An acknowledgment of an event is a projection of that event into a lower-dimensional representation. This work asks, across three parts, what such a projection necessarily loses, and shows the loss is both unavoidable and structured. Part I (The Entropy Gap) proves the classical result: when an occurrence's intrinsic dimension exceeds an acknowledgment's capacity, the conditional entropy of the occurrence given its acknowledgment is strictly positive, so the occurrence cannot be recovered from any record of it. The barrier is conditional — a sufficiently rich acknowledgment captures a sufficiently simple occurrence — but for physical events, whose intrinsic dimension is effectively unbounded, every finite record leaves a strictly positive residual. Part II (Quantum Extension) ports the result to quantum systems, where it sharpens. The gap exists precisely when the acknowledging channel is not sufficient in the sense of quantum relative entropy, witnessed by the failure of optimal (Petz) recovery to reach unit fidelity; the classical dimension condition emerges as the generic case of this finer, information-theoretic one. This part defines the quantum residual as the infidelity of optimal recovery, the object the third part requires. Part III (Residual Invariance) asks what the residual is, and finds it is an invariant. The unrecoverable remnant of a quantum occurrence carries a signature of its origin that is preserved under variation of the acknowledging channel: two lossy views of one occurrence leave residuals sharing this signature, two views of different occurrences do not, and — decisively — the shared signature persists when the two acknowledgments are taken on disjoint subsystems sharing no mutual information, which distinguishes genuine invariance from correlation routed through shared data. The scope is stated precisely throughout: the residual is an invariant, not a field; it is preserved under transformation, it does not propagate or interact; and the results establish that the remnant identifies its origin, not that it influences anything. Across all three parts one object recurs — the structure a bounded projection cannot reach — and the unifying claim is that an occurrence is characterized not by what any acknowledgment contains but by the invariance of what every acknowledgment omits. Each result carries a reproducible numerical verification.
An acknowledgment of an event is a projection of that event into a lower-dimensional representation. This work asks, across three parts, what such a projection necessarily loses, and shows the loss is both unavoidable and structured. Part I (The Entropy Gap) proves the classical result: when an occurrence's intrinsic dimension exceeds an acknowledgment's capacity, the conditional entropy of the occurrence given its acknowledgment is strictly positive, so the occurrence cannot be recovered from any record of it. The barrier is conditional — a sufficiently rich acknowledgment captures a sufficiently simple occurrence — but for physical events, whose intrinsic dimension is effectively unbounded, every finite record leaves a strictly positive residual. Part II (Quantum Extension) ports the result to quantum systems, where it sharpens. The gap exists precisely when the acknowledging channel is not sufficient in the sense of quantum relative entropy, witnessed by the failure of optimal (Petz) recovery to reach unit fidelity; the classical dimension condition emerges as the generic case of this finer, information-theoretic one. This part defines the quantum residual as the infidelity of optimal recovery, the object the third part requires. Part III (Residual Invariance) asks what the residual is, and finds it is an invariant. The unrecoverable remnant of a quantum occurrence carries a signature of its origin that is preserved under variation of the acknowledging channel: two lossy views of one occurrence leave residuals sharing this signature, two views of different occurrences do not, and — decisively — the shared signature persists when the two acknowledgments are taken on disjoint subsystems sharing no mutual information, which distinguishes genuine invariance from correlation routed through shared data. The scope is stated precisely throughout: the residual is an invariant, not a field; it is preserved under transformation, it does not propagate or interact; and the results establish that the remnant identifies its origin, not that it influences anything. Across all three parts one object recurs — the structure a bounded projection cannot reach — and the unifying claim is that an occurrence is characterized not by what any acknowledgment contains but by the invariance of what every acknowledgment omits. Each result carries a reproducible numerical verification.
The identity of a quantum subsystem over time, in the minimal sense in which the question can be given quantities, is what of its origin can still be recovered from it or from whoever inherited it. An earlier paper defined it by the coherent information from a reference maximally entangled with the subsystem at a root time, proved that its heir is unique and that inheriting is the complement of being recorded, and two notes gave the law of its loss to imperfect records and the realization theorem for what two disjoint regions can jointly inherit. This note completes that minimal theory with the two statements that remained: how identity is transformed, and how fast it is lost. Krakauer, Bertschinger, Olbrich, Flack and Ay proposed to measure the individuality of a system by how much of its future is determined by its own past rather than by its environment, decomposing the information its future carries about the past into an organismal and an environmental part. We give a coherent-information analogue of that program for a subsystem in a fixed tensor-product structure, using the same coherent information as the measure of heritage. Four exact statements follow, of which the first and the fourth are, to our knowledge, new in this form. (i) The heritage of a system's future decomposes exactly into an individual part, an environmental part and a non-negative heritage synergy equal to the conditional mutual information of the two pasts given the future, and under closed dynamics the total is fixed by the dimensions: what a system loses in individuality goes into synergy with its environment's past or into environmental determination. (ii) Quantum individuality is monogamous: a past has at most one heir, which has no classical counterpart, where a past can be copied into many futures. (iii) Records and individuality are complementary: the classical information a witness acquires about a basis of the system plus the individuality the system retains is at most log d, a corollary of the uncertainty principle with quantum memory; hence a non-inherited record is bounded by the identity that nobody keeps, and the entropy of the witness does not enter. (iv) The leading short-time loss of individuality is a t^2 log(1/t^2) with a the variance, over the state of the environment, of the fluctuation of the interaction about its mean field on the subsystem; for a maximally mixed environment a is the inverse squared scrambling time of Zanardi, Dallas, Andreadakis and Lloyd, so that the bipartite Gaussian scrambling coefficient underlying their operational mereology is recovered as the maximally-mixed-environment specialization of a state-dependent coefficient, and an environment with vanishing coupling variance has no leading-order loss of individuality, whereas one with non-zero coupling variance does. We relate these to the philosophy-of-physics literature on the individuality of quantum objects, where the question is ordinarily formulated through discernibility and identity conditions rather than through a diachronic quantitative measure of this kind, and state what is not claimed.
