Wave-Function Collapse as Conformal Relational Selection
Aligned with Penrose Objective Reduction

Thomas A. Smeenk, BA

 

Abstract

We reformulate wave-function collapse as conformal relational selection, aligning the Conformal Consciousness Hypothesis (CCH) with Penrose’s Objective Reduction (OR). Rather than a stochastic or dynamical collapse, we propose that collapse corresponds to non-computable selection constrained by a conformal invariant A = E/(hν). This preserves unitarity while providing a geometric and non-computational mechanism consistent with Penrose’s gravitationally induced OR.

1. Measurement Problem and Penrose OR

Penrose’s Objective Reduction proposes that quantum superpositions become unstable when spacetime curvature differences exceed a threshold. CCH complements this by identifying the invariant relational quantity A that constrains which branch becomes classically realized.

Figure 1. Schematic flow from unitary evolution through decoherence into effectively classical branches, followed by conformal relational selection constrained by the invariant condition A(R)=const.

2. Conformal Invariant A and OR Thresholds

In OR, collapse timescale τ ≈ ħ / ΔE_G depends on gravitational self-energy. CCH reframes this by noting that ΔE_G enters through relational energy-frequency ratios, leaving A invariant under conformal rescaling.

Figure 2. Under conformal rescaling g_ab → Ω² g_ab, energies and characteristic frequencies often scale identically (E → Ω⁻¹E, ν → Ω⁻¹ν) in many field-theoretic and cosmological contexts; the ratio A ≡ E/(hν) remains invariant.

3. Non-Computability and Awareness

Penrose argues OR is fundamentally non-computable. CCH identifies awareness as the relational structure accessing this non-computable selection, consistent with OR’s rejection of algorithmic collapse.

 

Figure 3. Conceptual separation between computable unitary dynamics (fully simulable) and a proposed non-computable selection constraint, aligned with Penrose’s OR motivations and expressed here as conformal relational alignment A(R)=const.

4. Experimental Topology

We propose an interferometric experiment where gravitational potential differences and frequency scaling are adjusted while preserving A. OR predicts collapse timing shifts; CCH predicts invariant relational outcome statistics.

Figure 4. Experimental topology (final, non-overlapping layout). Arm 1 introduces a tunable gravitational potential difference ΔΦ and corresponding gravitational self-energy ΔE_G relevant to Penrose Objective Reduction. Arm 2 implements controlled conformal rescaling of energy E and characteristic frequency ν while enforcing the invariant constraint A ≡ E/(hν) fixed. Outcome statistics are recorded at detectors D₁ and D₂.

5. Expected Statistical Outcomes and Global Calibration Protocol

This section specifies (i) the observables to be reported, (ii) the expected statistical behavior under the competing hypotheses, and (iii) a calibration protocol designed to permit replication across laboratories with different absolute scales. The intent is not to modify quantum mechanics, but to test whether outcome statistics remain invariant under controlled rescalings that preserve the dimensionless relational quantity A ≡ E/(hν).

5.1. Observables

Each laboratory shall report, at minimum, the following measured quantities with associated uncertainties:
(O1) Single-detector outcome probabilities P(D1), P(D2) (or generalizations to multi-detector configurations).
(O2) Interference visibility V, defined operationally as V = (C_max − C_min)/(C_max + C_min) for a scanned phase φ, where C(φ) is the coincidence (or count) rate.
(O3) Correlation functions. For a two-setting, two-outcome variant, report E(a,b) and the CHSH parameter S = |E(a,b)+E(a,b')+E(a',b)−E(a',b')|.
(O4) Any inferred collapse-time proxy (if implemented): e.g., the characteristic time at which interference visibility drops below a predefined threshold V_* under controlled ΔΦ or mass-separation settings.

5.2. Hypotheses and Statistical Predictions

Baseline (standard QM + decoherence): In the absence of additional collapse mechanisms, unitary dynamics combined with well-characterized environmental decoherence predicts that (P(Dk), V, E(a,b), S) depend on standard interferometric parameters and environmental couplings. Under changes that do not materially affect decoherence, these statistics remain stable within experimental error.

Penrose OR-aligned expectation: In OR-type models, a superposition of distinct spacetime geometries is associated with a gravitational self-energy ΔE_G, yielding a characteristic reduction timescale τ_OR ≈ ħ/ΔE_G. Operationally, increasing ΔE_G (e.g., by increasing mass m, separation d, or gravitational potential difference ΔΦ over relevant spacetime regions) should induce earlier suppression of interference, manifested as a reduction in V and/or a measurable change in the time-resolved statistics of outcomes. The precise mapping from τ_OR to measured V(t) depends on the physical implementation (photonic, optomechanical, matter-wave).

CCH conformal-relational expectation (as tested here): When experimental controls are adjusted so that a prescribed relational configuration preserves A within tolerance (ΔA/A ≤ ε_A), and when decoherence sources are held fixed (or measured and subtracted), the outcome statistics are predicted to be invariant under conformal rescalings of absolute energy and frequency scales. Concretely, for a scaling parameter Ω applied such that E → Ω⁻¹E and ν → Ω⁻¹ν (or the closest achievable laboratory proxy), the CCH prediction is:
  P_Ω(Dk) ≈ P(Dk),   V_Ω ≈ V,   E_Ω(a,b) ≈ E(a,b),   S_Ω ≈ S,
to within statistical uncertainty and systematic calibration error, provided A is matched and the relational geometry of the interferometer is preserved.

