We propose a minimal pre-bang framework in which physical reality emerges from a non-integrable null prestate. The inability of the null to admit a global measure necessitates the formation of a zero-dimensional relational construct (the “bridge”), whose dynamics consist of a bounded, non-convergent recursion. This recursion defines a primitive ordering parameter identified with fundamental time. Spacetime and fields arise as projections of successive bridge states, while experienced temporal ordering (“mass-time”) emerges only in subsystems capable of retaining records across recursions. The framework is structurally compatible with General Relativity, Quantum Mechanics, and Quantum Field Theory, addressing the problem of initial conditions and the origin of time without invoking a temporal singularity.
Definition 1 (Non-integrable Null)
Let $\mathcal{N}$ denote a pre-physical null prestate. We assume:
This does not mean $\mathcal{N}=0$. It means $\mathcal{N}$ cannot be globally integrated or normalized within a standard measure-theoretic framework.
Interpretation: the null cannot be treated as a closed totality. This assumption is the only ontological primitive of the model.
Definition 2 (Opposed Subsets)
We assume the null admits a decomposition into mutually opposed, unresolved subsets:
where $A$ and $\bar{A}$ are not complements in a Boolean sense, but mutually defining oppositions whose joint totality fails to close.
Definition 3 (Bridge State)
Define the bridge $\mathcal{B}$ as a minimal relational construct encoding the unresolved relation between $A$ and $\bar{A}$:
The bridge has no internal spatial extent and is therefore zero-dimensional in the geometric sense. It is not embedded in space; it precedes spatial structure.
Assumption 1 (Recursive Update)
We postulate a deterministic update operator:
acting on the space of admissible bridge states $\mathcal{B}$, such that:
Assumption 2 (No Fixed Point, Bounded Orbit)
We require:
and the orbit $\{B_n\}$ is bounded.
This implies the existence of a limit cycle or quasi-periodic orbit in $\mathcal{B}$. This condition can be satisfied by compact-state dynamics (e.g. rotations on $S^1$) and does not require chaos or divergence.
Definition 4 (Primitive Time)
Define real time $t_R$ as the recursion index:
No metric, duration, or continuum structure is assumed at this level. Time is an ordering parameter only.
The recursive dynamics of the bridge state $(B)$, defined by
admit no fixed point but are assumed to be bounded. Under these conditions, the system necessarily possesses an invariant associated with recurrence under iteration. We identify this invariant as phase.
Let $\{B_n\}$ denote the orbit of the bridge under recursive update. Define an equivalence relation:
Phase $(\phi)$ is defined as the label of the equivalence class of bridge states under this relation. Formally,
Phase therefore represents ordering modulo recurrence. It is not a spatial angle, not a temporal duration, and not an energetic quantity. It is the minimal invariant distinguishing successive recursive states in a system lacking convergence.
In a bounded recursive system with no fixed point, distinguishability between successive states requires a nontrivial invariant. Without such an invariant, the recursion would be observationally inert. Phase is the minimal structure sufficient to satisfy this requirement.
Phase arises prior to any projection into spacetime and does not depend on geometric, metric, or energetic assumptions. Its existence follows directly from the coexistence of boundedness and non-convergence in the bridge dynamics.
The phase space associated with bridge recursion must be compact, continuous, and without boundary, while supporting orientation (i.e., an ordering of states). The minimal manifold satisfying these conditions is the circle $(S^1)$.
Consequently, closure of phase is expressed as:
The appearance of $(\pi)$ here is not geometric but topological: it is the normalization constant required to represent closure of a continuous one-dimensional cycle. The irrationality of $(\pi)$ reflects the fact that such closure cannot be finitely represented using linear measure, despite being topologically complete.
Dimensional physical quantities arise only after projection introduces scale. In particular, when spacetime duration $(t)$ and energy scale $(E)$ become meaningful, phase may be expressed as:
where $(S)$ is action and $(\hbar)$ serves as a conversion constant preserving phase invariance. In this framework, phase is primary, while action and $(\hbar)$ are emergent constructs tied to projection-dependent units.
Phase constitutes the minimal structure required for distinguishability between recursive states. A physical event occurs when a change in phase produces a difference in projected observables. Without phase, neither events nor memory-bearing structures could arise.
The existence of phase as an invariant of bounded, non-convergent bridge recursion (§3.1) is not by itself sufficient to support coherent projected structure. In particular, if phase is permitted to vary arbitrarily between successive recursion steps, then no notion of adjacency, persistence, or stable projection can be defined. A further consistency condition is therefore required: phase variation must be bounded.
