Let I ⊆ ℝ be an interval of the ordering parameter λ . An IFR history is a map

where

Here G denotes the admissible class of information geometries, and P denotes the admissible class of observer families. The parameter λ is not identified with physical time. It is an ordering parameter on the space of possible interference-geometric configurations. Physical time will be derived later from the ordered sequence of stable objective centers.

The space of admissible histories is denoted

A physical IFR history is a stationary point of an action functional

Thus, δ S IFR = 0 is the fundamental dynamical condition.

The curvature term of the IFR action is evaluated as an integral over the comparison manifold, not as a scalar attached to the ordering parameter. We fix this, together with the regularity class of the manifold on which the metric variation of Section 3 is performed, before writing the action, since that variation presupposes a volume measure, a boundary prescription and a fixed dimensional bookkeeping.

Assumption II.1 (Comparison domain and its regularity).Let ( ℳ , g Γ ) be the information-geometric comparison space introduced in Part I of the series. Throughout the variational derivation of this paper we assume the following.

(M1) Smoothness. ( ℳ , g Γ ) is a C ∞ Riemannian manifold. In the canonical realisation ℳ is the interior Δ d ∘ of the simplex of probability vectors over d + 1 outcomes, or a smooth parametric family of distributions, on which the Fisher-Rao metric is real-analytic.

(M2) Dimension. We write

fixed along the history, so that d = ( d + 1 ) − 1 is the dimension of the canonical simplex realisation of (M1). The realisations employed in this series require d ≥ 5 ; the value d = 6 holds under the additional conditions stated in the gauge-sector analysis of the series. The symbol d is reserved for the dimension of the comparison manifold and is distinguished throughout from the number d + 1 of measurement outcomes entering the observer maps.

(M3) Signature. g Γ is positive definite. No Lorentzian structure is assumed on ℳ : the indefinite signature belongs exclusively to the emergent manifold ℳ eff and arises through the informational Wick mechanism developed in Part V. This is the precise point at which the general information-geometric framework and the smooth Riemannian realisation required by the curvature sector are to be distinguished: the kinematic framework of Part I fixes a comparison geometry Γ = ( ℳ , g , ∇ ) without imposing the dimensional curvature realisation adopted here, and admits non-smooth comparison structures, whereas everything stated in Sections 2-3 concerning ℛ Γ presupposes (M1)-(M7); see Remark II.1.

(M4) Domain and boundary conditions. The action is evaluated on a relatively compact, connected, geodesically convex open subset Ω ⊂ ℳ with piecewise-smooth boundary ∂ Ω . Metric variations obey the Dirichlet condition

δ g i j Γ | ∂ Ω = 0 , and the Gibbons-Hawking-York boundary term κ IFR − 1 ∮ ∂ Ω K d σ , with

K the trace of the second fundamental form of ∂ Ω , is understood to be included so that the variational problem for ℛ Γ is well posed. Where Ω is taken closed without boundary the term is absent.

(M5) Topology. Ω is contractible and may be taken to be a Fisher-Rao normal ball. No global topological assumption is placed on ℳ , and all correspondence results of Section 3 are local in Ω.

(M6) Two levels of the comparison geometry and the informational length scale. A distinction must be drawn between the dimensionless divergence level, on which the empirical instantiations of the series operate, and the dimensional metric level required by the curvature sector. At the divergence level the comparison distance is the dimensionless distance d ˜ Γ induced by g ˜ Γ : = ℓ Γ − 2 g Γ , which is realised as the Jensen-Shannon distance in the high-redundancy regime,

and is not equal to it off shell. The second-order expansion of d JS at q = p + δ is, with natural logarithms,

so that the dimensionless metric associated with d JS is 1 8 g FR . The dimensional

analysis of Section 2.4 requires [ ℛ Γ ] = L − 2 and hence [ g Γ ] = L 2 ; the two levels are therefore related, exactly and off shell, by a single positive informational length scale ℓ Γ ,

where ℛ ˜ Γ and V ˜ Γ ( Ω ) are the scalar curvature and the Riemannian volume of g ˜ Γ and are dimensionless. The Fisher-Rao form appears only in the high-redundancy regime,

and is not imposed off shell, where g ˜ Γ is any metric obeying (M1)-(M5). Distances at the two levels are related exactly by d g Γ = ℓ Γ d ˜ Γ , and only in the high-redundancy regime does this become

off shell no relation to d JS is asserted. Wherever a dimensionless norm is required, as in Remark II.2, it is g ˜ Γ that is understood. The identification of g ˜ Γ with the rescaled Fisher-Rao metric is a matching condition in the high-redundancy regime and not a kinematic constraint on Γ, which off shell is any metric obeying (M1)-(M5); see Remark II.3.

(M7) Reduction scale and the characteristic volume:a matching ansatz. The semiclassical matching of Section 2.4 is performed against the four-dimensional Einstein-Hilbert action and therefore involves a three-dimensional characteristic volume, [ V char ] = L 3 , whereas [ V Γ ( Ω ) ] = L d . We adopt as a matching ansatz that the two are related by the same scale,

so that κ red is the reduced coupling matched to κ eff and

every term of the action then carrying the dimension of energy. We state the epistemic status of this ansatz precisely. The relation κ IFR = ℓ Γ d − 3 κ red does not by itself determine ℓ Γ : it links two quantities neither of which is derived here. Once either ℓ Γ or κ IFR is fixed independently, the matching fixes the other. What is not supplied in this paper is the interpretation of ℓ Γ as the scale at which multi-observer redundancy performs the reduction from d to three effective dimensions; that identification belongs to the dimensional-reduction argument of Parts V and VI and is recorded as a limitation in Section 3.17.

