Foundations of Source-Projected Geometry in an Einstein-Locked OT–GKSL Framework: Reproducible Source/Readout Stress Tests from Low-Energy Protocols to Planck–DESI–Pantheon–KiDS1000 Growth–Shear Data

Historical low-energy tension and architectural motivation:

Relativity assigns physical meaning to lengths, durations, synchronization, and reference frames through clocks, rods, signals, observers, and comparison protocols. Einstein’s operational analysis of simultaneity already made clock synchronization part of the physical definition of time (Einstein, 1905), while later reconstructive approaches showed how substantial parts of spacetime geometry can be constrained by light propagation, free fall, and causal relations (Ehlers, Pirani & Schild, 1972; Hawking, King & McCarthy, 1976; Malament, 1977). Geometric invariants remain extraordinarily powerful, but their physical significance is expressed through relations that remain coherent across admissible observers and measurement procedures.

Geometry therefore does not disappear. Its status changes. It is no longer an absolute stage on which physics simply unfolds, but a structure that organizes and reconciles physically accessible measurements. One of the deepest lessons of relativity is precisely that geometric objectivity can survive without requiring an absolute background independent of all operational reference. General relativity deepens this displacement: matter and geometry no longer evolve as independent structures on a fixed Newtonian background, while causal propagation, proper duration, free fall, and spatial relation acquire their classical meaning within a Lorentzian geometry dynamically constrained by source content.

Nevertheless, the classical formalism already presupposes that events, clocks, local frames, causal order, and a differentiable spacetime description are meaningful. Quantum theory introduces a qualitatively different descriptive level. Its mathematical language is organized around states, density operators, amplitudes, observables, correlations, channels, instruments, and probabilities. The operational theory of quantum instruments makes explicit that a measurement is not specified only by an abstract observable but by a physical operation linking outcomes and state transformations (Davies & Lewis, 1970; Ozawa, 1984). Decoherence theory further shows that stable classical records and pointer structures require particular dynamical conditions rather than appearing as primitive labels (Zurek, 2003).

Measurement context is therefore not an incidental detail. Interference, polarization, which-path information, incompatible observables, entanglement, and basis-dependent readout all show that the relation between preparation, measurement protocol, and recorded outcome cannot in general be reduced to the passive revelation of a complete pre-existing set of classical properties. The tension with general relativity is consequently deeper than a disagreement between two dynamical equations.

OT–GKSL Foundations context — consolidated historical motivation:

General relativity organizes classical physics through a local Lorentzian geometry in which events, causal relations, proper times, and local frames are already available. Quantum theory provides a broader state- and operator-based structure in which coherence, correlations, measurement relations, and open-system dynamics need not themselves possess classical spacetime semantics.

A major family of responses to the quantum–gravitational problem promotes geometrical degrees of freedom themselves to quantum objects. Quantum geometrodynamics, loop and connection formulations, spin-network constructions, and related approaches exemplify different realizations of this strategy (see, e.g., Kiefer, 2012; Rovelli & Smolin, 1995). In loop quantum gravity in particular, spin-network states provide quantum states of geometry with area and volume operators.

That route is physically important, but it is not logically forced by the quantum character of matter alone. It incorporates an additional architectural choice: that geometrical variables themselves belong to the primitive structures to be quantized.

The OT–GKSL architecture investigates a different possibility. Rather than assuming that quantum matter already inhabits a fundamentally given classical geometry, or that the spacetime metric must itself be the primary quantum variable, it asks whether classical geometric readability can be reconstructed as a downstream physical structure of a more general quantum/open-state domain.

The native domain is therefore prelocal in the metric and spacetime sense, but it is not structureless. Here prelocal is an OT–GKSL architectural term: it does not mean absence of relations, subsystem distinctions, correlations, or algebraic structure. The native level contains density operators, observable and algebraic relations, correlations, internal spectral and gauge/spin structures, and open-system dynamics. The latter is described on the declared Markovian working sector by the standard GKSL structure of Gorini, Kossakowski and Sudarshan and Lindblad (1976). On the faithful detailed-balance dissipative sector, the Carlen–Maas construction supplies a positive noncommutative optimal-transport geometry in which the quantum Markov evolution is represented as a relative-entropy gradient flow (Carlen & Maas, 2017).

What this native level does not yet contain as primitive physical data are classical spatial distance, proper time, simultaneity, Lorentzian causal order, or an externally supplied classical coordinate system.

