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.
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.
Quantum Darwinism explains why the world agrees about a system: redundant classical records make its pointer observable objective. We ask the complementary question, what it would take for a region of the world to be the continuation of a system rather than a description of it, and organize the answer around one quantity, the coherent information of the history channel from an origin to a region, I(R > X)=S(X)-S(RX) on the Choi state of the dynamics. For a pure global state this is log d minus what the rest of the world knows about the origin, so inheriting is that everybody else be ignorant, and objectivity and individuality are complementary. From this one quantity, three standard facts (strong subadditivity, global purity, data processing) and four ways of cutting the world (space, time, body versus records, and large scale) we obtain the following. Uniqueness of the heir and its causal-clique shape in spacetime; the size of the minimal heir in terms of Darwinian redundancy, with the constant kappa(0.1)=0.026, its form in the currency of recording strength for unequal witnesses, where the bound needs no exchangeability hypothesis, and a shared-secret environment that separates objectivity from heritability; a classification of interactions by degradability, under which pointer interactions never produce an heir. A realization theorem for what two disjoint regions can jointly inherit: their maximal correctable algebras commute, every commuting pair with a multiplicity matrix is realized as the maximal pair of an explicit channel, and what the two regions hold in common is always abelian, of dimension the number of connected components of a bipartite graph that the channel fixes -- so that the shareable part of an ancestry is exactly its classical part. An exact Zeno law for bodies whose dynamics does not commute with what is recorded, whose extended-body form is a classical Markov chain with unistochastic transition matrix and whose criterion is quantum non-demolition, and, for imperfect records, the fact that a perfect record is NOT the fastest way to take a body's identity: the loss is fastest at an interior record strength, where its rate exceeds the perfect-record rate by a factor 2/theta in the rotation angle; a budget inequality under which a lineage that acts on the world spends itself, with a closed frontier for qubits by convexity and a non-convexity for qudits. An existence criterion via information-preserving structures; a de-ordering lemma for biographies in indefinite causal order; and a security formulation in which exclusivity is the privacy of the root's classical record. Most of the physics is other people's and every theorem is attributed in place; Section 1.3 lists one by one what is not, with the realization theorem at the head of that list, and two companion notes carry the two longest proofs. We report, with equal weight, what was tried and fell: a proposed dynamical law of non-inheritance, a dynamical selection of subsystems, and a localization diagnostic, all retracted after adversarial review, and we propose a measurement of the minimal heir on an existing superconducting quantum-Darwinism platform.
Quantum Darwinism explains effective classical objectivity through the selective proliferation of information about preferred quantum states into multiple environmental fragments. Its mature formulations operate within quantum theory, using a system-environment decomposition, decoherence or einselection, and information-theoretic or broadcast-structure criteria for redundant accessibility.This paper asks an upstream question from Registration Theory (RT): what makes a physical distinction a record before that record is described as information available to an observer? Canonical RT answers that retained registration is persistent organization constraining later evolution, that a distinction becomes a physical record only when it remains consequential, and that persistent physical organization is prior to information extracted from it.The paper assembles a theorem-backed finite-model chain now available in the RT corpus. Canonical ACRF dynamics supplies write-retain-read consequential history. The exact finite-amplitude ACRF-4 quotient identifies a nine-dimensional host-host-boundary state. The certified consequence-minimality theorem (DOI 10.6084/m9.figshare.34063464) shows that the retained relational boundary contributes irreducible future-consequence directions. The spectrum-preserving ACRF-8 redundant-record theorem (DOI 10.6084/m9.figshare.34063500) then proves an explicit local witness in which one quotient-valid source distinction is written into two disjoint retained fragments that are locally recoverable, counterfactually separable, and independently readable, giving RRT(q)≥2 under the stated criteria.The resulting distinction from Quantum Darwinism is precise but bounded. RT recordhood is consequence-based, while Darwinian objectivity is an accessibility-and-redundancy criterion formulated within quantum theory. The ACRF witness demonstrates redundant retained record formation without assuming Hilbert space, density operators, mutual information, decoherence, einselection, Born probabilities, or an observer-defined environment. Therefore, multiple persistent records of one distinction are not definitionally identical to Quantum-Darwinian redundancy.The paper does not claim that Quantum Darwinism is false or that RT has derived quantum mechanics, decoherence, Born probabilities, a mutual-information redundancy plateau, or macroscopic objectivity. Instead, it proposes a disciplined hierarchy: consequential distinction → retention → persistent record → branching → redundant retained records → redundant accessibility → environmental objectivity. The final two steps remain open in RT. A future bridge must dynamically select and scale multi-host redundancy and map consequence-bearing RT records into quantum environmental fragments that satisfy decoherence and strong objectivity conditions.