Remark II.1 (Compatibility with Part I).Part I of the series defines the comparison geometry as Γ = ( ℳ , g , ∇ ) with g positive definite, denotes by d Γ the geodesic distance of g , and writes the redundant dispersion functional as D red = ∑ a < b w a b d Γ 2 ( Π a [ Ψ ] , Π b [ Ψ ] ) . No dimensional assignment is made there, the treatment of Part I being kinematic. The present paper preserves that content without alteration, under the identification

so that every formula of Part I, and in particular the dispersion functional and the definition of the stable-center set Q , is unchanged. What is new here is only the dimensional metric g Γ = ℓ Γ 2 g ˜ Γ , introduced because the curvature sector requires [ ℛ Γ ] = L − 2 while a statistical metric is dimensionless; the tilde is therefore a notational refinement made necessary by the gravitational sector of the present paper and not a correction of Part I.

The identification propagates to the curvature, and this must be stated explicitly. Since the scalar curvature of a metric rescaled by a constant c is divided by c , the curvature built from the metric of Part I is the dimensionless one,

where ℛ Γ is the dimensional scalar used here. Preserving the curvature term of the action therefore requires β I ℛ ˜ Γ = β ℛ Γ , that is

This is not a correction of Part I, which assigns no dimensions: it is the dimensional refinement introduced here, and without it the identification would preserve g and d Γ but not the curvature derived from them. The positivity α , β > 0 asserted in Part I is unchanged, and the curvature entry of the reduced action is here the mean curvature ℛ ¯ Γ of Lemma II.1.

The reconstruction map requires the same care. Part I writes χ I : Q → ℳ eff , whereas the map of Theorem II.1 (H1) is defined on the event manifold. The former is not replaced but recovered as the equal-time restriction of the latter: with j t : Q 3 ( t ) ↪ ℰ the inclusion of the slice,

Part I commits to no dimension for its domain, and the present paper supplies one once the reduction is counted.

With Assumption II.1 in place, the IFR action is defined as the double integral

where the fraktur symbols denote the corresponding densities on Ω and

Equation (1) is the definition employed in the metric variation of Section 3; the reduced form used in the next subsection is a consequence of it and not an independent postulate.

Lemma II.1 (Zero-mode reduction).Define the mean informational curvature on the comparison domain,

Suppose the source densities are uniform on Ω at the order considered, that is, k int , v int , d red and l Π do not depend on q ∈ Ω . Then (1) reduces to

with K int = V Γ ( Ω ) k int and correspondingly for the remaining source terms, and with

Proof. The q -integration of a q -independent density multiplies it by V Γ ( Ω ) . For the curvature term no uniformity hypothesis is required: by the definition of ℛ ¯ Γ one has the identity β ^ ∫ Ω d μ Γ   ℛ Γ = β ^ V Γ ( Ω ) ℛ ¯ Γ = β ℛ ¯ Γ . The last equality of the statement is (M7). ☐

Three consequences should be noted. First, since Γ is itself varied, V Γ ( Ω ) is in general λ -dependent and so is β = β ^ V Γ ( Ω ) . The action to be varied is therefore always the double integral (1), in which the constant is β ^ = 1 / ( 2 κ IFR ) ; the reduced form of Lemma II.1 is a subsequent contraction, used for bookkeeping and for the semiclassical matching, and is never used to compute δ S / δ g Γ . Second, the reduced form carries the mean curvature ℛ ¯ Γ and not a pointwise value: the curvature sector is never assumed uniform, and the full q -dependence is retained in the metric variation of Section 3. Third, the characteristic volume of Section 2.4 is identified through (M7) with the reduced volume V Γ ( Ω ) / ℓ Γ d − 3 , so that it is a property of the comparison domain and of the informational length scale rather than a free normalisation constant. The zero-mode hypothesis on the source terms is the minimality assumption of the present paper; its relaxation, which promotes those terms to genuine fields on Ω, belongs to the renormalisation-group treatment of Part VI.

The minimal IFR action is defined as

where the dot denotes differentiation with respect to the ordering parameter λ .

The minimal Lagrangian is postulated to have the form

where the positive sign of the curvature coupling is fixed by the semiclassical matching of Section 2.4 and is used throughout. The curvature entry is the mean curvature ℛ ¯ Γ of Lemma II.1 and the coupling is β = V Γ ( Ω ) / ( 2 κ IFR ) ; this reduced form is used for bookkeeping and for the semiclassical matching only, the action varied in Section 3 being always the double integral (1). Thus,

(See Section 2.4 for the sign of β and the dimensional normalisation of K int .)

The terms have the following roles.

is the kinetic or propagating component of the interference state. It controls how interference configurations change along admissible histories.

is an informational potential. It encodes selection, instability, symmetry breaking, or self-organization of the interference structure.

is the redundant multi-observer dispersion. It penalizes configurations for which observer projections fail to contract toward a common representation.

is the scalar curvature of the information geometry. It supplies the geometric sector of the theory and is the precursor of effective gravitational curvature.

is an optional observer-kinematic or regularization term. It constrains the admissible variation of observer maps and prevents pathological observer families from dominating the variational problem.

