Causal Diamond Chains and Null--Modular Double Cover\\ in Computational Universe:\\ Discrete Causal Structure, Topological Time Phase,\\ and Self-Referential Parity Under Unified Time Scale

Causal Diamond Chains and Null–Modular Double Cover in Computational Universe: Discrete Causal Structure, Topological Time Phase, and Self-Referential Parity Under Unified Time Scale Haobo Ma1Wenlin Zhang2 1Independent Researcher 2National University of Singapore November 24, 2025 Abstract Previously on computational universe axiomatic framework Ucomp = (X, T,C,I) we have successively constructed discrete complexity geometry, discrete information geometry, control manifold (M, G) induced by unified time scale, task information manifold (SQ, gQ), time–information–complexity joint variational principle, multiobserver consensus geometry, causal diamonds and boundary computation, as well as topological complexity and undecidability. On other hand, on physical universe side, small causal diamonds and Null–Modular double cover structure play key role in unified time scale–boundary time geometry: phase–delay–entropy on Null boundary has natural Z2parity and double cover structure, used to characterize time direction, energy conditions, and self-referential feedback networks. This paper constructs at computational universe level completely discrete “causal diamond chains and Null–Modular double cover” theory, and proves its isomorphism with causal diamond chains and Null–Modular structure on physical side in limit under unified time scale and complexity geometry. Specifically, we do following: 1. Define on event layer E=X×Ncomplexity causal partial order and finitebudget causal diamonds ♢(ein, eout;T), and formalize “diamond chains” as ordered families {♢k}k∈Zsatisfying appropriate overlap conditions in partial order. We prove this family under natural conditions constitutes directed graph–chain complex with boundary operators, whose 1-skeleton characterizes “discrete timeline” under unified time scale. 2. Introduce Null–Modular double cover on diamond chains: assign to each diamond boundary state Z2-valued “mod 2 time phase” label, and construct for diamond chain double cover graph e D→D, such that existence or nonexistence of lifted path for closed diamond chain corresponds to parity of Z2 holonomy. We prove this double cover consistent with previously introduced self-referential parity invariant σ(γ)∈Z2in topological complexity. 1 3. Introduce unified time scale into diamond chains: define on each diamond chain edge discrete time increment ∆τk, given by unified time scale density κ(ω) and local scattering phase derivative. We prove in refinement limit, time interval and Z2holonomy of diamond chain jointly define class of “time direction field with Null–Modular structure” on control manifold, thus connecting discrete Null–Modular double cover with continuous unified time scale. 4. In multi-observer causal network, embed each observer’s “diamond world tube” into diamond chain double cover, constructing multi-observer Null– Modular consensus structure, proving Z2parity transition in self-referential scattering network and multi-observer consensus geometry can be viewed as holonomy invariant on diamond chain double cover. 5. In context of topological complexity and undecidability, we prove: on general constructible computational universe families, deciding “whether given diamond chain closed loop can lift to closed path in Null–Modular double cover” is undecidable, thereby giving “Null–Modular version of halting problem”. Simultaneously, we construct “time phase–complexity second law” compatible with complexity entropy: under joint constraints of unified time scale and Null–Modular double cover, joint invariant composed of self-referential parity and compressible complexity has monotonic structure along diamond chain coarse–graining evolution. Through above construction, this paper unifies discrete causal diamond chains, unified time scale, topological self-referential parity, and complexity second law in computational universe into Null–Modular double cover framework, and gives in appendices detailed proofs of main structures and rigorous construction of chain complex. Keywords: Computational universe; Causal diamond; Null–Modular double cover; Z2 holonomy; Self-referential parity; Unified time scale; Second law of complexity 1 Introduction In unified time scale–boundary time geometry construction of physical universe, small causal diamonds, Null boundaries, and Null–Modular double cover structure are fundamental building blocks connecting scattering phase, group delay, generalized entropy, and time direction. Particularly, when examining series