La Profilée is the structural architecture of determinate nature: Constitution identifies the whole, Transformation evaluates its existence and identity through change, and Coexistence evaluates relations among realized wholes. This paper applies the full architecture to quantum measurement without changing the quantum equations. Its physical subject remains definite outcomes, their relation to unitary evolution, the Born law, and the identity of the records that carry them. A physically completed coupling resolves an exhaustive exclusive record variable; LP determinability then gives one actual record value without an observer selecting it. Exclusive actuality does not entail projection of the globally retained quantum state. Rank-one calibration and repeatability fix the affirmative branch map. Statistical non-contextuality additionally requires identity of the complete input–event experiment and descent of its law to that identity class. Stable Boolean event frequencies then give the Gleason representation, and preparation faithfulness identifies its operator with the prepared state. Each inference is proved in the text. The argument determines the carrier and instrument before using these results. It proves that a unique instrument constitution does not select a run outcome, and that decoherence alone does not identify the physical record partition. Worked reversible-laboratory and relational cases distinguish an actual event from persistence of its record. Quantum recovery makes the distinction quantitative: a two-qubit copying channel preserves classical labels exactly, loses local full-qubit recoverability with optimal normalized diamond error one half, and preserves the full code jointly. Continuation-family and complementary-channel results connect those measurement records to information preservation, including evaporation. The resulting measurement theorem states its structural and physical requirements explicitly; it does not predict the individual outcome or infer multi-time consistency from a formal state–label pair alone.
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.
Redundancy-Limited Emergence of Classical Histories in Quantum Systems This work investigates how classical reality emerges from underlying quantum dynamics, focusing on the role of environmental information, noise, and fundamental entropy limits. In quantum theory, systems evolve into superpositions of many possible outcomes. However, in practice we observe a world with definite, classical histories. A widely accepted explanation is decoherence, in which interactions with the environment suppress interference between different quantum states. More recent developments, particularly in quantum Darwinism, emphasize that classical information becomes objective when it is redundantly recorded across many independent parts of the environment. This paper addresses a key unresolved question: how many distinct classical histories can physically exist? To answer this, we construct a minimal computational model in which a system’s history is encoded into multiple environmental fragments, each subject to noise. By computing how much information about the system can be recovered from subsets of these fragments, we identify a characteristic scale that determines how many fragments are required to reconstruct the system’s history. Our results reveal a simple and robust relationship: the ability to recover classical information depends on the entropy of the noise affecting environmental records. As noise increases, more fragments are required to reconstruct a history, reflecting a loss of distinguishability. This leads to the definition of a “recoverability factor,” which quantifies how efficiently entropy can be converted into usable, classical information. We then connect this information-theoretic result to fundamental bounds from gravitational physics. In particular, existing entropy bounds limit the total amount of information that can pass through a region of spacetime. We show that only a fraction of this entropy—determined by the recoverability factor—can contribute to distinguishable classical histories. This leads to a new principle: the number of classical histories is not limited only by total entropy, but by the efficiency with which that entropy can be encoded, preserved, and accessed. Applying this framework to cosmological entropy estimates, we derive explicit upper bounds on the number of distinguishable classical histories in the observable universe. While this number is extraordinarily large, it is nonetheless finite and strongly constrained by both entropy and noise. These results refine the standard picture of quantum branching. While quantum theory allows a vast number of possible outcomes, only a subset can become physically meaningful in the sense of being stable, redundantly recorded, and accessible to observers. More broadly, this work provides a quantitative bridge between decoherence theory, information theory, and gravitational entropy bounds. It suggests that classical reality is fundamentally constrained not just by physical laws governing entropy, but by the efficiency of information transfer within those constraints.
