# The Timeless Boundary and the Polar Sector ## A Wheeler-DeWitt extension of Causal Symmetry and a possible gravitational dark sector **Felix J. Ruch — July 2026 — research note** ## Status This note develops a new branch of the Causal Symmetry programme. It is not yet part of the validated physics core. It replaces the earlier consciousness-centred interpretation with a more precise hypothesis: the White Hole Observer is a non-local, non-temporal boundary condition on the total quantum state. The dark-matter connection is explicitly conjectural. Information complementarity alone does not generate gravitational mass. Any dark-sector claim must derive an effective gravitational source and confront lensing, structure formation, cluster dynamics, and cosmological constraints. ## 1. Central proposal The White Hole Observer should not be understood as an agent, measurement event, or process occurring at a moment in an external time. It is a constraint selecting globally admissible histories. In canonical quantum-gravity language, the total state is described schematically by \[ \widehat{\mathcal H}\,\Psi[g,\phi]=0, \] with no preferred external time variable. Causal Symmetry adds a boundary constraint \[ \widehat{\mathcal B}_{\rm CS}\,\Psi=0, \] which restricts the global state to histories in which the absorption sector, bounce dynamics, and emission sector satisfy the information-preserving boundary map. The refined map from the core programme is \[ A:\mathcal H_M\rightarrow\mathcal H_P,\qquad A^\dagger A=I_M, \] \[ E=WA^\dagger,\qquad U=EU_BA=W(A^\dagger U_BA). \] Here, absorption is an isometry into a Planck-region code subspace, emission is a partial isometry, and the total map is unitary only when the bounce dynamics preserves and legibly acts on the code subspace. Under the timeless interpretation, this is not necessarily information moving through a universal temporal channel. It is a relation among boundary sectors of one constrained quantum history. The appearance of transfer and succession arises internally through relational clocks and local thermodynamic arrows. ## 2. Why this improves the framework The earlier shorthand \[ \frac{dI_{\rm total}}{dt}=0 \] is useful locally but problematic globally because disconnected domains need not share a preferred time parameter. A more defensible formulation is: > The admissible global quantum state preserves the relevant information relations across its paired boundary sectors. This reframing: 1. removes the need for an external meta-cosmic clock; 2. aligns the WHO condition with constrained quantum dynamics; 3. separates global informational consistency from local temporal evolution; 4. makes the local arrow of time emergent and relational rather than fundamental; 5. turns the WHO condition into a selection rule on complete histories. ## 3. Polar decomposition A possible extension factorizes the total state space as \[ \mathcal H_{\rm total}=\mathcal H_{\rm visible}\otimes\mathcal H_{\rm polar}, \] with \[ |\Psi\rangle\in\mathcal H_{\rm visible}\otimes\mathcal H_{\rm polar}. \] Observers in the visible sector have access only to \[ \rho_{\rm visible}=\operatorname{Tr}_{\rm polar}|\Psi\rangle\langle\Psi|. \] The polar sector is defined initially by complementarity in the global constraint, not by ordinary positive or negative energy balance. It may contain degrees of freedom that are inaccessible electromagnetically while remaining correlated with visible geometry and matter. This construction by itself explains local mixedness, not dark matter. A gravitational interpretation requires an additional dynamical statement. ## 4. Three dark-sector realizations ### 4.1 Long-lived black/white-hole remnants The most conservative branch identifies part of the dark sector with stable or long-lived remnants produced by black-to-white-hole evolution. These are ordinary gravitational sources in our spacetime and can, in principle, be tested as compact dark-matter candidates. This branch belongs naturally to the existing late-transition scenario and does not require the polar sector to modify gravity. ### 4.2 A hidden but gravitationally active polar sector A stronger model allows the polar degrees of freedom to contribute an effective stress-energy tensor: \[ G_{\mu\nu}=8\pi G\left(T^{\rm visible}_{\mu\nu}+T^{\rm polar}_{\mu\nu}\right). \] To behave as dark matter, the polar contribution must be stable, approximately pressureless on galactic and cosmological scales, weakly self-interacting within observational bounds, and nearly electromagnetically inert. The central derivation problem is to obtain \(T^{\rm polar}_{\mu\nu}\) from the global constraint rather than inserting it by hand. ### 4.3 A geometric shadow of the boundary condition The most speculative possibility is that the boundary constraint alters effective gravitational dynamics: \[ G_{\mu\nu}+\Delta_{\mu\nu}[\Psi,\mathcal B_{\rm CS}] =8\pi G\,T^{\rm