This distinction is essential. A tensor-product decomposition, an observable algebra, a correlation, or an interaction channel may already distinguish physical subsystems at the quantum level without thereby assigning them a classical spatial separation. Work on quantum reference frames similarly demonstrates that position, orientation, velocity, and clock-reference information can themselves be treated relationally when the reference frame is a physical quantum system (Bartlett, Rudolph & Spekkens, 2007; Giacomini, Castro-Ruiz & Brukner, 2019).

Native relational structure therefore precedes certified spacetime localization in the OT–GKSL architecture. This last statement is not imported from the cited quantum-reference-frame literature; it is a framework-specific placement rule.

The central question consequently becomes:

Under what physical conditions can a native quantum/open-state distinction become a stable classical distinction, then an individuated event, and finally a geometric relation?

This is the central displacement introduced by the OT–GKSL architecture. Quantum source content and classical geometry are not treated as two primitive objects already belonging to the same descriptive level. Instead, the architecture constructs two typed descendants of a common native domain.

One descent produces matter/source variables, effective currents, endomorphisms, response terms, and other readable source structures. A second descent produces certified physical records, event/readout structure, locally independent coordinate records, a calibrated physical coframe, and ultimately a Lorentzian readout metric.

The two descendants are not identified. Their compatibility is assessed only after each has been independently constructed.

2. OT–GKSL Foundations context Consolidated historical motivation :
They are independently constructed and only later compared through controlled recovery conditions, including the Einstein residual on the gravitational branch.
This distinction is stronger and more precise than saying that geometry is merely a “projection of the source”. The native source dynamics does not algebraically define the metric. Rather, native domain → source/matter descendant, record/geometric descendant, (1) and the classical gravitational relation is tested only after both sides have been independently fixed. This architecture also gives a sharper role to information and physical records. A mathematical difference between two density operators is not automatically a classical event distinction. For a classical attribution to acquire physical meaning, the relevant difference must survive through a stable, accessible, calibrated record structure. The framework therefore distinguishes native state difference from certified classical distinguishability. This leads to a structural consistency rule that is implicit in the certified-readability architecture and is now made explicit: Classical geometry may not introduce a physical distinction that is absent from the  omplete certified event-record structure. If two admissible native representatives are indistinguishable with respect to the complete certified records used to individuate an event, then, at fixed calibration and modulo declared gauge freedoms, they cannot receive two physically distinct geometric event assignments:
ν1 ∼evt ν2 ⇒Gro(ν1) gauge = Gro(ν2) gauge. (2)
Equivalently, if the complete event-record map is denoted by q_rec, the local differential consistency condition is kerDq_rec ⊆ kerDG_ro. (3)
An infinitesimal native variation that changes none of the complete event records therefore cannot generate an independent physical geometric variation.
The content of this rule is simple but nontrivial: no certified event distinction =⇒ no physical geometric distinction. (4) Geometry is therefore not allowed to create physical information that has no record-level origin. This does not mean that correlation, entanglement, and geometry are the same object. Two systems can be operationally indistinguishable without being entangled; two systems can be correlated without defining a classical spatial relation; and entanglement does not constitute a
superluminal causal channel.

3. OT–GKSL Foundations context — Consolidated historical motivation

The claim is therefore one of physical ordering: native relational structure precedes certified records; certified records precede event individuation; and event individuation precedes geometric localization. This ordering is specific to the OT–GKSL architecture.

Relational descriptions of position, orientation, synchronization, and other reference-dependent quantities already have a substantial history in quantum information and quantum-reference-frame theory, where the reference system itself is treated as a physical system rather than as an absolute external structure (Bartlett, Rudolph & Spekkens, 2007; Giacomini, Castro-Ruiz & Brukner, 2019). The OT–GKSL proposal goes one step further in a specific direction: a native distinction is not yet a classical spatial distinction, and geometric localization is licensed only after the relevant distinction has become part of a certified event/readout structure.

The spatial separation of two classical events is therefore itself a physical readout statement. This also clarifies an important distinction inside the geometric descent.

The complete certified event-record structure need not have dimension four. The rank-D family

X = (X¹, ..., Xᴰ)

contains the locally independent coordinate records required for the geometric readout. On the observed spacetime branch, D = 4. But these four coordinate records need not exhaust all physically readable information associated with an event. Additional records may distinguish internal state, matter content, preparation history, measurement outcomes, orientation, gauge information, or other operational data without becoming additional spacetime coordinates.

Thus, complete event-record content is not identical to coordinate dimension. The four-dimensionality of the readout geometry is a statement about the local geometric organization of classical events, not a statement that a physical event can contain only four independent pieces of information.