5.3. Practical Realization of “Conformal Rescaling” in the Laboratory

Because a literal spacetime conformal rescaling is not an experimental control knob, laboratories shall implement a proxy scaling program that varies the absolute energy and characteristic frequency scales while preserving (a) dimensionless ratios, (b) interferometric geometry up to similarity transformations, and (c) the target A. Two realizations are recommended:

(R1) Optomechanical superposition realization (OR-sensitive): Prepare a mechanical resonator or levitated nanoparticle in a spatial superposition with controllable separation d and mass m. Let ν denote a dominant internal frequency of the prepared system (e.g., the trap frequency ν_trap or an internal mode). Adjust m and ν so that A = mc²/(hν) is held fixed across runs (e.g., scale m ∝ ν). Independently vary d or ΔΦ to modulate ΔE_G. Record V(t) and the inferred τ_* at which V crosses a threshold V_*. Compare τ_* across runs where A is fixed but ΔE_G is varied, and across runs where ΔE_G is held fixed but (m, ν) are rescaled with A fixed.

(R2) Photonic entanglement / interferometry realization (high-count-rate control): Use an entangled photon source and interferometric arms with adjustable phase and controlled gravitational potential differences (e.g., vertical separation; fiber delay lines routed at different heights). Here E=hν is tightly linked for each photon, making A=1 for single-photon energy-frequency pairing; therefore, the relevant “E” in A must be specified as an experimental energy scale associated with an internal mode (e.g., cavity/arm stored energy or a reference oscillator energy) while ν is taken as the associated oscillator frequency. Laboratories must explicitly document their operational definitions of E and ν. The statistical invariance test is then performed under matched A, with phase-scanned visibilities and CHSH parameters compared across rescaled configurations.

5.4. Data Collection and Analysis Plan

(D1) Pre-registration: Each laboratory should pre-register (i) the operational definition of E and ν used to compute A, (ii) the tolerance ε_A, (iii) target ranges for Ω, ΔΦ (or ΔE_G proxies), and (iv) primary endpoints (V, S, P(Dk)).

(D2) Acquisition: For each configuration (Ω, ΔΦ), collect counts in fixed acquisition windows T with randomized measurement setting order. Record environmental monitors (temperature, vibration, pressure, magnetic fields) and laser/source stability metrics. For photonic variants, correct for accidental coincidences and detector dead time. For optomechanical variants, characterize motional heating and residual gas collisions.

(D3) Statistical estimation: Compute P(Dk) by maximum-likelihood estimation for multinomial outcomes; compute V from sinusoidal fits to C(φ) with uncertainty via bootstrap resampling; compute CHSH S with standard error propagation from E(a,b). Report confidence intervals (95%) and Bayes factors where appropriate for model comparison between (i) invariance under scaling (CCH proxy prediction) and (ii) scaling-dependent deviations.

(D4) Primary test: For each endpoint X ∈ {P(Dk), V, S}, test the null hypothesis H0: X_Ω = X within tolerance δ_X determined by combined statistical and systematic error. A conservative criterion is |X_Ω − X| ≤ 2σ_X (two standard deviations). A consistent pattern of scaling-dependent deviations exceeding this bound, after systematic controls, constitutes evidence against the CCH invariance claim in this topology.

5.5. Global Calibration Protocol

To enable global replication, laboratories shall publish a calibration sheet containing:
(C1) A computation: A = E/(hν) with explicit operational definitions and measurement methods for E and ν.
(C2) Geometry: arm lengths (or equivalent delays), vertical height differences, optical path similarity transforms, and alignment tolerances.
(C3) Decoherence budget: quantified contributions from known decoherence channels with uncertainty.
(C4) Scaling map: the set of Ω values and how E and ν were adjusted to realize E→Ω⁻¹E, ν→Ω⁻¹ν (or proxy).
(C5) Raw data: time-stamped event records sufficient for independent reanalysis.

Recommended tolerance targets for cross-laboratory comparability are ε_A ≤ 10⁻³ for well-controlled oscillator references, and ≤10⁻² where E or ν are inferred indirectly. Laboratories should report systematic uncertainty separately from statistical uncertainty.

5.6. Limitations and Scope

This topology tests a constrained claim: invariance of specified outcome statistics under controlled rescalings that preserve A within tolerance, given well-characterized decoherence. It does not by itself establish a unique microscopic mechanism of selection. Conversely, failure of invariance—after controlling for systematic effects—would directly falsify the CCH prediction in this experimental form. Where OR-type predictions are tested simultaneously (via ΔE_G modulation), the experiment can also bound or discriminate OR parameterizations through the observed scaling of τ_* or visibility loss.

6. Conclusion

By aligning conformal relational selection with Penrose’s OR, CCH provides a geometric and awareness-linked interpretation of collapse that preserves quantum mechanics while enabling experimental scrutiny.

References

Penrose, R. (1989). The Emperor’s New Mind.
Penrose, R. (1996). On Gravity’s Role in Quantum State Reduction.
Penrose, R. (2010). Cycles of Time.
Gödel, K. (1931). On Formally Undecidable Propositions.
Zurek, W. H. (2003). Decoherence and the transition from quantum to classical—Revisited.