The constants α > 0 ,   β > 0 set the relative strength of objectivizing contraction and informational curvature.

In many formal developments, one may set ℒ Π = 0 and treat the observer family Π as an externally fixed admissible class. However, for a fully dynamical theory, Π may also be varied subject to operational constraints.

Earlier formulations of the framework wrote the curvature term of the action with a negative sign, − β ℛ Γ . A dimensional and sign analysis fixes both the orientation of this term and the scale of the kinetic term; we record the result here, as it is required for the semiclassical matching carried out in Part III of the series, and it is the convention adopted throughout the present series.

Sign of the curvature coupling.In the semiclassical regime treated in Part III of the series, the curvature term must reproduce the Einstein-Hilbert action

Matching − β ℛ Γ onto + 1 2 κ eff R would force β < 0 , i.e. a repulsive (wrong-

sign) gravitational sector. The correspondence limit in which this matching is performed is the one established in Theorem II.1 through the Wick correspondence, namely ℛ ℰ ( L ) → R ( g eff ) for the post-Wick scalar of the event manifold, and not a direct identification of the comparison scalar ℛ Γ with R ( g eff ) . The correct matching is obtained by adopting the curvature term with a positive sign,

The qualification is needed because the reduced action of Lemma II.1 integrates over the ordering parameter alone, the integration over the comparison domain having already been carried out: the unnormalised coupling is β = V Γ ( Ω ) / ( 2 κ IFR ) = V char / ( 2 κ red ) , with V char = V Γ ( Ω ) / ℓ Γ d − 3 and κ red = κ IFR / ℓ Γ d − 3 as fixed in Assumption II.1 (M7). The writing above records the ratio β / V char fixed by the semiclassical matching, as detailed in the dimensional analysis that follows. With this convention the emergent gravitational sector is attractive and Newton’s constant is recovered with the correct sign. All subsequent variational identities are understood with this orientation; the earlier − β ℛ Γ writing corresponds to the opposite curvature-sign convention and is not used in the semiclassical matching.

Dimensional consistency of the kinetic term.Writing [ ⋅ ] for physical dimension with [ S ] = M L 2 T − 1 (action) and [ λ ] = T , every term of ℒ IFR must carry dimension M L 2 T − 2 (energy). The dispersion term α D red with α = ℏ / τ dec and dimensionless D red gives M L 2 T − 2 ; the curvature term with β = ℏ c V char / ( 16 π ℓ P 2 ) and [ ℛ Γ ] = L − 2 likewise gives M L 2 T − 2 . Two remarks are needed on the status of this normalisation. First, the reduced IFR action integrates over the ordering parameter alone, whereas the Einstein-Hilbert action integrates over the four-volume; β therefore carries the characteristic volume V char of the region of the comparison manifold on which ℛ Γ is evaluated, which by Assumption II.1 (M7) is the reduced volume V Γ ( Ω ) / ℓ Γ d − 3 of the comparison domain, of dimension L 3 as the matching requires. Second, the coupling that appears in the informational field equation of Section 3 is κ IFR , defined on the comparison manifold, and not the emergent spacetime coupling κ eff . The defining relation is therefore

the limit being the one established in Theorem II.1 through the Wick correspondence, that is ℛ ℰ ( L ) → R ( g eff ) for the post-Wick scalar of the event manifold, and not a direct identification of the comparison scalar ℛ Γ with R ( g eff ) , where κ red = κ IFR / ℓ Γ d − 3 is the reduced coupling of Assumption II.1 (M7). It is the reduced coupling, and not κ IFR itself, that is matched to κ eff : for d ≠ 3 the two differ by the reduction factor. So that in the semiclassical regime

The semiclassical matching fixes the ratio β / V char = 1 / ( 2 κ eff ) in that limit; the dimensional argument fixes the overall scale. The identification κ red → κ eff is imposed by matching and is not derived here (Section 3.17). Throughout the series we write the matching in the form β / V char = 1 / ( 2 κ eff ) and never as β = 1 / ( 2 κ eff ) : the two are not the same quantity, since [ β ] = E   L 2 whereas [ κ eff − 1 ] = E   L − 1 . What the sign convention fixes is β > 0 ; what the semiclassical matching fixes is the ratio. The Fubini-Study kinetic term as written, K int = ℏ IFR 2 2 m Ψ g FS ( Ψ ˙ , Ψ ˙ ) , carries dimension M L 4 T − 4 : for a dimensionless normalised state [ g FS ( Ψ ˙ , Ψ ˙ ) ] = T − 2 , while [ ℏ IFR 2 / m Ψ ] = M L 4 T − 2 . Restoring the energy dimension

therefore requires division by a squared velocity, not by a squared length: division by a length scale L 0 2 alone would leave M L 2 T − 4 . We accordingly adopt the dimensionally corrected form

where c IFR is the informational light speed of the Lorentzian sector (Part V of the series) and L 0 is the IFR coherence length, that is the Compton length associated with the interferential inertia m Ψ . The two writings are identical, and the identification L 0 → ℓ P in the fundamental regime is no longer an independent stipulation: together with c IFR → c and ℏ IFR → ℏ it fixes m Ψ → m P = ℏ c / G . Combined with the variance identity (3) below, the kinetic term admits the transparent reading

that is, the energy variance of the interference state measured in units of its interferential rest energy. The potential term V int carries the dimension of energy directly, so that [ K int ] = [ V int ] = M L 2 T − 2 . With these assignments the action is dimensionally homogeneous, every term carrying the dimension of energy.

