Shannon meets Gödel-Tarski-Löb: Undecidability of Shannon Feedback Capacity for Finite-State Channels

1 Shannon meets G ¨ odel-T arski-L ¨ ob: Undecidability of Shannon Feedback Capacity for Finite-State Channels Angshul Majumdar , Senior Member , IEEE Abstract W e study the exact decision problem for feedback capacity of finite-state channels (FSCs). Gi ven an encoding e of a binary-input binary-output rational unifilar FSC with specified rational initial distrib ution, and a rational threshold q , we ask whether the feedback capacity satisfies C fb ( W e , π 1 ,e ) ≥ q . W e pro ve that this exact threshold problem is undecidable, even when restricted to a se verely constrained class of rational unifilar FSCs with bounded state space. The reduction is ef fectiv e and preserves rationality of all channel parameters. As a structural consequence, the exact threshold predicate does not lie in the e xistential theory of the reals ( ∃ R ), and therefore cannot admit a uni versal reduction to finite systems of polynomial equalities and inequalities ov er R . In particular, there is no algorithm deciding all instances of the exact feedback-capacity threshold problem within this class. These results do not preclude approximation schemes or solv ability for special subclasses; rather , they establish a fundamental limitation for exact feedback-capacity reasoning in general finite-state settings. At the metatheoretic level, the undecidability result entails corresponding G ¨ odel–T arski–L ¨ ob incompleteness phenomena for suf ficiently expressi ve formal theories capable of representing the thresh- old predicate. Index T erms Feedback capacity , finite-state channels, unifilar channels, undecidability , computational complexity , directed information. A. Majumdar is with Indraprastha Institute of Information T echnology , Delhi, India (e-mail: angshul@iiitd.ac.in). March 19, 2026 DRAFT 2 I . I N T R O D U C T I O N A. Exact feedback-capacity thresholding as a decision pr oblem Shannon’ s foundational work established channel capacity as the central operational quantity in communication theory [1], [2]. In many classical memoryless settings, capacity admits a clean v ariational characterization and a well-understood coding theorem. For channels with memory , ho we ver , exact capacity analysis is substantially more delicate: the operational quantity remains well-defined, but the associated optimization is typically history-dependent and asymptotic in nature [3]–[5]. This paper studies the follo wing exact decision problem for channels with memory and feedback. Given: • an effecti ve encoding e ∈ { 0 , 1 } ∗ of a binary-input binary-output rational unifilar finite-state channel W e together with its specified rational initial state distribution π 1 ,e , and • a rational threshold q ∈ Q , determine whether C f b ( W e , π 1 ,e ) ≥ q . As in Section III, we denote this predicate by C ap ( e, q ) and the associated language by LC ap . The emphasis is on exactness . W e are not studying a finite-horizon surrogate, an additiv e- approximation problem, or a promise-gap v ariant. The question is whether the e xact infinite- horizon feedback-capacity functional crosses a rational threshold. B. Historical context and the FSC feedback-capacity literatur e The broader context is the long line of work on channels with memory , finite-state channel (FSC) models, and feedback capacity . Classical information-theoretic treatments already make clear that memory changes the analytical character of capacity problems, e ven before feedback is introduced [3]–[5]. Finite-state models became a central abstraction because they retain a concrete stochastic structure while capturing temporal dependence, hidden channel modes, and input/state/output interactions. In the feedback setting, a ke y conceptual step was the introduction of dir ected information by Massey [6], which provides the correct information measure for causal communication o ver channels with memory . This perspective was developed further by Kramer [7] and underlies modern feedback-capacity formulations for channels with memory and state [5], [8], [9]. March 19, 2026 DRAFT 3 For finite-state channels specifically , sev eral milestone results established exact characteri- zations or computable formulations for important subclasses. In particular , the works of Chen– Berger [10], Y ang–Ka v ˇ ci ´ c–T atikonda [11], and Permuter –W eissman–Goldsmith [12] form a canon- ical core of the FSC feedback-capacity literature. These papers dev elop dynamic-programming and structural formulations, and they show that substantial progress is possible when one exploits channel-specific properties. A second major thread focuses on unifilar FSCs, where deterministic state e volution gi ven ( S t , X t , Y t ) mak es finer structural analysis possible. In this line, single-letter upper bounds and graph/context based formulations ha ve led to po werful computable bounds and exact results for important examples [13]. This thread is especially relev ant for the present paper because our undecidability theorem is proved within a rational unifilar FSC class, not outside it. T aken together , the literature sho ws both sides of the story: • there is deep and successful theory for structured FSC subclasses, and • exact feedback capacity remains intrinsically asymptotic and structurally delicate in general. This naturally raises the foundational question addressed here: whether a univer sal exact thresh- old decision procedure can exist for a broad ef fectiv ely specified FSC class. C. What is pr oved her e and how it is positioned The main result of this paper is a structural limitation theorem for exact feedback-capacity reasoning. a) (1) Undecidability of exact thr esholding.: W e prov e that the predicate C ap ( e, q ) is undecidable ev en for a sev erely restricted class of channels: binary-input, binary-output, rational, unifilar finite-state channels (with the bounded-state delayed-activ ation construction used later). Thus, undecidability already appears in a highly concrete and information-theoretically natural FSC family . b) (2) A barrier to univer sal r eal-algebr aic exact formulations.: W e further sho w that the exact threshold language LC ap is not in ∃ R (under the reduction notion fixed later). Equi valently , the exact predicate cannot be captured by a uni versal existential semialgebraic formulation. This places the problem outside the standard ETR/ ∃ R frame work used for many e xact geometric and algebraic decision problems [14]–[16]. c) (3) Computability-theor etic consequences for e xact certification.: From the undecidabil- ity theorem and the formal setup in Section III, we deri ve information-theoretic impossibility March 19, 2026 DRAFT 4 consequences: there is no uni versal e xact certified-bounding scheme and no uni versal finite auxiliary-state/finite-letter exact characterization valid across the full encoded class. d) (4) G ¨ odel–T ar ski–L ¨ ob consequences.: Finally , once exact capacity-threshold truth is encoded as an arithmetic predicate and shown undecidable, standard meta-theoretic consequences follo w for recursi vely axiomatizable theories extending the base entropic theory T 0 : incomplete- ness, undefinability barriers, and L ¨ ob-style limitations on uniform internal reflection for exact threshold truth [17]–[21]. This contribution is not a ne w computable upper bound, a ne w dynamic program, or a new single-letter ansatz. It is a boundary theorem explaining why no uni versal exact framew ork of that type can exist at the le vel of generality studied here. D. What this paper does not claim Because undecidability results are easy to misinterpret, we state the scope explicitly . First, the paper does not contradict the man y positi ve results for structured FSC subclasses. Exact formulas, computable bounds, and dynamic-programming characterizations for special channels remain v alid and important [10]–[13]. Second, the