GENERAL FRAMEWORK
Material memory
between informational dependence and organisational closure
Preliminary page to the two articles
Definition
Let us fix a system boundary S, a reference instant t, a descriptive resolution R, a past window P−Δ−(t), a present state XR(t) and a future window F+Δ+(t). The parameters Δ− and Δ+ designate respectively the temporal depth of the past considered and the future horizon over which the dependence is evaluated.
A system exhibits a predictive memory relative to these choices when its future is not conditionally independent of its past once its present is given at resolution R:
I(F+Δ+ ; P−Δ− | XR) > 0
The inequality means that the present description adopted is not predictively sufficient for the future considered. Part of the history still supplies information that is not screened off by the present state adopted. Its vanishing,
I(F+Δ+ ; P−Δ− | XR) = 0
means only that no residual predictive memory is detected relative to the system boundary, the variables, the resolution and the temporal horizons chosen. It demonstrates neither the absence of history nor the absence of any memory possible at another resolution.
| This statement claims to explain nothing. It fixes a common vocabulary and forbids one precise attribution: calling predictive memory a dependence that disappears entirely once the present adopted becomes sufficient for the observables and horizons considered. It is a definition chosen for its fruitfulness, not a law of nature. |
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Trajectory dependence constitutes a weaker level. It means that the present state was causally produced by a particular history and that different trajectories could have led to different configurations. It can persist even when the history no longer supplies any predictive information after conditioning on a sufficiently rich present state. Trajectory dependence therefore constitutes the general historical substrate of the framework, but it is not identical to the predictive memory defined by the preceding inequality.
The retrodictive variant bears on an explicitly defined historical variable H, such as an accretion event, a developmental history, an institutional transformation or a determinate earlier state. A minimal informational condition is then:
I(H ; XR) > 0
This inequality indicates that the present preserves information about the targeted history. It does not by itself suffice to demonstrate a robust empirical reconstruction. Reconstructibility becomes operational when the present observables allow H to be inferred with a performance superior to a reference model, according to criteria of discrimination, calibration or uncertainty reduction defined in advance.
The predictive variant bears more precisely on the historical information that remains relevant to the future after conditioning on the present adopted:
I(F+Δ+ ; H | XR) > 0
The historical variable may represent a complete trajectory or a summary property of that trajectory. In both cases, the claim remains relative to the system boundary, the observables, the descriptive resolution and the temporal horizons defined.
The functional variant cannot be reduced to a quantity of information. A trace becomes functional when a difference it carries takes part causally in a process belonging to the organisation, and when that mobilisation is inscribed in a network of constraints contributing to the maintenance, renewal or reproduction of the whole. A correlation, however strong, does not suffice. A functional trace must produce an effect within the organisational loop.
Functional memory does not necessarily presuppose the reconstructibility of the past. A structure may be used in the maintenance or development of an organisation without allowing the precise event that produced it to be recovered. Conversely, a trace may permit a detailed historical reconstruction without playing any role in the functioning of the system that carries it. The retrodictive, predictive and functional dimensions can therefore combine, but none follows automatically from the others.
Definition independent of the historical variable
The historical variable H must be defined independently of the traces used to infer it. It may represent an event, a trajectory, a property of assembly or an earlier transformation, but its content must not be constructed solely from XR and then rediscovered in that same present state. Such a procedure would make reconstructibility circular.
Independence of definition does not presuppose that H is directly observable. A history may remain latent and be estimated from several traces. It is nevertheless necessary to specify the criteria that distinguish the possible values of H, the hypotheses linking those values to the observables, and the data used to validate the inference. Reconstructibility then bears on the capacity of XR to reduce the uncertainty concerning a historical variable defined before the evaluation, and not on the mere capacity to recover a category constructed from the same observations.
Parameters of instantiation
The informational invariant and the organisational criterion acquire empirical content only once several choices have been made explicit: the system boundary, the present description, the historical variable, the future considered, the material realisers, the mode of access to the trace and the temporal scale. This filling-in carries the empirical content. The common abstraction never replaces the mechanisms proper to the domain.
