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When is a Physical System the Same System? - Identity, Dynamics, and Representative-Independent Evolution

Marc Maibom

Published Sep 10, 2026• 0 citations
Publisher

Abstract

Physical theories determine how states transform, but a same-system claim asks a different question: whether the transformed description concerns the same physical subject. La Profilée (LP) proposes a universal structural law of nature governing identity under transformation; here “universal” denotes a functional necessity claim about determinate same-subject judgments, not a derivation of domain-specific dynamics, constants, masses, or couplings. In LP, persistence is not an independent primitive alongside identity: it is diachronic identity, the positive consequence of preserving the relevant sameness relation through admissible transformation. This paper gives the physical identity formulation of that law and isolates a necessary dynamical condition: No Coreference Without Congruence. If two admissible representatives are identified as the same physical subject through the transformation under analysis, they must induce the same retained future; otherwise no representative-independent reduced evolution exists on the proposed identity quotient. Q1 certifies whether a determinate integrated bearer continues at all. Q2 is the identity verdict on that continuing bearer and, where the physical realization requires distinct carriers, is typed into Q2a, identity-class continuity, and Q2b, continuation-stable profile/history identity. A constrained bridge π_p:H_ℛ ⇀ Ā_ℛ relates those carriers, and Q2b ⇒ Q2a ⇒ Q1. The congruence requirement is stated for deterministic maps, linear quotients, CPTP channels, and quantum instruments. Greatest-fixed-point, bisimulation-like, and quotient-congruence techniques are established mathematical tools; the contribution is their typed integration into the physical identity law supplied by LP. Persistence is then a regime consequence of successful identity preservation, not a primitive used to define identity.