paper does not rule out approximation methods, finite-horizon optimization, or decidability under additional assumptions (for e xample, stronger structural restrictions beyond those imposed here). Our theorem concerns the e xact infinite-horizon threshold predicate over the encoded class. Third, the paper does not assert that all capacity-like quantities for channels with memory are undecidable. The statement is narro wer and sharper: the exact feedback-capacity threshold problem for the specified unifilar FSC class is undecidable. The correct interpretation is therefore constructiv e rather than pessimistic: the theorem identi- fies a rigorous boundary between successful structur ed exact theories and impossible universal exact theories. E. An inconspicuous logic bridge The main arguments are information-theoretic and computability-theoretic, but one later con- sequence is naturally proof-theoretic. Once exact capacity-threshold truth is formalized in the language Lent and shown to encode undecidable behavior , classical meta-mathematical tools March 19, 2026 DRAFT 5 become applicable. This is why the later sections include a brief G ¨ odel–T arski–L ¨ ob analysis [17]–[20]: not as a separate agenda, b ut as a precise formal e xpression of what undecidability implies for any putativ e complete proof system for e xact capacity-threshold statements. I I . P R E L I M I N A R I E S A N D N OTA T I O N This section fixes notation and background used throughout the paper . The purpose is purely stabilizing: to align the formal language in Section III, the structural obstruction in Section IV, the ∃ R route in V, and the G ¨ odel–T arski–L ¨ ob consequences in Section VI under a single set of conv entions. A. Basic notation and encodings W e write X = { 0 , 1 } and Y = { 0 , 1 } for the binary input and output alphabets. A generic finite state set is denoted by Sset (the delayed-activ ation construction in Section II uses a specific family of finite state sets). W e use Q and R for the rationals and reals, respectiv ely . A channel instance is encoded by a finite binary string e ∈ { 0 , 1 } ∗ . For each such encoding, W e denotes the corresponding finite-state channel, and π 1 ,e denotes its specified initial state distrib ution. Rational thresholds are denoted by q ∈ Q . W e fix, once and for all, a standard computable pairing function and write ⟨ e, q ⟩ for an effecti ve binary encoding of the pair ( e, q ) . The e xact feedback-capacity threshold predicate is the same predicate introduced in Defini- tion I.1: C ap ( e, q ) : ⇐ ⇒ C f b ( W e , π 1 ,e ) ≥ q , and the associated language is LC ap := {⟨ e, q ⟩ ∈ { 0 , 1 } ∗ : C f b ( W e , π 1 ,e ) ≥ q } . March 19, 2026 DRAFT 6 B. F inite-state c hannels and the unifilar subclass W e w ork with finite-state channels (FSCs) with feedback in the standard Shannon-theoretic sense. A broad background on channels with memory may be found in Gallager’ s classical text [3]; a modern treatment of feedback and directed-information based formulations appears in [5]. For this paper , an FSC is specified by: • a finite state space Sset , • binary alphabets X , Y , • an initial law π 1 ,e on Sset , • and a time-homogeneous kernel W e ( y t , s t +1 | x t , s t ) . a) F eedback model.: W e assume noiseless output feedback. At time t , the encoder may choose X t causally as a function of the message and the past outputs Y t − 1 = ( Y 1 , . . . , Y t − 1 ) . This is precisely the feedback setting in which directed information is the natural information quantity [5]. Definition II.1 (Restricted channel class) . Let C denote the class of finite-state channels with: • binary alphabets X = { 0 , 1 } and Y = { 0 , 1 } , • a finite state set Sset (encoded as part of e ), • a specified initial distribution π 1 ,e on Sset with rational probabilities, • a time-homogeneous kernel W e ( y , s ′ | x, s ) whose values are rational, • and the unifilar property of Definition II.2. All objects are given ef fecti vely by the encoding e ∈ { 0 , 1 } ∗ . Definition II.2 (Unifilar FSC) . A finite-state channel W ( y , s ′ | x, s ) is unifilar if there exists a deterministic update function f : Sset × X × Y → Sset such that for all ( s, x, y ) , W ( y , s ′ | x, s ) > 0 = ⇒ s ′ = f ( s, x, y ) . Equi v alently , conditioned on ( S t , X t , Y t ) the ne xt state is determined: S t +1 = f ( S t , X t , Y t ) almost surely . March 19, 2026 DRAFT 7 The e xact syntactic restrictions are formalized in the construction section, b ut the channels in C are all: 1) binary-input and binary-output, 2) finite-state, 3) rationally parameterized (transition/output probabilities), 4) equipped with a rational initial distribution. Thus, the negati ve results are not dri ven by irrational coefficients or infinite alphabets. b) Unifilar FSCs.: An FSC is unifilar if there exists a deterministic state-update map f : Sset × X × Y → Sset such that S t +1 = f ( S t , X t , Y t ) almost surely . Equi valently , once ( S t , X t , Y t ) is fixed, the next state is determined. The delayed- acti v ation family in Section IV is of this form. C. Dir ected information, finite-horizon values, and e xact feedbac k capacity Directed information was introduced by Masse y (already cited in the paper) and is the in- formation measure underlying finite-horizon feedback optimisation and exact feedback capacity . W e do not repeat Definitions II.2 and II.1 here; instead we only fix the notation used later . For each encoded channel e and horizon n ≥ 1 , the paper uses the finite-horizon normalized optimisation value V n ( W e , π 1 ,e ) , defined as the supremum of normalized directed information over causal input strategies. (This is the quantity appearing in Sections III. and IV) The exact feedback capacity is denoted by C f b ( W e , π 1 ,e ) , and the threshold language LC ap concerns this infinite-horizon quantity . The structural point exploited later is that finite-horizon agreement of the sequence { V n } n ≥ 1 up to an y fixed horizon does not force equality of the exact asymptotic quantity C f b . This finite-horizon versus exact-asymptotic distinction is standard in channels with memory and feedback, but here it becomes the piv ot of the impossibility results. March 19, 2026 DRAFT 8 D. Computability-theor etic notions W e use only basic computability notions; standard references include Rogers [22] and Soare [23]. Definition II.3 (Computable function) . A function f : { 0 , 1 } ∗ → { 0 , 1 } ∗ is computable if there exists a T uring machine that halts on e very input x ∈ { 0 , 1 } ∗ and outputs f ( x ) . Definition II.4 (Recursi vely enumerable set) . A set A ⊆ { 0 , 1 } ∗ is recursi vely enumerable (r .e.) if there exists a T uring machine that enumerates e xactly the elements of A (equiv alently , membership in A is semi-decidable). These notions will be used later to formalize: • undecidability of the exact threshold language LC ap , and • impossibility of uniform e xact-certification schemes for C f b o ver the full restricted class C . E. Pr oof-theor etic pr eliminaries for the GTL consequences Section VI deri ves G ¨ odel–T arski–L ¨ ob consequences from undecidability of exact capacity thresholds. For that section we need only a minimal amount of proof-theoretic terminology . Useful modern references for prov ability logic and arithmetization are Boolos [20] and Smory ´ nski [21]; the original G ¨ odel, T arski, and L ¨ ob papers are already cited in the manuscript. Definition II.5 (Recursi vely axiomatizable theory) . A first-order theory T is recursiv ely axiom- atizable if the set of G ¨ odel codes of its axioms is recursi vely enumerable. Definition II.6 (Prov ability predicate) . For a recursi vely axiomatizable theory T extending a weak arithmetic base, Pro v T ( x ) denotes the standard arithmetized pro v ability predicate express- ing that x is the G ¨ odel code of a sentence provable in T . The base entropic theory T 0 and its recursiv ely axiomatizable e xtensions (Definition