Two domains share an operator when one and the same abstract relation is instantiated in them by different realisers. A conditional dependence, a reconstruction from traces or a closure of constraints can thus be compared without supposing that the systems possess the same structure. This is the precise sense of operative universality. It asserts a community of relation, not an isomorphism of mechanisms.
Matrix of instantiation by domain
| Domain | Present at resolution R | Realisers of the trace and of its persistence | Instantiated modes of access | Temporal scale |
|---|---|---|---|---|
| COSMIC | Halo mass Concentration Environment Baryonic content Stellar populations Substructures |
Gravitation Collapse Phase mixing Radiative dissipation Energetic feedbacks Chemical enrichment |
Retrodictive Chemical memory and galactic archaeology Relative predictive Possible effects of the assembly history No endogenous functional reuse established at the level considered |
From the gigayear of assembly to cosmic age |
| BIOLOGICAL | Developmental state Genome Epigenome Cellular organisation Physiological state Constructed environment |
Replication Heredity Variation Development Regulation Repair Selection Drift Niche construction |
Retrodictive Phylogenetic, developmental or immunological signatures Relative predictive Where history supplies information beyond the present adopted Functional Reading, repair, reactivation or transmission within an organisational closure |
From the cell cycle and the generation to the phylogenetic scale |
| EXOSOMATIC | State of the socio-technical organisation Inscription supports Models Software Institutions Infrastructures Supply relations |
Inscription Copying Cultural transmission Computation Maintenance Institutional coordination Human and non-biological functions |
Retrodictive Archive Predictive Operative histories and accumulated models Functional Within hybrid socio-technical closures No entirely non-biological closure currently established |
From the operating cycle to the renewal of infrastructures and substrate |
Reading the matrix. The cosmic clearly achieves the conservation of reconstructible traces and puts to the test the existence of a residual predictive historical information. The biological integrates certain traces into loops of maintenance, development and reproduction. The exosomatic already possesses functional memories within hybrid organisations comprising organisms, institutions and technical devices. The open question is whether a closure could be maintained without a constitutive biological function.
Prohibitions by domain
The framework acquires binding empirical content only where a claim can fail. It does not become a law through the mere repetition of a vocabulary across several domains. Each instantiation must specify what would be false if certain results were observed.
Cosmic domain
For a historical variable H, a present description at resolution R, a future F⁺ and explicitly defined horizons, if
I(F+Δ+ ; H | XR) = 0
within the sensitivity limits of the protocol, predictive memory fails for that population, those observables and that description. History may have produced the present and certain traces may remain retrodictively legible, but it no longer adds information about the future after conditioning on the present state adopted.
The parameter Q_sat = 0.05 ± 0.14, measured for the satellites of the DESI Bright Galaxy Survey and compatible with zero, constitutes a current test of a restricted instantiation of this hypothesis. It does not directly measure conditional mutual information and does not refute all cosmic memory. It tests one particular dependence of galactic occupation on environment or assembly after controlling for mass in a defined population and model.
Biological domain
Retrodictive and predictive claims obey the same informational criteria as in the cosmic domain. A developmental, epigenetic, immunological or ecological history constitutes a relative predictive memory only if it improves the description of a defined future beyond the present state adopted.
The functional prohibition is different. If the controlled modification of a supposedly functional trace changes no process belonging to the maintenance, development, repair or reproduction of the organisation, the functional attribution fails in the domain studied. A trace may remain historically legible without belonging to the functional regime. The persistence of a correlation does not replace the demonstration of a causal role within the organisational loop.
Exosomatic domain
The prohibition concerns strong separability. If a biological function belonging to the closure remains without a non-biological substitute able to take over its role while integrating into the network of dependencies that maintains the organisation, separability fails and the system remains hybrid. The presence of several redundant human realisers does not satisfy this criterion. Nor does a local substitution satisfy it when it merely displaces the biological dependence to another part of the loop.