Remark II.2(Status of the ordering parameter in the dimensional analysis).Assigning [ λ ] = T is a choice of unit, not an identification of λ with physical time, and the two are distinguished by structure rather than by dimension. Writing λ = τ ⋆ ς with ς dimensionless and τ ⋆ > 0 a fixed interferential scale, the assignment [ λ ] = T records only that λ is measured in units of τ ⋆ ; solely the ratios λ / τ ⋆ enter the field equations. Sharing a physical dimension is not sharing an identity: proper time and coordinate time both carry the dimension T and are nonetheless distinct structures, as are arc length along a curve and a spatial coordinate.

What the framework denies is therefore not the dimension but the identification, and the denial is constructive rather than stipulative. Physical time is defined in Part III of the series as the Jensen-Shannon arc length along ordered sequences of stable Fréchet centers. Since d ˜ Γ = d JS is dimensionless at the divergence level of Assumption II.1 (M6), the arc length is converted into a duration by the interferential scale itself,

so that λ ↦ t is a degenerate, in general non-invertible reparametrisation. Three structural properties separate the two. First, t is defined only where the variational problem selects a center: for a single observer, N = 1 , the redundant dispersion vanishes identically, D red ≡ 0 , the minimisation defining Q ⋆ is completely degenerate—every configuration is a minimiser—and no curve λ ↦ Q ⋆ ( λ ) is selected, so that the arc length above is not defined and no temporal sector exists while the ordering parameter continues to run. Second, t is oriented by the monotone growth of redundancy along the history, whereas λ carries no arrow. Third, t is observer-relative, being constructed from the projections Π a , whereas λ is not. A parameter whose associated duration can fail to be defined at all is not the duration it is alleged to be; no contradiction therefore arises from the two statements that λ is not time and that λ is measured in units of time.

An equally consistent bookkeeping takes λ dimensionless and assigns the dimension of action to ℒ IFR ; the two conventions differ by the constant factor τ ⋆ and lead to the same field equations. We retain [ λ ] = T because it allows the semiclassical matching of Part III to be carried out term by term.

The minimal action introduced above contains a kinetic term K int [ Ψ , Ψ ˙ ; Γ ] that has not yet been specified. Any concrete realisation of K int must respect the foundational constraints of IFR: in particular Postulate I together with Section 3.1 of Part I, which prohibit the use of any differentiable, metric, or locally Euclidean structure on the pre-spatiotemporal configuration space X . A choice such as K int ∝ ‖ ∇ X Ψ ‖ 2 would presuppose a differential structure on X and is therefore inadmissible at this level. The kinetic term must be built exclusively from objects available on ℋ X , namely the inner product 〈 ⋅ , ⋅ 〉 X and the resulting projective structure on ℙ ( ℋ X ) .

This requirement selects a unique canonical choice. On the projective Hilbert space ℙ ( ℋ X ) , the Fubini-Study metric g FS is the unique (up to a positive multiplicative constant) Riemannian metric invariant under the full unitary group U ( ℋ X ) of the underlying Hilbert space [2][3]. This uniqueness mirrors Chentsov’s theorem on Δ n [14], established in Part VI of the series (the renormalisation-group flow): just as g FR is the unique monotone metric on the comparison space, g FS is the unique unitarily invariant metric on the primitive interference state space. The two are sibling structures, distinguished by the level at which they act ( ℙ ( ℋ X ) vs. ℳ ) and unified by their common information-geometric origin.

Remark II.3 (Chentsov uniqueness and the variability of Γ).An apparent tension must be recorded, since the IFR action varies Γ while Chentsov’s theorem asserts that the Fisher-Rao metric is, up to a constant factor, the unique monotone metric on the comparison space. Were Γ constrained to be Fisher-Rao kinematically, the variation δ S / δ   Γ = 0 would be over-determined and, on Δ d ∘ , the curvature ℛ Γ would be constant, so that the gravitational sector would carry no dynamics. The resolution is that monotonicity under Markov morphisms, the hypothesis of Chentsov’s theorem, is a sufficiency requirement, and sufficiency is meaningful in IFR only where multi-observer contraction has already taken place. Accordingly the admissible class G consists of Riemannian metrics on Ω satisfying monotonicity only asymptotically, in the limit D red → 0 ; off shell Γ is unconstrained beyond (M1)-(M5), and the Fisher-Rao form recorded in (M6) is the on-shell matching condition recovered as a fixed point of the Γ-dynamics rather than a kinematic constraint imposed on the metric that is to be varied. A testable corollary follows: in the high-redundancy regime the comparison geometry must flow towards the constant-curvature Fisher-Rao geometry, whose sectional curvature on the simplex is constant and positive, so that what the framework predicts is constancy of the informational curvature and not its vanishing. The corresponding flow is analysed in Part VI.