DOI
10.5281/zenodo.22687637

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Physical theories determine how states transform, but a same-system claim asks a different question: whether the transformed description concerns the same physical subject. La Profilée (LP) proposes a universal structural law of nature governing identity under transformation; here “universal” denotes a functional necessity claim about determinate same-subject judgments, not a derivation of domain-specific dynamics, constants, masses, or couplings. In LP, persistence is not an independent primitive alongside identity: it is diachronic identity, the positive consequence of preserving the relevant sameness relation through admissible transformation. This paper gives the physical identity formulation of that law and isolates a necessary dynamical condition: No Coreference Without Congruence. If two admissible representatives are identified as the same physical subject through the transformation under analysis, they must induce the same retained future; otherwise no representative-independent reduced evolution exists on the proposed identity quotient. Q1 certifies whether a determinate integrated bearer continues at all. Q2 is the identity verdict on that continuing bearer and, where the physical realization requires distinct carriers, is typed into Q2a, identity-class continuity, and Q2b, continuation-stable profile/history identity. A constrained bridge π_p:H_ℛ ⇀ Ā_ℛ relates those carriers, and Q2b ⇒ Q2a ⇒ Q1. The congruence requirement is stated for deterministic maps, linear quotients, CPTP channels, and quantum instruments. Greatest-fixed-point, bisimulation-like, and quotient-congruence techniques are established mathematical tools; the contribution is their typed integration into the physical identity law supplied by LP. Persistence is then a regime consequence of successful identity preservation, not a primitive used to define identity.
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.
A tensor factorization specifies a mathematical subsystem, but it does not by itself establish that its reduced state is a complete description of the physical bearer it claims to represent. This paper supplies an exact selection criterion and a quantitative certificate without altering the global physics. It retains the observable-induced algebraic formulation, the quotient-dynamics theorem, the strict two-qubit counterexample, the autonomy defect, additive error accumulation, nested persistence margins, the coupling–time estimate and the non-circular selection argument. The physical-cut theorem now distinguishes three independent descent requirements: complete constitution, retained existence and identity verdicts, and autonomous dynamics. A realization x=(c,ρ,η) separates the physical profile, quantum state and preparation/history/continuation data. This prevents a historical identity verdict from being silently treated as a function of a momentary density matrix. The CNOT-conjugated cut has identical reduced inputs and orthogonal reduced outputs at t=π/4. A second witness has perfectly closed baseline dynamics while a partial trace erases two physically distinguishable coupling profiles. Dynamic closure alone therefore does not establish physical completeness. The quantitative results bound unavoidable prediction error from below by half the fibre defect and, with an admissible lift in the chosen model class, from above by that defect. Reachable-state error bounds accumulate under CPTP comparison. A bound below an independently grounded Q1, Q2a or Q2b margin is a sufficient persistence certificate; loss of that inequality is not an identity-loss verdict. For a bounded interaction the specified decoupled predictor has the sharper error bound |λ|(t−s)‖V‖/ℏ, while the fibre defect has twice that bound. The resulting criterion links the physical cut to emergence, thermodynamic process closure and record memory without identifying those questions. All exact theorems below are finite-dimensional and relative to explicitly named preparation and continuation domains.
Modern physics relies on several highly successful mathematical frameworks — quantum theory, gauge theory, general relativity, and thermodynamics. These theories describe nature extremely well, yet their mathematical forms are usually introduced through independent postulates. This work asks a prior question: why do these mathematical structures appear at all? The framework developed here begins from two minimal structural requirements for persistence:coherent return (self-consistency under transformation) and triadic closure (the minimal structure required for such verification). Within clearly stated domain restrictions — independent composability, smooth relational locality, and finite resolution — these postulates impose strong constraints on how physical systems can be represented. From these structural constraints, several familiar elements of modern physics arise as minimal admissible forms: • complex Hilbert space and unitary quantum evolution• quadratic weighting (Born rule)• relational geometry and Einstein-type gravitational response• irreversible temporal ordering from non-invertible return• entropy and the Second Law from indistinguishability growth• compact unitary gauge structure The framework does not attempt to predict parameter values or replace existing physical theories. Instead, it explains why the mathematical forms used by those theories are structurally admissible once coherent persistence under finite resolution is required. As a numerical application, the Koide relation for charged-lepton masses is shown to arise as a geometric consequence of triadic equilibrium in quadratic mass-amplitude space, conditional on the charged-lepton family realizing a coherent triadic structure. The result is a structural program for physics: rather than proposing new dynamical laws, it constrains the mathematical forms that viable physical laws may take.
Classical theories of identity presuppose state distinguishability over time. Quantum mechanics appears to challenge this. The present paper establishes that LP’s structural persistence condition IR = R / (F · I · C) ≤ 1 is compatible with and illuminated by quantum mechanics. LP requires distinguishability of states, not constituents — satisfied in quantum mechanics at the system level. Superposition is a regime of structural indeterminacy (IR > 1 at the outcome level). Measurement is IR-reduction through environmental coupling. v3 strengthens Section 4a: the proof chain LP identity preservation → state-space isometry → Wigner symmetry → unitarity replaces the weaker injective-CPTP argument of v2. Identity is not merely compatible with unitary evolution — it structurally requires it.
This paper is Part II of The Pre-Emergence Origin of Probability trilogy. Part I develops the structural requirements for finite recurrent stability before spacetime. Part II asks when persistent boundary-readable structure can support numerical probability. Part III examines what probability-status remains after boundary loss, before numerical weighting, and under what narrower conditions Born-form weighting may follow. The paper analyzes how probability-measure status may arise within a closed deterministic description rather than being assumed as a primitive numerical rule. A closed description cannot appeal to an external observer, hidden weighting convention, retained label, post-outcome selection, or independently supplied probability measure to explain its own probability assignments. Numerical probability must instead be supported by structures and relations admitted within the described system. The analysis begins with the requirements for persistent structured identity. Persistent identity is treated as recoverable cross-state relational distinction rather than exact sameness, repeated naming, scalar aggregation, or externally preserved memory. Within the tested structural registry, persistent refinable identity requires ordered dependence, a sufficient stabilization architecture, distributed relational structure, recurrence, bounded recoverability, and noncollapse of the distinctions being preserved. Bounded recurrent relational correction is classified as weak oscillatory form in a deliberately structural sense. The classification does not establish a