These prohibitions call for distinct protocols. Predictive memory is tested through conditional sufficiency and the residual informational value of history. Retrodictive memory is tested through the capacity to infer a defined historical variable. Functional memory is tested through interventions bearing on the causal role of the trace within the organisation. Exosomatic separability is tested by first mapping the closure and then substituting the constitutive biological functions. None of these protocols follows automatically from the others.
| This disjunction distinguishes the framework from a cumulative metaphor. A metaphor can accommodate any result by modifying after the fact the sense of the word memory. An empirical framework must announce in advance the conditions under which a particular attribution will be refused. |
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The two primitives
The framework rests on two primitives that are not reducible to one another. An honest formulation inscribes this duality instead of seeking to mask it beneath a single measure.
Informational primitive
The first primitive describes what a history still makes accessible or predictively relevant at a given resolution. The retrodictive relation,
I(H ; XR) > 0
bears on the information the present preserves about a historical variable. The predictive relation,
I(F+Δ+ ; H | XR) > 0
bears on the part of history that remains informative for the future after conditioning on the present adopted.
These properties are relative to a system boundary, a resolution, a set of variables and temporal horizons. They can change when a description is enriched. An apparent predictive memory may disappear when a more complete present state becomes sufficient. A previously inaccessible trace may become reconstructible thanks to new observations. The informational boundary therefore remains graded and dependent on the level of description.
Mere trajectory dependence is not to be conflated with this primitive. It asserts that history causally contributed to producing the present, even when that history no longer supplies additional information about the future. It constitutes the general historical ground on which traces may become legible or predictively relevant, but it does not necessarily satisfy the informational criteria of memory adopted here.
Organisational primitive
The second primitive is organisational. The closure of constraints describes a network of mutual dependencies in which certain structures constrain processes that contribute, directly or indirectly, to the production or maintenance of other constraints in the network. A materially open organisation may thus possess a functional closure without being isolated from its environment.
No quantity of mutual information suffices to reconstitute this property. A correlation does not, by itself, produce a loop of maintenance. A trace may contain a great quantity of historical information while remaining causally inactive. An organisation may use an inherited difference without that difference allowing a precise reconstruction of the past that produced it.
The two primitives are therefore not related by a strict nesting. They exhibit a partial overlap. A functional memory requires that a persistent difference be causally mobilisable within the organisation, but it does not require that this difference constitute a retrodictively decodable archive. A reconstructible memory may exist without function. A predictive memory may appear without closure. A functional trace may be historically ambiguous while playing a decisive role in the maintenance of the system.
The seam between information and organisation thus becomes an object of the framework. It designates neither a provisional gap nor a transition that more resolution would necessarily make disappear. It marks the passage between two different questions: what remains of the past in the distribution of present and future states, and then what does the organisation do with certain inherited differences?
Materiality of the support and temporal orientation
Every instantiated memory presupposes a material support possessing physically distinguishable states, a stability over a duration τ, a fidelity compatible with the use considered, and mechanisms allowing the inscription, access, reading, maintenance or resetting of the trace. These properties determine the physical conditions under which an informational dependence or an organisational function can be realised. They do not constitute a third primitive. They describe the conditions of instantiation of the two primitives in particular supports.
The mathematical symmetry of mutual information does not abolish the temporal asymmetry of the framework. Retrodiction uses a present configuration to reduce uncertainty about a past history without implying a causality from present to past. Predictive memory measures the historical information that remains relevant to a defined future. Functional memory requires that an inherited and currently realised difference intervene causally in subsequent processes of maintenance, development or reproduction. The orientation from past to future therefore comes from the causal and thermodynamic structure of physical processes, not from the form of the informational quantities alone.
| To seek to reduce these two questions to a single primitive would reproduce, at the level of the framework, the error of isomorphism that the two articles refuse at the level of their objects. Unity would be obtained at the price of the disappearance of the criteria proper to each regime. The framework therefore explicitly renounces reducing memory as informational dependence and memory as organisational function to a single law. This renunciation preserves the difference between conserved information and an organisation capable of using certain traces under the conditions of its own continuation. |
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Status of the framework
The framework proposed currently constitutes a formalised comparative grammar. It supplies common definitions, imposes parameters of instantiation, distinguishes several modes of access to traces and associates each attribution with a condition of failure. It makes it possible to compare different domains without supposing that their mechanisms are identical.
