gives the horizon entropy

S_BH = (L−1)·ln8 = A/(4G) − ln8                         (5.5)

—an exact identity, with no higher-order correction terms: taking the horizon entropy in the Bekenstein–Hawking form [11,12] S = A/(4G) (here G is Newton's gravitational constant, to be distinguished from the notation for the charge space) is the external physical junction (belonging, together with the l_P input of Corollary 3, to the boundary of the single-weave zero-parameter principle); A/(4G) = L·ln8 is guaranteed exactly by the scale relation of the edge (bond), a_P² = 4G·ln8 (Corollary 3). The term −ln8 is the logarithmic correction, a testable prediction.

5.6 Corollary 2 (Entropy quantization)

Corollary 2: the bipartite entropy of the TV state takes only the discrete values (L−1)·ln8—the “charge-counting” structure of the entanglement entropy.

5.7 Corollary 3 (Physical scale of the edge (bond))

Corollary 3: the external Bekenstein–Hawking junction [11,12] S = A/(4G) of Corollary 1 and the horizon entropy S_BH = (L−1)·ln8 of Theorem 1 are exactly equal; under thermodynamic-limit saturation (Theorem 3) the leading term gives A/(4G) = L·ln8. Taking the horizon area A = L·a_P² (L edges (bonds), each of area a_P²) and substituting gives a_P² = 4G·ln8; in natural units (ħ = c = 1) Newton's constant is G = l_P², hence:

a_P = 2√(ln8)·l_P ≈ 2.884·l_P ≈ 4.66×10⁻³⁵ m            (5.6)

i.e. a_P² = 4·ln8·l_P². The dimensionless number 2√ln8 is determined entirely by the weave structure—ln8 comes from the quantum dimension of the edge (bond), D = 8, and the coefficient 4 from the horizon entropy formula—with no fitting parameters whatsoever. The weave outputs the ratio a_P/l_P; the absolute scale l_P is supplied by the external universe (the boundary of the single-weave zero-parameter principle).

5.8 Corollary 4 (Interlock of the gravitational constant G and the speed of light c)

Corollary: from the area law (Theorems 1 and 3) and the entropic-gravity junction [13], one obtains, on the weave side, the interlock relation between the gravitational constant G and the speed of light c.

Weave-side entropy density (area law S = (L−1)·lnD, horizon area A = L·a_P², area a_P² per edge (bond)):

η = c·lnD/a_P²                                          (5.7)

where c is the area-law saturation coefficient (approaching the thermodynamic saturation limit c_∞ = 1, Theorem 3). Equating this with the Bekenstein–Hawking entropy density η = 1/(4G) [11,12] (external physical junction, see Corollaries 1 and 3) gives

G = a_P²/(4c·lnD)                                       (5.8)

Natural units (l_P = 1, c = 1, a_P = 2√(lnD)) → G = 1. G is not an independent fundamental constant but a composite quantity of a_P.

The geometric identity of c (namely the speed of light): c itself is a purely geometric quantity—the saturation strength of the entropy–area coupling; as c approaches the thermodynamic saturation limit (c_∞ = 1) this geometric quantity becomes, in natural units, identical with the speed of light: that is, the upper-bound propagation in which information can traverse one a_P in unit time along any direction of the spacetime geometry.

Solving for c:

c = a_P²/(4G·lnD)                                       (5.9)

Natural units (l_P = 1, G = 1, a_P = 2√(lnD)) → c = 1.

Interlock: a_P (the physical scale of the edge (bond), Corollary 3) enters both the relation for G and that for c—G and c are locked by the same weave quantity a_P: once a_P is fixed, both G and c are fixed.

5.9 Corollary 5 (Constraint origin of entanglement)

Corollary: in the Weaving-Formula state |Ψ[W]⟩, the charge a_e ∈ Z₂³ of each leg is locked by the charge-conservation constraints of the vertices at its two ends (Constraint 1 of Definition 1): changing the charge of any one unit forces the charges of the remaining legs at the same vertex to change in step, and there is no allowed configuration in which a single unit changes while the others stay fixed. By Theorem 1, the Schmidt rank of a cut crossing this constraint is 8^(L−1) ≥ 8 > 1, so the two sides of the cut are necessarily entangled (the statement that the rank exceeds 1 requires only charge conservation: components with different boundary charges are mutually orthogonal). This nonlocal correlation, in which the state of one unit changes and the states of the other units change accordingly at once, is the mechanism of the phenomenon of quantum entanglement in the weave.

6. Numerical Verification

All the theorems of this paper have been verified by exact recomputation (with errors within machine precision). The relevant numbers can be reproduced by designing computer simulation experiments as described in the text:

1. Theorems 1 and 2 (area law f = L−1 and region intrinsicality f = E_B − F_B): on the three trivalent nets K4, the triangular prism and the cube, every cut was recomputed point by point, 18 data points in total; f = E_B − F_B and the Euler relation V_B − c₀(B) match in every case; the number of independent boundary charges was independently cross-checked by a rank computation over GF(2); f = L−1 is the special case of the complementary region (E_B = L, F_B = 1).
2. Theorem 3 (thermodynamic-limit saturation c_∞ = 1): the area-law coefficient c(L) = (L−1)/L is 2/3, 2/3 and 3/4 on the three nets, respectively, and tends monotonically to 1 as L increases; the entropy density obtained from the singular-value spectrum of the fixed-point tensor is exactly ln8.
3. Theorem 4 (universal mass-gap law): the mass gaps m* obtained from the fixed-point spectra of the groups Z₃, Z₅, Z₆, Z₇, Z₈ and S₃ agree with −ln((|G|−2)/(|G|−1)) to machine precision; the abelian group Z₆ and the nonabelian group S₃ of the same order give the same spectrum.
4. Ground state = Weaving-Formula state: the string-net Hamiltonian H(W) = −Σ_v A_v − Σ_p B_p is exactly diagonalized on K4 (4 vertices, 6 edges, 4 triangular faces, sphere) within its conserved subspace (dimension 512 = 8³), giving ground-state energy E₀ = −8 (= −(4 vertices + 4 faces)), a unique ground state (degeneracy 1 under the spherical topology), and a gap of 2 (a hallmark of topological order); the overlap of the ground state with the Weaving-Formula state |Ψ[W]⟩ (Definition 2) is 1.