Definition II.1 (Canonical IFR kinetic term).The canonical kinetic term for the interference state is

where m Ψ > 0 is the IFR interferential inertia, ℏ IFR > 0 is the fundamental IFR action quantum, and the prefactor ℏ IFR 2 / ( 2 m Ψ c IFR 2 ) = 1 2 m Ψ L 0 2 is fixed by the dimensional analysis of Section 2.4.

Proposition II.1(Properties of K int FS ).The kinetic term (2) satisfies:

1) Gauge invariance: K int FS is invariant under global and local phase transformations Ψ ( λ ) → e i θ ( λ ) Ψ ( λ ) , consistent with the projective nature of the state space (Postulate I and Section 3.2 of Part I of the series).

2) No structure on X : the expression uses only the inner product 〈 ⋅ , ⋅ 〉 X on ℋ X ; no manifold, metric, or differential structure on X is invoked.

3) Uniqueness: any positive-definite U ( ℋ X ) -invariant quadratic form on tangent vectors to ℙ ( ℋ X ) is a positive multiple of g [ Ψ ] FS .

4) Variance interpretation: if there exists an emergent self-adjoint operator H ^ on ℋ X such that Ψ ˙ = − ( i / ℏ IFR ) H ^ Ψ , then

5) Speed limit: the Anandan-Aharonov inequality

holds along admissible histories, providing an intrinsic IFR quantum speed limit (Mandelstam-Tamm bound in pre-spatiotemporal form).

Proof. (1) Direct computation: 〈 Ψ ˙ + i θ ˙   Ψ , Ψ ˙ + i θ ˙   Ψ 〉 − | 〈 Ψ , Ψ ˙ + i θ ˙ Ψ 〉 | 2 = | 〈 Ψ ˙ , Ψ ˙ 〉 − 〈 Ψ , Ψ ˙ 〉 | 2 , since cross terms cancel due to ‖ Ψ ‖ = 1 . (2) Immediate from the definition. (3) ℙ ( ℋ X ) is a homogeneous space U ( ℋ X ) / ( U ( 1 ) × U ( Ψ ⊥ ) ) whose isotropy representation on T [ Ψ ] ℙ ( ℋ X ) ≅ Hom ℂ ( ℂ Ψ , Ψ ⊥ ) ≅ Ψ ⊥ is complex-irreducible with commutant ℂ . By Schur’s lemma the space of invariant real symmetric bilinear forms on the tangent space is one-dimensional; hence every positive-definite invariant form is a positive multiple of g [ Ψ ] FS , which is the Fubini-Study metric underlying the statistical-distance and geometric-phase results of [2][3]. (4) Substituting Ψ ˙ = − ( i / ℏ IFR ) H ^ Ψ ,

whose difference is the variance. (5) The function λ ↦ d FS ( [ Ψ ( λ ) ] , [ Ψ ( 0 ) ] ) = arccos | 〈 Ψ ( λ ) , Ψ ( 0 ) 〉 | is 1-Lipschitz with respect to Fubini-Study arc length, so its derivative is bounded above by the Fubini-Study speed g [ Ψ ] FS ( Ψ ˙ , Ψ ˙ ) , which by (4) equals Δ E Ψ / ℏ IFR with Δ E Ψ = Var Ψ ( H ^ ) 1 / 2 . ☐

Remark II.4(Relation to the quadratic velocity functional of Part I).Part I of the series introduced, for the purpose of the regularity conditions (R1)-(R4) that render S IFR finite on admissible histories, the quadratic velocity functional

K int [ Ψ , ∂ λ Ψ ] = 1 2 ‖ ∂ λ Ψ ‖ ℋ X 2 [1]. That expression and (2) are not competing choices:

they coincide exactly in the horizontal gauge. Differentiating the normalisation constraint ‖ Ψ ‖ 2 = 1 gives Re 〈 Ψ , Ψ ˙ 〉 X = 0 , so that 〈 Ψ , Ψ ˙ 〉 X = i ω ( λ ) with ω real — the Berry connection along the history. Hence

and in the horizontal gauge ω ≡ 0 —always attainable by the phase redefinition Ψ → e i θ Ψ with θ ˙ = − ω —the Fubini-Study form reduces to ‖ Ψ ˙ ‖ 2 . The functional of Part I is therefore the canonical kinetic term written in a particular gauge

and in units ℏ IFR 2 / ( 2 m Ψ c IFR 2 ) = 1 2 . Since the inequality above is uniform, the regularity estimates of Part I remain valid verbatim for (2). What the canonical form

adds is manifest gauge invariance in every gauge, which is what the projective structure of the state space requires and what the variational problem needs in order to be well posed independently of the phase convention.

2.5.1. Resulting Interference Equation

With K int specified by (2), the constrained variation of the interference state (Section 2.9) produces an explicit second-order equation in λ for the interference state. Writing ℱ ( Ψ , Ψ ˙ ) : = 〈 Ψ , Ψ ˙ 〉 − | 〈 Ψ , Ψ ˙ 〉 | 2 and varying with respect to Ψ * in Wirtinger calculus subject to ‖ Ψ ‖ 2 = 1 :

Equation (5) is the IFR interference equation in canonical form. It is second-order in λ (not first-order, as in the Schrödinger equation), consistent with the fact that λ is an ordering parameter rather than physical time: a Schrödinger-type equation would already presuppose temporal evolution, which in IFR is emergent. The structure of (5) is instead Klein-Gordon-like on ℙ ( ℋ X ) , with the redundant-dispersion term acting as an objectivizing source.

