The Zero-Point Cut and the Generation of Time: A Formal SGCV-MC-Infinity Algebra Framework

The Zero-Point Cut and the Generation of Time A Formal SGCV–MC–Infinity Algebra Framework Antonios Valamontes Kapodistrian Academy of Science December 2025 Abstract The present moment is traditionally modeled as an infinitesimal coordinate on a pre-existing temporal axis. This work advances a complementary formulation in which the present is an operator—the zero-point cut—that generates temporal sequence rather than occupying a location within it. The construction is developed independently within three frameworks: the Superluminal Graviton Condensate Vacuum (SGCV), the Multifaceted Coherence (MC) continuum field, and the graded morphism structure of Infinity Algebra. Each yields an operator with four invariant properties: absence of temporal interior, intrinsic irreversibility, non-observability, and monotonic admissible evolution. A non-observability theorem and an irreversibility proposition are proved for each framework, and a cross-framework equivalence result demonstrates that these invariants arise universally in models of generative temporal dynamics. The analysis further shows that spacetime geometry emerges as the minimal structure required for consistency of the zero-point operator algebra. The results collectively identify the zero-point cut as a foundational pregeometric element underlying the generation of time. Keywords: zero-point cut; temporal generation; pregeometry; Superluminal Graviton Condensate Vacuum (SGCV); Multifaceted Coherence (MC); Infinity Algebra; irreversibility; non-observability; generative operators; operator algebras; emergent spacetime; causal structure; foundational physics. 1 1 Introduction In conventional physical theories, time is modeled as a pre-given parameter t∈R, and the “present” is represented as the evaluation of dynamical fields at an infinitesimal coordinate value t0. This view assumes that the temporal axis exists independently of the processes that occur along it. In contrast, the present work advances the thesis that the present is not a coordinate on a background temporal continuum, but the operator responsible for generating temporal order. The present is formalized as a zero-point cut, an operator that maps unrealized or pre-physical configurations into realized physical states. Let Hdenote the total configuration space associated with a pre-physical vacuum structure, and let Sdenote the space of physically instantiated states. A zero-point cut is an operator of the form Zx,t :H −→ S,Zx,t ≡“present at (x, t)”.(1) The key proposal of this paper is that time emerges as the ordered family {Zx,t}t≥0, rather than the operators emerging from time. Hence, the present is not inside time; rather, time is inside the sequence of present-moment cuts. To make this construction explicit, we examine the zero-point operator in three independent theoretical frameworks: (i) the Superluminal Graviton Condensate Vacuum (SGCV), in which Zx,t acts as a localized slicing of an underlying pre-geometric vacuum state; (ii) the Multifaceted Coherence (MC) field, where Zx,t is the infinitesimal update of a continuous coherence functional C(x, t); (iii) Infinity Algebra, where Zx,t is realized as an irreversible positive-grade ∞-morphism δt: F(t)→ F(t+dt). Although the internal mathematical machinery of these three frameworks differs, each yields a structure exhibiting the following invariant properties: 1. No temporal interior. The operator Zx,t has no definable interval of duration. It corresponds to a morphism without an associated measurable segment: Dom(Zx,t)∩Dom(Zx,t+dt)=∅.(2) 2. No invertible morphism. In all frameworks, an inverse operator Z−1 x,t is structurally forbidden. The inability to reverse Zx,t establishes an intrinsic arrow of time. 3. No direct observability. Any measurement functional Mrequires positive temporal extension, ∆t>0, and hence cannot act on the limit ∆t→0+where Zx,t resides. 