Reflection positivity (RP) is the single positivity statement through which a Euclidean path integral is promoted to a unitary real-time quantum theory (Osterwalder–Schrader reconstruction). Its classical proofs consult a geometric reflection — an involutive automorphism fixing a hyperplane — which a randomly grown geometry generically lacks. This preprint shows that on ensembles of sequentially assembled geometries the positivity structure survives, and that it is carried entirely by the measure: RP holds if and only if the growth dynamics is memoryless on isomorphism classes. Characterization. Class-level Markovianity, class-local product form with terminal normalization, and history-independent Gram realizability of the conditional chain forms are equivalent; the multiplicative cocycle identity is derived as the Markov property in multiplicative form, not hypothesized separately. Explicit spectrum. Under these conditions the chain form is block-diagonal over terminal classes with rank-one blocks; the positive spectrum equals the arrival measure of the class process, independent of the phase rule — phases act only on the eigenbasis, identifying the chain form with the decoherent diagonal. Sharpness. Outside the class, failure is not gradual: a history-dependent weight family violates the class Markov property with exact rational witnesses (1/60 ≠ 9/260, 1/30 ≠ 3/130), and no Gram realization exists. Physical anchoring. The covariant Rideout–Sorkin family of classical sequential growth dynamics for causal sets satisfies the hypotheses, so all theorems apply verbatim: its chain forms are positive semidefinite at every depth. A dictionary instantiates the framework in causal dynamical triangulations, group field theory and spin foams, tensor/matrix models, and causal sets. Verification. All statements verified by exhaustive exact enumeration in rational arithmetic through graph size eight (isomorphism classes 1–3–14–86–709–8534; full depth-two assembly over all 85,393 paths), across six weight families and three degree bounds.
Reflection positivity (RP) is the single positivity statement through which a Euclidean path integral is promoted to a unitary real-time quantum theory (Osterwalder–Schrader reconstruction). Its classical proofs consult a geometric reflection — an involutive automorphism fixing a hyperplane — which a randomly grown geometry generically lacks. This preprint shows that on ensembles of sequentially assembled geometries the positivity structure survives, and that it is carried entirely by the measure: RP holds if and only if the growth dynamics is memoryless on isomorphism classes. Characterization. Class-level Markovianity, class-local product form with terminal normalization, and history-independent Gram realizability of the conditional chain forms are equivalent; the multiplicative cocycle identity is derived as the Markov property in multiplicative form, not hypothesized separately. Explicit spectrum. Under these conditions the chain form is block-diagonal over terminal classes with rank-one blocks; the positive spectrum equals the arrival measure of the class process, independent of the phase rule — phases act only on the eigenbasis, identifying the chain form with the decoherent diagonal. Sharpness. Outside the class, failure is not gradual: a history-dependent weight family violates the class Markov property with exact rational witnesses (1/60 ≠ 9/260, 1/30 ≠ 3/130), and no Gram realization exists. Physical anchoring. The covariant Rideout–Sorkin family of classical sequential growth dynamics for causal sets satisfies the hypotheses, so all theorems apply verbatim: its chain forms are positive semidefinite at every depth. A dictionary instantiates the framework in causal dynamical triangulations, group field theory and spin foams, tensor/matrix models, and causal sets. Verification. All statements verified by exhaustive exact enumeration in rational arithmetic through graph size eight (isomorphism classes 1–3–14–86–709–8534; full depth-two assembly over all 85,393 paths), across six weight families and three degree bounds.
Reflection positivity is a positivity condition on a reflected half-space, and it is not inherited automatically by covariance pieces produced by coarse graining. We isolate a finite-dimensional obstruction for a common projection-type construction. Let H=H+⊕H− carry a unitary reflection Θ exchanging the two halves, let C=C∗ commute with Θ, and assume Osterwalder--Schrader (OS) positivity on H+. For a reflection-symmetric orthogonal projection Π that also preserves H+, we characterize exactly when the projection-sandwich residual C=C−ΠCΠ remains OS-positive: this holds if and only if the positive-half projection Π+ commutes with the OS Gram operator W=P+ΘCP+. Equivalently, the mixed positive/negative-time commutator block vanishes. This gives a direct algebraic test for projection-type coarse graining. We then record three consequences. First, in the scalar infinite-lattice setting a finite-range, fine-translation-invariant orthogonal projection is necessarily 0 or id, so nontrivial local projections must break fine translation invariance or leave this class. Second, a null-escape mechanism shows that a projection which sends an OS-null vector to a strictly positive OS-energy direction forces the residual to violate reflection positivity. For the massive free covariance with bond reflection we identify the OS null cone explicitly and obtain its dimension. Third, a complete 4×4 rational example gives a standalone exact failure certificate with positive original covariance and a negative residual OS direction. A larger audited 8×6 block-harmonic computation is retained only as supplementary provenance because the closure archive does not contain the complete projector table required for independent recomputation. The obstruction is deliberately narrow: it applies to projection-sandwich residuals and does not exclude non-projective Markov kernels, conditional expectations, completely positive maps, nonlinear coarse graining, or continuum constructions.
