Rank, Fibers, and Falsifiability: A Rank-Aware Possibilistic Cramér–Rao Bound for Evidence-Driven Contraction

1. Introduction

The Possibilistic Cramér–Rao Bound (PCRB) is the load-bearing information bound of the TEAG corpus [1,2,4]. It asserts that no epistemically admissible filter can delete admissible microstates faster than the Choquet-aggregated surprisal of the current observation resolves:

$$H_{\pi }^{k|k}\ge H_{\pi }^{k|k-1}+{\displaystyle \frac{n}{2}}log(1-I_k),$$

(1)

where $H_{\pi }={\int }_0^{1}log\mathrm{Vol}\left({\mathcal{C}}_{\alpha }\right)d\alpha$ is the possibilistic entropy of the admissible support in ${\mathbb{R}}^n$ and $I_k=1-e^{-{\overline{S}}_k}\in [0,1)$ is built from the Choquet integral ${\overline{S}}_k$ of per-hypothesis surprisal against the prior possibility capacity [2].

The bound (1) carries the state dimension n. This paper argues, and then proves, that the correct constant is the measurement rank m, and that the discrepancy is not a technical refinement but the visible edge of a structural fact with consequences of its own. The fact is this: possibilistic entropy is a state-space quantity, but evidence does its deforming work in evidence space. The two are connected by the state-to-measurement map h, and h is not injective — many states produce the same evidence. A single observation of rank m therefore cannot compare states along the $(n-m)$-dimensional fibers of h, and where there is no comparison there can be no information: no deformation of the impossibility field, no deletion of admissible volume, no reduction of ignorance. The floor in (1) charges the observation for n dimensions of deletion capacity it does not possess.

Three results follow from taking the non-injectivity seriously.

First, an exact factorization identity (Lemma 1): for a linear rank-m measurement, the per-step entropy change equals the admissibility-integral of the log base-mass retained,

$$H_{\pi }^{k|k}-H_{\pi }^{k|k-1}={\int }_0^{1}log{\mu }_{\alpha }\left({\mathcal{B}}_{w\left(\alpha \right)}\right)d\alpha ,$$

where ${\mu }_{\alpha }$ is the fiber-volume-weighted base marginal of the prior $\alpha$-cut and ${\mathcal{B}}_w$ is the evidence-compatibility region in the m-dimensional base. The fibers contribute no ratio at all. This identity is unconditional and replaces the approximate volume-scaling steps of the earlier proof with an equality.

Second, the fiber conservation theorem (Theorem 1): a single observation leaves the fiber-direction geometry of every $\alpha$-cut invariant, and between observations the propagation phase can only expand it. Ignorance in unobserved directions is conserved under evidence and grows under dynamics — honestly, visibly, in the epistemic ledger. This is the Principle of Comparative Information [2] promoted from an invariance statement to a conservation law: entropy is conserved along every direction in which the evidence performs no comparison.

Third, the rank-aware PCRB (Theorem 2): under an explicit base-marginal coupling condition (Assumption 1) that formalizes the earlier innovation–state isotropy assumption in its correct home — the m-dimensional base, where the measurement whitening actually lives — the floor becomes

$$H_{\pi }^{k|k}\ge H_{\pi }^{k|k-1}+{\displaystyle \frac{m}{2}}log(1-I_k).$$

(2)

Since $m\le n$ and $log(1-I_k)\le 0$, the floor (2) is higher than (1): the rank-aware bound is a strictly stronger theorem whenever $m<n$, and recovers the original at full rank. A rank-2 radar pass cannot delete seven dimensions of ignorance, and the bound now says so.

Finally, passing from one observation to ℓ of them yields the falsifiability–observability theorem (Theorem 3): a direction of state space is falsifiable if and only if some observation, pulled back through the dynamics, compares along it; the state is totally falsifiable if and only if the accumulated row spaces span ${\mathbb{R}}^n$ — the classical observability condition of [10], rederived here as a statement about which comparisons exist. Its redundancy corollary (Corollary 1) characterizes when observations can falsify each other: joint total falsification — the abstention boundary $S_{\pi }=⌀$, the filter’s model-stress signal — is geometrically reachable if and only if the stacked system is overdetermined. Comparison is the source of all information; redundancy is the source of all integrity.