4. Monotonicity. The composition law Zx,t2◦ Zx,t1is defined only for t2> t1,(3) induces a natural order Zt1≺ Zt2, which is not imposed externally but arises from the operator structure itself. 2 The appearance of the same four structural features across three independent theoretical environments suggests that the zero-point cut is not an artifact of a particular model but an invariant operator underlying the generation of temporal order. This provides a unified mathematical explanation for why the present cannot be observed, reversed, subdivided, or halted: it is not a temporal object but a generative morphism that gives rise to the possibility of temporal objects in the first place. 2 Zero-Point Cut in SGCV The SGCV framework posits a pre-geometric vacuum state |ΩSGCV⟩characterized not by metric structure but by an undifferentiated graviton-condensate amplitude. Physical spacetime events arise only after a local extraction of structure from this vacuum. Accordingly, a realized configuration at (x, t)is produced by a vacuum-slicing operator that selects one coherent component of |ΩSGCV⟩ and maps it into the physical state sector. Let HΩdenote the pre-physical SGCV Hilbert space and Sthe space of realized field configurations. We consider a localization functional Λx,t :HΩ−→ C(4) which extracts vacuum amplitudes compatible with the event (x, t). Definition 2.1 (SGCV Cut Operator).A zero-point cut at (x, t)is the operator Kx,t =ι◦Λx,t,(5) where Λx,t :|ΩSGCV⟩7→λx,t ∈Cis the localization functional and ι:C→ S embeds the extracted amplitude into a realized configuration |Ψ(x, t)⟩=ι(λx,t).(6) The operator Kx,t has no definable temporal interior: there exists no ∆t>0for which Kx,t decomposes into suboperators localized in [t−∆t, t + ∆t]. The lack of temporal interior reflects the fact that the SGCV cut corresponds to the boundary between unrealized and realized structure, not an evolution within realized time. Locality and Orthogonality of Successive Cuts The vacuum-slicing operations at different times must satisfy an orthogonality condition: Λx,t1⊥Λx,t2, t2> t1,(7) which follows from the requirement that no two cuts act on overlapping temporal domains. This orthogonality ensures that: Kx,t2◦ Kx,t1is defined only if t2> t1.(8) Thus the order of cuts induces a canonical temporal ordering without assuming a pre-existing temporal axis. 3 Observational Constraints A measurement functional requires nonzero duration. Let Mdenote an admissible SGCV measurement defined on realized state histories: M[{|Ψ(x, t)⟩}t∈[a,b]].(9) For Mto be defined, one requires b−a>0. An isolated zero-point cut Kx,t corresponds to a=b=t, for which no measurement functional exists. Thus no observer—classical or quantum—can access the zero-point operation directly; only sequences of cuts can be observed. Irreversibility We now formalize the irreversibility of the SGCV cut. Proposition 2.1 (SGCV Irreversibility).There exists no operator K−1 x,t satisfying K−1 x,t ◦ Kx,t = idHΩor Kx,t ◦ K−1 x,t = idS.(10) Hence the SGCV cut enforces an intrinsic forward direction of time. Proof. Assume an inverse exists. Then for any realized configuration |Ψ(x, t)⟩∈S, |ΩSGCV⟩=K−1 x,t |Ψ(x, t)⟩.(11) But |Ψ(x, t)⟩results from the evaluation of the localization functional Λx,t, which projects onto a single complex amplitude λx,t. Projection maps are non-invertible: Λ−1 x,t (λx,t)contains a full preimage subspace. (12) Thus no left-inverse exists. Similarly, the embedding ι:C→ S is not surjective, so no right-inverse can exist. Consequently, no operator K−1 x,t can satisfy the stated conditions. The irreversible nature of the SGCV cut is therefore structural and not the result of entropy considerations, thermodynamic arguments, or informational loss. The temporal asymmetry is a direct consequence of the projection Λx,t and the embedding ιthat define the cut operator. 