of nested or intersecting small causal diamonds, phase–delay data on boundaries has natural Z2parity structure: some closed diamond chains lift to closed paths on Null–Modular double cover, others produce “odd-parity jump”, leaving topological holonomy. This holonomy closely related to self-referential feedback networks, self-identity, and complexity second law. On other hand, in this series on “computational universe” works, we have already constructed in completely discrete abstract axiomatic framework:  Complexity graph Gcomp = (X, E, C) and complexity distance dcomp;  Task information manifold (SQ, gQ,ΦQ) and discrete information geometry;  Control manifold (M, G) induced by unified time scale and geodesic structure;  Time–information–complexity joint variational principle; 2  Causal diamonds and boundary computation operator K♢;  Multi-observer consensus geometry and causal network;  Topological complexity, self-referential loops, and Z2self-referential parity σ(γ). Natural question is: can we completely simulate on discrete causal structure of computational universe structure of “small causal diamond chains + Null–Modular double cover” from physical universe? If yes, is its topological Z2holonomy consistent with previous self-referential parity invariant? Does unified time scale naturally stratify on diamond chain into combination of “time step + parity transition”? How does self-referential feedback of multi-observers in causal network manifest in this structure? Answer of this paper is affirmative, we will show: 1. On event layer E=X×Nof computational universe, can define completely discrete “causal diamond chains”, whose chain complex and boundary operators naturally correspond to discrete time steps of unified time scale; 2. On diamond chains can construct Null–Modular double cover, whose Z2holonomy isomorphic to previous self-referential parity σ(γ); 3. Multi-observer world tubes can be viewed as family of “lifted paths” in diamond chain double cover, multi-observer consensus geometry and self-referential feedback network become geometric–topological structure on this double cover; 4. In this framework, can define “Null–Modular version of halting problem”, prove its undecidability, and simultaneously give “time phase–complexity second law” compatible with complexity entropy. Paper structure: Section 2 constructs event layer causal diamond chains and chain complex; Section 3 defines Null–Modular double cover and Z2holonomy on diamond chains; Section 4 connects unified time scale with diamond chains and double cover; Section 5 introduces multi-observer Null–Modular consensus geometry; Section 6 discusses Null–Modular version of halting problem and complexity second law; Appendices give detailed proofs of chain complex construction, double cover existence, holonomy–selfreferential parity correspondence, undecidability, and second law prototype. 2 Event Layer Causal Diamonds and Diamond Chains This section constructs causal diamonds and diamond chains on computational universe event layer, giving discrete chain complex structure. 2.1 Event Layer and Causal Partial Order Consider computational universe Ucomp = (X, T,C,I), event layer defined as E=X×N, e = (x, k). One-step update relation TE={((x, k),(y, k + 1)) : (x, y)∈T}. 3 Define causal reachability relation: for e, e′∈E e⪯e′ if there exists finite path Γ : e=e0→e1→ · · · → en=e′, where (ei, ei+1)∈TE. This is partial order (in reachable subset) on event layer. Merging steps and cost: define event layer complexity cost CE((x, k),(y, k + 1)) = C(x, y), path cost CE(Γ) = n−1 X i=0 CE(ei, ei+1), event layer complexity distance dE(e, e′) = inf Γ:e→e′ CE(Γ). 2.2 Causal Diamonds and Boundaries Given two events ein = (xin, kin), eout = (xout, kout), kout > kin, and complexity budget T > 0. Define causal diamond under budget T ♢(ein, eout;T) = J+ T(ein)∩J− T(eout), where J+ T(e) = {e′:e⪯e′, dE(e, e′)≤T}, J− T(e) = {e′:e′⪯e, dE(e′, e)≤T}. Denote V♢=♢as vertex set, edge set as E♢={(e, e′)∈TE:e, e′∈V♢}. Boundary defined as ∂♢={e∈V♢:∃e′/∈V♢,(e, e′)∈TEor (e′, e)∈TE}. Further decompose ∂−♢={e∈∂♢:∃e′/∈V♢,(e, e′)∈TE}, ∂+♢={e∈∂♢:∃e′/∈V♢,(e′, e)∈TE}. ∂−♢is incoming boundary, ∂+♢is outgoing boundary. 