2.5.2. Conditional Schrödinger Correspondence

Equation (5) is second order in λ , whereas Schrödinger evolution is first order. We state the conditions under which the two are related, and we are explicit about which of them are established here and which are not.

A preliminary observation excludes the reduction that first suggests itself. Since the physical configuration is the ray [ Ψ ] ∈ ℙ ( ℋ X ) and the kinetic term is the Fubini-Study form, a λ -dependent global phase Ψ ↦ e − i θ ( λ ) Ψ is pure gauge: it leaves [ Ψ ] , K int and the whole action invariant. A fast global phase can therefore not be used as a physical carrier from which a term linear in ∂ λ is extracted; any such term produced by substitution is cancelled by the corresponding transformation of the projected terms 〈 Ψ , Ψ ¨ 〉 Ψ , 〈 Ψ , Ψ ˙ 〉 Ψ ˙ and | 〈 Ψ , Ψ ˙ 〉 | 2 Ψ appearing in (5). The reduction must come from geometry, not from a phase.

The relevant geometry is available. The projective space ℙ ( ℋ X ) is Kähler, carrying the compatible triple ( g FS , ω FS , J ) , and on such a manifold the Hamiltonian flow of the expectation function h ( Ψ ) = 〈 Ψ , H ^ Ψ 〉 is the Schrödinger equation generated by H ^ , provided the symplectic form is normalised to carry the interferential quantum of action. Since g FS and hence ω FS are dimensionless, whereas h has the dimension of energy, we fix once and for all the physicalsymplecticform

where ω FS is the Kähler form of g FS in the convention ω FS ( ⋅ , ⋅ ) = g FS ( J ⋅ , ⋅ ) , and the Hamiltonian vector field is defined by ι X h ω IFR = d h . The factor 2 is not a convention but is fixed by the normalisation of g FS already adopted for the kinetic term. Indeed, for horizontal v , w one has g FS ( v , w ) = ℜ 〈 v , w 〉 and

ω FS ( v , w ) = ℑ 〈 v , w 〉 , while d h ( w ) = 2 ℜ 〈 w , ( H ^ eff − 〈 H ^ eff 〉 ) Ψ 〉 for

h = 〈 Ψ , H ^ eff Ψ 〉 ; the Schrödinger vector field X h = − i ℏ IFR ( H ^ eff − 〈 H ^ eff 〉 ) Ψ then gives ω FS ( X h , w ) = ℏ IFR − 1 ℜ 〈 w , ( H ^ eff − 〈 H ^ eff 〉 ) Ψ 〉 , so that ι X h ω IFR = d h requires

precisely ω IFR = 2 ℏ IFR ω FS . Without this declaration the definition of X h could not produce ℏ IFR , with the correct coefficient, on the left-hand side of the resulting evolution equation. A first-order quantum evolution is therefore recovered exactly when the second-order interference dynamics restricts, on the stable-state manifold Q ˜ introduced in (S2) below, to such a Hamiltonian flow.

Assumption II.2 (Conditional Schrödinger correspondence).

(S1) High redundancy, in value and in gradient. The history is restricted to a neighbourhood of the stable-center manifold Q , on which D red = O ( ϵ red ) with ϵ red ≪ 1 , and on which the associated force term satisfies ‖ δ D red / δ   Ψ ‖ = O ( ϵ red ) in the norm of ℋ Q . The second condition does not follow from the first and is assumed separately: it is the smallness of the gradient, not of the value, that makes the objectivizing term a regular perturbation of the reduced vector field and justifies the residual O ( ϵ red ) in Proposition II.2.

(S2) Stable-state manifold and Kähler inheritance. The stable centers Q live on the comparison manifold, not on the state space, so a distinct object must be introduced: let

be the stable-state manifold, with ι : Q ˜ ↪ ℙ ( ℋ X ) the inclusion. We assume Q ˜ is a complex submanifold, so that it inherits the compatible triple ( g FS , ω FS , J ) by restriction and ι * ω FS is non-degenerate on it.

(S3) Existence of a first-order invariant manifold. The second-order dynamics defined by (5) admits, on Q ˜ , an invariant slow manifold on which it reduces to the Hamiltonian flow of the expectation function h ( [ Ψ ] ) = 〈 Ψ , H ^ eff Ψ 〉 / 〈 Ψ , Ψ 〉 with respect to ι * ω IFR as normalised in (6). This is a hypothesis; no construction of such a manifold is supplied in the present paper.

(S4) Effective generator. H ^ eff , the generator whose expectation function is h , is densely defined and self-adjoint on the reduced Hilbert space ℋ Q whose projectivisation contains Q ˜ .

(S5) Branch selection. Of the two branches admitted by a second-order system the positive-energy branch is retained; this projection is supplied by the informational Wick mechanism of Part V, which also fixes the orientation of the resulting evolution.

(S6) Adiabaticity of the reduction data. The embedding ι λ ( Q ˜ ) and the generator H ^ eff ( λ ) vary slowly over the scale of the reduction. No adiabaticity of Γ is invoked here: g FS is a fixed structure on ℙ ( ℋ X ) and does not depend on the comparison geometry.

