There exists an abstract configuration space

whose elements are not assumed to be spacetime events, particle positions, field values on a manifold, or classical states. Rather, X denotes a pre-spatiotemporal space of possible informational configurations.

The primitive dynamical object is an interference state

where ℋ X is a complex Hilbert space associated with X . In a representation where elements of X can be labeled by x , the state may be written formally as

The quantity Ψ ( x ) is not interpreted as the amplitude of an event occurring at a spacetime point. It is instead the amplitude associated with a primitive informational configuration. Interference, rather than localization in spacetime, is therefore the primary structural feature of the theory.

Figure 1. Conceptual pipeline of Interference-First Reality. The pre-observational interference substrate Φ is mapped by an observer projection Π O into a Born-normalized probability distribution p O ∈ Δ ∘ on the interior of the probability simplex. A multi-observer ensemble { p i } i = 1 N , compared in Jensen-Shannon geometry ( Δ ∘ , d JS ) , contracts toward an empirical Fréchet barycenter p ( N ) ; its large- N stabilization toward a stable center p ⋆ (Theorem 1) defines operational objectivity as a stable geometric invariant, while the cumulative Jensen-Shannon arc length defines emergent time. The diagram is a conceptual illustration; the corresponding formal objects are defined in Sections 3 - 4.

This postulate distinguishes IFR from approaches in which spacetime, fields, or particles are taken as fundamental. In IFR, spacetime localization, particle-like persistence, and field-theoretic descriptions arise only after stable patterns in Ψ become redundantly accessible to operational observers.

An observer is defined operationally as any physical, instrumental, environmental, or coarse-graining structure capable of extracting a stable representation from the underlying interference state.

Mathematically, an observer O a is represented by a map

where ℳ a is the representation space accessible to observer a . The observed representation is

The index a may label a detector, an environmental fragment, a measurement context, an instrumental configuration, a smoothing scale, a temporal epoch, a catalog partition, or any operational subsystem that produces a reproducible projection of Ψ. No appeal to consciousness or subjective awareness is involved.

The observer map Π a may include coarse-graining, noise filtering, basis selection, dimensional reduction, statistical estimation, or physical interaction with a subsystem. The essential requirement is that Π a [ Ψ ] be an operationally accessible representation.

Thus,

This postulate makes observer dependence explicit without making the theory subjective. Different observers may produce different representations, but IFR will define objectivity through the existence of stable structures common to these representations.

The representation spaces ℳ a are not merely sets. They are equipped, locally or globally, with structures that quantify distinguishability between observational representations.

For each observer space ℳ a , there exists an information-geometric structure

which induces a distance or divergence function

When representations from different observers are compared, IFR assumes the existence of a common comparison geometry

together with embeddings or alignment maps

into a common comparison space ℳ . For notational simplicity, we often absorb these embeddings into the definition of Π a and write

The distance between two observer projections is then written

The geometry Γ may be Riemannian, metric, statistical, Wasserstein, Fisher-Rao, Bures, Fréchet, or more general, depending on the class of representations under consideration. In the case of a parametric statistical model p ( y | θ ) , a canonical local choice is the Fisher information metric [3]

The third postulate is therefore

Spacetime geometry is not assumed at this level. It may emerge only if Γ admits an effective low-dimensional metric representation.

Objectivity is not a primitive property of a state. It is a relational stability property of multiple observer projections.

Given a family of observer maps { Π a } a ∈ A and a common information geometry Γ, define the redundant dispersion functional

where w a b ≥ 0 are the weights satisfying

when normalization is required.

A configuration is said to exhibit redundant contraction when D red is small relative to an appropriate null or reference dispersion. A structure is objective when the observer projections admit a stable common center.

More precisely, define the Fréchet functional

with w a ≥ 0 and ∑ a w a = 1 . A Fréchet center is any minimizer

The structure Q ⋆ is objective at resolution ε > 0 if

and if the minimizer is stable under admissible perturbations of the observer maps,

Thus,

This postulate replaces the assumption of observer-independent objects with a precise criterion for the emergence of observer-invariant structures.

The physical configurations of IFR are selected by a variational principle. There exists an action functional

defined over interference states, information geometries, and admissible observer maps, such that physically realized configurations satisfy

In its minimal form, the action is

where λ is an ordering parameter, K int describes interference dynamics, V int encodes informational selection or instability, D red enforces redundant contraction, and ℛ Γ is an intrinsic curvature scalar associated with the information geometry.

The constants

control, respectively, the strength of objectivizing contraction and the contribution of informational curvature.