Let $\phi_n := \phi(B_n)$ denote the phase associated with the bridge state $B_n$. We impose a bound on the per-step phase change:
where $|\cdot|_{S^1}$ denotes the minimal arc distance on $S^1$, and $\Phi_{\max}\in(0,\pi]$ is a finite constant. This constraint is pre-geometric: it does not presuppose spatial distance, duration, or energetic scale. It is a statement solely about admissible transitions in the recursive state evolution.
Without a finite bound $\Phi_{\max}$, the recursion permits arbitrarily large phase discontinuities between adjacent recursion steps. In that regime:
Thus bounded phase variation is required for the existence of coherent projected dynamics and for the subsequent emergence of persistent structures.
The bound induces an adjacency structure on the orbit of bridge states. Define a neighborhood relation $\mathcal{N}_\phi$ by:
This provides a minimal notion of locality in the pre-geometric regime: permissible transitions are restricted to a finite neighborhood in phase.
Once projection introduces extended configurations $\Sigma_n$, bounded phase variation lifts to a constraint on how rapidly projected observables can change. Specifically, for any observable functional $\mathcal{O}:\mathcal{S}\to\mathbb{R}$, bounded phase variation implies the existence of finite bounds of the form:
for some observable-dependent constant $K_{\mathcal{O}}$, whenever $\mathcal{O}$ depends continuously on $\phi$ under the projection map. This establishes the minimal precondition for propagation.
In later regimes where the projection admits effective spatial and temporal metric structure, bounded phase variation corresponds to finite bounds on phase gradients with respect to emergent coordinates:
When phase is expressed in terms of action via $\phi=S/\hbar$ (§3.1e), such gradient bounds correspond to a finite maximum propagation speed in the effective description.
The specific form of the adjacency condition follows uniquely from the structures already established. Since phase is the only invariant available prior to projection, admissibility of transitions can depend only on relational differences in $\phi$, not on absolute values. Because phase is defined on the compact manifold $S^1\cong\mathbb{R}/2\pi\mathbb{Z}$, the difference between two phases must be taken as the shortest arc distance on the circle. Explicitly, for any two phases $\phi_1,\phi_2$, define:
This definition ensures continuity across the $2\pi$ identification and removes any artificial boundary in phase space. The use of an absolute bound ensures orientation-independence and avoids introducing a preferred direction or temporal arrow (which arises only with record-bearing subsystems in §8). An inequality (rather than equality) avoids imposing a rigid step size or hidden metric structure. The adjacency condition:
is therefore the minimal and unique admissibility criterion compatible with bounded, non-convergent recursion and phase closure.
Sections 3.1–3.2 establish (i) a bounded, non-convergent recursion of bridge states and (ii) a phase invariant $(\phi\in S^1)$ with bounded per-step variation. At this stage, however, the bridge dynamics remain purely global: the system has an ordering parameter and a compact invariant, but no internal degrees of freedom. Without further structure, the recursion cannot support distinguishable substructure, adjacency relations among subcomponents, or record-bearing persistence. A minimal representational expansion is therefore required. We formalize this requirement as the necessity of a projection from bridge states into a configuration space admitting multiple effective degrees of freedom.
Any effective theory with local records requires a representational space with $m>1$ degrees of freedom; we denote this representational expansion as a projection.
Let $\mathcal{B}$ denote the space of admissible bridge states and let $\phi:\mathcal{B}\to S^1$ be the phase functional defined in §3.1. A representation containing only the global invariant $\phi_n$ is insufficient for distinguishability, adjacency, and persistence: it admits no internal factorization into distinguishable degrees of freedom. Accordingly, the model requires a configuration space whose states have nontrivial internal structure beyond the single global phase coordinate.
Let $\mathcal{S}$ denote a space of effective configurations and define a projection map:
Minimal admissibility conditions for $\mathcal{P}$ are:
We require the projected configuration admit a nontrivial factorization:
where each $\sigma_n^{(i)}$ represents an effective degree of freedom and the collection is not reducible to a single scalar invariant without loss of information.
Thus, projection is not an additional primitive physical postulate. It is a minimal representational requirement implied by bounded non-convergent recursion, bounded phase variation, and the necessity of distinguishable substructure for event formation and record-bearing persistence.
Let $\mathcal{B}$ denote the space of admissible bridge states and let $\mathcal{S}$ denote a space of effective configurations. Define:
The projection $\mathcal{P}$ is generally many-to-one: distinct bridge states may map to indistinguishable or equivalent configurations in $\mathcal{S}$. This reflects the fact that $\mathcal{S}$ is coarse-grained.