Proposition II.2 (Schrödinger sector).Under Assumption II.2 the reduced motion on Q ˜ is the Hamiltonian flow

In the horizontal lift, characterised by 〈 Ψ , ∂ λ Ψ 〉 = 0 , Equation (7) reads

and the familiar form i ℏ IFR ∂ λ Ψ = H ^ eff Ψ is recovered by the phase lift that reinstates the dynamical phase exp ( − i ℏ IFR ∫ 〈 H ^ eff 〉 Ψ d λ ) . The two lifts describe the same curve in ℙ ( ℋ X ) .

Proof. By (S2) the restriction of ( g FS , ω FS , J ) to Q ˜ is a Kähler structure with ι * ω FS , and therefore ι * ω IFR , non-degenerate, so the Hamiltonian vector field X h of (6) is well defined. By (S3) the reduced motion is X h , with h the expectation function of the self-adjoint operator supplied by (S4). On a Kähler manifold the ω IFR -Hamiltonian flow, with ω IFR = 2 ℏ IFR ω FS as in (6), of the expectation function of a self-adjoint operator is the flow generated by J applied to its g FS -gradient; computing that flow in a homogeneous representative and imposing the horizontal-gauge condition 〈 Ψ , ∂ λ Ψ 〉 = 0 , which is admissible because the global phase is gauge, gives (8), the subtraction of 〈 H ^ eff 〉 Ψ being exactly the horizontality constraint. The branch is fixed by (S5) and the residual O ( ϵ red ) is the contribution of the objectivizing term admitted by (S1). ☐

The flow (7) is a flow in the ordering parameter, not in physical time. Where a temporal sector exists, that is where d t / d λ > 0 in the sense of Remark II.2, the chain rule d / d t = ( d λ / d t ) d / d λ converts it into an evolution in t generated by the rescaled operator H ^ eff ( t ) = ( d λ / d t ) H ^ eff . The ordering parameter therefore remains an ordering parameter throughout, and no temporal structure is smuggled into the reduction.

Three points must be stated plainly. First, the imaginary unit in (8) originates in the complex structure J of the Kähler geometry of ℙ ( ℋ X ) ; it is produced neither by a global phase, which is gauge, nor by an analytic continuation of the ordering parameter. Analytic continuation enters only through (S5), where it selects the branch. Second, the correspondence is conditional on (S3), which asserts the existence of a first-order invariant manifold and is not established here; a construction of that manifold, or a proof that it does not exist, is an open problem of the programme and is recorded in Section 3.17. Third, the correspondence degenerates where Q does not exist: for a single observer no stable-center manifold is selected, H ^ eff is undefined and no Schrödinger sector is obtained. The quantum-mechanical limit is therefore neither postulated nor obtained by rotating a parameter: it is the Hamiltonian sector of the IFR variational structure on the manifold of stable objective centers, under hypotheses that are here stated rather than proved.

Remark II.5 (Status of ℏ IFR and m Ψ ).The constants ℏ IFR and m Ψ are not free parameters of the framework. Their physical identification follows the same correspondence chain as κ eff in Section 3 and c IFR in the Lorentzian sector of Part V of the series: in the semiclassical limit ℏ IFR → ℏ (ordinary Planck constant) and m Ψ sets the scale of interferential inertia for the underlying objective structures. The matching conditions are stated explicitly in Part III of the series.

This canonical choice closes a structural gap in the original formulation: the IFR action is now specified up to the potential V int and the geometric curvature term β ℛ ¯ Γ , with all kinetic structure determined by the Fubini-Study geometry of ℙ ( ℋ X ) . The variational problem is therefore well-posed at the fundamental level, without smuggling in any pre-existing spacetime structure.

The kinetic term having been fixed, the informational potential V int remains the principal element of the action that is not determined from first principles. It is not, however, an arbitrary functional: symmetry and regularity restrict it to a low-order family, and the interference states themselves admit a concrete parametrised finite-dimensional realisation. We record both, since without them the variational principle would remain a template rather than a physical model. Neither step is a derivation, and we mark the residue explicitly: the higher-order coefficients of the potential, the dynamics of Γ and the observer-kinematic term remain open.

Admissible potentials. Three requirements act on V int . It must be invariant under the global phase Ψ ↦ e i θ Ψ , since the state space is projective. It must be invariant under the IFR gauge group U ( A ′ ) , the unitary group of the commutant of the observer algebra A generated by { Π a } , because transformations in A ′ are by construction unobservable to every member of the observer family. And it must be bounded below and satisfy the regularity conditions (R1)-(R4) inherited from Part I, so that S IFR is finite on admissible histories. We accordingly restrict the analysis to the gauge-invariant class

where E A is the conditional expectation onto the observer algebra. We do not claim that gauge invariance forces this dependence for every algebra type: for a single block of A the orbit of a pure state under U ( A ' ) is determined by ρ A , while relative phases between blocks are further invariants; the class above is a restriction, adopted here, and not a theorem.