This distinction is specific to the OT–GKSL readout architecture, although it is compatible with the broader lesson of quantum measurement and reference-frame theory that physically accessible information may be carried by degrees of freedom whose role is not reducible to that of spacetime coordinates (Davies & Lewis, 1970; Bartlett, Rudolph & Spekkens, 2007).

The difficulty becomes particularly clear when the sources themselves are strongly quantum. A coherent or collective quantum source is not fundamentally described as a classical point particle carrying a unique position, trajectory, orientation, and proper time. Its native description involves a state or density operator, coherences, correlations, internal algebraic structure, interactions, and open-system channels.

Classical position and proper time become meaningful only after suitable records and reference structures have stabilized. Decoherence can contribute to this transition, but decoherence alone is not identified here with spacetime emergence. Decoherence and environment-induced superselection explain how particular pointer structures can become robust, while environmental interactions can also stabilize and redundantly encode selected records (Zurek, 2003; Ollivier, Poulin & Zurek, 2004). These results support the physical possibility of stable classical records, but they do not by themselves derive a Lorentzian spacetime geometry.

A classical reference frame is therefore not defined by the disappearance of quantum structure altogether, but by the existence of sufficiently stable and accessible record-bearing degrees of freedom. This is consonant with quantum-reference-frame approaches in which the reference frame is explicitly a physical, and potentially quantum, system (Bartlett, Rudolph & Spekkens, 2007; Giacomini, Castro-Ruiz & Brukner, 2019).

This makes the measuring reference itself part of the physical problem. A clock, ruler, detector, or coordinate frame is not treated as an ontologically external classical object. It is a physical system, or an effective sector of one, whose record-bearing role must itself be certified. The quantum character and finite physical resources of clocks and reference systems have long been recognized, from Salecker and Wigner’s analysis of quantum limitations on spacetime measurements to relational-clock constructions such as Page and Wootters (Salecker & Wigner, 1958; Page & Wootters, 1983).

The same logic extends from ordinary measurement to spacetime readout. A clock is a record-bearing structure capable of supporting stable temporal discrimination. A spatial frame is a record-bearing structure capable, within the OT–GKSL geometric descent, of supporting a nondegenerate calibrated coframe.

4. OT–GKSL Foundations context — Consolidated historical motivation:

A spacetime readout requires the compatibility of these records with Lorentzian calibration, bridge control, section stability, and the other certification conditions attached to the claimed classical structure. These requirements are specific to the OT–GKSL architecture. Standard measurement theory, decoherence theory, and quantum-reference-frame theory provide part of the upstream physical motivation, but they do not by themselves supply this complete certification hierarchy.

This distinction also prevents a possible confusion concerning time. The native GKSL trajectory is parametrized by an internal evolution parameter ξ. It is not identified with physical proper time.

Relational approaches to quantum time have independently emphasized that observable temporal information may be tied to physical clock systems rather than to an externally observable coordinate-time parameter (Page & Wootters, 1983), while earlier analyses established that clocks themselves are finite physical systems subject to quantum limitations (Salecker & Wigner, 1958).

OT–GKSL makes a more specifically typed distinction. In the faithful detailed-balance dissipative sector, relative-entropy production may define a monotone native ordering variable t_ent. This supplies an intrinsic ordering of certain state-space processes, but it is still not a clock. Physical temporal readout t_ro appears only when an appropriate clock-like record has been stabilized, discriminated, and calibrated.

The architecture therefore maintains the structural distinction

ξ ≠ t_ent ≠ t_ro.

The native domain is consequently not a classical spacetime metric manifold. It is a quantum/open-state domain carrying open-system dynamics and, where the required hypotheses hold, quantum optimal-transport geometry. On the declared Markovian working sector,

ρ_ξ ∈ D°(H_eff),  dρ_ξ/dξ = L*(ρ_ξ),

with L* of GKSL type. The mathematical structure of completely positive Markovian quantum dynamics is supplied by the independent results of Gorini, Kossakowski and Sudarshan and of Lindblad (Gorini, Kossakowski & Sudarshan, 1976; Lindblad, 1976).

On the faithful reversible detailed-balance dissipative component, the Carlen–Maas construction provides the corresponding noncommutative optimal-transport gradient-flow structure. Writing

F(ρ) = D(ρ || π),

one has, on that declared sector,

L*_db(ρ) = − grad_gOT F(ρ),

with the associated detailed-balance entropy production:

σ_db(ρ) = − D F_ρ [L*_db(ρ)] = ||grad_gOT F(ρ)||²_gOT ≥ 0.