Section 7 derives the practical consequences: the PCRB-admissible basin’s information factor corrects from ${(1-I_k)}^{1/2}$ to ${(1-I_k)}^{m/\left(2n\right)}$ in isotropic-equivalent form, and the correction generates a falsifiable prediction about over-pruning in rank-deficient tracking that existing numerical evidence appears to support; this is consistent with the geometry-driven design imperative in set-membership and ellipsoidal filtering [8,9]. Section 8 states what remains open — including tightness, which the rank refinement inherits and does not resolve.

2. Setup

2.1. Primitives

Let $H={\left\{{\chi }^{\left(i\right)}\right\}}_{i=1}^M\subset {\mathbb{R}}^n$ be a finite admissible support with normalized possibility field [20] $\pi :H\to [0,1]$, ${sup}_i{\pi }^{\left(i\right)}=1$; hypotheses with $\pi =0$ are falsified. For $\alpha \in (0,1]$ the $\alpha$-cut is ${\mathcal{C}}_{\alpha }=\{{\chi }^{\left(i\right)}:{\pi }^{\left(i\right)}\ge \alpha \}$ [16,17,18]. Where continuum statements are needed we work with an admissible region $K\subset {\mathbb{R}}^n$ (compact, non-empty interior) carrying an upper-semicontinuous possibility field, of which the support cloud is the computational discretization; all identities below are stated for the continuum and hold for the discretization by the counting convention of Remark 1.

The possibilistic entropy is

$$H_{\pi }={\int }_0^{1}log\mathrm{Vol}\left({\mathcal{C}}_{\alpha }\right)d\alpha ,$$

(3)

where $\mathrm{Vol}$ denotes n-dimensional Lebesgue volume of the cut (see Remark 2 for the relation to the MVEE-based form used in the ESPF).

The observation model is $y=h\left(x\right)+e$ with whitened compatibility $\mathrm{Comp}\left(x\right)=e^{-{\displaystyle \frac{1}{2}}{\parallel L_e^{-1}(y-h\left(x\right))\parallel }^2}$, where ${\Pi }_e=L_e{L}_e^{\top }$ is the MVEE shape matrix of the predicted measurement support. The surprisal field is $\psi \left(x\right)=-log\mathrm{Comp}\left(x\right)=\frac{1}{2}{\parallel L_e^{-1}(y-h\left(x\right))\parallel }^2$, and the canonical update is the conjunctive contraction ${\pi }^{+}=min({\pi }^{-},\mathrm{Comp})$, equivalently the tropical update ${\Phi }^{+}=max({\Phi }^{-},\psi )$ on the impossibility field $\Phi =-log\pi$ [1].

The aggregate epistemic surprisal is the Choquet integral of $\psi$ against the prior possibility capacity, in layer-cake form

$${\overline{S}}_k={\int }_0^{\infty }{\Pi }_{k|k-1}\left(\{x:\psi (x)\ge t\}\right)dt,I_k=1-e^{-{\overline{S}}_k}\in [0,1),$$

(4)

with ${\Pi }_{k|k-1}\left(A\right)={sup}_{x\in A}{\pi }_{k|k-1}\left(x\right)$ [2,6]. Two facts from [2] are used repeatedly: the modal prior hypothesis satisfies $\psi \left(x_{\mathrm{mode}}\right)\le {\overline{S}}_k$, and every hypothesis satisfies ${\pi }_{k|k-1}\left(x\right)\psi \left(x\right)\le {\overline{S}}_k$.

2.2. The Linear Rank-M Measurement

Throughout Section 3, Section 4 and Section 5 the measurement map is linear, $h\left(x\right)=Hx+b$ with $H\in {\mathbb{R}}^{p\times n}$ of rank $m\le min(p,n)$. Nonlinear maps are discussed in Section 8; the rank structure below applies to their linearizations.

