Chapter 12 — Three Computational Axes

Prerequisites: Chapter 11 (Cech framework), Chapter 6 (Door).

12.1 Overview#

Once the world's constitution is fixed, invariants are extracted via three computational axes:

The sequential protocol extends:

Each axis depends only on the preceding ones.

12.2 Axis 1: Fruit Topology $H^{\ast }(K_{\alpha };A)$#

Definition 12.1 (Fruit complex)#

Given fruit $F\in {\mathfrak{F}}_t$, construct the clique complex $K_{\alpha }$ of the induced graph ${\mathcal{G}}_t[F]$: a $k$-simplex for each $(k+1)$-clique.

Theorem 12.1 (Finiteness of the nerve)#

$\dim (\mathrm{Nerve}(K_{\alpha }))\le |F|-1$.

Proof. The maximum clique size in $F$ is at most $|F|$. $\square$

Computation#

where $D_K^k=(B_{k+1}^K{)}^T$ is the coboundary matrix built from the boundary matrices of $K_{\alpha }$.

Lemma 12.2 ($U(1)$ case)#

For $G=U(1)$ and abelian coefficients $A=\mathbb{Z}$: ${\check{H}}^{\ast }(K_{\alpha };U(1))\cong \mathrm{Hom}(H_{\ast }(K_{\alpha };\mathbb{Z}),U(1))$ by the universal coefficient theorem.

12.3 Axis 2: Pair Cohomology $H^{\ast }(K_{\alpha },L_{\alpha };A)$#

Definition 12.2 (Simplicial pair)#

• $K_{\alpha }$: the fruit complex (Definition 12.1).
• $L_{\alpha }\subset K_{\alpha }$: the door subcomplex, generated by all simplices containing at least one door node.

Theorem 12.3 (Long exact sequence of a pair)#

For each degree $k$, there is a long exact sequence:

Proof. At cochain level, the short exact sequence

holds, where $C^k(K_{\alpha },L_{\alpha };A):=\{\phi \in C^k(K_{\alpha };A):\phi {|}_{C_k(L_{\alpha })}=0\}$.

The coboundary operators commute with inclusion $J_k$ and restriction $R_k$: $D_K^k\circ J_k=J_{k+1}\circ D_{\mathrm{rel}}^k$ and $R_{k+1}\circ D_K^k=D_L^k\circ R_k$. The snake lemma yields the long exact sequence. $\square$

Theorem 12.4 (Matrix computation of relative cohomology)#

For the relative coboundary matrix $D_{\mathrm{rel}}^k=P_{k+1}{D}_K^k{I}_k$ (where $I_k$ injects relative basis into $C^k(K)$ and $P_{k+1}$ projects to relative basis):

Proof. Rank--nullity theorem on finite-dimensional cochain complexes. $\square$

Lemma 12.5 (Bridge to removed-space cohomology)#

Under Hypothesis 11.1 (continuous enrichment), excision and retraction give:

This separates topological assumptions from algebraic computation: the computation is performed on the finite pair $(K_{\alpha },L_{\alpha })$; the geometric interpretation is optional.

Algorithm 12.1 (Axis 2 computation)#

Input: fruit complex $K_{\alpha }$, door subcomplex $L_{\alpha }$.

Output: $\dim H^k(K_{\alpha },L_{\alpha };A)$ for each $k$.

1. Build simplex bases for $K_{\alpha }$ and $L_{\alpha }$ by degree.
2. Construct boundary matrices $B_k^K$, $B_k^L$ and coboundary matrices $D_K^k$, $D_L^k$.
3. Build relative coboundary: $D_{\mathrm{rel}}^k=P_{k+1}{D}_K^k{I}_k$.
4. Compute $\dim H^k$ by rank/nullity.
5. Verify cochain SES: $R_k{J}_k=0$, $\mathrm{im}(J_k)=\ker (R_k)$, $R_k$ surjective.
6. Connecting map: for $[{\alpha }_L]\in H^k(L)$, extend to $\tilde{\alpha }\in C^k(K)$ with $R_k\tilde{\alpha }={\alpha }_L$; then $\delta ([{\alpha }_L])=[D_K^k\tilde{\alpha }]\in H^{k+1}(K,L)$.