For the variational principle δ S IFR = 0 to be mathematically meaningful, the functionals appearing in S IFR must be specified on a precise class of admissible histories and must satisfy minimal regularity conditions; these are stated in full in Section and summarized here. An admissible history is a curve λ ↦ ( Ψ ( λ ) , Γ ( λ ) , Π ( λ ) ) , λ ∈ I = [ λ 0 , λ 1 ] ⊂ ℝ , with Ψ ∈ H 1 ( I , ℋ X ) (i.e. both Ψ and its weak derivative ∂ λ Ψ are square-integrable in λ with values in ℋ X , and ‖ Ψ ( λ ) ‖ X = 1 ), Γ ( λ ) ∈ G a continuous family of complete geodesic information geometries, and Π ( λ ) ∈ P satisfying the admissibility conditions of Section 3.3 uniformly in λ . The kinetic term is taken to be the quadratic velocity functional

which is finite and Fréchet differentiable on H 1 ( I , ℋ X ) . The potential V int : ℋ X → ℝ is assumed continuous, bounded below, and Gâteaux differentiable on ℋ X (or on the dense domain of an associated self-adjoint operator, when V int arises from such an operator). Together with the regularity of D red (Section 3.6) and of ℛ Γ (Section 3.9), these conditions ensure that the integrand ℒ IFR ( λ ) = K int − V int − α D red + β ℛ Γ is integrable on I , so that S IFR is finite on admissible histories. Under these conditions, δ S IFR = 0 is to be understood as the vanishing of the first Fréchet variation of S IFR with respect to fixed-endpoint variations δ   Ψ ∈ H 0 1 ( I , ℋ X ) and variations δ   Γ , δ   Π tangent to G and P , respectively. The explicit Euler-Lagrange form of this problem, and the further structural assumptions needed to render it dynamically predictive, are developed in Part II.

The corresponding formal generating functional is

This is not yet a conventional path integral over fields on spacetime. It is a formal sum over interference structures, comparison geometries, and observer projections. A conventional path integral over matter fields and spacetime metrics can arise only after an effective spacetime representation has emerged.

The fifth postulate is therefore

Gravitation is not introduced as a primitive force acting on matter in spacetime. Nor is the spacetime metric assumed to exist at the fundamental level. Instead, IFR postulates that gravitational geometry is the macroscopic, observer-stable manifestation of curvature in the information geometry Γ.

Persistent objective structures constrain the surrounding space of admissible informational transitions. Such constraints deform the geometry of distinguishability and accessibility for other structures. This deformation is encoded by the curvature term

When the information geometry admits an effective spacetime representation, there exists a map

and an effective metric

such that informational distances and transition constraints are approximated by spacetime metric relations. In that regime, informational curvature induces effective spacetime curvature:

where R ( g eff ) is the Ricci scalar of the emergent metric.

The gravitational postulate may be written as

This postulate ensures that gravity is not an external addition to IFR. It is one of its central emergent sectors.

The parameter λ appearing in the variational formulation is not physical time. It is an ordering parameter used to describe families of possible configurations.

Physical time emerges only when the theory produces a sequence of stable objective structures

that admits an intrinsic order relation. IFR defines such an order through asymmetric informational dependence, memory, or monotonic stabilization.

Let ℳ mem ( Q j ⋆ | Q i ⋆ ) denote the amount of recoverable information about Q i ⋆ encoded in Q j ⋆ . A temporal ordering relation may be defined by

if

Equivalently, temporal order may be associated with a monotonic sequence of stable minima of redundant dispersion,

The seventh postulate is

Thus, IFR does not assume time as an external parameter. It derives temporal order from the asymmetric organization of objective structures.

Spacetime and ordinary field-theoretic descriptions arise when the information geometry Γ admits a stable, low-dimensional, approximately local representation.

Specifically, if there exists an effective manifold ℳ eff and a map

such that

then ℳ eff is interpreted as emergent space, or spacetime once temporal ordering has also emerged.

In the high-redundancy semiclassical limit, the IFR action must reduce to an effective action of the form

possibly with correction terms controlled by residual dispersion, nonlocal informational curvature, or incomplete observer contraction.

The final postulate is therefore

The postulates may be summarized as follows:

I. Reality is grounded in a pre-spatiotemporal interference state Ψ.

II. Observers are operational projection maps Π a .

III. Observer representations are compared through information geometry Γ.

IV. Objectivity is stable redundant contraction toward a common center Q ⋆ .

V. Physical configurations are selected by δ S IFR = 0 .

VI. Gravitation is emergent observer-stable informational curvature.

VII. Time is the emergent order of stable objectivizations.

VIII. Spacetime and field theory arise as effective high-redundancy limits.

These postulates define the conceptual and mathematical domain of IFR. The next section develops the corresponding formal structure in detail.

Let

be an abstract measurable space equipped with a sigma-algebra Σ X and a reference measure μ X . We write this as

No manifold structure is assumed at this level. In particular, X is not required to be differentiable, locally Euclidean, finite-dimensional, metric, or temporally ordered. Its elements

represent primitive informational configurations rather than spacetime events.