Fix once and for all a splitting adapted to H: let $U=\mathrm{row}\left(H\right)\subset {\mathbb{R}}^n$ (dimension m) and $V=kerH$ (dimension $n-m$), and write $x=(u,v)$ in coordinates adapted to ${\mathbb{R}}^n=U\oplus V$, chosen with unit Jacobian. The defining property of the splitting is

$$\psi \left(x\right)=\tilde{\psi }\left(u\right),$$

(5)

the surprisal is the pullback of an m-dimensional field: it depends only on the base coordinate u and is constant on every fiber $\left\{u\right\}\times V$. Many states produce the same evidence; all states on a fiber produce exactly the same evidence, and the evidence cannot tell them apart.

For $w\ge 0$, the base compatibility region is

$${\mathcal{B}}_w=\{u\in U:\tilde{\psi }\left(u\right)\le w\},$$

(6)

an ellipsoidal region of U (the sublevel set of a positive semidefinite quadratic restricted to U, on which it is positive definite by the rank condition).

Remark 1

(Counting convention). All volumes are quantized at the hypothesis-space cell resolution $v_{\mathrm{cell}}>0$ of [2]: Boltzmann counts $W_{\alpha }=\mathrm{Vol}\left({\mathcal{C}}_{\alpha }\right)/v_{\mathrm{cell}}\ge 1$ for non-empty cuts, so $H_{\pi }$ is finite for every non-empty surviving support and every log-mass appearing below is bounded below by the single-cell floor. We suppress $v_{\mathrm{cell}}$ from the notation; it cancels from every entropy difference.

Remark 2

(Lebesgue cuts and the MVEE bridge). The ESPF evaluates $H_{\pi }$ through MVEE volumes of the cuts [2]. Every identity in Section 3 and Section 4 is exact for Lebesgue cut volumes. For the MVEE form, John’s sandwich [19] $\mathrm{Vol}\left(K\right)\le \mathrm{Vol}\left(\mathrm{MVEE}\left(K\right)\right)\le n^{n/2}\mathrm{Vol}\left(K\right)$ (for centrally symmetric K; $n^n$ in general) bounds the discrepancy per cut; for the MVEE-self-similar class of [5], whose cuts are themselves ellipsoids, the bridge is exact and the two forms of $H_{\pi }$ coincide. We state theorems for the Lebesgue form and flag the bridge where it matters.

3. The Factorization Identity

Lemma 1

(Fubini factorization of the update). Let the prior α-cut be ${\mathcal{C}}_{\alpha }^{-}\subset {\mathbb{R}}^n$ with $\mathrm{Vol}\left({\mathcal{C}}_{\alpha }^{-}\right)>0$, and let the observation have linear rank-m structure (5). Write $w\left(\alpha \right)=-log\alpha$. Define the fiber-slice volume and the fiber-weighted base marginal

$$f_{\alpha }\left(u\right)={\mathrm{Vol}}_{n-m}\left(\{v:(u,v)\in {\mathcal{C}}_{\alpha }^{-}\}\right),{\mu }_{\alpha }\left(A\right)={\displaystyle \frac{{\int }_A{f}_{\alpha }\left(u\right)du}{{\int }_U{f}_{\alpha }\left(u\right)du}},A\subseteq U.$$

Then under the canonical update:

(i) 

the posterior cut is the cylinder intersection ${\mathcal{C}}_{\alpha }^{+}={\mathcal{C}}_{\alpha }^{-}\cap \left({\mathcal{B}}_{w\left(\alpha \right)}\times V\right)$;

(ii) 

the cut volumes satisfy the exact ratio identity

$${\displaystyle \frac{\mathrm{Vol}\left({\mathcal{C}}_{\alpha }^{+}\right)}{\mathrm{Vol}\left({\mathcal{C}}_{\alpha }^{-}\right)}}={\mu }_{\alpha }\left({\mathcal{B}}_{w\left(\alpha \right)}\right);$$

(iii) 

the per-step entropy change is

$$H_{\pi }^{k|k}-H_{\pi }^{k|k-1}={\int }_0^{1}log{\mu }_{\alpha }\left({\mathcal{B}}_{w\left(\alpha \right)}\right)d\alpha .$$

Proof. 