(i) Phase Preservation
(ii) Compatibility with Bounded Phase Variation
If $|\phi_{n+1}-\phi_n|_{S^1}\le\Phi_{\max}$ holds at the bridge level, then for any observable $\mathcal{O}:\mathcal{S}\to\mathbb{R}$ that depends continuously on $\phi$ under projection, there exists a finite bound:
(iii) Support for Decomposition
(iv) Absence of Presupposed Geometry
At this stage, $\mathcal{S}$ is not assumed to possess metric, topological, or causal structure beyond what is minimally required to represent distinct degrees of freedom and bounded variation. No assumptions are made regarding distance, dimensionality, curvature, or coordinate systems.
Spacetime is not identified with $\mathcal{S}$ itself but arises as an effective description of families of projected configurations $\{\Sigma_n\}$ that exhibit regularity under recursion. When $\mathcal{S}$ admits stable coarse-graining, it becomes possible to interpret subsets of $\mathcal{S}$ as spatially extended regions and the recursion index as a temporal ordering parameter.
Consider two adjacent projected degrees of freedom $\sigma^{(i)}$ and $\sigma^{(j)}$ whose adjacency is defined by admissible phase variation between corresponding bridge states. Because phase is the only invariant available at this level, any influence between adjacent degrees of freedom must be mediated by phase-consistent recursion.
Proposition (Local Update Constraint)
There exists a minimal integer $\Delta n_{\min}\ge 1$ such that for any adjacent degrees of freedom $\sigma^{(i)}\sim\sigma^{(j)}$:
No influence is admissible for $\Delta n < \Delta n_{\min}$.
Once projection introduces a characteristic spatial scale $\ell$ and associates a duration $\tau$ with a single recursion step, the constraint is expressed as:
The recursion index $t_R$ advances uniformly by construction. Mass-time $t_M$ emerges only where persistent records exist (§8). The density of mass-time reflects the rate at which record persistence accumulates relative to recursion steps.
Consequently, the invariant throughput $c$ remains fixed with respect to recursion time, while its expression in terms of mass-time may vary with local mass-time density. This distinction accounts for relativistic time dilation without modifying the underlying propagation bound.
The present framework determines the existence, necessity, and structural role of minimal temporal and spatial scales, as well as a finite invariant propagation bound. It does not, by itself, determine numerical values for these quantities. Numerical identification requires empirical anchoring of the real-time recursion rate relative to physical units.
At the bridge level, recursion produces ordered states but no events. Events become definable only after projection introduces multiple effective degrees of freedom (§4). We formalize the minimal conditions under which a physical event may be said to occur.
Let $\mathcal{S}$ denote the space of projected configurations, and let $\mathcal{O}:\mathcal{S}\to\mathbb{R}$ be any observable functional. Observable functionals are defined on projected configurations $\Sigma_n=\mathcal{P}(B_n)$; there are no observables defined directly on bridge states.
Definition (Event). An event occurs at recursion step $n$ iff there exists at least one observable functional $\mathcal{O}$ such that
Equivalently, an event is present when two successive projections are not observationally equivalent under the full set of admissible observables.
Because projection preserves phase (§4.2) and adjacency (§3.2e), any admissible event must respect bounded phase variation. However, the presence of an event does not imply causality, irreversibility, or temporal asymmetry. These arise only when records persist (§8).
At this stage, events do not yet form a causal chain or a time-ordered history. The recursion index provides ordering but not an arrow. Sequences of events acquire temporal structure only when memory-bearing subsystems exist.
Energy does not appear as a primitive quantity in the pre-geometric framework. At the bridge level there is no motion, no spatial extension, and no temporal duration. Nevertheless, the recursive dynamics encode a persistent asymmetry: the unresolved opposition between the subsets $A$ and $\bar{A}$. We formalize energy as the minimal scalar quantity associated with this unresolved relational structure.
Let $B_n\in\mathcal{B}$ be a bridge state at recursion step $n$. Define a scalar functional $\mathcal{Q}:\mathcal{B}\to\mathbb{R}_{\ge 0}$ by:
where $D$ is a non-negative, normed measure of unresolved opposition encoded by $B_n$. Required properties: non-negativity, vanishing under resolution (which does not occur globally), and recursion dependence.
Energy emerges only after projection introduces distinguishable degrees of freedom (§4). Under projection $\Sigma_n=\mathcal{P}(B_n)$, the scalar $\mathcal{Q}(B_n)$ appears as a distributed quantity over projected configurations and is interpreted as energy density.
