1. Scope and logical status
Digital immortality can mean preservation of records, continued execution of a cognitive process, or survival of a particular person. These are different specifications. We study the implication from indefinite persistence to increasing storage density, and identify the hypotheses under which it is valid. We make no assumption that a human mind is digitally realizable, and no prediction of a technological singularity.
We work externally in classical set theory with choice and appropriate universes. The higher-categorical ambient category is an arbitrary infinity-topos $\mathcal X$: an accessible left-exact localization of a presheaf infinity-category of spaces. All categorical limits, colimits, and pullbacks are homotopy coherent. A point of $S\in\mathcal X$ means a global section $1\to S$; local inhabitation need not supply such a point. Numerical storage budgets are external resource assignments, not intrinsic cardinalities of arbitrary objects of $\mathcal X$.
2. Exact retention and its resource requirements
Definition 2.1 (archive). At stage $n$, let $H_n$ be a finite nonempty set of admissible records. An exact archive is an encoder $e_n:H_n\to M_n$ and decoder $d_n:M_n\to H_n$ satisfying $d_ne_n=\mathrm{id}_{H_n}$. A $b_n$-bit memory has $|M_n|\le 2^{b_n}$. Every record-dependent resource available to the decoder, including external storage or a varying codebook, is included in $M_n$. A fixed decoder may be shared among all records.
Theorem 2.2 (exact archive bound). An exact $b_n$-bit archive exists abstractly if and only if $$b_n\ge K_n:=\lceil\log_2|H_n|\rceil.$$ Proof. The decoding identity makes $e_n$ injective, so $|H_n|\le|M_n|\le2^{b_n}$. Conversely, enumerate $H_n$ and encode its elements by distinct binary strings of length $K_n$. Define a decoder arbitrarily on unused strings. This establishes representability, not an efficient or physically realizable implementation. $\square$
Corollary 2.3 (density requirement). If usable capacity is at most $\rho_nV_n$ bits, exact retention requires $\rho_nV_n\ge K_n$. If $\liminf_{n\to\infty}K_n/n\ge h>0$ and $V_n\le V_0<\infty$, then, for every $0<\epsilon<h$, eventually $$\rho_n\ge\frac{(h-\epsilon)n}{V_0}.$$ Proof. Apply Theorem 2.2 and the definition of the lower limit. $\square$
For $H_n=H_0\times A^n$, with $|A|=q\ge2$, one has $K_n=\lceil\log_2|H_0|+n\log_2q\rceil$. This gives linear memory growth for freely combinable distinguishable inputs. Positive information growth is a sufficient condition for unbounded memory, not a consequence of elapsed time alone.
Proposition 2.4 (bounded-error retention). Let $W$ be uniform on $N\ge2$ records, let the stored variable $Z$ take at most $2^b$ values, and let a deterministic decoder have error probability $p$. Then $$b\ge\log_2N-h_2(p)-p\log_2(N-1),$$ where $h_2(p)=-p\log_2p-(1-p)\log_2(1-p)$ with $0\log 0=0$. In particular, $p\le\epsilon<1$ implies $b\ge(1-\epsilon)\log_2N-1$. Proof. Let $E$ indicate decoding error. Given $Z$ and $E=0$, $W$ is known; given $Z$ and $E=1$, at most $N-1$ alternatives remain. The entropy chain rule gives $H(W\mid Z)\le h_2(p)+p\log_2(N-1)$. Hence $\log_2N=I(W;Z)+H(W\mid Z)\le b+h_2(p)+p\log_2(N-1)$. Use $h_2(p)\le1$ for the final inequality. $\square$
This is the elementary Fano-type bound. Neither lossless compression nor a fixed nonunit tolerated error removes the divergence when $\log_2|H_n|\to\infty$. These are coding bounds, not estimates of human experience.
3. Continued behavior without complete history
Definition 3.1 (task-relative memory). For a finite input alphabet $A$ and output set $O$, fix a task $\tau:A^*\to O$. Define $$u\sim_\tau v\quad\Longleftrightarrow\quad \forall w\in A^*,\ \tau(uw)=\tau(vw).$$ A deterministic realization consists of a state set $S$, initial state $s_0$, updates $\delta:S\times A\to S$, and observation $o:S\to O$ with $o(\delta^*(s_0,u))=\tau(u)$. The empty word is denoted $\varepsilon$.