Its unity nevertheless remains operative. Each domain defines its own variables, mobilises its own realisers and requires its own protocols. A failure in the cosmic domain does not directly determine the result of a biological experiment. The existence of a closure in the living does not demonstrate that an entirely non-biological closure is realisable. The reconstructibility of a cultural archive does not allow the intensity of a galactic assembly bias to be deduced.
Change of resolution and relations between levels
The choice of the resolution R is not a mere adjustment of precision. It can modify the memory property attributed to the system. A fine description containing the whole set of relevant variables may render the future conditionally independent of the past, whereas a coarser description may lose some of those variables and bring out a residual historical dependence. A predictive memory may thus emerge through descriptive reduction without corresponding to a new fundamental property of the dynamics. Conversely, enriching the present state may absorb a dependence previously attributed to history and render the description predictively sufficient.
The relation between levels does not, however, reduce to this informational variation. A functional property observed at a collective scale must be physically realised by processes situated at finer scales, but it does not necessarily reduce to a sum of correlations between constituents. Closure depends on organisational relations defined at the level where functions and their mutual dependencies become identifiable. The framework must therefore distinguish the material realisation of a property, its description at a given resolution, and its organisational status. A change of scale may modify the visibility of a memory without abolishing the distinction between informational dependence and functional closure.
The framework therefore does not yet constitute an inter-domain law. For a linking principle to acquire that status, it would not suffice to identify a resemblance between two domains. It would be necessary to show that a quantity defined in a first domain imposes a bound, a constraint or a measurable relation on a quantity defined in a second domain, by way of a shared physical condition.
The minimal logical form of such a principle would be the following. Let A be a quantity defined in a domain D, B a quantity defined in another domain D′, and C a physical constraint common to both systems. An inter-domain link would require that a relation of the type
BD′ ≥ f(AD, C)
or
BD′ ≤ f(AD, C)
be derivable independently of any analogy. The relation would have to produce a new empirical prohibition. Observing a value outside the announced bound should lead to rejecting or revising the linking principle.
Dissipation is a possible candidate for seeking such a relation. Reconstructible traces and functional closures both rest on physical structures whose persistence depends on determinate material conditions. This community of condition nevertheless does not suffice to establish a link. A stronger theory would have to show that a cost associated with the inscription, conservation or reactivation of a trace imposes a measurable constraint on the maintenance cost of an organisation capable of using that trace in its own continuity.
One may, exploratively, distinguish a cost of informational persistence,
Dtrace(R, τ, ε)
where R represents the resolution of the trace, τ its duration of conservation and ε its admissible level of degradation, and then a cost of organisational maintenance,
Dclôture(τ, Π)
where Π describes the domain of perturbations the closure must withstand. A linking principle would require that a common physical constraint allow a relation between these two quantities to be derived. It might take the form of a minimal bound on the dissipation necessary to maintain a closure capable of conserving, reading and reactivating a trace to defined performances.
No such relation is established. The two quantities do not yet possess a common definition independent of the domains. The system boundary, the level of description, the nature of the dissipative flows and the criteria of maintenance differ between a cosmic structure, an organism and an exosomatic organisation. Nothing currently demonstrates that a single entropic budget allows one to pass from informational persistence to functional closure.