The absence of an assumed spacetime structure is essential. IFR requires that spatial localization, temporal ordering, and metric geometry be derived from later stability conditions rather than inserted into the ontology at the beginning. Thus, the symbols x , y ∈ X should not be interpreted as spacetime points.

The minimal mathematical requirement on X is that it supports a complex Hilbert space of interference states. In the simplest representation, one may define

with inner product

More generally, ℋ X may be any separable complex Hilbert space associated with X , including Fock-type constructions, direct integrals, or Hilbert bundles over X . For the foundational development, separability is assumed unless explicitly stated otherwise:

This assumption ensures that standard spectral, variational, and measure-theoretic tools may be applied.

A fundamental IFR state is a normalized vector

In a representation over X , this state is written as

where Ψ ( x ) is an amplitude associated with a primitive informational configuration.

The phase structure of Ψ is physically relevant. IFR therefore treats the interference pattern encoded in relative phases, overlaps, and superpositions as more fundamental than any classical localization. For two states Ψ , Φ ∈ ℋ X , their interference overlap is

The corresponding transition intensity is

At the primitive level, this expression should not yet be interpreted as a probability of transition between spacetime events. It is a measure of compatibility between interference configurations. Probabilistic and spacetime interpretations become available only after observer projections and stable objectivity conditions have been introduced.

The projective Hilbert space

is the physically relevant state space whenever global phase is unobservable. Thus, states related by

represent the same primitive ray.

When needed, the natural distance between rays is the Fubini-Study distance

This distance is defined on the primitive interference-state space. It is distinct from the later observer-level distance d Γ , which compares projected representations.

Let A be an index set labelling operational observers. Each observer a ∈ A is represented by a map

where ℳ a is the representation space accessible to observer a .

In general, Π a need not be linear. It may include nonlinear reconstruction, coarse-graining, statistical estimation, thresholding, dimensional reduction, environmental encoding, or detector response. However, for the variational theory to be well-defined, IFR restricts attention to an admissible class

satisfying the following minimal conditions.

Measurability. Each observer map is measurable with respect to the sigma-algebra on ℋ X and the measurable structure on ℳ a :

for every measurable set B ⊆ ℳ a .

Finite accessibility. The representation m a = Π a [ Ψ ] contains only finite operational information. This condition may be expressed by requiring either finite-dimensional ℳ a , finite effective rank, finite entropy, or finite statistical resolution. Formally, one may impose

where C denotes an operational complexity, entropy, or description-length functional.

Stability. Small perturbations of the interference state should not generically produce unbounded changes in the observed representation. Thus, for each admissible observer, there exists a local constant L a such that

in a neighbourhood of physically relevant states, where d a is a distance on ℳ a and d ℋ is a Hilbert-space or projective distance.

Operational reproducibility. For repeated application under the same observational conditions, the map Π a must define a reproducible representation up to a noise model η a :

or, more generally,

where P a is an observer-specific response distribution.

The deterministic notation

should therefore be understood as either an exact projection in the idealized case or the expectation of an observational response in the stochastic case:

These assumptions make the observer concept broad enough to include physical detectors, environmental fragments, data-processing choices, catalog partitions, and coarse-grained reconstructions, while still being mathematically controlled.

Observer outputs may initially belong to different representation spaces:

To compare them, IFR assumes the existence of a common comparison space

together with admissible embedding or alignment maps

The common representation of observer a is therefore

For notational economy, we define

and usually write

with the embedding understood.

The comparison space ℳ is the mathematical arena in which objectivity is evaluated. It is not necessarily spacetime. It may be a statistical manifold, a space of probability distributions, a space of signals, a Wasserstein space, a manifold of density matrices, a metric-measure space, or a more general geodesic metric space.

The key requirement is that ℳ be equipped with a distance or divergence allowing the comparison of observer projections.

Let

denote the information-geometric structure on ℳ . Depending on the context, Γ may specify a Riemannian metric, an affine connection, a divergence, an optimal-transport geometry, or a more general metric structure.

In the Riemannian case,

where g is a positive-definite metric tensor and ∇ is a connection [3]. The distance between two points p , q ∈ ℳ is the geodesic distance

where the infimum is taken over all smooth curves γ : [ 0 , 1 ] → ℳ such that

In the statistical case, if ℳ is a parametric family of probability distributions

the canonical local metric is the Fisher information metric

If ℳ is a space of density matrices, a natural choice may be the Bures metric. If ℳ is a space of probability measures, a natural choice may be a Wasserstein metric. IFR does not require a unique geometry for all applications; rather, it requires that the chosen Γ be operationally justified by the class of observer representations under comparison.

The distance induced by Γ will be denoted

For the main theoretical results, we assume that

is a complete geodesic metric space. When uniqueness of Fréchet centers is required, additional convexity assumptions will be imposed.