Carlen and Maas established this gradient-flow representation for an appropriate class of finite-dimensional quantum Markov semigroups satisfying quantum detailed balance, with the Riemannian metric playing the role of a noncommutative analogue of Wasserstein geometry (Carlen & Maas, 2017). The restriction to the detailed-balance component is essential: Hamiltonian contributions and declared non-detailed-balance remainder terms of the full GKSL generator retain their separate status.

This supplies a controlled geometric description of dissipation on state space. The positive native transport metric g_OT and the Lorentzian readout metric g_ro are deliberately different mathematical objects with different physical roles. There is no signature-change operation transforming one into the other.

The native transport geometry organizes motion and geometric structure on the state manifold. The readout geometry is constructed later through certified records, the vertical/horizontal decomposition, a local representative section, a calibrated soldering or frame-transfer map, and a physical coframe e^a. The Lorentzian metric is then constructed as:

g_ro = η_ab e^a ⊗ e^b.

The relationship between native OT geometry and Lorentzian readout geometry is therefore a controlled bridge, not an identity.

This shift has an important consequence for gravity. The OT–GKSL architecture retains an Einstein lock on the observed gravitational branch.

5. OT–GKSL Foundations context — Consolidated historical motivation

On the observed gravitational branch, the two-derivative kinetic sector is retained in Einstein–Hilbert form with a universal constant G0G_0. No freely varying G(x)G(x), no arbitrary phenomenological Geff(z,k)G_{\rm eff}(z,k), and no preparation-dependent multiplicative factor is inserted in front of R[g]R[g]. This choice is the OT–GKSL Einstein lock: it selects the classical gravitational kinetic structure to be recovered rather than deriving that structure as a unique consequence of the native GKSL dynamics.

There is an established mathematical context for privileging the Einstein tensor in four-dimensional second-order metric gravity. Under the assumptions studied by Lovelock, the metric and Einstein tensors exhaust the relevant symmetric divergence-free rank-two concomitants of the metric and its first two derivatives in four dimensions (Lovelock, 1971). This provides useful structural context, but the OT–GKSL Einstein lock is not presented as a consequence of Lovelock’s theorem; it is a branch-selection and recovery condition of the architecture.

Preparation- and state-sensitive structure is instead assigned to independently constructed source, response, gauge, interface, higher-order, and certification sectors. The gravitational comparison is made only after the readout geometry, the effective source, and the interface terms have been fixed. The relevant residual is

CE=Gμν[g_ro]−κ0Tμν_eff−Δμν_if.

A successful gravitational branch is therefore a recovery and compatibility statement, not a definition of the geometry from the source or of the source from the geometry.

This organization retains the universal normalization of the Einstein–Hilbert kinetic sector and keeps the contracted Bianchi identity as a genuine consistency condition on the total readable source/response/interface structure. Since ∇μGμν=0 is a geometric identity of the Einstein tensor, the independently constructed right-hand side must satisfy the corresponding covariant balance on an accepted gravitational branch.

The architecture is also designed to remain compatible with the standard equivalence-principle structure of the certified Einstein regime rather than replacing it with an explicitly preparation-dependent gravitational coupling. The distinction is important: universality of G0G_0 helps prevent one obvious source of composition- or state-dependent gravitational coupling, but equivalence-principle compatibility still depends on the detailed matter, source, interface, and readout couplings and therefore remains subject to independent checks. The experimental and theoretical status of the Einstein equivalence principle and universal metric coupling is reviewed extensively by Will (2014).

Nor should the Einstein lock be interpreted as a claim that the architecture has solved every possible pathology associated with quantum gravity or effective gravitational theories. The narrower statement is that the present construction does not introduce arbitrary state-dependent curvature prefactors or higher-derivative modifications into the locked two-derivative gravitational kinetic block. Possible higher-order readout corrections, source effects, interfaces, or effective operators have their own declared status and must be assessed separately.

The broader matter sector follows the same typed discipline. Fermionic CAR/Fock structure, internal gauge data, state-dependent endomorphisms, collective variables, source channels, and geometric records occupy different mathematical levels of the architecture. Controlled Dirac, Maxwell/Yang–Mills, Poisson/Newtonian, and Einstein recoveries are therefore evaluated separately, with their own domains, coefficients, bridge conditions, calibration choices, and residuals.

The architecture does not claim that full QED or QCD follow automatically from GKSL dynamics alone. Standard fermionic and gauge structures are imported where their hypotheses and algebraic content are required; the OT–GKSL contribution lies in their placement inside a common native/source/readout dependency structure and in the requirement that downstream classical sectors be independently reconstructed and tested rather than silently identified.