12.4 Axis 3: Door Set Topology + Persistence Barcode#

Definition 12.3 (Persistence barcode)#

The barcode of the door set $\Sigma$ is the persistence diagram of the energy-filtered simplicial complex:

where $a_i$ is the birth energy (when a new cohomological feature appears) and $b_i$ is the death energy.

Construction#

1. Order doors by energy: $e_{p_1}\le e_{p_2}\le \cdots \le e_{p_L}$.
2. Build filtration: $L_0\subset L_1\subset \cdots \subset L_L=L_{\alpha }$ by adding door nodes in energy order.
3. Compute persistent cohomology across the filtration.

Theorem 12.5 (Persistence--cohomology correspondence)#

Proof sketch. Standard result in persistent cohomology: the number of infinite bars in degree $k$ equals the Betti number ${\beta }_k(L_{\alpha })$. $\square$

Definition 12.6 (Hausdorff dimension of the barcode)#

where ${\ell }_i=b_i-a_i$ is the lifespan of bar $i$.

• ${\dim }_H=0$: isolated doors (finitely many, finite barcode).
• $0<{\dim }_H<1$: fractal door structure.
• ${\dim }_H=1$: maximally complex.

Theorem 12.7 (Finite door set, $U(1)$)#

If $\Sigma =\{p_1,\dots ,p_n\}$ is finite and $A=U(1)$:

Proof. A finite discrete set has trivial higher cohomology. $\square$

12.5 Integration: The Three-Axis Exact Sequence#

Theorem 12.8 (Three-axis exact sequence)#

For each degree $k$, the cochain short exact sequence

induces the long exact sequence in cohomology:

This is the rigorous relationship among the three axes. $\square$

Interpretation: The three axes are not independent; they are linked by the connecting homomorphism $\delta$. Knowing any two determines constraints on the third.

12.6 Worked Example: 5-Node Graph, $U(1)$#

(Continuing from Chapter 10, Section 10.2.)

Fruit $F=\{4,5\}$, door $\Sigma =\{4\}$, kernel $F^{\circ }=\{5\}$.

Fruit complex $K_{\alpha }$: edge $(4,5)$ gives a 1-simplex. $K_{\alpha }=\{[4],[5],[4,5]\}$.

Door subcomplex $L_{\alpha }$: vertex $\{4\}$. $L_{\alpha }=\{[4]\}$.

Axis 1: $H^0(K_{\alpha };\mathbb{Z})=\mathbb{Z}$ (connected), $H^{k>0}=0$ (contractible).

Axis 2: $C^0(K,L)=\mathbb{Z}$ (generated by $[5]$), $C^1(K,L)=\mathbb{Z}$ (generated by $[4,5]$). Relative coboundary $D_{\mathrm{rel}}^0:[5]\mapsto [4,5]$ is an isomorphism. So $H^0(K,L)=0$, $H^1(K,L)=0$.

Axis 3: $H^0(L;\mathbb{Z})=\mathbb{Z}$ (single point), $H^{k>0}=0$.

Barcode: single bar $[0,e_4)=[0,1)$, ${\dim }_H=0$.

Exact sequence check ($k=0$):

Consistent. The map $H^0(K)\to H^0(L)$ is the restriction (isomorphism since $K$ is connected and $L$ is a single point). $✓$

12.7 Computation Protocol (Meta-Algorithm)#

function ComputeWorldAxes(F, Sigma, [A_infty], A):
    // Build complexes
    K = CliqueComplex(InducedGraph(F))
    L = DoorSubcomplex(K, Sigma)
 
    // Axis 1
    H1 = SimplicialCohomology(K, A)
 
    // Axis 2
    H2 = RelativeCohomology(K, L, A)
 
    // Axis 3
    barcode = PersistentCohomology(L, energy_filtration, A)
    H3 = CohomologyFromBarcode(barcode)
    dim_H = HausdorffDimension(barcode)
 
    // Verify exact sequence
    assert LongExactConsistency(H1, H2, H3, delta)
 
    return (H1, H2, H3, barcode, dim_H)

Complexity: $O(|F{|}^3)$ (dominated by matrix rank computations on the simplicial pair).