Given an interference state Ψ, observer projections

and pairwise weights w a b ≥ 0 , the redundant dispersion functional is

Equivalently,

When the observer set is continuous rather than discrete, with observer measure ν on A , the natural generalization is

where w ( a , b ) ≥ 0 is a symmetric pairwise weight kernel.

Choice and constraints on the weights. The weights w a (and w a b ) are not free parameters fitted independently for each application. IFR fixes them by the following operational rule. For each observer a , let σ a 2 denote the effective variance (or, more generally, an operational uncertainty functional) of the response distribution P a ( ⋅ | Ψ ) of Section 3.3. The baseline single-observer weight is the inverse-uncertainty weight

so that more reliable (lower-variance) observers contribute more strongly to the Fréchet functional. Pairwise weights combine the single-observer weights with an operational independence factor ρ a b ∈ [ 0 , 1 ] , estimated from the empirical correlation of the observer residuals η a , η b of Section 3.3:

Strongly correlated observers ( ρ a b → 1 ) are downweighted, since they add little independent redundancy; fully independent observers ( ρ a b = 0 ) recover w a b ∝ w a w b . This rule determines w a and w a b from operationally measurable quantities ( σ a 2 , ρ a b ) once the observer family is specified, and removes the underdetermination otherwise present in the objectivity criterion of Postulate IV. The uniform choice w a = 1 / N , w a b = ( N 2 ) − 1 is recovered as the special case of equal reliability and mutual independence, which is the convention adopted in the cross-domain analyses of [8]-[10].

The functional D red is the first central scalar quantity of IFR. It measures the degree to which observer-dependent projections fail to agree. A low value indicates that independent observational channels encode mutually compatible representations. A high value indicates disagreement, fragmentation, or absence of a stable common structure.

To make this quantity dimensionless in empirical applications, one may introduce a reference or null dispersion

and define the normalized contraction index

Positive values indicate contraction relative to the null model:

Although pairwise dispersion is useful, the definition of objectivity is most naturally expressed through a common center [11][12].

For a fixed Ψ and Γ, define the Fréchet functional

where

A Fréchet center is any minimizer

If the minimizer is unique, we write

where bar Γ denotes the barycenter induced by the geometry Γ.

The distinction between the exact center Q ⋆ and an empirical estimator Q ^ is important. The exact center is a mathematical object defined by the variational problem above. An empirical estimator is constructed from finite, noisy, or sampled observations:

In general,

The estimator Q ^ converges to Q ⋆ only under consistency assumptions on the observer noise, sampling procedure, geometry, and optimization method. IFR therefore treats the exact Fréchet center and empirical overlap estimators as conceptually distinct.

This distinction is essential for maintaining mathematical rigor and for avoiding ambiguity between variational objectivity and data-based reconstruction.

Objectivity requires not only agreement, but stability. A common center that disappears under arbitrarily small changes of the observers cannot be regarded as physically objective.

Let the observer maps be perturbed as

where δ   Π a belongs to an admissible perturbation class and ϵ is small.

The perturbed Fréchet functional is

with minimizer

The center Q ⋆ is stable if

A stronger Lipschitz stability condition is

for some finite constant C .

This condition ensures that the objective structure is not an artifact of a particular observer choice, coordinate representation, smoothing scale, or reconstruction convention.

The gravitational sector of IFR requires that the information geometry Γ possess a curvature quantity. In the Riemannian case, this is standard. If

with ∇ the Levi-Civita connection of g , one defines the Riemann curvature tensor

the Ricci tensor

and the scalar curvature

In this case ℛ Γ is the intrinsic curvature scalar of the observer-comparison geometry.

If ℳ is not smooth, ℛ Γ must be replaced by an appropriate generalized curvature quantity. Possible choices include Alexandrov curvature bounds, Ollivier-Ricci curvature, Bakry-Émery curvature, Forman curvature, or curvature derived from optimal-transport convexity. The choice depends on the mathematical category of ℳ .

In the formal theory, ℛ Γ denotes any curvature scalar satisfying three requirements:

(i) ℛ Γ is intrinsic to the information geometry;

(ii) ℛ Γ controls the deformation of distinguishability relations;

(iii) ℛ Γ → R ( g eff ) in the effective spacetime limit.

The third condition is the gravitational correspondence condition of IFR.

The variational formulation requires histories of the primitive state and geometry. Let

be an ordering parameter. An IFR history is a map

where

The set G denotes admissible information geometries, and P denotes admissible families of observer maps.

The parameter λ is not physical time. It is an ordering parameter used to describe variation, histories, and stationary principles. Physical time emerges only when a sequence of objective centers

admits an intrinsic order relation of the type introduced in Postulate VII (Section 2.7), whose full development is deferred to Part III of the series.

Thus, a history in λ is not yet a temporal evolution. It is a path through the space of interference-geometric configurations.