Once spacetime geometry, duration, and interaction structure are established, $\mathcal{Q}$ admits multiple effective decompositions (kinetic, potential, field energy). These are emergent; the invariant origin is unresolved opposition.
Assumption (Finite Projection Capacity). For any projected configuration $\Sigma_n\in\mathcal{S}$ and any identifiable subregion $\mathcal{R}\subset\Sigma_n$, there exists a finite capacity $C_{\mathcal{R}}<\infty$ such that:
Here $\mathcal{Q}_{\mathcal{R}}$ denotes the integrated contribution of $\mathcal{Q}$ over degrees of freedom comprising $\mathcal{R}$. No assumption is made regarding additivity, locality, or integration form; only finiteness is required.
Definition (Threshold Transition). Let $\mathcal{Q}_{\text{crit}}(\mathcal{R})\le C_{\mathcal{R}}$ be a critical value. If $\mathcal{Q}_{\mathcal{R}} \ge \mathcal{Q}_{\text{crit}}(\mathcal{R})$, then $\Sigma_n$ is no longer admissible under $\mathcal{P}$ and the system must transition to $\Sigma_{n+1}$ in which the tension is redistributed or reduced.
This replaces singular behavior: instead of unbounded accumulation, admissibility enforces reconfiguration when critical limits are reached.
Once spacetime and locality are established, threshold-induced reconfiguration admits effective interpretations including phase transitions, particle creation, and high-energy structural reorganization.
The recursion index provides ordering but not a temporal arrow. Directionality arises only when record persistence introduces asymmetry between successive recursion steps. We formalize record persistence and define mass-time as the partial order induced by record persistence.
Let $\Sigma_n\in\mathcal{S}$ denote the projected configuration at recursion step $n$. Associate with certain subsystems a record functional $R_n\in\mathcal{R}$ representing physically retained structural information.
Definition (Record Update Rule). Records evolve by:
with monotonic retention:
Definition (Mass-Time Ordering). Define a partial order $\prec_M$ on recursion indices by:
This ordering is not total. Where defined, it constitutes a genuine temporal arrow. We refer to this emergent ordering as mass-time $t_M$, distinct from $t_R$.
Record persistence requires stability against reconfiguration (§7), which demands nonzero unresolved differential tension (§6). Only subsystems carrying sufficient $\mathcal{Q}$ can support persistent records. Mass and record persistence are inseparable: mass provides energetic conditions for records; records provide the mechanism for temporal direction.
Mass-time is subsystem-dependent and does not replace $t_R$. Causality arises only where mass-time exists and interactions propagate under the local update constraint (§4–5). Prior to record persistence, neither causality nor temporal direction is defined.
The Null–Bridge framework is not proposed as a replacement for existing physical theories, but as a pre-geometric substrate from which their effective structures may arise. We outline points of structural correspondence with General Relativity, Quantum Mechanics, and Quantum Field Theory, without assuming modification of their established formalisms.
In General Relativity, spacetime geometry is dynamical and determined by matter–energy content. Here, geometric degrees of freedom arise through projection into $\Sigma$, yielding effective metric structure only after projection (§4). The role of time as a parameter mirrors $t_R$, while $t_M$ emerges locally and is not globally defined, consistent with constraint-based formulations. Classical singularities correspond to regions where unresolved differential tension exceeds local projection capacity (§7); threshold-induced reconfiguration provides a non-singular mechanism.
In Quantum Mechanics, time enters as an external parameter governing unitary evolution, aligning with the recursion index $t_R$. Measurement corresponds to establishment of persistent records (§8), yielding local mass-time ordering and providing a basis for measurement asymmetry without modifying unitary dynamics. Decoherence corresponds to proliferation and stabilization of records across interacting subsystems. No observer-dependent collapse is required.
QFT treats the vacuum as an active structure supporting fluctuations and excitations. Here, vacuum activity corresponds to unresolved oscillatory dynamics of the bridge under projection. Field excitations arise as stable projected configurations sustained by local differential tension (§6). Phase transitions correspond to threshold-induced reconfigurations (§7). Zero-point fluctuations are interpreted as manifestations of unresolved bridge oscillation that do not accumulate into persistent records.
These correspondences are structural rather than derivational. The framework does not alter the predictive content of GR, QM, or QFT within their established domains. It provides a minimal pre-geometric substrate from which shared assumptions can be understood as emergent.
This framework is falsified if:
The Null–Bridge framework offers a logically minimal pre-bang ontology in which time, space, and energy emerge from a single unresolved recursive structure. It removes the need for an initial temporal boundary while remaining structurally compatible with existing physical theories. Its validity depends on whether effective dynamics and observational constraints can be recovered.