The fourth-order coefficient of the mean relative-entropy disturbance of a reduced encoded state under Gaussian coherent noise was assembled in closed form in previous work, but with two objects left as sums over spectral indices: the quartic commutator channel C, given there only as a four-index contraction of the third divided difference of the logarithm, and the diagonal part b_1 of the mixed channel. Both are closed here by one method, and the consequences are read off. The method. Substitute the integral representation of the divided differences before performing the index sums, which turns every Kronecker delta into a coincidence of resolvent labels, and then integrate by parts in the ring Q[d_1][t, m_1, ..., m_4] carrying the single derivation dm_k/dt = -k m_(k+1), where m_k(t) = sum_u (p_u + t)^(-k). Two structural facts make it usable, and ignoring either costs the solution. First, the integrand of chi = sum_u 1/p_u is exact, m_2 = -d(m_1)/dt, so chi is not an independent integrand but a boundary value; including it in the ansatz leaves its own coefficient undetermined. Second, no per-monomial convergence bound may be imposed on the antiderivative: only the total need converge, and the correct antiderivative contains four individually divergent monomials whose divergences cancel. Imposing the bound cuts the ansatz space from 111 monomials to 55 and the system then has no solution at all. The results are C = 1 - 2 Xi + (d_1^2/2 + 1/3) chi - (2/3) T - (d_1/6) S_w, b_1 = d_1^2 (Xi - chi), each with an explicit antiderivative, so no coefficient is fitted: the three coefficients of C emerge from the linear system as unknowns, and the additive constant and the chi term emerge from the two boundary values. No index contraction beyond the four functionals Xi, chi, T and S_w survives anywhere in the fourth-order coefficient. The functional S_w = sum_uv w_uv is not new to the theory: the connected channel already needs it at order d_2^2, in the form Conn-A_22 = 1 - Xi + D(Xi - 3)/4 + d_1 d_2^2 S_w/2. What is new is that the same functional returns in the d_2-independent part through C, that the two weights collect into (d_1/6)(3 d_2^2 - 1) S_w, that the coefficient needs exactly four spectral functionals and no fifth, and that S_w is not a linear combination of the other three. The last is proved for every d_1 >= 3, analytically: a degeneration lemma, sending one eigenvalue to zero and separating the coefficient space by the functional form of the log-coefficient, confines the kernel to two dimensions; an expansion about the degenerate spectrum then closes it, with discriminant d_1 (d_1 - 2)^2 / 4, which vanishes exactly at d_1 = 2. The same statement proves the irreducibility of the third functional T, which the preceding paper had established by least squares; and at d_1 = 2 the two relations S_w = chi + 2 Xi - 4 and S_w = -2T - 2chi are exhibited, which is why four functionals collapse to two there. The geometry. All four functionals are contractions of the Kubo-Mori (Bogoliubov-Kubo-Mori) metric g and of the Bogoliubov-Kubo-Mori skewness tensor tau, the Amari-Chentsov tensor of quantum information geometry, on two canonical directions: the identity 1 and the maximal-coherence operator J of the Schmidt eigenbasis, which is d_1 times the projector on the flat Schmidt superposition. Explicitly chi = g(1,1) = g(1,J), S_w = g(J,J), Xi = g(rho.J, J) and T = l_2[rho.J, J, J], with rho.J the Jordan product, and the numerical coefficients are contractions too: g(rho,rho) = 1 and g(rho,1) = d_1, the rank. The normalisation is fixed, not assumed, by the relative-entropy expansion S(rho + sX || rho) = s^2 g(X,X)/2 - s^3 tau(X,X,X)/6 + O(s^4). The directions are not a choice. In the two index patterns in which it enters, the mode two-point function of the invariant ensemble equals 2 (rho.J)_uv J_vu - 2 rho_uv rho_vu and 2 (rho.J)_uv 1_us - 2 rho_uv rho_vs identically in the spectrum, for every d_1; and a free index sum is a pairing with J since sum_uv X_uv = Tr[X J], while a delta-identified pair is a pairing with 1. Those two facts do not depend on the order of the expansion, so any contraction of the mode two-point function against any kernel l_r, at any order, lands in the span of {1, rho, J, rho.J}. In that language the identity and the pure coherence operator K = J - 1 are Kubo-Mori orthogonal, g(1,K) = 0, and S_w - chi = norm(K)^2: the fourth functional is the squared Kubo-Mori length of the coherence direction and chi is its classical Fisher part. Coherence enters through two functionals rather than one, since