I.4 and Definition I.5) fit e xactly into this frame work. As in Section VI, we assume soundness under the intended semantics and the usual Hilbert–Bernays deriv ability conditions when in voking L ¨ ob-style arguments. March 19, 2026 DRAFT 9 F . Real-algebr aic decision backgr ound and ∃ R V compares the exact threshold language LC ap with the e xistential theory of the reals. W e record the minimum background here. An ETR instance is an existential first-order sentence o ver ( R , + , · , ≤ ) of the form ∃ z 1 , . . . , z m ∈ R : Ψ( z 1 , . . . , z m ) , where Ψ is a quantifier-free Boolean combination of polynomial equalities and inequalities with rational coef ficients. Equiv alently , ETR asks whether a semialgebraic set is nonempty; see, e.g., [24]. The comple xity class ∃ R consists of all languages polynomial-time many-one reducible to ETR. In addition to the references already cited in Section V, classical complexity-theoretic background on the first-order theory of the reals is giv en by Reneg ar [15]. The ke y contrast used later is that ETR is decidable (indeed in PSP A CE ), whereas the exact threshold language LC ap is shown to be undecidable. This completes the common notation and background used in the remaining sections. I I I . D E C I S I O N P R O B L E M S , F I N I T E - L E T T E R L A N G U AG E S , A N D R E P R E S E N TA B I L I T Y A. Capacity threshold pr oblem Recall from Definition III.1 that for each encoded channel e ∈ { 0 , 1 } ∗ , the feedback capacity C fb ( W e , π 1 ,e ) is a well-defined finite real number . Definition III.1 (Feedback capacity threshold problem) . Gi ven a channel encoding e ∈ { 0 , 1 } ∗ and a rational number q ∈ Q , the capacity thr eshold pr oblem asks whether C fb ( W e , π 1 ,e ) ≥ q . W e denote this predicate by Cap( e, q ) . This formulation follo ws the standard reduction of real-valued functionals to rational threshold decision problems. March 19, 2026 DRAFT 10 B. Dir ected information and finite-horizon optimisation Directed information was introduced by Massey [6] and is defined in Definition III.2. Definition III.2 (Directed information) . Let X n := ( X 1 , . . . , X n ) and Y n := ( Y 1 , . . . , Y n ) be random variables on finite alphabets. The dir ected information from X n to Y n is I ( X n → Y n ) := n X t =1 I ( X t ; Y t | Y t − 1 ) , where I ( · ; · | · ) is conditional mutual information under the joint law of ( X n , Y n ) . In the feedback setting, the joint law is induced by a causal input policy p ( x n ∥ y n − 1 ) := n Y t =1 p ( x t | x t − 1 , y t − 1 ) , together with the channel law (and initial state, when present). W e write I ( X n → Y n ) for the directed information computed under this induced joint distribution. For feedback channels, capacity can be e xpressed via directed information; see, e.g., [7]. For fixed horizon n , recall from Definition III.3: Definition III.3 (Finite-horizon directed-information value) . Fix an encoded channel instance ( W e , π 1 ,e ) and a horizon n ∈ N . Define the finite-horizon value V n ( W e , π 1 ,e ) := sup p ( x n ∥ y n − 1 ) I ( X n → Y n ) , where the supremum is o ver all causal input policies p ( x n ∥ y n − 1 ) and I ( X n → Y n ) is computed under the joint distrib ution induced by the polic y , the channel W e , and the initial distrib ution π 1 ,e . W e also use the normalized quantity 1 n V n ( W e , π 1 ,e ) when con venient. For each fixed n , this is a finite-dimensional continuous optimisation o ver a compact semial- gebraic set (Remark III.4). Remark III.4 (Compact semialgebraic policy space) . For fixed horizon n and finite alphabets, the set of causal policies p ( x n ∥ y n − 1 ) = Q n t =1 p ( x t | x t − 1 , y t − 1 ) can be identified with a finite product of probability simplices. Equi v alently , it is a closed and bounded subset of a Euclidean space defined by finitely many polynomial equalities and inequalities: nonne gati vity constraints and normalization constraints P x t p ( x t | x t − 1 , y t − 1 ) = 1 for each ( x t − 1 , y t − 1 ) . Hence the feasible set is compact and semialgebraic, and the optimization defining V n ( W e , π 1 ,e ) is a finite-dimensional continuous optimization problem. March 19, 2026 DRAFT 11 C. F ormal language for finite-letter entr opic expr essions W e now formalise the syntactic fragment that captures all finite-letter entropic e xpressions. Logical language: Let L ent be a first-order language extending the language of ordered fields by: • Function symbols for rational arithmetic, • For each finite alphabet A , a function symbol H A ( · ) interpreted as Shannon entropy , • For each finite collection of random variables, function symbols representing I ( · ; · ) and I ( · ; · | · ) . Semantically , these symbols are interpreted using the Shannon entropy definition [1]: H ( P ) = − X a ∈A P ( a ) log P ( a ) . F inite-letter fr agment: Definition III.5 (Finite-letter entropic functional) . A functional Φ on the restricted channel class C is called finite-letter entr opic if for each encoding e , Φ( e ) = sup u ∈U e F e ( u ) , where: 1) U e ⊂ R d ( e ) is a compact set defined by finitely many polynomial equalities and inequalities with rational coef ficients; 2) F e ( u ) is an L ent -term constructed using: finitely man y rational constants, finitely many conditional probability v ariables, finite sums and products, and finitely many entropy or mutual-information operators e v aluated on finite alphabets. Importantly , Definition III.5 e xcludes an y limit operations or infinite recursions. D. F inite-letter r epr esentability of feedback capacity Definition III.6 (Finite-letter representability) . Feedback capacity is finite-letter r epr esentable ov er C if there exists a finite-letter entropic functional Φ such that for all encodings e , Φ( e ) = C fb ( W e , π 1 ,e ) . This definition is purely semantic: it asserts equality as real numbers for all encoded channels. March 19, 2026 DRAFT 12 E. Entr opic pr oof systems W e now define the proof-theoretic frame work used later . Definition III.7 (Base theory T 0 ) . Let T 0 be a sound first-order theory in the language L ent capable of reasoning about: 1) Rational arithmetic, 2) Polynomial equalities and inequalities, 3) Finite sums and products, 4) Shannon entropy and mutual information over finite alphabets. Definition III.8 (Entropic theory) . An entr opic theory T is a recursiv ely axiomatizable extension of T 0 such that: 1) T is sound under the standard real-number semantics; 2) For each fix ed n , T can reason about the quantity V n ( W e , π 1 ,e ) ; 3) T can reason about finite maximisations o ver compact semialgebraic sets as in Defini- tion III.5. F . Pr ovability of capacity thr esholds Definition III.9 (Prov able threshold) . For entropic theory T , we write T ⊢ Cap( e, q ) if the sentence C fb ( W e , π 1 ,e ) ≥ q is deriv able in T . The distinction between semantic truth and syntactic pro vability will be central in Section VI. I V . A S T R U C T U R A L B A R R I E R : N O U N I F O R M F I N I T E - H O R I Z O N C O L L A P S E This section establishes a structural obstruction that is independent of computational complex- ity: ev en within the restricted rational unifilar class C (Definition II.1), feedback capacity cannot be recovered from any fixed finite-horizon directed-information optimisation. The construction and proofs in this section are fully explicit. March 19, 2026 DRAFT 13 A. A delayed-activation unifilar family Fix the binary alphabets X = { 0 , 1 } and Y = { 0 , 1 } . For each inte ger N ≥ 1 , define a finite state set § N := { 0 , 1 , 2 , . . . , N } ∪ { ⋆ } , and an initial distribution π 1 ,N concentrated at S 1 = 0 . W e define two channels in C : Ch ( N ) goo d and Ch ( N ) bad , both unifilar with the same deterministic update map f N described belo w , but with different output kernels once the channel reaches the activ e state ⋆ . Deterministic state update.: Define f N : § N × X × Y → § N by f N ( s, x, y ) :=            s + 1 , s ∈ { 0 , 1 , . . . , N − 1 } , ⋆, s = N , ⋆, s = ⋆. Thus the state increments deterministically for N steps and then enters the absorbing acti ve state ⋆ forever . This update does not depend on ( x, y ) . Output kernels.: For s ∈ { 0 , 1 , . . . , N } , define the same output kernel for both channels: P ( y | x, s ) = 1 { y = 0 } , ∀ x ∈ X . (1) Thus, during the first N + 1 states (the “delay phase”), the output is deterministically 0 . At the acti ve state s = ⋆ , define: Ch ( N ) goo d : P ( y | x, ⋆ ) = 1 { y = x } , (2) Ch ( N ) bad : P ( y | x, ⋆ ) = 1 2 1 { y = 0 } + 1 2 1 { y = 1 } . (3) All probabilities are rational, hence these are members of C . By Definition II.2, each channel can be written as W ( y , s ′ | x, s ) = P ( y | x, s ) 1 { s ′ = f N ( s, x, y ) } . March 19, 2026 DRAFT 14 B. Exact finite-horizon values W e now compute V n exactly for both channels. Lemma IV .1 (Zero information during the delay phase) . F ix N ≥ 1 . F or either c hannel Ch ( N ) goo d or Ch ( N ) bad , and for any causal strate gy p ( x n ∥ y n − 1 ) , we have Y t = 0 a.s. for all t ≤ N + 1 . Consequently , I ( X t ; Y t | Y t − 1 ) = 0 for all t ≤ N + 1 , and hence I ( X n → Y n ) = n X t = N +2 I ( X t ; Y t | Y t − 1 ) . Pr oof. By construction, S 1 = 0 a.s. and S t deterministically equals t − 1 for t ≤ N + 1 , so S t ∈ { 0 , 1 , . . . , N } for t ≤ N + 1 . Then (1) gi ves Y t = 0 a.s. for all t ≤ N + 1 . If Y t is almost surely constant given Y t − 1 , then H ( Y t | Y t − 1 ) = 0 , hence I ( X t ; Y t | Y t − 1 ) ≤ H ( Y t | Y t − 1 ) = 0 . Lemma IV .2 (Finite-horizon value for the bad channel) . F or Ch ( N ) bad , V n ( Ch ( N ) bad ) = 0 ∀ n ≥ 1 . Pr oof. For t ≤ N + 1 , Lemma IV .1 gi ves I ( X t ; Y t | Y t − 1 ) = 0 . For t ≥ N + 2 , the state is ⋆ deterministically , and under (3), Y t is independent of X t (indeed, independent of all X t ). Thus I ( X t ; Y t | Y t − 1 ) = 0 for all t ≥ N + 2 . Hence I ( X n → Y n ) = 0 for all strate gies, and therefore V n = 0 for all n . Lemma IV .3 (Finite-horizon value for the good channel) . F or Ch ( N ) goo d , V n ( Ch ( N ) goo d ) =      0 , 1 ≤ n ≤ N + 1 , n − ( N + 1) n , n ≥ N + 2 . Pr oof. If n ≤ N + 1 , Lemma IV .1 giv es I ( X n → Y n ) = 0 for all strate gies, so V n = 0 . No w let n ≥ N + 2 . For t ≥ N + 2 , the state is ⋆ deterministically and the channel is noiseless: Y t = X t by (2). Conditioned on Y t − 1 , we ha ve Y t = X t and Y t − 1 is a function of X t − 1 , hence I ( X t ; Y t | Y t − 1 ) = I ( X t ; Y t | Y t − 1 ) = H ( Y t | Y t − 1 ) , March 19, 2026 DRAFT 15 since Y t is a deterministic function of X t and Y t − 1 . Moreo ver , because Y t = X t , we can choose a strate gy that makes X t independent of Y t − 1 and uniform on { 0 , 1 } for all t ≥ N + 2 . Under this strategy , H ( Y t | Y t − 1 ) = H ( X t ) = 1 , hence I ( X t ; Y t | Y t − 1 ) = 1 for all t ≥ N + 2 . Combining with Lemma IV .1 yields I ( X n → Y n ) = n X t = N +2 1 = n − ( N + 1) , and therefore V n ≥ n − ( N + 1) n . On the other hand, for any strate gy and any t ≥ N + 2 , I ( X t ; Y t | Y t − 1 ) ≤ H ( Y t | Y t − 1 ) ≤ H ( Y t ) ≤ 1 , since Y t ∈ { 0 , 1 } . Hence I ( X n → Y n ) = n X t = N +2 I ( X t ; Y t | Y t − 1 ) ≤ n X t = N +2 1 = n − ( N + 1) , so V n ≤ n − ( N + 1) n . (4) Thus equality holds. Theorem IV .4 (Undecidability of exact feedback-capacity thresholds) . Ther e is no algorithm that, given (i) an encoding e ∈ { 0 , 1 } ∗ of a binary-input binary-output unifilar finite-state channel W e fr om the r estricted rational class C (Definition II.1), tog ether with its specified rational initial distribution π 1 ,e , and (ii) a rational thr eshold q ∈ Q , decides whether C fb ( W e , π 1 ,e ) ≥ q . Equivalently , the languag e LCap := {⟨ e, q ⟩ ∈ { 0 , 1 } ∗ : C fb ( W e , π 1 ,e ) ≥ q } is undecidable (even when r estricted to encodings e of channels in C ). March 19, 2026 DRAFT 16 Pr oof of Theor em IV .4. W e argue by many-one reduction from the fixed undecidable source problem used in Section I-A (denote its instance space by I and its YES-language by L ⊆ I ). Step 1 (effectiv e mapping). By the construction in Section I-A, there exists a total computable function F : I → { 0 , 1 } ∗ × Q , i 7→ ( e ( i ) , q ( i )) , such that e ( i ) encodes a channel instance ( W e ( i ) , π 1 ,e ( i ) ) in the restricted class C (Definition II.1) and q ( i ) ∈ Q is the threshold specified by the reduction. The ef fectivity of F follows because e very component of the channel description—finite state set, unifilar update function, and rational transition/output probabilities—is computed explicitly from the finite description of i . Step 2 (correctness of the reduction). The correctness lemmas prov ed in Section I-A establish the equiv alence i ∈ L ⇐ ⇒ C fb ( W e ( i ) , π 1 ,e ( i ) ) ≥ q ( i ) . (5) (Here the forw ard direction is the “YES-case” analysis of the delayed-activ ation construction, and the re verse direction is the “NO-case” analysis; together they yield (5).) Step 3 (undecidability). Assume for contradiction that there exists an algorithm A deciding C ap ( e, q ) for all encodings e of channels in C and all q ∈ Q ; i.e., on input ( e, q ) it halts and outputs Y E S iff C fb ( W e , π 1 ,e ) ≥ q . Then we could decide the source language L as follows: on input i ∈ I , compute ( e ( i ) , q ( i )) = F ( i ) and run A on ( e ( i ) , q ( i )) . By (5), A returns Y E S iff i ∈ L . This yields a decision procedure for L , contradicting the undecidability of the source problem. Therefore no such algorithm A exists, and C ap ( e, q ) is undecidable e ven when restricted to encodings e of channels in C . Equiv alently , the language LCap is undecidable on this class. C. Differ ent feedback capacities with identical short-horizon behaviour Pr oposition IV .5 (Arbitrarily long identical initial behaviour , different capacities) . For e very N ≥ 1 , there exist encodings e ( N ) goo d and e ( N ) bad in { 0 , 1 } ∗ encoding Ch ( N ) goo d and Ch ( N ) bad respecti vely , such that V n ( W e ( N ) goo d , π 1 ,e ( N ) goo d ) = V n ( W e ( N ) bad , π 1 ,e ( N ) bad ) = 0 for all 1 ≤ n ≤ N + 1 , (6) March 19, 2026 DRAFT 17 but C fb ( W e ( N ) goo d , π 1 ,e ( N ) goo d ) = 1 , C fb ( W e ( N ) bad , π 1 ,e ( N ) bad ) = 0 . Pr oof. The statements about V n for n ≤ N + 1 follow from Lemmas IV .2 and IV .3. For capacities, by Definition III.1 and Lemma IV .2, C fb ( Ch ( N ) bad ) = lim sup n →∞ 0 = 0 . For the good channel, Lemma IV .3 gi ves V n ( Ch ( N ) goo d ) = n − ( N + 1) n − − − → n →∞ 1 , hence C fb ( Ch ( N ) goo d ) = 1 . D. What this does (and does not) imply Proposition IV .5 shows that no fixed finite horizon can determine feedback capacity , ev en within C . This yields an unconditional barrier against any attempted characterisation of C fb that depends only on finitely many V n ’ s with a horizon bound that is uniform o ver channels. Definition IV .6 (Uniform finite-horizon characterisation) . A functional Ψ : C → R is a uniform finite-horizon char acterisation if there exists N ∈ N and a (possibly channel-dependent) mapping G : R N → R such that for all encodings e , Ψ( e ) = G  V 1 ( W e , π 1 ,e ) , . . . , V N ( W e , π 1 ,e )  . Corollary IV .7 (No uniform finite-horizon characterisation) . Ther e is no uniform finite-horizon char acterisation Ψ such that Ψ( e ) = C fb ( W e , π 1 ,e ) for all e . Pr oof. Assume such Ψ e xists with horizon bound N . Apply Proposition IV .5 with that N : there exist e ( N ) goo d and e ( N ) bad for which V n ( · ) = 0 for all 1 ≤ n ≤ N + 1 (hence also for 1 ≤ n ≤ N ), but C fb dif fers. Then Ψ takes the same input v ector (0 , . . . , 0) ∈ R N on both channels, so Ψ( e ( N ) goo d ) = Ψ( e ( N ) bad ) , contradicting Ψ ≡ C fb . Remark IV .8 (Scope of this section) . Corollary IV .7 does not by itself rule out a general finite- letter representability statement as in Definition III.6, because a finite-letter expression may