4 2.3 Diamond Chains and Chain Complex Structure Definition 2.1 (Causal Diamond Chain).A causal diamond chain is sequence {♢k}k∈Z, where each ♢kis causal diamond defined under some pair (ek, ek+1) and budget Tk, satisfying: 1. Time-ordering consistency: there exists event sequence {ek}k∈Zsuch that ek∈ ∂+♢k∩∂−♢k+1; 2. Complexity overlap: ♢k∩♢k+1 =∅, and ♢k∩♢k+l=∅for |l| ≥ 2; 3. Budget consistency: Tksatisfy appropriate upper bound, such that each diamond only covers finite time–complexity window. Viewing all ♢kas “basic elements” on 1-chain, defining 2-cells at their overlaps, can construct one-dimensional chain complex with 2-cells D=D({♢k}), whose 1-skeleton is diamond chain, 2-cells characterize local relations of diamond gluing. Proposition 2.2 (1-Skeleton of Diamond Chain and Discrete Timeline).If {♢k}satisfies above conditions, and under unified time scale each ♢kcorresponding time interval ∆τk has unified lower bound ∆τmin >0, then natural ordering k7→ ♢kon 1-skeleton can be viewed as discrete timeline, each single step corresponding to finite-budget causal diamond. Proof see Appendix A.1: core is using ekpartial order and budget overlap to construct local substitution for “same time layer”. 3 Null–Modular Double Cover and Z2Holonomy This section constructs Null–Modular double cover structure on diamond chains, defines Z2holonomy, and connects with self-referential parity invariant. 3.1 Mod 2 Time Phase on Diamond Boundaries Under unified time scale and scattering framework, each causal diamond ♢kcan be associated with local scattering operator S♢k(ω) and group delay matrix Q♢k(ω), whose trace gives unified time scale density increment on this diamond. Consider frequency interval Ωkand weight wk(ω), define diamond average phase increment ∆φk=ZΩk wk(ω)φ′ ♢k(ω) dω, and corresponding time increment ∆τk=ZΩk wk(ω)κ♢k(ω) dω. We introduce structure of “phase mod 2π” into Z2label. 5 Definition 3.1 (Mod 2 Time Phase Label).For each diamond ♢kon each event e∈∂+♢k at its outgoing boundary, assign label ϵ(e)∈Z2, defined as parity class of ∆φk/π: ϵ(e) = ∆φk πmod 2. On diamond chain, we require outgoing boundary label compatible with next diamond’s incoming boundary label, i.e., ϵ(ek) only depends on local structure of diamond ♢k. 3.2 Construction of Null–Modular Double Cover Definition 3.2 (Null–Modular Double Cover).Given diamond chain complex D, construct double cover graph e Das follows: 1. For each diamond vertex vk(representing ♢k) introduce two copies ev(0) k,ev(1) k; 2. For each chain edge (vk, vk+1), if corresponding phase parity ϵ(ek) = 0, then in double cover connect ev(i) kwith ev(i) k+1; if ϵ(ek) = 1, then connect ev(i) kwith ev(1−i) k+1 , where i∈ {0,1}; 3. Projection π:e D→Dmaps ev(i) k7→ vk. Thus, walking around diamond chain once, whether endpoint and starting point of lifted path on double cover are identical depends on accumulation of phase parity along way. Proposition 3.3 (Double Cover and Z2Holonomy).Let γ= (vk0, vk1, . . . , vkm=vk0)be closed loop on diamond chain, let eγbe its lifted path on double cover. Then: 1. If Pm−1 j=0 ϵ(ekj)≡0 mod 2, then there exists closed lifted path eγsuch that ev(i) km=ev(i) k0; 2. If Pm−1 j=0 ϵ(ekj)≡1 mod 2, then any lifted path starting from ev(i) k0ends at ev(1−i) k0, no closed lift exists. Therefore, Z2holonomy of closed loop given by phase parity sum Pϵ(ek), completely consistent with double cover structure. Proof see Appendix A.2. 3.3 Correspondence with Self-Referential Parity Invariant In previous topological complexity work, we defined for self-referential loop γself-referential parity σ(γ)∈Z2. Here, we can re-express self-referential loop on diamond chain as some closed diamond chain γ♢, whose self-referential operation corresponds to some local feedback structure on diamond chain. 6 Theorem 3.4 (Consistency of Self-Referential Parity and Null–Modular Holonomy). Under appropriate encoding, each self-referential loop γcorresponds to closed diamond chain loop γ♢, such that σ(γ) = X k∈γ♢ ϵ(ek) mod 2, i.e., self-referential parity equals Z2holonomy on diamond chain double cover. Proof see Appendix A.3: construct local scattering phase–time delay associated with self-referential feedback network, and translate into parity transition on diamond chain using Null–Modular double cover rules. 4 Implementation of Unified Time Scale on Diamond Chains This section introduces unified time scale density κ(ω) into diamond chain structure, and recovers continuous time parameter on control manifold in refinement limit. 4.1 Time Increment on Diamonds and Local Scattering For each diamond ♢k, assume its corresponding local scattering process S♢k(ω) satisfies unified time scale master formula: κ♢k(ω) = 1 2πtr Q♢k(ω), Q♢k(ω) = −iS♢k(ω)†∂ωS♢k(ω). Define diamond time increment ∆τk=ZΩk wk(ω)κ♢k(ω) dω. On diamond chain, cumulative time is τN= N−1 X k=0 ∆τk. 4.2 Limit from Diamond Chain to Control Manifold Worldline Consider discrete control parameter θk∈ M and diamond chain index k. Suppose there exists embedding map θ(k) = Θ(τk), where Θ : [τ0, τN]→ M is continuous curve on control manifold. If diamond size ∆τk→0 and chain becomes dense, then 1-skeleton of diamond chain converges in Gromov–Hausdorff sense to image of Θ. 7 Proposition 4.1 (Time–Geometric Limit of Diamond Chains).Under unified time scale and local scattering regularity assumptions: if diamond chain {♢k}satisfies supk∆τk→0, and each diamond on control manifold corresponds to one geodesic step in local coordinate neighborhood, then complexity distance of diamond chain converges in limit to control manifold geodesic distance dG, time parameter τis unified time scale parameter. Proof see Appendix B.1: using standard discrete–continuous geodesic approximation and scattering–group delay relation of unified time scale. 