The details of the numerical setup are as follows:

6.1 Weave tensor and fixed-point spectrum

Under the trivalent setting (§4.1 G2) the vertex tensor Tabc satisfies charge conservation δ{a⊕b⊕c,0}; coarse-graining two vertex tensors yields the four-leg tensor A[i,j,k,l], and charge conservation is preserved—it is nonzero only if i⊕j⊕k⊕l = 0, i.e. the charge of the fourth leg is determined by the first three (l = i⊕j⊕k). The fixed-point spectrum is given by the coarse-graining iteration of this tensor, and the mass gap is taken as the negative logarithm of the largest nonzero eigenvalue of the fixed-point spectrum.

6.2 Exact diagonalization

For K4 (4 vertices, 6 edges, 4 triangular faces, sphere) the conserved configurations are generated by the face constraints: the 6 edge charges are subject to 4 face constraints, and after subtracting 1 global redundancy there remain 3 independent constraints, so the dimension of the conserved subspace is 8³ = 512; dense exact diagonalization is then performed within this subspace.

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7. Boundaries

1. The minimality of the assumptions G1–G4 means “no redundancy inside the formula”; it does not answer “why Z₂³”—that choice is supplied by the upstream Taiji Evolution Cosmology [1];
2. The continuum limit (edge (bond) scale → continuum field theory) is not addressed in this paper—that is an open problem of weave dynamics.

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8. Conclusion

This paper turns the “weave” from a philosophical claim into mathematical theorems: a weave is a charged network W = (V, E, T); the Weaving Formula defines its quantum state; the assumptions G1–G4 are minimal (G = Z₂³ is the sole structural choice); the formula directly yields four fundamental theorems—the area law (f = L−1), region intrinsicality (f = E_B − FB), thermodynamic-limit saturation (c∞ = 1), and the universal mass-gap law (m* = −ln((|G|−2)/(|G|−1)), |G| ≥ 3)—all verified to exact numerical precision. Starting from the Weaving Formula, variant Weaving Formulas for gauge fields can in principle be derived, together with a unified derivation of the gravitational equation, Maxwell's equations, the Schrödinger equation and the Yang–Mills equations, and the geometric origin of the cosmological constant. We name the theory developed from the Weaving Formula the Weaving Theory.

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References

[1] Xinkuang Liu. Taiji Evolution Cosmology: A Philosophical Framework of Monistic Cosmology[Z]. Zenodo. DOI: 10.5281/zenodo.22669633.
[2] Michael A. Levin, Xiao-Gang Wen. String-net condensation: A physical mechanism for topological phases[J]. Physical Review B, 2005, 71(4): 045110. DOI: 10.1103/PhysRevB.71.045110.
[3] Vladimir G. Turaev, Oleg Y. Viro. State sum invariants of 3-manifolds and quantum 6j-symbols[J]. Topology, 1992, 31(4): 865–902. DOI: 10.1016/0040-9383(92)90015-A.
[4] Xiao-Gang Wen. Quantum order from string-net condensations and the origin of light and massless fermions[J]. Physical Review D, 2003, 68(6): 065003. DOI: 10.1103/PhysRevD.68.065003.
[5] Robbert Dijkgraaf, Edward Witten. Topological gauge theories and group cohomology[J]. Communications in Mathematical Physics, 1990, 129(2): 393–429. DOI: 10.1007/BF02096988.
[6] F. P. Bundy, J. S. Kasper. Hexagonal diamond—a new form of carbon[J]. The Journal of Chemical Physics, 1967, 46(9): 3437–3446. DOI: 10.1063/1.1841236.
[7] Alexei Kitaev, John Preskill. Topological entanglement entropy[J]. Physical Review Letters, 2006, 96(11): 110404. DOI: 10.1103/PhysRevLett.96.110404.
[8] Michael Levin, Xiao-Gang Wen. Detecting topological order in a ground state wave function[J]. Physical Review Letters, 2006, 96(11): 110405. DOI: 10.1103/PhysRevLett.96.110405.
[9] Michael Levin, Cody P. Nave. Tensor renormalization group approach to two-dimensional classical lattice models[J]. Physical Review Letters, 2007, 99(12): 120601. DOI: 10.1103/PhysRevLett.99.120601.
[10] Alexei Yu. Kitaev. Fault-tolerant quantum computation by anyons[J]. Annals of Physics, 2003, 303(1): 2–30. DOI: 10.1016/S0003-4916(02)00018-0.
[11] Jacob D. Bekenstein. Black holes and entropy[J]. Physical Review D, 1973, 7(8): 2333–2346. DOI: 10.1103/PhysRevD.7.2333.
[12] Stephen W. Hawking. Particle creation by black holes[J]. Communications in Mathematical Physics, 1975, 43(3): 199–220. DOI: 10.1007/BF02345020.
[13] Ted Jacobson. Thermodynamics of spacetime: The Einstein equation of state[J]. Physical Review Letters, 1995, 75(7): 1260–1263. DOI: 10.1103/PhysRevLett.75.1260.