Regularity conditions for S I F R . We collect here the hypotheses (R1)-(R4) under which the action S IFR of Postulate V is well-posed:

(R1) Ψ ∈ H 1 ( I , ℋ X ) , with ‖ Ψ ( λ ) ‖ X = 1 for all λ ∈ I ;

(R2) V int is continuous, bounded below, and Gâteaux differentiable on ℋ X ;

(R3) Γ ( λ ) is a continuous family of complete geodesic metric structures on ℳ , admitting the curvature scalar ℛ Γ ( λ ) of Section 3.9 as an L 1 ( I ) function of λ ;

(R4) the observer family Π ( λ ) ∈ P satisfies the measurability, finite-accessibility, stability, and reproducibility conditions of Section 3.3 uniformly for λ ∈ I , so that D red [ Ψ ( λ ) , Γ ( λ ) ] ∈ L 1 ( I ) .

Under (R1)-(R4) one has S IFR [ Ψ , Γ , Π ] < ∞ for every admissible history, and the first variation δ S IFR is well-defined for variations δ   Ψ ∈ H 0 1 ( I , ℋ X ) and δ   Γ , δ   Π tangent to G , P . This is the precise sense in which δ S IFR = 0 selects physical configurations in Postulate V.

Combining the previous definitions, the complete kinematic state of IFR is an element of

An element of this space is written

The physical content of such a state is not given directly by Ψ alone. It is given by the triple:

Ψ determines primitive interference,

Γ determines distinguishability and curvature,

Π determines operational accessibility.

The observer-accessible content of s IFR is the family

and the objective content, when it exists, is the corresponding stable Fréchet center

An effective spacetime description becomes possible only under additional embedding and locality conditions.

Let Q ⊂ ℳ denote the subset of stable objective centers:

An effective manifold exists if there is a differentiable manifold ℳ eff and an embedding

such that, for stable centers Q i ⋆ , Q j ⋆ ∈ Q ,

where d eff is the metric distance induced by an effective metric tensor g μ ν eff and ϵ eff measures the residual non-geometric distortion.

If

over the domain of interest, the objective structures admit an approximately spacetime description.

In this case,

is not fundamental. It is an emergent representation of stable informational relations.

The mathematical structure introduced above must recover standard physical kinematics in an appropriate limit. IFR therefore imposes the following correspondence condition.

In the high-redundancy, low-dispersion, stable-center limit,

and under the existence of an effective manifold representation, the observer-independent centers Q ⋆ behave as ordinary physical events, fields, or localized structures on an emergent spacetime:

The information geometry reduces to an effective spacetime geometry:

and its curvature scalar reduces to

Thus, the kinematic limit of IFR reproduces the usual ingredients of relativistic field theory:

events, fields, metric, curvature.

The dynamics of this limit will be specified in the subsequent sections through the IFR action and its variational equations.

Let

be the pre-spatiotemporal configuration space introduced in Section 3, and let

be a normalized interference state.

Let

be a finite family of operational observers. Each observer is represented by an admissible projection map

After alignment into a common information-geometric comparison space ℳ , the observer-dependent representation is written as

The collection

is called the observational orbit of Ψ with respect to the observer family { Π a } a = 1 N .

The objectivity problem is then the problem of determining whether the orbit O Ψ admits a stable common center in the information geometry Γ.

Let

be a complete metric space induced by the information geometry Γ. Given non-negative pairwise weights

define the redundant dispersion functional

Equivalently,

This functional measures the failure of observer projections to agree. If

then all projections coincide pairwise, modulo the identifications induced by d Γ . If

then the observer projections are dispersed in the comparison geometry.

A necessary condition for objectivity at resolution ε > 0 is

However, pairwise contraction alone is not sufficient. IFR also requires the existence of a stable center.

Let

be observer weights. For fixed Ψ and Γ, define the Fréchet functional

A Fréchet center is an element

When the minimizer is unique, we write

The value

is the centered redundant dispersion. It measures how tightly the observer representations concentrate around their common center.

The pairwise dispersion and centered dispersion are related, but not identical. In Euclidean geometry with suitable weights, they are proportional. In general metric or curved information-geometric spaces, they must be distinguished.

The exact Fréchet center

is a mathematical object defined by the geometry and the observer projections. It is not the same as a finite-sample estimator obtained from data. This distinction is essential for the theoretical consistency of IFR.

We now give the central definition.

Definition 1 (IFR-objective structure). Let Ψ ∈ ℋ X be an interference state, let { Π a } a = 1 N be an admissible observer family, and let ( ℳ , d Γ ) be a complete information-geometric comparison space. A structure

is called IFR-objective at resolution ε > 0 if the following conditions hold:

1. Existence of a center:

2. Redundant concentration:

3. Observer stability: for every admissible perturbation

with perturbation norm

the corresponding perturbed center

satisfies

4. Non-triviality relative to a null model: there exists a reference dispersion

such that

The null condition prevents arbitrary agreement from being interpreted as objectivity when the observer family is degenerate, artificially constrained, or informationally empty.