Theorem 3.2 (minimal task memory). The quotient $Q=A^*/{\sim_\tau}$ has a well-defined realization with initial state $[\varepsilon]$, update $[u]\mapsto[ua]$, and observation $[u]\mapsto\tau(u)$. Every realization has at least $|Q|$ reachable states. Consequently a finite-state realization exists exactly when $Q$ is finite. Proof. If $u\sim_\tau v$, taking $w=\varepsilon$ gives equal observations, and taking continuations $aw$ shows $ua\sim_\tau va$. This proves well-definedness and correctness. If two histories reach the same state in any realization, every continuation produces equal task outputs, so those histories are equivalent. The map from reachable states to their history classes is therefore a well-defined surjection. $\square$
This is the Moore-output version of the Myhill-Nerode argument [2], not a new minimization theorem. For binary inputs and $\tau(u)$ equal to parity, there are exactly two classes. One bit supports arbitrarily many logical updates while discarding most of the history. By contrast, the task $\tau(u)=u$ distinguishes all histories. Abstract indefinite execution therefore does not imply an ever-growing archive. A finite autonomous deterministic system eventually cycles; an open finite-state system can have nonperiodic outputs supplied by its environment. Neither fact proves physical eternal operation.
4. Higher-categorical dynamics and migration
Let $BA^*$ be the free category with one object and one generating loop for each $a\in A$, regarded as an infinity-category. Composition is successive input processing, so the action maps satisfy $f_{uv}\simeq f_vf_u$. Fix an observation object $O\in\mathcal X$.
Definition 4.1 (observed dynamics). Put $$\mathcal D_{A,O}(\mathcal X)= \operatorname{Fun}(BA^*,\mathcal X) \times_{\mathcal X}\mathcal X_{/O},$$ using evaluation at the unique object and the domain functor. An object is a coherent action on a state object $S$ together with $o:S\to O$. A migration $m:D\to D'$ is a coherent natural transformation whose underlying $m:S\to S'$ also satisfies $o'm\simeq o$. Thus $mf_a\simeq f'_am$, with the coherences required by the functor category. For ordinary sets the same data are a coalgebra $S\to O\times S^A$ [3]. A chosen initial state is a point of $S$, not necessarily a fixed point of the action.
Lemma 4.2 (preservation under migration). A migration preserves observations after every finite input word. If its underlying state map is an equivalence, it is an equivalence in $\mathcal D_{A,O}(\mathcal X)$. Proof. Naturality for each word gives $mf_w\simeq f'_wm$; composing with $o'$ gives $o'f'_wm\simeq of_w$. Equivalences in a functor infinity-category are objectwise, and equivalences in a slice are detected on their underlying arrows. Equivalences are also detected componentwise in the displayed pullback. $\square$
Theorem 4.3 (coherent migration). Suppose $D:\mathbb N\to\mathcal D_{A,O}(\mathcal X)$ is a coherent sequence all of whose transition maps are equivalences. Its colimit $D_\infty$ exists, and every canonical map $D_n\to D_\infty$ is an equivalence. A point of $S_0$ is transported to a point of the colimit state object with the same finite-word observations. Proof. Colimits of actions are formed objectwise. Colimits in $\mathcal X_{/O}$ are created on domains, so these colimits give compatible colimits in the defining pullback. The arrows $D_0\to D_n$ form an objectwise equivalence from the constant diagram at $D_0$ to $D$. The nerve of the indexing poset $\mathbb N$ is contractible; hence its constant-diagram colimit is $D_0$. The same argument applies to every cofinal tail. Point and observation preservation follow from the colimit cocone and Lemma 4.2. $\square$
This theorem concerns a diagram already supplied with coherent migrations. It does not construct future hardware, guarantee a physical transfer, or identify a colimit with an event at an infinite time. Pairwise equivalences chosen independently are not the full specified diagram. Equivalence is a sufficient preservation condition; behavioral quotients such as Theorem 3.2 may preserve selected observations while losing state information.
5. Descent and compatible continuation
Theorem 5.1 (descent for observed dynamics). Let $U\to1$ be an effective epimorphism in $\mathcal X$, and let $U_\bullet$ be its Cech nerve, so $U_n=U^{\times(n+1)}$. Write $O_U=O\times U\to U$. Then $$\mathcal D_{A,O}(\mathcal X)\simeq \lim_{[n]\in\Delta} \mathcal D_{A,O_{U_n}}(\mathcal X_{/U_n}).$$ Consequently local dynamics with a full coherent descent datum determine global dynamics, uniquely up to a contractible space of choices. Proof. Effective descent in an infinity-topos gives $\mathcal X\simeq\lim_\Delta\mathcal X_{/U_n}$ [4, Theorem 6.1.3.9 and Section 6.2.3]. Forming functor categories with fixed source $BA^*$ preserves limits. Forming slices over the compatible family $O_{U_n}$ also preserves this limit: objects, arrows to that family, and their coherences are computed componentwise. Taking the pullback in Definition 4.1 proves the equivalence. The statement about choices is the homotopy-fiber characterization of an equivalence. $\square$
Example 5.2 (local sections do not supply a global point). In the infinity-topos of space-valued sheaves on $S^1$, take the discrete-valued sheaf $P$ of local sections of $z\mapsto z^2$. It is locally inhabited, so $P\to1$ is an effective epimorphism. A global point would give a continuous section $s$ of this double covering. On fundamental groups it would satisfy $2s_*=\mathrm{id}_{\mathbb Z}$, which is impossible. Thus global objects obtained by descent must not be confused with global sections of those objects. Theorem 5.1 requires coherent compatibility; it does not assert that arbitrary local choices glue. Only Cech descent is used, with no hypercompleteness assumption.