3 Zero-Point Cut in Multifaceted Coherence In the MC framework, physical evolution is governed by a coherence field C:M×R≥0−→ R,(x, t)7→ C(x, t),(13) defined on spacetime manifold M. Unlike the SGCV vacuum, which is pre-geometric, the MC field presupposes a continuum structure but not a predetermined temporal axis. Temporal ordering arises from the manner in which Cupdates infinitesimally along the time parameter. 4 Let MMC denote the MC configuration space. A temporal update law is specified by a connectionlike operator Dt:MMC −→ TMMC,(14) encoding how the MC field changes across an infinitesimal step dt. Infinitesimal Dynamics For fixed spatial coordinate x, the temporal evolution of the MC field is ∂C ∂t (x, t) = DtC(x, t),(15) interpreted as the projection of the MC tangent vector along the temporal direction. The zero-point cut corresponds to the value of this derivative in the limit as the interval shrinks to zero. Definition 3.1 (MC Zero-Point Update Operator).The zero-point update of the MC field at (x, t0) is the limit ZMC x,t0= lim ∆t→0+ C(x, t0+ ∆t)− C(x, t0) ∆t,(16) provided the limit exists. Equivalently, ZMC x,t0=DtC(x, t0).(17) As in the SGCV formalism, ZMC x,t0is not a field value but a temporal morphism that cannot be subdivided into smaller temporal components. Temporal Interior and Locality Consider a small interval [t0, t0+ε]. We may integrate the MC field along this interval: C[x;t0, t0+ε] = Zt0+ε t0 C(x, t)dt. (18) The MC update operator satisfies a locality condition: ZMC x,t0depends only on lim ε→0+C(x, t0+ε),(19) not on any finite interval. Thus the zero-point update has no temporal interior in the MC sense. Measurement Functionals and Temporal Width Any admissible measurement of the MC field must evaluate a nonzero interval. Let M:MMC −→ R(20) be such a measurement functional. To be physically realizable, Mmust satisfy: 5 M[C]=FZt0+ε t0 W(t)C(x, t)dt, ε > 0,(21) where W(t)is a windowing kernel with nonzero support. Thus measurement requires an interval of positive length: no measurement functional is defined at ε= 0. Non-Observability as a Structural Theorem The above considerations lead to Theorem 3.1 (Zero-Point Non-Observability in MC).No admissible measurement functional M can determine the value of ZMC x,t0exactly. In particular, M[C]=M[C′]whenever C(x, t)=C′(x, t)for all t > t0,(22) even if ∂tC(x, t0)=∂tC′(x, t0). Proof. All admissible Mhave nonzero temporal support. Thus M[C]=FZt0+ε t0 W(t)C(x, t)dt, ε > 0.(23) Replacing Cby another field C′with identical values for t>t0but different right-derivative at t0 does not change the integral. Hence no measurement can isolate or infer the limit lim ∆t→0+ C(x, t0+ ∆t)− C(x, t0) ∆t.(24) Thus ZMC x,t0is not a measurable quantity. Irreversibility of MC Zero-Point Dynamics The MC update operator is inherently direction-selective. Suppose a reverse update existed: R:C(x, t0)−→ C(x, t0−dt).(25) For this to be well-defined, one would require: ∂tC(x, t0−dt)=−ZMC x,t0.(26) But the MC framework imposes forward-coherence monotonicity: C(x, t2)≥ C(x, t1)for t2> t1.(27) Thus negative temporal derivatives violate MC admissibility, and no such Rcan exist. This establishes that MC evolution contains an intrinsic arrow of time, arising from the structure of the update operator itself rather than from entropy or statistical arguments. 6 4 Zero-Point Cut in Infinity Algebra In the Infinity Algebra (IA) framework, physical configuration space is modeled as a graded manifold F(t) = M n∈Z Fn(t),(28) where each Fn(t)is a grade-ncomponent encoding nth-order interactions, flows, or coherence structures. Temporal evolution corresponds to morphisms between successive graded manifolds: Mor∞F(t),F(t+dt).