March 19, 2026 DRAFT 18 depend on channel parameters in states that are only reachable after long time. The stronger non-representability claim will therefore be established later , by combining structural properties with a computability-theoretic barrier . V . ∃ R R O U T E : W H A T I T W O U L D I M P L Y , A N D W H Y I T C A N N O T H O L D H E R E This section formalizes the “ ∃ R route” and prov es a clean neg ati ve conclusion: the undecid- ability theorem for exact feedback-capacity thresholds (Theorem IV .4) rules out membership in ∃ R for the exact predicate Cap( e, q ) . A. Existential theory of the r eals and the class ∃ R Definition V .1 (Existential theory of the reals (ETR)) . An ETR-instance is a sentence of the form ∃ z 1 , . . . , z d ∈ R : Ψ( z 1 , . . . , z d ) , where Ψ is a quantifier-free Boolean combination of polynomial equalities and inequalities with rational coefficients. Definition V .2 (The class ∃ R ) . A language L ⊆ { 0 , 1 } ∗ lies in ∃ R if there exists a polynomial- time many-one reduction mapping x 7→ φ x such that x ∈ L ⇐ ⇒ φ x is a true ETR-instance . Remark V .3 . ETR is equiv alent to non-emptiness of a semialgebraic set. A modern complexity- theoretic account of ∃ R is given in [16]. B. Decidability of ETR Definition V .4 (The class PSP ACE ) . PSP A CE := PSP A CE denotes the class of decision prob- lems solvable by a deterministic T uring machine using polynomial space. Theorem V .5 (Decidability of ETR; PSP A CE upper bound [14]) . The existential theory of the r eals is decidable. Mor eover , ∃ R ⊆ PSP ACE . Remark V .6 . See [14] and [] for real-algebraic decision procedures. March 19, 2026 DRAFT 19 C. Capacity threshold language Recall Definition III.1. Define L Cap := {⟨ e, q ⟩ ∈ { 0 , 1 } ∗ : C fb ( W e , π 1 ,e ) ≥ q } , where ⟨ e, q ⟩ is an y fixed computable pairing encoding of ( e, q ) . D. Non-membership in ∃ R Theorem V .7 (Exact capacity thresholds are not in ∃ R ) . Assuming Theor em IV .4, L Cap / ∈ ∃ R . Pr oof. Suppose L Cap ∈ ∃ R . Then by Definition V .2 there exists a polynomial-time reduction mapping ⟨ e, q ⟩ to an ETR-instance φ e,q such that ⟨ e, q ⟩ ∈ L Cap ⇐ ⇒ φ e,q is true . By Theorem V .5, truth of φ e,q is decidable. Hence L Cap would be decidable. This contradicts Theorem IV .4, which states that the exact predicate Cap( e, q ) is undecidable e ven under se vere channel restrictions. Corollary V .8. Ther e e xists no polynomial-time r eduction of the exact feedback-capacity thr esh- old pr oblem to ETR. E. T akeaway Any attempt to reduce the exact feedback capacity threshold problem to a decidable real- algebraic framew ork such as ∃ R is impossible. The obstruction is purely computability-theoretic. V I . L O G I C A L C O N S E Q U E N C E S O F U N D E C I D A B I L I T Y : G ¨ O D E L – T A R S K I – L ¨ O B I M P L I C A T I O N S Sections I-A – VI established that the exact feedback-capacity threshold language L Cap = {⟨ e, q ⟩ : C fb ( W e , π 1 ,e ) ≥ q } is undecidable, ev en o ver the restricted binary unifilar FSC class. W e now deri ve formal conse- quences for any effecti ve axiomatization of exact capacity statements. This section is meta-theoretic and does not introduce ne w information-theoretic quantities. March 19, 2026 DRAFT 20 A. F ormal framework Let L be a first-order language extending arithmetic, containing symbols suf ficient to encode: 1) natural numbers and rational numbers, 2) finite FSC encodings e ∈ { 0 , 1 } ∗ , 3) rational thresholds q , 4) the predicate Cap( e, q ) . W e consider a theory T over L satisfying: 1) T is recursi vely axiomatizable; 2) T is consistent; 3) T is sound with respect to the intended Shannon-theoretic semantics; 4) T interprets Robinson arithmetic Q . Definition VI.1 (The formula schema Cap( e, q ) and its semantics) . Fix the encoding con ventions from Section III-C. F or each binary string e ∈ { 0 , 1 } ∗ and rational q ∈ Q , let Cap( e, q ) denote the closed L -sentence obtained by substituting the numerals/encodings of e and q into the capacity- threshold formula schema fixed in Definition III.1. Its intended semantics is: Cap( e, q ) is true ⇐ ⇒ C fb ( W e , π 1 ,e ) ≥ q . B. G ¨ odel incompleteness Theorem VI.2 (Incompleteness for exact capacity) . Under the above assumptions, T cannot decide all instances of Cap( e, q ) . Pr oof. Suppose T decided ev ery instance: for each ( e, q ) , either T ⊢ Cap( e, q ) or T ⊢ ¬ Cap( e, q ) . Because T is recursiv ely axiomatizable, the set of theorems is recursi vely enumerable. Since we assumed T decides ev ery instance, e xactly one of T ⊢ Cap( e, q ) or T ⊢ ¬ Cap( e, q ) holds for each ( e, q ) , so the do vetailing procedure halts on e very input. Thus, gi ven ( e, q ) , one can dov etail o ver all proofs in T until either Cap( e, q ) or its negation appears, yielding a decision procedure for L Cap . This contradicts Theorem IV .4. Therefore T cannot be complete for L Cap . Corollary VI.3. No sound, r ecursively axiomatizable entr opic theory is complete for exact feedback-capacity thr eshold statements over the r estricted FSC class. March 19, 2026 DRAFT 21 This is a direct instance of G ¨ odel’ s first incompleteness theorem [17] applied to the arithmetic representation of L Cap . C. T arski semantic separation Definition VI.4 (Internal truth predicate) . Let Sent L ( x ) be the arithmetical predicate e xpressing “ x is the G ¨ odel code of a closed L -sentence”. An internal truth predicate for L in T is a L -formula T rue( x ) such that for ev ery closed L -sentence φ , T ⊢ T rue( ⌜ φ ⌝ ) ↔ φ. Theorem VI.5 (T arski undefinability , specialized) . Under the assumptions of Section VI, no internal truth pr edicate for L e xists in T . Pr oof. This is the classical undefinability of truth theorem of T arski [25], applicable because T interprets Q and is consistent. T arski’ s undefinability theorem states that no suf ficiently expressi ve consistent theory can define its o wn semantic truth predicate [25]. Corollary VI.6 (No definable truth predicate e ven for Cap -instances) . Ther e is no L -formula T rueCap( e, q ) suc h that for all e ∈ { 0 , 1 } ∗ and q ∈ Q , T ⊢ T rueCap( e, q ) ↔ Cap( e, q ) , and T rueCap( e, q ) is corr ect with r espect to the intended semantics of Definition VI.1 for all ( e, q ) . Pr oof. If such T rueCap existed, then the mapping x = ⌜ Cap( e, q ) ⌝ 7→ T rueCap( e, q ) w ould yield an internal truth predicate for the infinite family of closed L -sentences Cap( e, q ) , uniformly in ( e, q ) . Moreov er , the set { ⌜ Cap( e, q ) ⌝ : e ∈ { 0 , 1 } ∗ , q ∈ Q } is infinite and effecti vely enumer- able from the pairing/encoding. This produces an internal truth predicate fragment, contradicting Theorem VI.5. Thus semantic capacity truth strictly exceeds formal deriv ability within T . March 19, 2026 DRAFT 22 D. L ¨ ob obstruction Because T interprets Q , it admits an arithmetized prov ability predicate Pro v T ( · ) . Assume further that T satisfies the Hilbert–Bernays deriv ability conditions (as is standard for such theories; see [19]). Theorem VI.7 (L ¨ ob’ s theorem for capacity formulas) . F or any fixed ( e, q ) , if T ⊢ Prov T ( ⌜ Cap( e, q ) ⌝ ) → Cap( e, q ) , then T ⊢ Cap( e, q ) . Pr oof. This is an instance of L ¨ ob’ s theorem [19]. Corollary VI.8. Ther e is no consistent r ecursively axiomatizable theory that pr oves the uniform r eflection principle ∀ e, q  Pro v T ( ⌜ Cap( e, q ) ⌝ ) → Cap( e, q )  . Pr oof. If such uniform reflection were prov able, then by Theorem VI.7, T would prove ev ery true instance Cap( e, q ) , contradicting incompleteness (Theorem VI.2) and consistency . E. Consequences for this paper Sections I-A – VI established algorithmic limitations: no decision procedure and no uniformly certified conv ergence scheme e xist. The present section sho ws the logical counterpart: • Exact capacity undecidability forces incompleteness of any sound ef fecti ve entropic calculus (G ¨ odel). • Semantic capacity truth cannot be internally defined