(i) By the $\alpha$-cut intersection identity of the canonical update [1], ${\mathcal{C}}_{\alpha }^{+}={\mathcal{C}}_{\alpha }^{-}\cap \{\psi \le w\left(\alpha \right)\}$, and by (5) the constraint set is $\{(u,v):\tilde{\psi }\left(u\right)\le w\left(\alpha \right)\}={\mathcal{B}}_{w\left(\alpha \right)}\times V$.

(ii) Fubini in the adapted coordinates: $\mathrm{Vol}\left({\mathcal{C}}_{\alpha }^{+}\right)={\int }_{{\mathcal{B}}_{w\left(\alpha \right)}}{f}_{\alpha }\left(u\right)du$ and $\mathrm{Vol}\left({\mathcal{C}}_{\alpha }^{-}\right)={\int }_U{f}_{\alpha }\left(u\right)du$; the ratio is ${\mu }_{\alpha }\left({\mathcal{B}}_{w\left(\alpha \right)}\right)$ by definition. The fiber-slice volumes $f_{\alpha }$ appear identically in numerator and denominator on the retained base region: the update truncates the base measure and touches no fiber.

(iii) Integrate the logarithm of (ii) against $d\alpha$ and use the definition (3). □

The identity in (iii) is the structural heart of this paper. The entropy change of an evidence-driven contraction is entirely a base quantity: the admissibility-integral of the log-fraction of fiber-weighted base mass that survives. Nothing about the $(n-m)$-dimensional fiber geometry enters the ratio. The earlier proof’s approximate John-ellipsoid volume scalings in ${\mathbb{R}}^n$ are replaced by an equality in ${\mathbb{R}}^m$.

4. Fiber Conservation

Theorem 1

(Fiber conservation). Under a single linear rank-m observation:

(i) 

for every α and every retained base point $u\in {\mathcal{B}}_{w\left(\alpha \right)}$, the posterior fiber slice equals the prior fiber slice: $\{v:(u,v)\in {\mathcal{C}}_{\alpha }^{+}\}=\{v:(u,v)\in {\mathcal{C}}_{\alpha }^{-}\}$;

(ii) 

the fiber-conditional entropy profile $u\mapsto log{f}_{\alpha }\left(u\right)$ is invariant on the retained base, for every α;

(iii) 

no possibilistic quantity of the update distinguishes two hypotheses on the same fiber: ψ, $\mathrm{Comp}$, the posterior possibility ordering, and membership in every posterior α-cut are all fiber-constant.

Between observations, the propagation phase (set-valued dynamics with process bounds) can only enlarge fiber slices. Consequently, ignorance in unobserved directions is conserved under evidence and monotonically non-decreasing under dynamics until a subsequent observation, pulled back through the flow, compares along it.

Proof. 

(i) is Lemma 1(i): the deleted set is a full cylinder, so retention is a property of u alone. (ii) is immediate from (i). (iii): $\psi$ is fiber-constant by (5); $\mathrm{Comp}=e^{-\psi }$ and the conjunctive update $min({\pi }^{-},\mathrm{Comp})$ modify ${\pi }^{-}$ by a fiber-constant field, so any two hypotheses on one fiber have their possibilities reduced by the same evidentiary term; posterior cut membership is (i). The propagation statement is the Minkowski-sum expansion of the ESPF’s Jaynesian phase [3], which contains the identity map. □

Remark 3

(The Principle of Comparative Information as a conservation law). The Principle of Comparative Information [2] states that every information-bearing quantity of the filter is a comparison. Theorem 1 gives the principle its conservation form:entropy is exactly conserved along every direction in which the evidence performs no comparison.No comparison, no information; no information, no deletion; no deletion, no entropy change. The contrapositive is the operational content of the rank-aware bound of the next section: whatever deletion occurs must be paid for in the m directions where comparison happens, and the price is floored by the Choquet surprisal.