Definition 5.3 (compatible horizons). Let $R_n$ be the set of admissible realizations through stage $n$, with restriction maps $r_n:R_{n+1}\to R_n$. An indefinitely extendible realization in this model is an element of $\lim_nR_n$.
Proposition 5.4 (finite horizons and compactness). Nonemptiness of every $R_n$ does not generally imply nonemptiness of the inverse limit. It does imply it when every $R_n$ is finite. Proof. For the negative result take $R_n=\{m\in\mathbb N:m\ge n\}$ and inclusion maps. A compatible sequence would have a common value at least every natural number. For the positive result form the tree of compatible finite tuples $(x_0,\ldots,x_n)$. Every level is nonempty, since an element of $R_n$ restricts to such a tuple. Each level is finite. Starting at a root with arbitrarily long extensions, at least one of its finitely many children has arbitrarily long extensions; otherwise their finite height bounds would bound the parent. Repeating produces an infinite branch, hence a compatible sequence. This is the classical finite-branching compactness argument. $\square$
6. Physical capacity and reliability
Assumption 6.1 (bounded accessible capacity). A specified resource envelope admits at most $2^{B_*}$ perfectly recoverable classical records, for some finite $B_*$. For bounded-error claims we additionally require $I(W;Z)\le B_*$ for every allowed record ensemble and decoder-accessible observation $Z$. The envelope includes all record-dependent decoder resources. A classical memory with at most $2^{B_*}$ states satisfies both clauses; the exact clause alone need not imply the information clause.
Theorem 6.2 (bounded-resource archive obstruction). Under the exact clause of Assumption 6.1, exact archives for all $n$ are impossible if $\sup_nK_n=\infty$. Under its information clause, uniform records cannot be retained with error bounded by a fixed $\epsilon<1$ when $\log_2|H_n|\to\infty$. Proof. Exact recovery would give $\log_2|H_n|\le B_*$ for every $n$, contradicting unbounded $K_n$. For bounded error, the proof of Proposition 2.4 uses $I(W;Z)\le B_*$ in place of $H(Z)\le b$, giving $(1-\epsilon)\log_2|H_n|-1\le B_*$. $\square$
The theorem is independent of a particular physical bound. A conditional motivation is the Bekenstein entropy bound for a complete weakly self-gravitating system of circumscribing radius $R$ and total energy $E$ [6]. If accessible classical information is bounded by maximal entropy in bits, one may take $$B_*=\frac{2\pi ER}{\hbar c\ln2}.$$ This is an additional physical applicability assumption, not a theorem about all cosmological spacetimes or all information tasks. The operational distinctions studied in [7] caution against equating every information quantity with storage capacity. At bounded $E$ and $R$ in the stated regime, endlessly denser media cannot accommodate an unbounded exact archive. Distributed expansion may enlarge the resource envelope; it does not refute the theorem.
Capacity and dissipation are different budgets; reversible computation forbids assigning an erasure cost to every logical operation [8,9].
Theorem 6.3 (cumulative failure). Let $L_n$ be the event of no fatal loss through stage $n$, with $L_0$ certain and $L_n\subseteq L_{n-1}$. Set $p_n=1-\Pr(L_n\mid L_{n-1})$, assuming $0\le p_n<1$. Then $$\Pr\left(\bigcap_{n\ge1}L_n\right)=\prod_{n\ge1}(1-p_n).$$ This probability is positive exactly when $\sum_np_n<\infty$, and equals one exactly when every $p_n=0$. Proof. Iterated conditioning gives $\Pr(L_N)=\prod_{n=1}^N(1-p_n)$, without independence. Continuity of probability for decreasing events gives the infinite product. Since $-\log(1-p)\ge p$, a divergent hazard sum forces product zero. If the sum converges, eventually $p_n\le1/2$ and $-\log(1-p_n)\le2p_n$; the finite initial factors are positive, so the product is positive. A single positive hazard makes it strictly less than one. $\square$
For $p_n=1/(n+1)$ survival through $N$ stages is $1/(N+1)$; for $p_n=1/(n+1)^2$ it is $(N+2)/(2(N+1))$, tending to $1/2$. Improving reliability can therefore give either zero or positive indefinite survival probability. Positive probability is not certainty. These are illustrative hazards, not measurements of a proposed technology.