Writing σ : = ρ A − 𝟙/(d +1) for the deviation from the maximally mixed state, so that tr   σ 2 = tr   ρ A 2 − 1 / ( d + 1 ) ≥ 0 , a smooth, bounded-below and unitarily covariant F admits the Landau-type expansion

with H ^ 0 = H ^ 0 † ∈ A and g ∈ ℝ . Since the stationarity condition minimises V int along the objectivizing directions, the sign of g has a definite meaning: g > 0 stabilises the symmetric, maximally mixed configuration and penalises purity of the observer-accessible sector, whereas g < 0 destabilises it and drives the symmetry breaking that selects a preferred stable center, the resulting instability being arrested by the higher-order terms. This does not amount to a derivation of V int , and we do not present it as one; it establishes that the admissible potentials form a low-order gauge-invariant family indexed by ( H ^ 0 , g ) together with the retained higher-order coefficients, rather than an unconstrained choice of functional.

Aconcreteparametrisedfinite-dimensionalrealisation. The interference states of the framework are not left abstract. The minimal realisation used in the empirical instantiations of the series is the following. Here d = dim ℳ as in Assumption II.1 (M2), so that the comparison manifold is the interior of the simplex over d + 1 outcomes:

(I1) ℋ X = ℂ d + 1 with d ≥ 5 , so that Ψ ∈ ℙ ( ℂ d + 1 ) ;

(I2) a finite observer family A = { 1 , ⋯ , N } , N ≥ 2 , each observer a carrying a rank-one positive operator-valued measure { E a ( k ) } k = 1 d + 1 with ∑ k E a ( k ) = 𝟙;

(I3) observer maps given by the induced pushforward onto the comparison manifold,

(I4) comparison geometry given by the dimensionless distance d ˜ Γ of (M6), realised in the high-redundancy regime as the Jensen-Shannon distance, d ˜ Γ ≃ d JS , with natural logarithms;

(I5) potential (9) with H ^ 0 diagonal in a reference basis.

Given a truncation of the expansion (9) and a choice of ℒ Π , every object entering S IFR is then a computable function of the parameters

and the stable Fréchet centers, the redundant dispersion and the embedding diagnostics of Proposition II.3 below are directly evaluable. We do not claim that the realisation is complete: the truncation and the choice of ℒ Π are inputs, recorded as such in Section 3.17. The estimators, null tests and resampling protocol by which this realisation is confronted with detector, cosmological and physiological ensembles are developed in Part IV of the series and were introduced in [12][13]; the present paper establishes the variational structure on which they operate.

For a finite observer family A = { 1 , ⋯ , N } , the redundant dispersion term is

The pairwise weights w a b ≥ 0 are not free parameters of the variational problem. They are fixed operationally, as in Part I [1], by the inverse-uncertainty rule w a ∝ σ a − 2 combined with an independence factor ρ a b ∈ [ 0 , 1 ] estimated from the empirical correlation of the observer residuals, so that w a b ∝ w a w b ( 1 − ρ a b ) ;

the uniform choice w a b = ( N 2 ) − 1 is the special case of equal reliability and mutual

independence. This removes the underdetermination that would otherwise affect D red as a term of the action.

Here d ˜ Γ is the dimensionless distance induced by the rescaled comparison metric g ˜ Γ = ℓ Γ − 2 g Γ of Assumption II.1 (M6). Two points must be kept distinct. The dimensionless level is the one that enters D red , which is thereby dimensionless as required by the dimensional analysis of Section 2.4; were the dimensional distance d g Γ used instead, D red would carry dimension L 2 . And d ˜ Γ remains a functional of Γ off shell: its identification with the Jensen-Shannon distance [15][16], with the logarithm base declared explicitly, is the high-redundancy realisation used in the empirical instantiations of the series, not a fixed kinematic structure. This distinction is not cosmetic. If d ˜ Γ were frozen to d JS independently of Γ, then δ D red / δ   Γ = 0 and the redundancy contribution T i j red to the informational stress tensor—the specifically IFR part of the gravitational source—would vanish identically.

For a continuous observer family ( A , ν ) , it becomes

The appearance of D red in the action means that objectivity is not imposed after the dynamics. Instead, the tendency toward multi-observer agreement contributes directly to the selection of physical configurations.

The variational sign convention − α D red may be understood as follows. In a stationary-phase formulation, configurations with smaller redundant dispersion contribute coherently to the effective physical sector, whereas configurations with large observer disagreement are suppressed or fail to stabilize as objective structures.

Equivalently, in an Euclideanized or statistical version of the theory, one may write

with

In this form, high-dispersion configurations are exponentially suppressed. The Lorentzian and Euclidean forms therefore encode the same principle: stable physical reality is associated with low redundant dispersion.

The formal generating functional of IFR is

This expression is analogous in form to a path integral, but its domain is not the space of fields on a given spacetime. The integration is over Ψ, the primitive interference states; Γ, the information geometries; and Π, the observer projection structures.

Thus the IFR generating functional is not

but rather

Conventional field-theoretic path integrals are expected to emerge only in a limiting regime in which Γ admits an effective spacetime representation and the stable objective centers behave as fields or events on that emergent spacetime.

We first consider variations of the interference state, Ψ ↦ Ψ + ϵ   δ   Ψ , with δ   Ψ ∈ T Ψ ℋ X , subject to the normalization constraint

The constrained variation can be implemented by introducing a Lagrange multiplier Λ ( λ ) and defining