within such a calculus (T arski). • No such theory can uniformly certify correctness of its own capacity-threshold proofs (L ¨ ob). These conclusions follow solely from the undecidability results prov ed earlier and require no additional information-theoretic assumptions. V I I . L I M I T A T I O N S A N D F U T U R E D I R E C T I O N S This paper prov es an impossibility result for the exact infinite-horizon threshold predicate C ap ( e, q ) : ⇐ ⇒ C f b ( W e , π 1 ,e ) ≥ q March 19, 2026 DRAFT 23 ov er the encoded class of rational unifilar FSCs considered in the paper . The scope is intentionally narro w and should be read exactly as such. The theorem concerns e xact threshold truth; it does not settle approximation, promise-gap, or finite-horizon v ariants. A. Exact thresholding versus appr oximation The main open direction is the status of approximation versions of the problem. Our un- decidability theorem does not imply undecidability of an y fixed-gap or approximate threshold problem. In particular , it remains open whether there exists a uniform procedure for a promise-gap decision problem of the form C f b ( W e , π 1 ,e ) ≥ q + ε versus C f b ( W e , π 1 ,e ) ≤ q − ε, for fixed rational ε > 0 , on the same encoded class. Lik e wise, the theorem does not rule out ef fecti ve approximation schemes under additional assumptions, nor does it preclude certified finite-horizon surrogates when one can control the gap between V n ( W e , π 1 ,e ) and C f b ( W e , π 1 ,e ) . The present result therefore isolates a boundary for exact asymptotic thresholding, not for approximation methodology . B. Structural r estrictions and the decidability fr ontier Although the channel class studied here is already heavily restricted (binary alphabets, rational parameters, unifilar state ev olution), it is still broad enough to encode undecidable behavior . A natural next step is to identify stronger restrictions under which decidability returns. Promising directions include subclasses with verifiable contraction/mixing properties, effecti ve finite-context reductions, or additional monotonicity/re gularity conditions that force a computable finite description of the asymptotic optimization. The positiv e FSC feedback-capacity literature suggests that such structure can be decisi ve; our theorem shows only that no single exact procedure can cov er the full class considered here. C. Complexity r efinements be yond undecidability The paper pro ves undecidability and the corresponding ∃ R barrier , but it does not attempt a finer recursion-theoretic classification of LC ap . In particular , we do not classify the language as r .e., co-r .e., or complete for any standard degree under stronger reductions. March 19, 2026 DRAFT 24 These are legitimate follow-up questions, b ut they require a dif ferent lev el of coding analysis than is needed for the present structural impossibility theorem. The same applies to complexity classification of promise-gap variants, should such variants turn out to be decidable. D. Logical extensions and r estricted exact theories Section VI deriv es G ¨ odel–T arski–L ¨ ob consequences for recursi vely axiomatizable extensions of the base entropic theory T 0 . Those consequences are stated only at the lev el required by the undecidability result. W e do not attempt a full proof-theoretic classification of information- theoretic formal systems. A natural future direction is to isolate restricted fragments (for example, bounded-horizon fragments or structurally constrained channel classes) for which exact threshold statements admit complete axiomatizations or stronger internal reflection principles. This w ould complement the present paper by locating islands of exact formal tractability inside the broader impossibility landscape. E. Methodological implication The theorem should not be read as a ne gati ve verdict on FSC feedback-capacity research. Its implication is methodological: univ ersal exact threshold procedures are impossible at the generality studied here, so progress must continue to come from explicit structure, subclass assumptions, and approximation principles. That is already how the field has adv anced; the present result explains wh y this is not merely a matter of current technique, but a fundamental boundary . V I I I . C O N C L U S I O N W e studied the exact threshold decision problem for feedback capacity in finite-state channels and proved that the predicate C ap ( e, q ) : ⇐ ⇒ C f b ( W e , π 1 ,e ) ≥ q is undecidable e ven over a sev erely restricted class of binary-input, binary-output, rational, unifilar FSCs. Thus, there is no algorithm that decides this exact threshold predicate uniformly ov er the encoded class considered in the paper . March 19, 2026 DRAFT 25 W e then established a complementary barrier: the exact threshold language LC ap does not lie in the e xistential-real frame work used for exact semialgebraic decision problems. In particular , there is no uni versal e xact reduction of this predicate to an existential theory-of-the-reals instance. From the same formal setup, we deriv ed computability-theoretic consequences for exact certifica- tion and finite auxiliary-state characterizations, and finally the corresponding G ¨ odel–T arski–L ¨ ob limitations for recursi vely axiomatizable e xtensions of the base entropic theory T 0 . The result is a boundary theorem. It does not contradict positi ve exact or computable results for structured FSC subclasses, and it does not rule out approximation methods or finite-horizon analysis. Its point is more precise: for the e xact infinite-horizon threshold predicate on the encoded unifilar FSC class considered here, no universal exact decision procedure exists. This boundary is informativ e rather than discouraging. It clarifies why successful FSC feedback- capacity theory relies on structural assumptions, and it identifies the right direction for future work: characterize the subclasses and approximation regimes where e xact or certified computa- tion is still possible. R E F E R E N C E S [1] C. E. Shannon, “ A mathematical theory of communication, ” Bell System T echnical Journal , vol. 27, no. 3, pp. 379–423, 623–656, 1948. [2] ——, “The zero error capacity of a noisy channel, ” IRE T ransactions on Information Theory , vol. 2, no. 3, pp. 8–19, 1956. [3] R. G. Gallager, Information theory and reliable communication . New Y ork: John W iley & Sons, 1968. [4] T . M. Cov er and J. A. Thomas, Elements of information theory , 2nd ed. Hoboken, NJ: Wiley- Interscience, 2006. [5] A. E. Gamal and Y .-H. Kim, Network information theory . Cambridge: Cambridge Univ ersity Press, 2011. [6] J. L. Massey , “Causality , feedback and directed information, ” in Proceedings of the International Symposium on Information Theory and Its Applications (ISITA) , W aikiki, Hawaii, 1990, pp. 303–305. [7] G. Kramer, “Directed information for channels with feedback, ” Ph.D. dissertation, ETH Z ¨ urich, 1998, diss. ETH No. 12656. [8] S. T atikonda, “Control under communication constraints, ” Ph.D. dissertation, Massachusetts Institute of T echnology , Cambridge, MA, USA, 2000. [9] S. T atikonda and S. Mitter , “The capacity of channels with feedback, ” IEEE T ransactions on Information Theory , vol. 55, no. 1, pp. 323–349, 2009. [10] J. Chen and T . Berger , “The capacity of finite-state markov channels with feedback, ” IEEE T ransactions on Information Theory , vol. 51, no. 3, pp. 780–798, 2005. [11] S. Y ang, A. Kav ˇ ci ´ c, and S. T atikonda, “Feedback capacity of finite-state machine channels, ” IEEE T ransactions on Information Theory , vol. 51, no. 3, pp. 799–810, 2005. [12] H. H. Permuter , T . W eissman, and A. J. Goldsmith, “Finite-state channels with time-in variant deterministic feedback, ” IEEE T ransactions on Information Theory , vol. 55, no. 2, pp. 644–662, 2009. March 19, 2026 DRAFT 26 [13] O. Sabag, H. H. Permuter, and H. D. Pfister, “ A single-letter upper bound on the feedback capacity