physical wave, sinusoidal motion, exact periodicity, frequency, propagation, dynamics, mechanism operation, or physical carrier. It identifies a recurrent recoverability architecture that a later realization would have to preserve. The paper then develops the transition from persistent structure to boundary observability. Representation-sensitive bulk descriptions may depend on phase origin, refinement history, decomposition, internal labels, subpart identity, or other distinctions that do not survive admissible boundary readout. Under admissible-observable indistinguishability, boundary-readable observables factor through an emergence-boundary quotient that preserves only the distinctions carried by the admitted observable family and surviving the declared transport relation. The boundary quotient does not itself provide probability. It supplies a local architecture of boundary-readable alternatives and permits the admission of candidate nonnegative contribution scalars. Neither the alternative carrier nor a candidate scalar establishes scalar uniqueness, scalar closure, additivity, measure realization, normalization, or probability interpretation. Numerical probability-measure status becomes available only under a narrower cumulative condition package. The package requires an exact finite carrier, a finite positive event measure, a normalized scalar component on the same carrier, exact scalar-measure compatibility, one-scalar factorization of every admitted weight-bearing map, exclusion of independent weight-bearing invariants, pre-outcome assignment, absence of an external weighting source, and a declared scalar-only predictive regime. Additivity, commensurability, regularity, positive scale, finite nonzero total, and normalization remain separately controlled conditions within that route. Under those conditions, normalized scalar weights may receive conditional probability-measure status inside the declared local regime. This is not an unconditional derivation of probability from determinism. The result does not establish unrestricted probability, universal scalar representation, a global probability law, objective chance, stochastic dynamics, collapse, measurement theory, physical realization, or a final physical theory. Within the trilogy, Part II therefore supplies the middle structural layer: persistent recurrent structuretoboundary-readable alternativestocandidate scalar contenttoconditional numerical probability-measure status. Part III begins from a different question. It examines what remains when a boundary-readable description loses the ability to guarantee which admissible alternative obtains, before any numerical scalarization is supplied. Part II provides the scalar, measure, normalization, compatibility, and realization-burden architecture inherited by that later analysis, but it does not pre-establish the boundary-loss results of Part III. ### Revision note This revision is a substantial reconstruction and trilogy integration rather than a minor editorial update. The derivational architecture, definitions, condition packages, failure boundaries, dependency order, and claim-scope controls have been revised throughout. The manuscript has also been rewritten in new prose to present the argument as Part II of The Pre-Emergence Origin of Probability trilogy. Major changes include a clearer separation of persistent identity, ordered dependence, stabilization, distributed relational recurrence, weak oscillatory form, boundary observability, candidate scalar status, scalar closure, measure realization, normalization, and conditional probability-measure status. The emergence-boundary quotient and transport-survival architecture have been clarified, and the numerical-probability result has been narrowed to an expressly declared scalar-only predictive regime satisfying the complete carrier, measure, compatibility, factorization, pre-outcome, and independent-invariant exclusion conditions. The revision also strengthens the distinction between candidate weights, normalized weights, conditional measures, and probability interpretation. It adds more precise failure localization, preserves the distinction between conditioned failure and universal impossibility, and expands the realization-burden architecture separating mechanism satisfaction, physical scalar selection, boundary enforcement, branch generation, refinement implementation, physical realization, theory-specific realization, empirical validation, and final physical-theory status. The paper is now explicitly positioned between Part I, which establishes the structural requirements for finite recurrent stability before spacetime, and Part III, which develops boundary-unresolved and pre-numerical probability-status after recoverability loss. The central research program remains mechanism-neutral. No physical mechanism, collapse process, specific realization theory, unrestricted probability law, or empirical validation is claimed.
When the conditions under which a record is formed remain fixed, a physical theory may represent that record completely by a unary law of the state observed. This paper asks what structure is forced when the same independently certified physical identity gives different complete records under different formation conditions. The initial datum distinguishes two typed empirical roles, while the state identity of the formation side is derived from its complete behavior. Failure of factorization through the first projection proves that the formation-side role cannot be deleted. The complete row and column behaviors of the primitive law then determine two kernel pairs and their coequalizers. Unique descent retains every observable distinction while removing every redundant label, and the resulting realization is terminal among all exact role-preserving realizations. The product pairs the reduced roles; a pullback extracts the combinations on which records can actually be formed. Only after these steps does exponential transposition identify each surviving formation class with one complete response map. The resulting bijection supplies the state criterion and establishes the observation-state object. Compatibility with the antecedent theory preserves the independently given physical state space and yields the physical–observation law in which a fact is generated jointly by the state observed and the state of observation. The same construction gives exact criteria for conditional, label-free, and joint identifiability and selects the same-source pullback on which transport may subsequently be defined. A nonzero coherent response on that pullback yields the phase-transport fundamental equation, the first fundamental equation written directly on observation space. Quantum theory subsequently enriches this canonical interface. Coherent response and full circular phase select the complex field; reversible binary mixing yields CP1 ∼= S2 ; independent composition creates the four-dimensional event domain; response-event identity glues probabilities across complete contexts; and Gleason representation with marginal consistency gives the Born trace pairing. The two-qubit correlation tensor then yields the singlet cosine law, the CHSH inequality, the Tsirelson bound, and the Horodecki spectral criterion. Bell violation is thereby identified with the absence of a positive global answer section across incompatible observation states. The quantum formalism is thereby closed into one architecture: complex state geometry describes coherent physical possibility, complete observation states specify the ways in which facts may form, the Born interface joins them, and quantum instruments carry a realized fact into subsequent dynamics. Fixed observation sections recover classical unary laws, while smooth equivariant enrichment lifts relativistic frame geometry and internal gauge geometry to the same joint domain. The central result is ontological. Physobser Theory admits observation into objective physical reality not as a mind, an external readout, or a perturbation, but as an irreducible state-bearing axis; a fact is generated at its lawful interface with physical state. Category theory is the proof instrument. The discovered physical content is the dual-axis constitution of facts, which supplies a common foundation for quantum, classical, relativistic, and gauge physics.