The associated dimensionless contraction index is

Thus,

is a necessary empirical signature of redundant contraction.

The previous definition admits degrees of objectivity. It is useful to distinguish weak, strong, and asymptotic objectivity.

Definition 2 (Weak IFR objectivity). A structure Q Ψ , Γ ⋆ is weakly objective at resolution ε if it satisfies existence and redundant concentration:

Weak objectivity means that the observer projections agree, but does not guarantee robustness under perturbations.

Definition 3 (Strong IFR objectivity). A structure Q Ψ , Γ ⋆ is strongly objective at resolution ε if it is weakly objective and stable under all admissible perturbations of the observer maps, weights, and comparison geometry:

In strong objectivity, the center is not an artifact of a particular representation, weighting scheme, or geometric convention.

Definition 4 (Asymptotic IFR objectivity). Let { A N } N ∈ ℕ be an increasing sequence of observer families. A structure Q ⋆ is asymptotically objective if

and

Asymptotic objectivity is the ideal limit in which infinitely many independent observational channels converge to the same informational structure.

The existence of a Fréchet center is not automatic in arbitrary metric spaces. IFR therefore imposes mathematical conditions when exact objectivity is required.

Proposition 1 (Existence of Fréchet centers).Let ( ℳ , d Γ ) be a proper complete metric space,meaning that closed bounded subsets are compact. Let

be observer projections with finite weights w a ≥ 0 and ∑ a w a = 1 . Then the Fréchet functional

admits at least one minimizer.

Proof. Since F ( Q ) ≥ 0 , let { Q k } be a minimizing sequence. Choose any observer point m 1 . Because

is bounded along the minimizing sequence, and because at least one weight is positive, the distances d Γ ( Q k , m a ) remain bounded for at least one a and therefore, by the triangle inequality, { Q k } lies in a bounded closed subset of ℳ . Properness implies that a convergent subsequence exists:

Continuity of d Γ 2 implies

Thus Q ⋆ is a minimizer.

This proposition establishes that objective centers exist under standard compactness assumptions.

Existence alone is insufficient for a rigorous theory of objectivity. If multiple incompatible centers exist, the observer family does not select a unique objective structure. IFR therefore requires uniqueness for strong objectivity.

Proposition 2 (Uniqueness under strict geodesic convexity).Let ( ℳ , d Γ ) be a geodesic metric space. Suppose that for every observer projection m a ,the function

is strictly geodesically convex on a convex domain U ⊆ ℳ . If all observer projections lie in U and at least two non-identical projections have positive weight, then the Fréchet functional

has at most one minimizer in U .

Proof. Assume, for contradiction, that Q 0 and Q 1 are two distinct minimizers in U . Let γ : [ 0 , 1 ] → U be a geodesic with

By strict geodesic convexity,

for t ∈ ( 0 , 1 ) and for at least one term with positive weight. Summing over a gives

Since Q 0 and Q 1 are both minimizers,

Therefore,

which is impossible. Hence the minimizer is unique. □

In smooth Riemannian settings, strict geodesic convexity is guaranteed locally in sufficiently small geodesic balls, and globally under stronger curvature assumptions. Thus IFR objectivity is naturally local when the comparison geometry is curved.

Strong objectivity requires that the Fréchet center depend continuously on the observer projections. The following statement gives a sufficient condition.

Proposition 3 (Stability under perturbations). Let ( ℳ , d Γ ) be a geodesic metric space, and suppose the Fréchet functional

has a unique minimizer Q ⋆ . Let perturbed observer projections

satisfy

Assume that the perturbed functionals

are equicoercive and converge uniformly to F on compact sets as δ → 0 . Then any sequence of minimizers

satisfies

Proof. Uniform convergence on compact sets and equicoercivity imply convergence of minimizers for variational problems with unique limiting minimizer. More explicitly, any sequence Q δ ⋆ has a compactly convergent subsequence by equicoercivity. Let its limit be Q ˜ . Uniform convergence implies

Thus Q ˜ is a minimizer of F . Since the minimizer is unique,

Every convergent subsequence has the same limit; hence

□

This result formalizes the intuition that an objective structure must persist under small changes of the observational channel.

A central distinction in IFR is the distinction between exact theoretical centers and empirical estimators. Let

be the exact Fréchet center associated with the ideal observer projections

In empirical applications, one observes noisy finite data

or more generally,

where P a is an observer-specific response distribution. An empirical estimator is then defined by

where

In general,

A consistency condition may be stated as

as observational noise decreases, sample size increases, and the estimated geometry satisfies

This distinction prevents the theory from confusing an estimator-based overlap with the variational object that defines IFR objectivity. In particular, an empirical geometric overlap is evidence for objectivity only if it can be shown to approximate a stable Fréchet center.