of unifilar finite-state channels, ” IEEE T ransactions on Information Theory , vol. 63, no. 3, pp. 1392–1409, 2017. [14] J. F . Canny , “Some algebraic and geometric computations in PSP A CE, ” in Pr oceedings of the 20th Annual A CM Symposium on Theory of Computing (STOC) . A CM, 1988, pp. 460–467. [15] J. Renegar , “On the computational complexity and geometry of the first-order theory of the reals. I. introduction. preliminaries. the geometry of semi-algebraic sets. the decision problem for the existential theory of the reals, ” Journal of Symbolic Computation , vol. 13, no. 3, pp. 255–299, 1992. [16] M. Schaefer , J. Cardinal, and T . Miltzo w , “The existential theory of the reals as a complexity class: A compendium, ” arXiv pr eprint arXiv:2407.18006 , 2024. [17] K. G ¨ odel, “ ¨ Uber formal unentscheidbare s ¨ atze der Principia Mathematica und verwandter Systeme I, ” Monatshefte f ¨ ur Mathematik und Physik , vol. 38, pp. 173–198, 1931. [18] A. T arski, “The concept of truth in formalized languages, ” in Logic, semantics, metamathematics: P apers fr om 1923 to 1938 , 2nd ed., J. Corcoran, Ed. Indianapolis: Hackett, 1983. [19] M. H. L ¨ ob, “Solution of a problem of leon henkin, ” Journal of Symbolic Logic , vol. 20, no. 2, pp. 115–118, 1955. [20] G. S. Boolos, The logic of pro vability . Cambridge: Cambridge Univ ersity Press, 1993. [21] C. Smory ´ nski, “The incompleteness theorems, ” in Handbook of mathematical logic , J. Barwise, Ed. Amsterdam: North- Holland, 1977, pp. 821–865. [22] J. Hartley Rogers, Theory of r ecursive functions and effective computability . New Y ork: McGraw-Hill, 1967. [23] R. I. Soare, Recur sively enumerable sets and degr ees: A study of computable functions and computably g enerated sets , ser . Perspectiv es in Mathematical Logic. Berlin: Springer , 1987. [24] J. Bochnak, M. Coste, and M.-F . Roy , Real algebraic geometry , ser . Ergebnisse der Mathematik und ihrer Grenzgebiete. Berlin: Springer, 1998, vol. 36. [25] A. T arski, “Der Wahrheitsbegriff in den formalisierten Sprachen, ” Studia Philosophica , vol. 1, pp. 261–405, 1936. [26] T . M. Cov er and J. A. Thomas, Elements of Information Theory , 2nd ed. W iley-Interscience, 2006. [27] M. Fannes, “ A continuity property of the entropy density for spin lattice systems, ” Communications in Mathematical Physics , vol. 31, no. 4, pp. 291–294, 1973. [28] K. M. R. Audenaert, “ A sharp continuity estimate for the von neumann entropy , ” Journal of Physics A: Mathematical and Theor etical , vol. 40, no. 28, pp. 8127–8136, 2007. [29] K. W eihrauch, Computable Analysis: An Intr oduction , ser . T exts in Theoretical Computer Science. Springer , 2000. [30] K.-I. Ko, Complexity Theory of Real Functions , ser . Progress in Theoretical Computer Science. Birkh ¨ auser , 1991. A P P E N D I X This appendix strengthens the finite-horizon frame work of Sections II–III by proving that the supremum in the definition of V n is attained and that V n (hence V n /n ) admits a uniform effecti ve approximation algorithm from the encoding and horizon. These results are self-contained and do not require any changes to the main paper . March 19, 2026 DRAFT 27 A. P olicy space and induced laws (finite alphabets) Fix an instance ( W e , π 1 ,e ) from the restricted encoded class of Section II and a horizon n ≥ 1 . As in Section III, a causal input policy has the f actorization p ( x n ∥ y n − 1 ) = n Y t =1 p ( x t | x t − 1 , y t − 1 ) , (7) with finite alphabets X , Y and finite state set S (Section II). Let P e,n denote the set of all such causal policies. Under the product-of-simplices parametrization described in Remark III.4, P e,n is compact. For any p ∈ P e,n , let P p denote the induced joint pmf of ( X n , Y n , S n +1 ) under ( W e , π 1 ,e ) . Lemma A.1 (Multilinearity of the induced joint pmf) . F or fixed ( e, n ) and any ( x n , y n , s n +1 ) , the quantity P p ( x n , y n , s n +1 ) is a multilinear polynomial in the policy coor dinates { p ( x t | x t − 1 , y t − 1 ) } , with coefficients that ar e rational and ef fectively determined by the encoding e . Pr oof. Unrolling the finite-horizon la w (as in Section II) gi ves P p ( x n , y n , s n +1 ) = π 1 ,e ( s 1 ) n Y t =1 p ( x t | x t − 1 , y t − 1 ) W e ( y t , s t +1 | x t , s t ) . For fixed ( x n , y n , s n +1 ) , this is a product of one policy factor per time t multiplied by rational channel/state coef ficients, hence multilinear in the policy coordinates with rational coef ficients. Ef fecti veness follo ws because W e and π 1 ,e are giv en by a finite encoding with rational entries (Section II). B. Attainment of the finite-horizon supr emum Define V n ( W e , π 1 ,e ) as in Definition III.3: V n ( W e , π 1 ,e ) = sup p ∈P e,n I p ( X n → Y n ) , I p ( X n → Y n ) = n X t =1 I p ( X t ; Y t | Y t − 1 ) . (8) Theorem A.2 (Attainment of V n ) . F or every encoding e (Section II) and horizon n ≥ 1 , ther e exists an optimal causal policy p ⋆ ∈ P e,n such that V n ( W e , π 1 ,e ) = I p ⋆ ( X n → Y n ) . Pr oof. By Remark III.4, P e,n is compact. It suf fices to sho w that the objecti ve p 7→ I p ( X n → Y n ) is continuous on P e,n . March 19, 2026 DRAFT 28 Fix t . Since alphabets are finite, I p ( X t ; Y t | Y t − 1 ) can be written as a finite linear combination of entropies of marginals of the induced pmf P p . By Lemma A.1, each such marginal probability is a polynomial function of the polic y coordinates, hence continuous. Shannon entropy is con- tinuous on the finite probability simple x (see, e.g., [26, Ch. 2]), and so are conditional entropies and conditional mutual informations. Therefore each p 7→ I p ( X t ; Y t | Y t − 1 ) is continuous, and their finite sum is continuous. A continuous real-v alued function on a compact set attains its maximum, hence the supremum in (8) is achieved by some p ⋆ ∈ P e,n . C. Continuity modulus for directed information via total variation W e use standard continuity bounds for entropy on finite alphabets. Let ∥ P − Q ∥ 1 := P a | P ( a ) − Q ( a ) | . Lemma A.3 (Entropy continuity (F annes–Audenaert)) . Let P , Q be pmfs on a finite alphabet A with ∥ P − Q ∥ 1 ≤ δ ≤ 1 / 2 . Then | H ( P ) − H ( Q ) | ≤ δ log ( |A| − 1) + h 2 ( δ ) , wher e h 2 ( δ ) = − δ log δ − (1 − δ ) log(1 − δ ) . Pr oof. This is a standard inequality; see [27], [28]. Lemma A.4 (Conditional mutual information continuity) . F ix finite alphabets for ( U, V , W ) . Ther e exists an e xplicit function ω U V W ( δ ) with ω U V W ( δ ) → 0 as δ → 0 such that whenever ∥ P U V W − Q U V W ∥ 1 ≤ δ ≤ 1 / 2 ,   I P ( U ; V | W ) − I Q ( U ; V | W )   ≤ ω U V W ( δ ) . Pr oof. Use the identity I ( U ; V | W ) = H ( U, W ) + H ( V , W ) − H ( W ) − H ( U, V , W ) and apply Lemma A.3 to each entrop y term (all alphabets finite). Each marginal dif fers in ℓ 1 by at most δ when the joint dif fers by δ , yielding an e xplicit ω U V W ( δ ) obtained by summing the four bounds. March 19, 2026 DRAFT 29 D. An explicit computable Lipschitz bound fr om policy coordinates to induced pmf Let d e,n denote the number of free polic y coordinates in P e,n (product-of-simplices represen- tation from Remark III.4). Let θ ( p ) ∈ [0 , 1] d e,n denote the coordinate v ector (with normalization constraints). Define the ℓ 1 distance on coordinates by ∥ θ − θ ′ ∥ 1 = P d e,n i =1 | θ i − θ ′ i | . Lemma A.5 (Explicit Lipschitz bound for induced joint laws) . F ix ( e, n ) . Ther e exists a constant L e,n , computable fr om e and n , such that for all p, p ′ ∈ P e,n , ∥ P p − P p ′ ∥ 1 ≤ L e,n ∥ θ ( p ) − θ ( p ′ ) ∥ 1 . One valid choice is L e,n = |X | n |Y | n |S | n +1 · n. Pr oof. Fix an outcome ( x n , y n , s n +1 ) . From the unrolled product form, P p ( x n , y n , s n +1 ) = C ( x n , y n , s n +1 ) n Y t =1 p ( x t | x t − 1 , y t − 1 ) , where C ( · ) ∈ [0 , 1] depends only on ( W e , π 1 ,e ) and the trajectory . F or any two sequences of scalars a t , b t ∈ [0 , 1] , the elementary inequality    n Y t =1 a t − n Y t =1 b t    ≤ n X t =1 | a t − b t | holds (expand telescopically and use | a | ≤ 1 ). Hence | P p ( · ) − P p ′ ( · ) | ≤ n X t =1   p ( x t | x t − 1 , y t − 1 ) − p ′ ( x t | x t − 1 , y t − 1 )   . Summing o ver all ( x n , y n , s n +1 ) and upper bounding each policy-coordinate dif ference by ∥ θ ( p ) − θ ( p ′ ) ∥ 1 yields ∥ P p − P p ′ ∥ 1 ≤ |X | n |Y | n |S | n +1 · n · ∥ θ ( p ) − θ ( p ′ ) ∥ 1 , which proves the claim with the stated e xplicit L e,n (computable since the alphabets/state sizes are part of the finite encoding model in Section II). E. A uniform computable appr oximation algorithm for V n W e now formalize computability in the standard T ype-2 sense: a real number z is computable if there is an algorithm which, on input k ∈ N , outputs a rational r k such that | r k − z | ≤ 2 − k . March 19, 2026 DRAFT 30 Theorem A.6 (Uniform computability of V n ) . Ther e exists a single algorithm Appro xV such that for every input ( e, n, k ) , it halts and outputs a rational r satisfying   r − V n ( W e , π 1 ,e )   ≤ 2 − k . Mor eover , the algorithm is uniform in ( e, n ) . Pr oof. Fix ( e, n, k ) . By Theorem A.2, there e xists p ⋆ attaining V n . Step 1: Build a finite rational net of policies. Let η > 0 be specified later . Because P e,n is a finite product of simplices, for any integer M ≥ 1 the set of policies with coordinates in { 0 , 1 / M , 2 / M , . . . , 1 } is finite and can be enumerated ef fecti vely . Choose M so that the resulting grid forms an η -net in ℓ 1 for P e,n , i.e., for e very p there e xists ˆ p on the grid with ∥ θ ( p ) − θ ( ˆ p ) ∥ 1 ≤ η . Denote this finite set by N η . Since P e,n is a finite product of simplices of total coordinate dimension d e,n , choosing M ≥ d e,n /η ensures that the rational grid with step 1 / M forms an η -net in ℓ 1 ov er P e,n . Step 2: Control objectiv e variation via an explicit continuity modulus. Let δ := L e,n η , where L e,n is from Lemma A.5. Then for an y p and its net approximation ˆ p , ∥ P p − P ˆ p ∥ 1 ≤ δ . Apply Lemma A.4 with ( U, V , W ) = ( X t , Y t , Y t − 1 ) to obtain, for each t ,   I p ( X t ; Y t | Y t − 1 ) − I ˆ p ( X t ; Y t | Y t − 1 )   ≤ ω t ( δ ) , where ω t ( δ ) → 0 as δ → 0 and is explicit from Lemma A.3. Hence   I p ( X n → Y n ) − I ˆ p ( X n → Y n )   ≤ n X t =1 ω t ( δ ) . Choose η (equi valently M ) ef fectiv ely so that P n t =1 ω t ( L e,n η ) ≤ 2 − ( k +2) . This is possible since each ω t is explicit and tends to 0 at 0 . Step 3: Ev aluate dir ected information on the finite net to sufficient precision. F or each ˆ p ∈ N η , the induced probabilities are rational combinations of rational channel coefficients and rational polic y coordinates (Lemma A.1), hence rational. Directed information is a finite sum of terms in volving log of rationals in (0 , 1] with the con vention 0 log 0 := 0 . The real function log is computable (see, e.g., [29], [30]); thus for each ˆ p one can compute a rational e I ˆ p such that   e I ˆ p − I ˆ p ( X n → Y n )   ≤ 2 − ( k +2) . No w compute r := max ˆ p ∈N η e I ˆ p , March 19, 2026 DRAFT 31 which is a maximum ov er a finite set of rationals and is therefore exact. Step 4: Error bound. Let p ⋆ attain V n and pick ˆ p ⋆ ∈ N η with ∥ θ ( p ⋆ ) − θ ( ˆ p ⋆ ) ∥ 1 ≤ η . Choose η so that P n t =1 ω t ( L e,n η ) ≤ 2 − ( k +2) and ev aluate each I ˆ p ( X n → Y n ) to accuracy 2 − ( k +2) (instead of 2 − ( k +2) as written abov e, keep this same value consistently). Then V n = I p ⋆ ( X n → Y n ) ≤ I ˆ p ⋆ ( X n → Y n ) + 2 − ( k +2) ≤ e I ˆ p ⋆ + 2 − ( k +2) + 2 − ( k +2) ≤ r + 2 − ( k +1) . Con versely , for ev ery ˆ p ∈ N η we hav e e I ˆ p ≤ I ˆ p ( X n → Y n ) + 2 − ( k +2) ≤ V n + 2 − ( k +2) , hence r ≤ V n + 2 − ( k +2) . Combining both inequalities yields | r − V n | ≤ 2 − k . Corollary A.7 (Uniform computability of normalized values) . Let a n ( e ) := 1 n V n ( W e , π 1 ,e ) . Ther e exists a uniform algorithm Appro xA such that on input ( e, n, k ) it halts and outputs a rational r with | r − a n ( e ) | ≤ 2 − k . Pr oof. Apply Theorem A.6 and di vide by n . Section III defines the exact-capacity threshold problem and Section IV proves its undecidabil- ity (Theorem IV .4). Section V (part C) notes that the paper does not classify the exact predicate as r .e. or co-r .e. This section adds a rigorous arithmetical-hierarchy upper bound for the e xact threshold language, stated and prov ed in a form that av oids any non-decidable comparison of computable reals. F . A lim sup thr eshold equivalence Define (as in the main text) the normalized finite-horizon values a n ( e ) := 1 n V n ( W e , π 1 ,e ) , C fb ( W e , π 1 ,e ) = lim sup n →∞ a n ( e ) . Let q ∈ Q be the threshold. Lemma A.8 ( lim sup with rational slack) . F or any r eal sequence { a n } n ≥ 1 and any rational q , lim sup n →∞ a n ≥ q ⇐ ⇒ ( ∀ k ∈ N )( ∃ n ∈ N ) a n > q − 2 − k . Pr oof. ( ⇒ ) If lim sup a n ≥ q , then for each k there exist infinitely many n such that a n > q − 2 − k . ( ⇐ ) If for ev ery k there exists n k with a n k > q − 2 − k , then lim sup k →∞ a n k ≥ q , hence lim sup n →∞ a n ≥ q . March 19, 2026 DRAFT 32 G. A recur sive matrix predicate using appr oximation certificates Recall the exact-threshold language (Section III and Section V, part C): LC ap := {⟨ e, q ⟩ : C fb ( W e , π 1 ,e ) ≥ q } . T o place LC ap in the arithmetical hierarchy we must write membership in the normal form ( ∀ k )( ∃ m ) R ( · ) with a r ecursive (decidable) predicate R . W e do this by using approximation certificates produced by Corollary A.7. Let Appro xA ( e, n, M ) denote a fix ed total algorithm that outputs a rational r satisfying | r − a n ( e ) | ≤ 2 − M . Such an algorithm exists by Corollary A.7. Define the decidable predicate R ( e, q , k , n, M ) : ⇐ ⇒ Appro xA ( e, n, M ) > q − 2 − k + 2 − M . This predicate is decidable because it asserts that a specific T uring machine halts with an output rational satisfying a strict rational inequality (all of which is checkable by simulating the machine and comparing rationals). Lemma A.9 (Certificate implication) . If R ( e, q , k , n, M ) holds, then a n ( e ) > q − 2 − k . Pr oof. Let r = App ro xA ( e, n, M ) . If r > q − 2 − k + 2 − M and | r − a n ( e ) | ≤ 2 − M , then a n ( e ) ≥ r − 2 − M > q − 2 − k + 2 − M − 2 − M = q − 2 − k . Thus a n ( e ) > q − 2 − k . Lemma A.10 (Existence of certificates when strict inequality holds) . If a n ( e ) > q − 2 − k , then ther e e xists M ∈ N such that R ( e, q , k , n, M ) holds. Pr oof. Let ∆ := a n ( e ) − ( q − 2 − k ) > 0 . Choose M so that 2 − M < ∆ / 2 . Let r = App roxA ( e, n, M ) with | r − a n ( e ) | ≤ 2 − M . Then r ≥ a n ( e ) − 2 − M > ( q − 2 − k ) + ∆ − 2 − M > ( q − 2 − k ) + 2 − M , i.e., r > q − 2 − k + 2 − M , hence R ( e, q , k , n, M ) holds. H. A Π 0 2 upper bound for LC ap W e now state the hierarchy placement in a fully arithmetical (certificate-based) form. March 19, 2026 DRAFT 33 Theorem A.11 ( LC ap ∈ Π 0 2 (arithmetical-hierarchy upper bound)) . The language LC ap belongs to Π 0 2 . Specifically , ⟨ e, q ⟩ ∈ LC ap ⇐ ⇒ ( ∀ k ∈ N )( ∃ n ∈ N )( ∃ M ∈ N ) R ( e, q , k , n, M ) , wher e R is the decidable pr edicate defined above. Pr oof. By Lemma A.8, ⟨ e, q ⟩ ∈ LC ap ⇐ ⇒ ( ∀ k )( ∃ n ) a n ( e ) > q − 2 − k . For fixed ( e, q , k , n ) , Lemmas A.9 and A.10 sho w that a n ( e ) > q − 2 − k ⇐ ⇒ ( ∃ M ) R ( e, q , k , n, M ) . Substituting yields ⟨ e, q ⟩ ∈ LC ap ⇐ ⇒ ( ∀ k )( ∃ n )( ∃ M ) R ( e, q , k , n, M ) . Since ( ∃ n )( ∃ M ) can be merged into a single existential quantifier o ver N (e.g., via a standard pairing function), this is a Π 0 2 definition with a recursive matrix predicate. a) Relation to the main impossibility r esults.: Theorem A.11 provides a precise arithmetical upper bound that complements the undecidability result in Section IV (Theorem IV .4) and the discussion in Section V (part C). It does not imply decidability; rather , it locates the e xact threshold language within a standard low lev el of the arithmetical hierarchy . March 19, 2026 DRAFT