Reconstructions derive quantum theory from operational principles about states, measurements and probabilities. This paper asks what relational distinguishability alone forces before these exist, answering with a machine-checked construction: one admitted unit of distinguishability forces a recurring tick. Its one axiom: existence is relational distinguishability; every later object is generated from it, each numeral a theorem, no parameter is free. Persistence, that the unit remain re-identifiable as the same unit, is a forced standing requirement generated before any runtime succession of actual occurrences exists. Four theorems follow. A persistent closed history must differ from its reversal, forcing three channels. Their transfers generate a discrete internal geometry with residual share Ω = 1 − 𝜋/(3 √ 2). Handed to each owner, that share is carried without erasure while its unresolved account cannot terminate, so its tick recurs with no input of time or scheduling; the residual reaching its threshold posts an escapement, the complete closure that re-identifies it. The process has a fixed direction: records are retained, irrationality makes the residual recover the history, no reverse step licensed. Under interfaces proposed in two appendices, the derived counting structure reproduces, without probabilities, two reconstructions’ counts: quantum theory’s degrees of freedom at dimension three and its elementary system’s parameters. Physical measurement may be available in subsequent work, so long as the construction is specified for its application and quantities are read as dimensionless relations to one chosen standard candle; if that identification fails, the theory as a physical theory fails as a whole.
A colloquial interpretation of entropy is that it is the knowledge gained upon learning the outcome of a random experiment. Conditional entropy is then interpreted as the knowledge gained upon learning the outcome of one random experiment after learning the outcome of another, possibly statistically dependent, random experiment. In the classical world, entropy and conditional entropy take only non-negative values, consistent with the intuition that one has regarding the aforementioned interpretations. However, for certain entangled states, one obtains negative values when evaluating commonly accepted and information-theoretically justified formulas for the quantum conditional entropy, leading to the confounding conclusion that one can know less than nothing in the quantum world. Here, we introduce a physically motivated framework for defining quantum conditional entropy, based on two simple postulates inspired by the second law of thermodynamics (non-decrease of entropy) and extensivity of entropy, and we argue that all plausible definitions of quantum conditional entropy should respect these two postulates. We then prove that all plausible quantum conditional entropies take on negative values for certain entangled states, so that it is inevitable that one can know less than nothing in the quantum world. All of our arguments are based on constructions of physical processes that respect the first postulate, the one inspired by the second law of thermodynamics.
The Origin of Probability: A Complete Derivation in Three Parts is the collected edition of a three-part theoretical investigation into the structural origin of probability within a deterministic closed-description framework. The work asks a deliberately narrow but foundational question: under what conditions can a system whose underlying description remains deterministic acquire a lawful probability-bearing description without inserting primitive randomness, probabilistic axioms, or the Born rule at the beginning of the construction? The trilogy develops that question cumulatively. Each part establishes a distinct layer of the argument, and later results depend on structures and limitations established earlier. The collected edition therefore preserves the full dependency order rather than treating the three papers as independent essays. It also includes a narrative overview and terminology guide intended to make the architecture of the complete derivation easier to follow. Part I, Finite Recurrent Stability Before Spacetime: Collapse, Emergence, Horizons, and Recurrence as Recoverability Boundaries, develops the realization-neutral structural foundation. It begins from persistent relational distinction and recoverability rather than assuming spacetime, metric distance, clock time, energy, probability, or quantum measurement. Within a declared finite-stability regime it distinguishes contraction, correction, carry-through, collapse, emergence, recurrence, cycle structure, and horizon-like boundaries by their effects on recoverable relational structure. Collapse and emergence appear as distinct lower- and upper-bound failures of one finite recurrent-stability architecture. Weak oscillatory form is subsequently obtained only at structural scope, without promoting recurrence into a physical wave, temporal oscillation, or cosmological mechanism. Part II, Stability, Boundary Observability, and Emergent Probability in Deterministic Systems, develops the transition from deterministic structural dynamics to probability-bearing description. Its central problem is not whether deterministic evolution can simply be renamed probabilistic, but whether loss of recoverability at an admissible boundary can support a lawful reduced description of unresolved alternatives. The paper therefore separates Boundary Loss from probability itself and distinguishes quotient formation, reduced observability, scalar closure, finite measure, normalization, and predictive interpretation as separate derivational burdens. A normalized set of branch weights is not treated as probability merely because it sums to one. Probability status is admitted only after the additional structural and interpretive conditions required for predictive use have been isolated. Part III, Boundary Loss and the Born Rule: The Origin of