Agreement among observer projections is physically meaningful only relative to an appropriate null hypothesis. Otherwise, trivial projections or shared preprocessing may artificially reduce dispersion.

Let

denote a null ensemble of observer projections generated under the hypothesis that no common underlying structure is shared. This may be constructed by randomization, permutation, independent phase scrambling, surrogate generation, noise-only simulations, or model-specific independence assumptions.

The null redundant dispersion is

The observed contraction index is

A statistically significant positive contraction,

supports the existence of a shared structure. A value near zero indicates no detectable redundant objectivity. A negative value indicates that the observer projections are more dispersed than expected under the null.

The normalized index is not itself the definition of objectivity. It is an empirical diagnostic of the contraction component of objectivity.

IFR does not define objectivity as independence from all observers. Rather, it defines objectivity as invariance across an admissible family of observers. This distinction is important.

A structure may be objective relative to a class

of observational maps, while failing to be objective relative to a broader or incompatible class. Thus objectivity is always defined relative to:

This relativity does not make objectivity subjective. Instead, it makes explicit the operational domain within which objectivity is claimed.

In the asymptotic limit of increasingly rich, independent, and mutually non-degenerate observer families, objectivity approaches observer invariance:

This is the IFR analogue of classical observer-independent reality.

Classicality is interpreted in IFR as a high-redundancy regime. A structure behaves classically when it is:

(i) redundantly accessible to many observers;

(ii) stable under perturbations of observer maps;

(iii) localized in an emergent effective geometry;

(iv) persistent across an ordered sequence of objective centers.

Thus classicality is not imposed as a fundamental approximation. It is derived from the combined limits

together with the emergence of an effective metric manifold.

In this regime, the observer-dependent representations become approximately interchangeable:

The structure Q ⋆ may then be treated as an ordinary physical object, event, or field value in an effective spacetime.

We now state the central objectivity theorem of IFR.

Theorem 1 (Emergence of objective structure). Let ( ℳ , d Γ ) be a proper complete geodesic metric space, and let

be observer projections of an interference state Ψ. Suppose that:

1. All m a lie in a geodesically convex domain U ⊆ ℳ ;

2. The squared distance functions

are strictly geodesically convex on U .

3. The Fréchet functional

is coercive on U .

4. The centered dispersion satisfies

5. The observer projections are stable under admissible perturbations.

Then there exists a unique stable center

such that

and this center is IFR-objective at resolution ε .

Proof. Properness and coercivity imply existence of a minimizer. Strict geodesic convexity implies uniqueness. The dispersion bound gives

Stability follows from the assumed perturbative stability of the observer projections and the stability proposition above. Therefore Q Ψ , Γ ⋆ satisfies all conditions in the definition of IFR-objectivity.

This theorem is the formal expression of the principle that objective reality emerges when observer-dependent projections contract toward a unique stable informational center.

The mathematical content of this section can be summarized as follows:

The role of the interference state Ψ is to generate the observer-dependent projections. The role of the observer maps Π a is to define finite operational access. The role of the information geometry Γ is to make comparison and contraction meaningful. The role of the Fréchet center Q ⋆ is to define the objective structure that emerges from the observer family.

This formulation avoids three common ambiguities.

First, objectivity is not reduced to subjective agreement. The observer maps are physical or operational structures, and the center is defined by a variational problem.

Second, objectivity is not identified with a single detector, representation, or reconstruction. It requires redundancy across an admissible family of projections.

Third, objectivity is not confused with empirical overlap. The exact center Q ⋆ and the estimator Q ^ are mathematically distinct.

This provides the foundation for the dynamical theory developed in the next section, where the emergence of objective structures is incorporated into the IFR action through the redundant dispersion term.

IFR shares vocabulary with several established programs, and it is therefore important to state precisely where its specific content lies. We distinguish IFR from four families of approaches.

Decoherence and environment-induced superselection. Decoherence explains the suppression of interference for a system coupled to an environment, and selects a preferred (pointer) basis through interaction with that environment. It presupposes a background spacetime, a fixed system/environment split, and an a priori Hilbert-space tensor structure. IFR does not assume any of these: there is no background manifold, no privileged system/environment partition, and observers are general projection maps Π a rather than environmental couplings. Where decoherence accounts for the loss of coherence given a structure, IFR specifies the conditions under which an objective structure (a stable Fréchet center) exists at all.

Quantum Darwinism. Quantum Darwinism characterizes classicality through the redundant proliferation of records of a pointer observable into many environmental fragments, quantified by redundancy and mutual information. IFR generalizes the redundancy idea in two ways. First, redundancy is not restricted to environmental fragments but extends to any admissible observer family (detectors, coarse-grainings, epochs, catalog partitions). Second, IFR replaces the existence of multiple records with a sharper criterion: the existence, uniqueness, and stability of a Fréchet center Q ⋆ in a specified information geometry (Definition 1, Theorem 1). Mere multiplicity of records is necessary but not sufficient for IFR-objectivity.