visible}_{\mu\nu}. \] Here, the inferred dark component would be a geometric response to global boundary data rather than a particle population. This branch functions as modified gravity and must reproduce galactic rotation curves, gravitational lensing, cluster collisions, cosmic microwave background peak structure, and large-scale growth with one covariant mechanism. ## 5. Black holes as coupling interfaces Black holes are natural places to test whether visible and polar sectors interact because they already force the framework to specify: - the absorption isometry \(A\); - the code-subspace projector \(\Pi=AA^\dagger\); - the bounce dynamics \(U_B\); - the emission partial isometry \(E=WA^\dagger\); - the exterior coupling \(\Gamma(\omega)\). The polar-sector hypothesis asks whether the degrees of freedom orthogonal to the visible code subspace are merely discarded from the daughter-domain description or remain gravitationally active in the parent domain. A candidate decomposition is \[ \mathcal H_P=\operatorname{ran}(A)\oplus\mathcal H_{\rm polar}, \] but this is only meaningful if the decomposition is invariant or approximately invariant under \(U_B\). Otherwise, the visible/polar distinction is basis dependent and has no physical content. ## 6. Non-negotiable constraints A viable theory must satisfy all of the following: 1. **Covariance:** no preferred external time or foliation may be introduced without justification. 2. **Boundary consistency:** the timeless constraint and the operational map \(U=EU_BA\) must be shown to be compatible. 3. **No double counting:** information transferred to a daughter domain cannot be independently copied into a polar sector. 4. **Gravitational derivation:** dark behaviour must follow from a stress-energy contribution or modified field equation. 5. **Cosmological viability:** the model must confront CMB, nucleosynthesis, structure-growth, lensing, and cluster constraints. 6. **Local tests:** it must not violate precision tests of general relativity and the equivalence principle. 7. **Clear falsification:** parameter ranges or qualitative outcomes must be identified that would rule out each branch. ## 7. Immediate calculations ### Task A — timeless reformulation Replace the global-time balance equation with a constraint or transition-amplitude formulation. Determine whether the WHO clauses can be written as projector conditions on physical states satisfying the Hamiltonian constraint. ### Task B — relational clock model Construct a toy constrained system with a clock degree of freedom and two boundary sectors. Recover an effective unitary transfer map in relational time and show precisely when the operational description agrees with the timeless one. ### Task C — polar-sector invariance Test whether a decomposition \[ \mathcal H_P=\operatorname{ran}(A)\oplus\mathcal H_{\rm polar} \] is dynamically stable under candidate \(U_B\). Quantify leakage in both directions. ### Task D — semiclassical backreaction Given a reduced visible state, derive the semiclassical source \[ \langle T_{\mu\nu}\rangle_{\Psi} \] and determine whether tracing or conditioning on the polar sector can produce an additional conserved gravitational contribution. ### Task E — phenomenological discrimination Compare three branches: 1. compact remnant dark matter; 2. particle-like hidden polar matter; 3. boundary-induced modified gravity. List observables that distinguish them, rather than fitting all three under one label. ## 8. Falsification criteria The polar-sector conjecture should be abandoned or sharply narrowed if: - the global constraint cannot be formulated without a preferred meta-time; - the visible/polar split is not invariant under the bounce dynamics; - no covariantly conserved effective gravitational source can be derived; - the required source conflicts with lensing or structure-growth data; - the construction duplicates information already carried by the daughter-domain map; - the only surviving version is indistinguishable from an arbitrary dark component inserted by hand. ## 9. Working statement > Causal Symmetry may be formulated as a timeless boundary constraint on the total wavefunction of geometry and matter. The White Hole Observer is the condition selecting globally admissible histories in which absorbed and emitted information occupy complementary sectors of a single constrained state. Temporal transfer is the relational description available from within a domain rather than a process occurring in an external universal time. A speculative extension identifies the complementary sector as a polar gravitational sector. If this sector generates a covariantly conserved effective stress-energy contribution, modifies the semiclassical field equations, or is populated by long-lived black/white-hole remnants, it may exhibit dark-matter phenomenology. Information balance alone does not imply this result; the gravitational effect must be derived.