The conservation law also renders the honest-divergence behavior of the filter structural rather than incidental. In directions the accumulated evidence has not compared, the ESPF’s ledger shows ignorance conserved by updates and grown by propagation — the epistemically truthful state of affairs — rather than confidence manufactured through model correlations. Silent overconfidence in unobservable directions is not merely avoided; it is impossible within the update’s geometry.

5. The Rank-Aware PCRB

The factorization identity reduces the bound to a single question: how small can the retained base mass ${\mu }_{\alpha }\left({\mathcal{B}}_w\right)$ be, given that the observation’s Choquet surprisal is ${\overline{S}}_k$? The answer requires one assumption — the correct, m-dimensional home of the earlier innovation–state isotropy condition.

Assumption 1

(Base-marginal coupling). There is a constant $c\in (0,1]$ such that for every $\alpha \in (0,1]$ and every $w\ge \tilde{\psi }\left(u_0\right)$, where $u_0$ is the base coordinate of a modal prior hypothesis,

$${\mu }_{\alpha }\left({\mathcal{B}}_w\right)\ge c·min\left\{1,\left({\displaystyle \frac{{\mathrm{Vol}}_m\left({\mathcal{B}}_w\right)}{{\mathrm{Vol}}_m\left(P_U{\mathcal{C}}_{\alpha }^{-}\right)}}\right)\right\},$$

where $P_U$ denotes the base projection. In words: the fiber-weighted base marginal assigns the evidence-compatibility region at least a fixed fraction of the mass that the uniform base measure would assign it. The condition is exact with $c=1$ when $f_{\alpha }$ is constant on the base projection (product-structured cuts), and holds for the MVEE-self-similar class of [5] with a c depending only on the profile function.

Assumption 1 is where the surprisal–volume coupling of the framework lives: because $L_e$ whitens by the MVEE of the predicted measurement cloud itself — the pushforward of the support through H — the base region ${\mathcal{B}}_w$ is measured against the support’s own extent, and a mild aggregate surprisal cannot coexist with the deletion of most of the base mass unless the fiber-weighted marginal is adversarially concentrated away from the evidence. The assumption forbids exactly that adversarial concentration, and nothing else.

Theorem 2

(Rank-aware PCRB). Let the observation be linear of rank m, let $I_k$ be the Choquet information content (4), and let Assumption 1 hold with constant c. Then

$$H_{\pi }^{k|k}\ge H_{\pi }^{k|k-1}+{\displaystyle \frac{m}{2}}log(1-I_k)+logc.$$

(7)

In particular, under exact coupling ($c=1$),

$$H_{\pi }^{k|k}\ge H_{\pi }^{k|k-1}+{\displaystyle \frac{m}{2}}log(1-I_k),$$

which recovers the state-dimension floor at full rank $m=n$ and is a strictly stronger bound whenever $m<n$. The fiber directions contribute zero to the entropy change identically (Theorem 1); the floor is a statement about the base alone.

Proof. 

By Lemma 1(iii) it suffices to bound ${\int }_0^{1}log{\mu }_{\alpha }\left({\mathcal{B}}_{w\left(\alpha \right)}\right)d\alpha$ from below.

Step 1 (modal base point survives). Let $x_{\mathrm{mode}}$ be a modal prior hypothesis with base coordinate $u_0$. By the Choquet modal bound, $\tilde{\psi }\left(u_0\right)=\psi \left(x_{\mathrm{mode}}\right)\le {\overline{S}}_k$, so $u_0\in {\mathcal{B}}_w$ for every $w\ge {\overline{S}}_k$, i.e. for every $\alpha \le e^{-{\overline{S}}_k}=1-I_k$. The base compatibility region is never empty on the admissibility range $(0,1-I_k]$.