Likewise, vacuum-like, CDM-like, or late-growth effective sectors are conditional descendants defined on their declared working branches. Their existence as mathematically consistent reduced sectors does not by itself establish that Nature realizes those branches.

Classical geometry consequently acquires a meaning more precise than either “fundamental background” or “quantized metric.” Within OT–GKSL it is a certified relational reconstruction. Its objectivity lies in the possibility of obtaining a coherent and stable geometric readout across admissible physical records, frames, calibrations, and representative choices, rather than in assuming spacetime points as primitive labels independent of all physical access.

To say that two physical systems are “at different places” is therefore already to make a downstream physical statement. The certified event-record structure must support their individuation as distinct events, and the geometric readout must consistently assign them distinct locations. This is the direct geometric consequence of the record-consistency principle introduced above.

6. OT–GKSL Foundations context — Consolidated historical motivation

Conversely, if no certified physical distinction exists between two candidate event representatives, geometry has no independent license to separate them. This is the operational content, within the OT–GKSL architecture, of saying that classical spacetime is a readout rather than a primitive native label.

The current cosmology-facing tests have a correspondingly precise and limited status. They are not a validation of the entire OT–GKSL architecture. They expose particular downstream source/readout branches to established observational data and likelihood structures while keeping the native-to-readout assumptions and calibrations frozen before comparison.

The external observational context is itself well established. Planck provides high-precision CMB constraints and lensing information; DESI supplies baryon-acoustic-oscillation and large-scale-structure measurements; Pantheon+ provides a calibrated Type-Ia-supernova distance sample; and KiDS-1000 provides weak-lensing cosmic-shear constraints (Planck Collaboration, 2020; DESI Collaboration, 2024/2025; Brout et al., 2022; Scolnic et al., 2022; Asgari et al., 2021).

Growth-, lensing-, BAO-, supernova-, and shear-facing tests therefore constrain the particular descendants propagated into those observables. If such a branch fails against the frozen likelihood and calibration contract, that failure weakens the cosmology-facing viability of the branch concerned. If it survives, the result establishes compatibility with the tested observations; it does not elevate the whole native-to-readout architecture to an empirically established fundamental theory.

The same logic applies to low-energy laboratory protocols. A null signal, bridge failure, calibration failure, constitutive null, holonomic null, spectral-support failure, or loss of record certification constrains the particular descendant whose assumptions have failed. It is not reinterpreted as disappearance of the upstream GKSL state dynamics. This claim-dependent falsifiability is essential to the architecture because it prevents a failed downstream realization from being absorbed by unconstrained reinterpretation of another layer.

The OT–GKSL architecture is therefore not simply an attempt to build “one more quantum gravity” by quantizing the metric. It changes the level at which the problem is posed.

The central question becomes:

Under what physical conditions can open quantum source dynamics support stable physical records, individuated events, calibrated clocks and frames, and ultimately a Lorentzian geometric readout whose independently reconstructed matter/source sector satisfies controlled Einstein and low-energy recovery tests?

In this perspective, the emergence of classical spacetime is first a problem of physical readability and event individuation, and only subsequently a problem of metric geometry.

The geometry of spacetime is not discarded. It is reconstructed at the descriptive level at which its operational and relational physical meaning has been certified.

• Lovelock, D. (1971), “The Einstein Tensor and Its Generalizations,” Journal of Mathematical Physics 12, 498–501, DOI 10.1063/1.1665613.
• Will, C. M. (2014), “The Confrontation between General Relativity and Experiment,” Living Reviews in Relativity 17, 4, DOI 10.12942/lrr-2014-4.
• Planck Collaboration (2020), “Planck 2018 results. VI. Cosmological parameters,” Astronomy & Astrophysics 641, A6, DOI 10.1051/0004-6361/201833910.
• DESI Collaboration, “DESI 2024 VI: Cosmological Constraints from the Measurements of Baryon Acoustic Oscillations,” arXiv:2404.03002; the corresponding DR1 products and cosmology chains are published by DESI.
• Brout, D. et al. (2022), “The Pantheon+ Analysis: Cosmological Constraints,” Astrophysical Journal 938, DOI 10.3847/1538-4357/ac8e04; Scolnic, D. et al. (2022), “The Pantheon+ Analysis: The Full Data Set and Light-curve Release,” DOI 10.3847/1538-4357/ac8b7a.
• Asgari, M. et al. (2021), “KiDS-1000 Cosmology: Cosmic shear constraints and comparison between two point statistics,” Astronomy & Astrophysics 645, A104, DOI 10.1051/0004-6361/202039070.

Historical low-energy tension:

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