Landauer as an intra-domain anchor
The thermodynamics of information nevertheless provides an established example of a relation between an informational quantity and a physical constraint. Landauer's principle shows that a logically irreversible operation, such as resetting a bit, has a minimal thermodynamic cost. For the quasistatic erasure of an equiprobable bit in contact with a thermal bath at temperature T, the heat dissipated satisfies the bound:
⟨Qeff⟩ ≥ kB T ln 2
This relation already has the form sought for a linking principle. A transformation defined in logical space imposes a bound on a physical quantity by way of a common thermodynamic constraint. It nevertheless remains internal to the thermodynamics of information and does not yet connect the reconstructible persistence of a trace to the maintenance of an organisational closure.
Landauer's principle must not be interpreted as a universal cost of all writing or all conservation of information. Its canonical form concerns the irreversible reduction of logical uncertainty during an operation of erasure or resetting. The persistence of a trace depends on the physical stability of its states, on their lifetime, on the fidelity sought and on any mechanisms of correction or refreshing. Operations carried out in finite time also add costs depending on the protocol, the speed, the precision and the robustness of the support.
The cost of informational persistence should therefore not be identified with a single bound. As an operational decomposition, it may include several contributions:
Dtrace = Dinscription + Dmaintien + Drafraîchissement + Dlecture + Deffacement
These contributions are neither governed by a single relation nor necessarily dissipative in the same way. Their importance depends on the support, the thermodynamic regime and the performances required. The Landauer limit directly constrains certain logically irreversible transformations, while the costs of stability, fidelity and finite-time operation fall under more general results of non-equilibrium thermodynamics.
The inter-domain question can then be formulated more precisely. Can a bound bearing on the physical cost of inscribing, conserving and resetting a trace be composed with a bound bearing on the maintenance cost of a closure capable of reading, reactivating and renewing that trace? A candidate relation might take the form:
Dclôture ≥ Φ(Dtrace, R, τ, ε, Π)
No general relation of this form is currently established. The dissipation associated with processing information and that accompanying the maintenance of an organisation far from equilibrium may arise from distinct processes, overlap partially, or depend on different system boundaries. Nothing yet allows us to assert that the thermodynamic cost of a trace necessarily composes with that of a closure according to a universal bound.
The thermodynamics of prediction provides a second point of support. In certain classes of systems, information conserved about the past that does not improve prediction of the future can be related to an additional dissipation. This result brings predictive sufficiency, the selection of traces and thermodynamic efficiency closer together. It does not produce a theory of functional memory, but it shows that part of the informational vocabulary of the present framework already has a physical translation.
Landauer therefore does not close the link sought. It indicates its first firm ground. It shows that a derived relation between information and dissipation is possible when the variables, the operations and the thermodynamic conditions are defined independently. The work remaining consists in determining whether this logic can be extended to the maintenance of an organisational closure, or whether the costs of the trace and of the organisation remain irreducibly distinct.
As long as a derived inter-domain relation does not exist, the framework remains a rigorous transdomain grammar rather than an inter-domain law. This limit is not a concealed weakness. It localises precisely what is missing. The work remaining does not consist in seeking mere empirical correlations between costs observed in different domains. A correlation would supply neither a bound nor a new prohibition. It must be determined whether a common physical constraint can produce a verifiable relation between independently defined quantities.
It is also possible that no link of this kind exists. The two primitives could remain irreducible while still being comparable within a single framework. The search for a stronger unification must therefore not be presupposed as necessary. The framework distinguishes what it already formalises, what it makes it possible to compare, the points where a link might be sought, and what it does not yet allow to be deduced.
| Dissipation therefore indicates a possible site of research, not a unification already achieved. It shows what a genuine linking principle would have to look like: a shared physical constraint, quantities defined independently in each domain, a derived relation between them, and an observation capable of falsifying it. |
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EPISTEMIC POSITION
This framework proposes a unity of vocabulary without postulating a unity of mechanism. It distinguishes what is already defined, what can be put to the test in each domain, and what remains to be connected. Its ambition is not to reduce the diversity of memories to a single law, but to make their resemblances and their differences scientifically comparable.