Step 2 (base volume ratio). In measurement-whitened base coordinates, ${\mathcal{B}}_w$ is the ball of radius $\sqrt{2w}$ about the evidence-consistent locus, and the base projection of the prior cut lies within the whitened unit ball of the predicted measurement support (the defining property of $L_e$). For $\alpha \le 1-I_k$, i.e. $w\left(\alpha \right)\ge {\overline{S}}_k$, the containment of $u_0$ and the radius monotonicity give

$${\displaystyle \frac{{\mathrm{Vol}}_m\left({\mathcal{B}}_{w\left(\alpha \right)}\right)}{{\mathrm{Vol}}_m\left(P_U{\mathcal{C}}_{\alpha }^{-}\right)}}\ge min\left\{1,{\left(e^{-{\overline{S}}_k}\right)}^{m/2}\right\}={(1-I_k)}^{m/2},$$

by the m-dimensional volume scaling of whitened balls whose radius ratio is at least $e^{-{\overline{S}}_k/2}={(1-I_k)}^{1/2}$: the compatibility ball at the level matching the modal survivor has whitened radius $\sqrt{2{\overline{S}}_k}$ measured against the unit-normalized extent of the predicted cloud, and each of the m base semi-axes carries the factor ${(1-I_k)}^{1/2}$.

Step 3 (coupling). Assumption 1 converts the uniform-measure ratio of Step 2 into a fiber-weighted mass bound: ${\mu }_{\alpha }\left({\mathcal{B}}_{w\left(\alpha \right)}\right)\ge c{(1-I_k)}^{m/2}$ for all $\alpha \le 1-I_k$.

Step 4 (top band by counting). For $\alpha \in (1-I_k,1]$, the retained mass is bounded below by the single-cell counting floor (Remark 1): every non-empty posterior cut retains at least one occupied cell, so $log{\mu }_{\alpha }\ge log(v_{\mathrm{cell}}/\mathrm{Vol}\left({\mathcal{C}}_{\alpha }^{-}\right))$, and the band’s contribution is bounded below by the same expression that bounds the total entropy from below in [2]; in the falsification regime, where the modal hypothesis survives at possibility $\ge 1-I_k$, the band $(1-I_k,1]$ has posterior cuts obtained from prior cuts of possibility level at least $1-I_k$, and the Step 2–3 bound extends to it with the same constant by monotonicity of $w\left(\alpha \right)$. No unaccounted absorption is invoked.

Step 5 (integrate). Combining,

$${\int }_0^{1}log{\mu }_{\alpha }\left({\mathcal{B}}_{w\left(\alpha \right)}\right)d\alpha \ge {\displaystyle \frac{m}{2}}log(1-I_k)+logc,$$

which with Lemma 1(iii) gives (7). □

Remark 4

(What became of the dimensional constant). The $n^{-n/2}$ factor of the earlier proof arose from a John-ellipsoid inner approximation performed in ${\mathbb{R}}^n$. Here the volume comparison of Step 2 is between whitened balls in ${\mathbb{R}}^m$ under the same whitening, and no inner-ellipsoid approximation is needed; the residual looseness is confined to the coupling constant c, which is explicit, equals 1 in the product-structured case, and multiplies the bound additively as $logc$ rather than compounding dimensionally.

Remark 5

(Tightness is inherited, not resolved). Whether any epistemically admissible filter saturates the floor (7) with equality remains open, exactly as for the earlier bound [2]. The rank refinement changes what tightness would mean — saturation is now an m-dimensional question about base-mass deletion — and the ESPF’s per-step slack retains its survivor-reserve component $log(M/M_{\mathrm{surv}})$. The refinement does, however, falsify tightness of theoldbound whenever $m<n$: no filter can saturate a floor that sits strictly below the true one.

6. Falsifiability, Observability, and Redundancy

Consider now ℓ observations at steps $k=1,\dots ,\mathsf{\ell }$ with linear maps $H_k$ of ranks $m_k$, connected by linear dynamics with transition matrices $F_k$ (flow from step 0: ${\Phi }_{k,0}=F_{k-1}\dots F_0$, ${\Phi }_{0,0}=I$).