4.3 Time Direction Field Under Null–Modular Double Cover Combining Null–Modular double cover structure, time parameter τtogether with selfreferential parity σdefine “time direction field” on control manifold:  Advancing along Θ(τ), corresponds to diamond chain {♢k};  On double cover e D, whether path lift returns or flips after going around once defines Z2time parity;  This parity structure can be viewed as Z2principal bundle holonomy on control manifold, consistent with Null–Modular double cover structure. In physical universe unified time scale–boundary time geometry language, this corresponds to comprehensive structure of mod 2πphase and Z2module on Null boundary. This paper gives its discrete–chain complex realization on computational universe side. 5 Multi-Observer Null–Modular Consensus Geometry This section embeds multi-observer consensus geometry into diamond chain double cover, constructing multi-observer Null–Modular consensus structure. 5.1 Multi-Observer World Tubes and Diamond Chain Embedding For observer family {Oi}i∈I, each observer Oihas “world tube” {(i, x(i) k, k)}kon event layer, corresponding to set of diamond chains embedded in event layer {♢(i) k}k, e.g., each diamond covers from (i, x(i) k, k) to (i, x(i) k+1, k + 1). These diamond chains can be viewed as subfamily of diamond chain complex D, each observer has lifted path eγion double cover e D. Definition 5.1 (Multi-Observer Null–Modular Consensus Graph).Define on diamond chain double cover graph CNM = (VNM, ENM), where vertex set VNM is all observer lifted path diamond chain nodes {ev(i) k}, edge set is combination of “space–time–information adjacency”: including both time-direction chain edges and consensus edges on information manifold. 8 On this graph, can define quantity similar to previous multi-observer consensus energy, except here each path also carries Z2holonomy, representing accumulation of selfreferential parity on observation chain. 5.2 Null–Modular Consensus and Self-Referential Parity Alignment In multi-observer scenario, different observers may have different self-referential structures: some observers’ internal models self-flip after one time period, others do not flip. On Null–Modular double cover, this corresponds to whether their lifted paths close or flip. Proposition 5.2 (Null–Modular Consensus Alignment Condition).If there exists multiobserver collaborative strategy such that in long-term limit t→ ∞ 1. All observers’ information states ϕi(t)converge on SQto same point or same orbit; 2. All observers’ lifted path Z2holonomies are same, i.e., σ(γi) = σ(γj)for all i, j; then on diamond chain double cover, multi-observer world tubes constitute “Null– Modular consensus cluster”, whose overall holonomy is common Z2value, representing unified self-referential parity under this task. This condition gives topological–geometric level “deep consensus” concept: not only information states reach consensus, self-referential structures also reach agreement. 6 Null–Modular Version of Halting Problem and Time Phase–Complexity Second Law This section defines Null–Modular version of halting problem based on previous structures, and gives second law prototype compatible with complexity entropy. 6.1 Null–Modular Version of Halting Problem Consider diamond chain closed loop γ, whose self-referential parity σ(γ)∈Z2and fundamental group homotopy class [γ]∈π1(D) already defined. Question we care about is: Given γ, does it have closed lifted path on Null–Modular double cover e D? This equivalent to σ(γ) = 0, i.e., holonomy is trivial element. Problem 6.1 (Null–Modular Halting Decision Problem). Input: Finite description of diamond chain complex Dand closed loop γin it. Question: Decide whether γhas closed lifted path on Null–Modular double cover e D. Using topological undecidability results from Section 4, we can prove this problem undecidable in general computational universe families: can encode halting problem as closure of certain class of self-referential diamond chains and their holonomy parity, thereby reducing halting to Null–Modular halting decision. 9