7. Semantic equivalence and personal identity
Let $\mathcal P$ be a category of physical realizations and allowed transformations, and let $F:\mathcal P\to\mathcal D_{A,O}(\mathcal X)$ be a chosen semantic functor. This functor is a modeling assumption; the paper does not construct it for brains. For a specification $D$, define its space of framed realizations by $$\operatorname{Real}(D)=\mathcal P^{\simeq} \times_{\mathcal D_{A,O}(\mathcal X)^{\simeq}}\{D\},$$ where the superscript denotes the maximal infinity-groupoid. Its points are realizations with specified equivalences to $D$.
Proposition 7.1 (nonuniqueness of realizers). A unique chosen semantic specification need not have a contractible realization space, even when every realization has exactly equivalent dynamics. Proof. Take $\mathcal P=\mathcal D_{A,O}(\mathcal X)\times K$, where $K$ is the discrete two-point category, and let $F$ be projection. Then $\operatorname{Real}(D)\simeq K$. The two realizations have identical semantics and distinct invariant tags, with no equivalence between them in $\mathcal P$. $\square$
The example is a logical countermodel to an implication, not evidence for a particular metaphysics. Univalence identifies equivalent types within its specified universe [5]; it does not make the fiber of an arbitrary realization functor contractible. In particular it does not identify two distinct elements merely because a map sends them to the same value.
Proposition 7.2 (scope of survival claims). If a satisfiable formal theory of realizations and semantics has no axioms involving an added binary predicate $\operatorname{Survives}$, then its structural axioms alone cannot determine whether that predicate holds of a specified pair of realizations. Proof. For any model and specified pair, expand the model once with the predicate interpreted as the full relation and once as the empty relation. Both expansions satisfy the original axioms and disagree at that pair. $\square$
This modest model-theoretic observation does not prove that personal survival is unknowable or nonphysical. It identifies a missing bridge assumption. The distinct questions of consciousness and identity under uploading are analyzed in [10, Sections 9-10]. Neither a compressed archive nor a coherent semantic migration supplies that bridge by definition.
8. Consequences and limitations
Ever-increasing storage density is necessary for an ever-growing retained information requirement within a fixed spatial budget; shrinking that budget strengthens the requirement. Indefinite execution alone has no such implication. Complete retention, bounded accessible capacity, and unbounded information growth cannot all hold. Coherent migration and descent establish structural preservation and compatibility, while the inverse-limit and hazard results isolate additional temporal obligations.
The model specifies exact dynamics and selected observations. It gives no neural encoding, consciousness test, working upload, engineering forecast, or cosmological survival proof. Established counting, minimization, descent, and compactness results are used with attribution; the contribution is their explicit separation and application to digital persistence. The accompanying appendices discuss approximate migrations, observational adequacy, and reproducibility.
Authorship and assistance. Anonymous author. AI assistance was used to draft the exposition, develop and check arguments, locate references, and generate the accompanying files. The manuscript supplies mathematical proofs, but no proof-assistant certification or empirical validation of digital immortality is claimed. Editorial acceptance and independent review are not presumed.
References
[1] C. E. Shannon. A mathematical theory of communication. *Bell System Technical Journal* 27 (1948), 379-423, 623-656. full text.
[2] A. Nerode. Linear automaton transformations. *Proceedings of the American Mathematical Society* 9 (1958), 541-544. DOI.
[3] J. J. M. M. Rutten. Universal coalgebra: a theory of systems. *Theoretical Computer Science* 249 (2000), 3-80. author's institutional repository.
[4] J. Lurie. *Higher Topos Theory*. Annals of Mathematics Studies 170, Princeton University Press, 2009. Author-maintained text.
[5] The Univalent Foundations Program. *Homotopy Type Theory: Univalent Foundations of Mathematics*. Institute for Advanced Study, 2013. Book and corrections.
[6] J. D. Bekenstein. How does the entropy/information bound work? *Foundations of Physics* 35 (2005), 1805-1823. preprint.
[7] P. Hayden and J. Wang. What exactly does Bekenstein bound? *Quantum* 9 (2025), 1664. DOI; preprint.
[8] R. Landauer. Irreversibility and heat generation in the computing process. *IBM Journal of Research and Development* 5 (1961), 183-191. DOI.
[9] C. H. Bennett. Notes on Landauer's principle, reversible computation, and Maxwell's Demon. *Studies in History and Philosophy of Modern Physics* 34 (2003), 501-510. Full text and publication metadata.
[10] D. J. Chalmers. The singularity: a philosophical analysis. *Journal of Consciousness Studies* 17, nos. 9-10 (2010), 7-65. Author's text.