Theorem 3

(Falsifiability–observability). Say a direction $v\in {\mathbb{R}}^n\setminus \left\{0\right\}$ isfalsifiable by the evidence systemif there exist a step k and hypotheses x, $x^{\prime }=x+tv$ ($t\ne 0$) whose surprisals at step k differ for some admissible evidence realization. Then:

(i) 

v is falsifiable if and only if $v\in {\sum }_{k=1}^{\mathsf{\ell }}\mathrm{row}\left(H_k{\Phi }_{k,0}\right)$, the accumulated pulled-back row space;

(ii) 

every direction is falsifiable — the state istotally falsifiable— if and only if the accumulated row spaces span ${\mathbb{R}}^n$, i.e. the observability matrix $\mathcal{O}=[H_1{\Phi }_{1,0};\dots ;H_{\mathsf{\ell }}{\Phi }_{\mathsf{\ell },0}]$ has rank n;

(iii) 

along any unfalsifiable direction, entropy is conserved by every update of the sequence (Theorem 1) and non-decreasing under every propagation; the cumulative floor over the sequence is ${\sum }_{k=1}^{\mathsf{\ell }}\frac{m_k}{2}log(1-I_k)$, and it constrains only the falsifiable subspace.

Proof. 

(i) At step k, surprisal in initial-state coordinates is a function of $H_k{\Phi }_{k,0}x$; two hypotheses differing by $tv$ receive identical surprisal for every evidence realization iff $H_k{\Phi }_{k,0}v=0$ for all k, i.e. iff $v\in {\bigcap }_{k}ker\left(H_k{\Phi }_{k,0}\right)$, whose orthogonal statement is (i). (ii) is (i) quantified over all v. (iii) applies Theorems 1 and 2 stepwise, with the fiber splitting at step k taken adapted to $H_k{\Phi }_{k,0}$. □

Observability is thus not an imported control-theoretic condition but a theorem about which comparisons exist: a state space is fully falsifiable exactly when the evidence system, pulled back through the dynamics, compares along every direction. With a static state, spanning requires ${\sum }_k{m}_k\ge n$ outright; with dynamics, even a single rank-m geometry can span, because the flow rotates the fibers into view — which is why orbit determination from range and range-rate is possible at all.

Corollary 1

(Redundancy: when evidence can falsify evidence). Stack the sequence into $\mathcal{H}:{\mathbb{R}}^n\to {\mathbb{R}}^P$, $P={\sum }_k{p}_k$, with $r=\mathrm{rank}\mathcal{O}\le n$ in initial-state coordinates, and consider the noiseless consistency question: does there exist a state reproducing the stacked evidence exactly?

(i) 

If $\mathrm{rank}\mathcal{H}=P$ (no redundancy: the stacked map is surjective onto evidence space), theneverystacked evidence vector is exactly attainable; the joint exact-compatibility set is a non-empty affine subspace, joint total falsification is geometrically unreachable, and inconsistency among the observations is undetectable in principle.

(ii) 

If $\mathrm{rank}\mathcal{H}<P$ (redundancy: the image is a proper subspace of evidence space), then generic stacked evidence lies at positive distance from the image, the joint minimum surprisal is strictly positive, and joint total falsification — the abstention boundary $S_{\pi }=⌀$, the filter’s model-stress signal — is reachable. Inconsistency, bias, and unmodeled dynamics become falsifiable events.

The degrees of redundancy, $P-\mathrm{rank}\mathcal{H}$, count the independent comparisons the observations make witheach otherrather than with the hypotheses.

Proof. 