Probability, carries the construction into a bounded physical realization framework. Boundary Loss supplies unresolved alternatives but does not itself supply numerical probability. The derivation therefore proceeds through source-grounded scalar residues, finite positive total measure, normalization, local predictive admission, and independent source-state and event bridges into Hilbert-space representation. A minimum compatibility condition is then isolated, explicitly adopted at the appropriate stage, and applied separately. At the resulting local source-linked scope, the construction reaches the exact Born form p_i = ⟨J(Ψ), P_i J(Ψ)⟩ = ||P_i J(Ψ)||². The significance of the trilogy lies as much in the separation of these stages as in the terminal equation. Boundary Loss is not equated with probability. Scalarization is not assumed to be probability. Normalization is not treated as sufficient for probability. Hilbert-space representation is not allowed to retroactively supply the earlier structural derivation. Compatibility required for the final Born-form application is distinguished from structures already derived. The dependency chain is therefore designed so that later mathematical convenience cannot be silently imported backward as an earlier premise. Across the three parts, the argument develops the following bounded sequence: ordered dependence → recoverable distinction → finite recurrent stability → collapse/emergence boundary structure → boundary observability → Boundary Loss → unresolved alternatives → scalar closure → finite measure → normalization → predictive probability → Hilbert-space source and event bridges → minimum compatibility → exact local source-linked Born form. The collected edition is intended to function as the canonical integrated research record for this derivation. It contains the trilogy overview followed by Part I Version 4, Part II Version 3, and Part III Version 2. Internal bookmarks and navigation link the overview directly to the title page of each constituent work, while the individual papers remain separately citable publications with their own persistent identifiers. Constituent works: Part I: Finite Recurrent Stability Before Spacetime: Collapse, Emergence, Horizons, and Recurrence as Recoverability BoundariesDOI: 10.5281/zenodo.19966230 Part II: Stability, Boundary Observability, and Emergent Probability in Deterministic SystemsDOI: 10.5281/zenodo.19966289 Part III: Boundary Loss and the Born Rule: The Origin of ProbabilityDOI: 10.5281/zenodo.20102172 Taken as a whole, the trilogy proposes a disciplined route from recoverability and deterministic boundary structure to probability and, at its final declared scope, to the Born form. Its central methodological requirement is that every transition be earned at the layer where it is used, with unresolved burdens, adopted conditions, nonclaims, and realization boundaries kept explicit rather than compressed into the final result.
The concept of entropy rst arose in the study of thermodynamics and statistical me- chanics. In information theory the so-called Shannon entropy plays a signi cant role, which measures how much information we gain, on average, when we learn the value of a random variable. An alternative interpretation is that the entropy of the random variable quanti es the amount of uncertainty about the random variable before we learn its value. The concept was introduced by Claude E. Shannon in his 1948 paper "A Mathematical Theory of Communication". In the rst part of the present work we consider this fundamental measure of information and show some of its properties, such as non-negativity. The relative entropy or the Kullback-Leibler divergence is a very useful measure of the distance between two prob- ability distributions. After introducing its de nition and other notions (joint entropy, conditional entropy or mutual information) we can prove additional assertions about the Shannon entropy and present statements about the relations between the above men- tioned quantities. In quantum information theory, there is a quantity, namely von Neumann entropy, which corresponds to Shannon entropy and originates from it. In the section Von Neu- mann entropy we will introduce this quantity which is a continuous functional on the states, or on their mathematical representatives, the density operators, and it ful ls in- equalities such as subadditivity and strong subaddititvity. Just as with the Shannon entropy, it is extremely useful to de ne a quantum version of the relative entropy, which is one of the most important numerical quantities appearing in quantum information theory. It is used as a measure of distinguishability between quantum states. Among others its properties are jointly convexity, monotonicity which are speci cally convenient. Transformations on structures which preserve operations, quantities or relations among elements appear in many areas of mathematics. These are the so-called preservers which show up also in physics and even in chemistry where they are usually called symmetries. Therefore, the study of preserver transformations is an important area of research. The third section of this work is based on a paper, in which Lajos Molnár described the structure every bijective map on the space of all density operators on a given Hilbert space, which preserves the relative entropy. This result is closely related to Wigner's famous theorem on the form of quantum mechanical symmetry transformations. The above theorem of Molnár can be extended into two directions. On the one hand, the result holds also in the case when the transformation under consideration acts on the space of all positive semide nite operators not only on density operators (i.e., we omit the condition of unit trace). On the other hand the statement remains valid even if we omit the strongly restrictive condition that the transformation is bijective. The second result is a joint work with Prof. Lajos Molnár which appeared in the journal Linear Algebra and Its Applications.