(29) Let Gdenote the graded morphism algebra, with decomposition G=G+⊕G0⊕G−,(30) where G+contains positive-grade forward morphisms, G0contains grade-preserving symmetries, and G−contains negative-grade transformations corresponding to reverse-time or reverse-coherence operations. Only G+morphisms are physically admissible. Temporal Evolution in the Infinity Framework A temporal update at time tis represented by an ∞-morphism δt:F(t)→ F(t+dt),(31) which decomposes as δt= ∞ X n=0 δ(n) t,(32) where δ(n) tmaps k-tuples of fields into the grade ncomponent and satisfies the full ∞-algebra coherence relations. Definition 4.1 (Temporal ∞-Morphism / Zero-Point Cut).The zero-point cut in the Infinity Algebra framework is the unique forward-directed morphism ZIA t≡δt∈G+, δt:F(t)→ F(t+dt),(33) with no admissible inverse morphism in G. This operator has no temporal interior: there exists no decomposition δt=δt+ϵ◦δt−ϵ(34) for any ϵ > 0that remains inside G+. 7 Positive Temporal Cone and Admissibility The space of forward morphisms forms a pointed cone: G+={γ∈G: grade(γ)>0},(35) with the property γ2◦γ1∈G+⇔γ1, γ2∈G+.(36) Backward evolution would require elements of G−: γ−1∈G−={γ∈G: grade(γ)<0}.(37) Such morphisms violate IA admissibility and cannot be physically realized. Non-Factorizability of the Zero-Point Cut The operator δtcannot be factored into smaller temporal components: δt=γt2◦γt1, t1< t < t2,(38) unless both γt1and γt2are trivial (grade 0symmetries). Proposition 4.1 (Non-Factorization Lemma).If δtis a nontrivial temporal ∞-morphism, then there do not exist γt1, γt2∈G+such that δt=γt2◦γt1, t1< t < t2.(39) Proof. Assume such a factorization exists. Then grade(δt) = grade(γt2) + grade(γt1).(40) Since δtcorresponds to an infinitesimal step dt, its grade must be minimal among positive-grade morphisms. Any nontrivial factorization into two positive-grade morphisms yields strictly greater total grade: grade(γt2) + grade(γt1)>grade(δt),(41) contradiction. Thus the factorization cannot occur. This mirrors the “no temporal interior” property already noted in SGCV and MC. Irreversibility of Temporal Morphisms We now formalize the impossibility of reversing time evolution in the IA framework. Proposition 4.2 (Morphismic Irreversibility).If δt∈G+is a physically admissible temporal ∞- morphism, then no inverse morphism δ−1 t∈Gexists such that δ−1 t◦δt= id, δt◦δ−1 t= id.(42) 8 Proof. An inverse morphism would require δ−1 t∈G−,(43) since reversing time corresponds to a negative temporal grade. But G−contains no admissible physical morphisms. Furthermore, the composition δ−1 t◦δtmixes positive and negative grades, violating the coherence constraints of the ∞-algebra: grade(δ−1 t◦δt) = grade(δ−1 t) + grade(δt)<0+ϵ. (44) No such morphism corresponds to a physical transformation. Therefore, no inverse δ−1 texists. Irreversibility is thus not a statistical phenomenon but a structural property of the graded morphism space itself. 5 Synthesis Across Frameworks Although the SGCV, MC, and Infinity Algebra frameworks originate from distinct mathematical and physical considerations, each produces a zero-point structure with the same invariants. We now show that the SGCV cut operator Kx,t, the MC update operator ZMC x,t , and the IA temporal ∞-morphism δtbelong to a single equivalence class of generative temporal operators. Common Structural Invariants Across all frameworks, the zero-point cut satisfies: 1. Absence of temporal interior. ∄ϵ > 0 : Zx,t =Zx,t+ϵ◦ Zx,t−ϵ.(45) No decomposition into sub-intervals is admissible. 2. Irreversibility. Z−1 x,t does not exist in the admissible operator class.(46) In SGCV this follows from the projection–embedding structure; in MC from coherence monotonicity; in IA from the positivity of temporal grades. 3. Non-observability. Measurement requires ∆t > 0, whereas Zx,t = lim ∆t→0+update over ∆t(47) has zero measure. 4. Monotonicity and temporal ordering. Zx,t2◦ Zx,t1is admissible iff t2> t1.(48) The ordered family of cuts induces the forward arrow of time. These invariants emerge independently in the three frameworks, suggesting that they arise from a deeper structural constraint rather than from the specifics of each model. 9 A.6 Generation of Temporal Continuity The temporal axis is recovered as the closure of the operator family: R≥0={t:Zx,t ∈Z}.(64) Hence continuity of time emerges from the density of the zero-point cuts. 16