(i) Surjectivity gives an exact solution for every right-hand side; the solution set is a coset of $ker\mathcal{H}$, on which joint compatibility is 1. (ii) A proper image subspace has Lebesgue-null evidence-space measure; for evidence off it, the minimum joint whitened residual is the positive distance to the image, so every hypothesis carries strictly positive joint surprisal, and driving the falsification threshold to that level empties the surviving support. □

Remark 6

(Comparison at both levels; two discounts). The Principle of Comparative Information now operates at two levels. With $\mathrm{rank}\mathcal{H}=P\le n$, evidence compares only hypotheses against observations. With redundancy, evidence also compares observations against observations — and only then do integrity events (bias, faults, maneuvers) become falsifiable, the possibilistic generalization of the redundancy requirement in receiver-autonomous integrity monitoring [11]. Correspondingly, two independent discount mechanisms govern the worth of repeated evidence:rank counts directions— ℓ same-geometry observations of a static state stack to rank m no matter how large ℓ grows, geometrically saturating what they can ever delete — whilethe Choquet capacity counts credibility within directions, discounting each repetition against the already-contracted prior possibility of the hypotheses it re-surprises [2,7]. Independent geometries escape both discounts at once: new rank, and fresh high-possibility hypotheses to surprise. Sensor diversity earns information twice over, and both mechanisms are now theorems.

7. Consequences

7.1. The PCRB-Admissible Basin

The basin radius of the companion papers couples an information factor to a survivor-stability reserve, $r_k^{★}=r_k^{-}·{(1-I_k)}^{1/2}{(M/M_{\mathrm{surv}})}^{1/n}$ [1,4]. The rank-aware floor corrects the information factor: the volume floor ${(1-I_k)}^{m/2}$ lives in the m-dimensional base, so its isotropic-equivalent expression on an n-dimensional radius is

$$r_k^{★}=r_k^{-}·{(1-I_k)}^{m/\left(2n\right)}{\left({\displaystyle \frac{M}{M_{\mathrm{surv}}}}\right)}^{1/n},$$

(8)

with the honest anisotropic statement being stronger still: contraction by ${(1-I_k)}^{1/2}$ per whitened base semi-axis, and none along the fibers.

7.2. A Falsifiable Prediction: Over-Pruning at Rank Deficiency

At the reference scenario’s $m=2$, $n=7$, the corrected information factor is ${(1-I_k)}^{1/7}$ where the implemented factor has been ${(1-I_k)}^{1/2}$: the fielded threshold contracts the basin roughly $3.5\times$ too fast in exponent, predicting systematic over-falsification in rank-deficient tracking — violent per-step thinning of the support with the diversity floors persistently active. This signature was in fact reported in the stress-run numerics of [2] and attributed there to a high-pruning regime. The rank-aware basin (8) predicts that the pathology attenuates substantially under the corrected exponent, at unchanged $I_k$. This is a falsifiable claim about the filter’s own behavior, checkable in a single re-run.

7.3. Corpus and Implementation Patches

The corrected constant propagates to: the PCRB statements of [1,2,4] (with m the per-step measurement rank and full-rank recovery noted); the capstone theorem’s contraction clause (deletion at most $\frac{m}{2}{\overline{S}}_k$ nats per step); the Hamilton–Jacobi kinetic term of [4]; and the basin contraction of the reference implementations, where $\mathtt{sqrt}(\mathtt{1}-\mathtt{I})$ becomes $(\mathtt{1}-\mathtt{I})(\mathtt{m}/(\mathtt{2}\mathtt{n}\left)\right)$ with $m=\mathrm{rank}\left(H_k\right)$ evaluated per step.

8. Open Problems

1.

The coupling constant. Sharp values of c in Assumption 1 for concrete cut families beyond the product-structured and self-similar cases; whether $c=1$ holds for all convex cuts under centered evidence.

2.

Tightness. Whether any admissible filter saturates the rank-aware floor; the ESPF’s slack decomposes as coupling slack plus the survivor reserve $log(M/M_{\mathrm{surv}})$, and saturation would require both to vanish.

3.

Nonlinear maps. The fiber structure of a nonlinear h is a foliation with curved leaves; the factorization identity holds leafwise via the coarea formula, at the price of a Jacobian factor whose control is the nonlinear form of Assumption 1.

4.

Adversarial verification. A numerical program that optimizes deletion masks over the admissible class at fixed $I_k$ and fixed rank, to probe the constant $\frac{m}{2}$ and the coupling condition directly.