Xi - d_1 = g(rho.K, K) is purely off-diagonal as well. Every channel is then a rational-weighted sum of metric and skewness contractions on the four directions, with d_2 the only number entering from outside the geometry: G = g(Lambda,Lambda)/2, A_13 = -(5/6) D sum_a g(J_a,J_a), and b_1 = d_1^2 g(rho.J - 1, J). Sharper still, the Amari-Chentsov tensor enters the whole coefficient exactly once, at order d_2^0 and with weight -1/6, which is precisely the weight the relative-entropy expansion assigns to its own cubic term; everything else, at every order in d_2, is metric. Corollaries. The chain d_1^2 <= Xi <= d_1 sqrt(chi) <= chi holds for every spectrum, with equality at any one place exactly at a degenerate spectrum; the left link is from the preceding paper, the other two are new, and the consequence is that b_1 <= 0 always and vanishes only at degeneracy. The coherence length is bounded sharply, norm(K)^2 >= d_1^2 (d_1 - 1) and hence S_w >= d_1^3, saturated at the degenerate spectrum and nowhere else. The d_2-independent bracket collapses to (d_1^2/2 - 1/6) chi + T/3 - (d_1/6) S_w, in which Xi and the additive constant cancel exactly, so the constant term is independent of the functional that carries the whole second-order theory. At a degenerate spectrum C = 1 - (5/3) d_1^2 + (2/3) d_1^4, which gives 40 at d_1 = 3, and at d_1 = 2 the bracket equals 2 + (chi - Xi) and is therefore at least 2. Certification. Three independent routes to the full coefficient agree at every configuration tested: the raw five-channel sum evaluated from the mode objects on an explicit orthonormal Hermitian basis, the assembled equation of the preceding paper as printed, and the fully explicit form derived here. Three numbers published before these derivations existed are reproduced by them: C = 7.0492 at d_1 = 2 with p = (0.7, 0.3), C = 40.000 at d_1 = 3 with a degenerate spectrum, and B_1 = -10.2986 at d_1 = 2, d_2 = 2. The reduction of the four-index sum is an exact symbolic identity in the spectrum and in the integration variable; the antiderivative identity holds as a polynomial, not numerically; and the geometric form of C agrees with the four-index sum to 10^(-79). Two limitations are stated. Everything here is for the unitarily invariant ensemble, and nothing survives for a structured family such as a Pauli family satisfying the Knill-Laflamme conditions on a stabilizer code, where the first-order mode objects vanish and C = 0 identically. And one identity, an exact relation among the five functions holding on geometric spectra at d_1 = 3 and explaining why a least-squares test of reducibility is weak there, is established numerically with exact rational coefficients and held-out parameter values rather than symbolically; nothing else depends on it. One evaluative sentence of the preceding paper is corrected: the coefficient needs four spectral functionals, not three, and the fourth is printed in that paper's own assembled equation. The undercount is in the count. No equation, no theorem, no table entry and no value of that paper is affected, because C was computed there from its four-index sum directly and b_1 from its defining double sum directly. An expectation for order eps^6 is recorded and labelled as one: the directions are already forced for the part of the expansion built from first-order mode objects alone, while the two-point structures of the higher mode objects and the count of contraction patterns available to l_4 and l_5 remain open. Nothing at sixth order is computed here. This note is posted here for the first time. It closes the two residual index sums of Closed Forms for the Remaining Fourth-Order Channels of Gaussian Coherent Noise, and a Definitional Correction (Zenodo DOI 10.5281/zenodo.23684111), and corrects one evaluative sentence in it. The functional S_w it identifies geometrically was introduced in A Complete Fourth-Order Expansion of Reconstruction and Logical Errors for Arbitrary Gaussian Coherent Noise (Zenodo DOI 10.5281/zenodo.23644208), and the functionals Xi and chi in A chi-squared divergence law for the fourth-order breakdown of entanglement wedge reconstruction in Gaussian holographic codes (Zenodo DOI 10.5281/zenodo.23621356). The scripts reproducing every number in the note are deposited as metric_skewness_scripts.zip. They require Python 3.10+ with NumPy, SciPy, mpmath and SymPy; no other dependencies, CPU only. Five of them are reused unmodified from the deposit of the preceding paper, which is deliberate: the new closed forms are checked against code written before they existed.