Relational and informational interpretations. Relational quantum mechanics and related views hold that physical quantities are defined only relative to an observer or reference system, and decline to assign observer-independent values. IFR agrees that single-observer projections are relational, but does not stop there: it defines an intersubjective invariant, the Fréchet center, and treats objectivity as the stable contraction of relational projections toward that invariant. Objectivity in IFR is thus relational at the level of Π a but observer-invariant at the level of Q ⋆ , with the relativity made explicit through the class

of Section 4.11.

Emergent-gravity programs. Thermodynamic and entropic-gravity programs derive gravitational dynamics from coarse-grained or holographic degrees of freedom, typically assuming an underlying spacetime or causal/horizon structure. IFR locates gravitation one level deeper, in the curvature ℛ Γ of an information geometry over observer projections, with the spacetime metric itself emergent (Postulate VI, Section 3.9). The dynamical realization of this identification is the subject of Part II; here we only note that the carrier of gravitational information in IFR is the comparison geometry Γ, not a pre-existing manifold.

In summary, the specific content of “interference-first” is that objectivity,time,and spacetime are all defined as stability properties of observer projections of a pre-spatiotemporal interference state, rather than presupposed. This single criterion—stable, unique, perturbation-robust Fréchet contraction relative to an explicit null model—is what distinguishes IFR from a restatement of the programs above.

IFR is intended to be discriminating, not merely interpretive. We list concrete signatures that would support IFR over nearby frameworks, together with the outcomes that would count as failure.

(F1) Positive,null-tested contraction. IFR predicts a statistically significant positive contraction index Δ IFR > 0 (Section 3.6) for genuinely redundant observer families, exceeding an appropriate null ensemble N (Section 4.10). Failure: Δ IFR ≤ 0 , or contraction indistinguishable from the null, for families that are independently known to share structure.

(F2) Observer-class invariance of the center. The recovered center Q ^ must be stable under enlargement and perturbation of the admissible observer class

(Definition 1, condition 3; Proposition 3). Failure: the center drifts beyond the stability bound under admissible changes of smoothing scale, instrumental mode, or catalog partition, indicating a protocol artifact rather than an objective structure.

(F3) Geometry-selection consistency. Because objectivity is defined relative to a comparison geometry Γ, IFR predicts that the ranking of structures by contraction is stable across operationally admissible geometries (e.g. Jensen-Shannon versus Bures versus Wasserstein), even when absolute values differ. Failure: qualitatively different, geometry-dependent conclusions for the same data under equally justified Γ, with no geometry-independent invariant surviving.

(F4) Convergence with observer number. Asymptotic objectivity (Definition 4) predicts F Ψ , Γ ( N ) ( Q N ⋆ ) → 0 and Q N ⋆ → Q ⋆ as the number of independent observers grows. Failure: no convergence, or convergence to incompatible centers, as independent channels are added.

(F5) Emergent-metric correspondence (deferred test). In the high-redundancy regime, IFR predicts the existence of an effective metric embedding with small residual distortion ϵ eff ≪ 1 (Section 3.12). Failure: persistent, irreducible non-metric distortion of the stable centers, precluding any effective spacetime description. The quantitative form of this criterion is developed in the dynamical parts of the series.

Criteria (F1)-(F4) are testable with the cross-domain estimators already employed in [8]-[10]; (F5) is theoretical at the level of Part I and becomes operational once the IFR action and its emergent-metric limit are specified.

The objectivity results of this section are conditional, and we state the conditions explicitly. First, IFR-objectivity is defined relative to a chosen comparison geometry Γ; existence and uniqueness (Propositions 1 - 2) rely on completeness, properness, and strict geodesic convexity of ( ℳ , d Γ ) , which hold locally in geodesic balls but may fail globally in curved geometries. Objectivity is therefore generically a local statement, and the admissible class of geometries has not yet been fully characterized. Second, the criterion depends on the null model N : the contraction index Δ IFR is meaningful only relative to a justified null ensemble, and a poorly chosen null can either mask or manufacture apparent objectivity. Third, objectivity is defined relative to an admissible observer class

(Section 4.11); a structure objective for one class need not be objective for a broader or incompatible class, so all claims are conditional on the operational domain. These conditional features are not defects to be hidden but the precise content of the IFR notion of objectivity: the framework makes its dependence on geometry, null model, and observer class explicit rather than implicit. Narrowing these dependencies—constraining the admissible geometries, deriving canonical null models, and characterizing maximal observer classes—is left to the subsequent parts of the series.

Continuation of the series. Part II develops the IFR variational principle and the emergence of gravitation from informational curvature.