This record contains the manuscript Quantum Mechanics as the Unique Lorentzian Continuation of an Osterwalder–Schrader Positive Theory by Chast K. Wolfe. The paper proves that the full mathematical structure of quantum mechanics—complex Hilbert space, linear and unitary time evolution, the Born rule, canonical (anti-)commutation relations, Poincaré invariance, the spin–statistics correspondence, and the existence of a unique vacuum—arises uniquely from the standard Osterwalder–Schrader (OS) axioms: reflection positivity, Euclidean covariance, locality, regularity, and the Markov semigroup property. Starting from a reflection-positive Euclidean field theory, the work establishes a monotone chain of implications showing that OS positivity forces:(1) a unique complex structure on the reconstructed Hilbert space,(2) linear unitary Lorentzian time evolution generated by a self-adjoint Hamiltonian,(3) the Born rule via Gleason’s theorem,(4) CCR/CAR relations in the Gaussian case, and(5) full OS → Wightman reconstruction, including Poincaré covariance, TCP analyticity, spin–statistics, and vacuum uniqueness. The paper also proves that nonlinear, real, or quaternionic variants of quantum theory are incompatible with OS positivity together with Euclidean covariance and locality. Thus, within the class of OS-positive models, standard complex quantum mechanics is the unique Lorentzian continuation. This Zenodo deposit includes:• the full manuscript (PDF),• a license file (CC BY-NC-ND 4.0),• a bibliography file, and• a README describing the archive contents.
This record is part of the Mass, Proper Time, and Gravity series. Full series community: https://zenodo.org/communities/mass-proper-time-gravity/ In causal-set sequential growth, general covariance requires physical quantities to be independent of the birth order in which the causal set is grown. This note gives an elementary construction inside the standard labeled framework: an explicit matter-carrying action phase, built from an order invariant, that is exactly covariant at every finite stage and spectator-independent at the amplitude level. The point is a four-line theorem: an element's depth (its longest ending chain) depends only on its past, and in growth an element is born with its past complete, so depth-at-birth equals final depth. Hence the phase exp[-(i/hbar) m c^2 tau_0 sum_x depth(x)] is birth-order invariant, exactly and at every n. Because spectators do not enter the past of the competing births, the phase ratio is spectator-independent in the Bell sense at the amplitude-phase level. A sharp dichotomy fixes admissible matter: element-intrinsic, past-measurable fields keep covariance, while birth-index fields break it. Given the standard requirement that a mass phase accumulate along proper time, depth is singled out among the standard past-invariants. All statements are verified by exhaustive enumeration; three reproducible Python scripts are included. Scope: this is kinematic and selects no dynamics. Whether the resulting quantum measure makes manifold-like orders dominate is pursued separately in Parts LXIX and LXX.
This paper gives a measure-theoretic account of stochastic processes obtained by projecting an upper Modal Triplet Theory evolution. Given a selected probability law on upper initial data, measurable upper dynamics, and an observable map, the projected histories carry a unique path-space law and admit regular conditional kernels. Projection can make the observed process non-Markovian even when the upper process is Markovian or deterministic. The paper separates failure of the observed Markov property, absence of finite Markov order, failure of a chosen classical composition law, and quantum CP-indivisibility. Every classical path law still factorizes sequentially through history-dependent kernels and becomes Markovian on a history state space. An exact binary example has infinite observed Markov order while its memory compresses to one posterior scalar. Exponential action weights define valid kernels when their action and reference measure are supplied, but rewriting a positive kernel in Gibbs form is parameterization rather than prediction. A commutative path-space algebra does not derive quantum observables, Born probabilities, process tensors, or instruments. The remaining MTT task is to select the upper probability source and physical intervention structure from the same geometry.
Why do local actions and exponential Euclidean weights arise so universally in classical, statistical, and quantum theories? We offer a structural explanation from minimal constraints on finite descriptions of admissible histories. Assume that histories admit finite, self-delimiting (prefix-free) generative codes that can be decoded sequentially in a single forward pass. These purely syntactic requirements define a minimal descriptive cost, interpretable as a smoothed minimal program length, that is additive over local segments. First, any continuous local additive cost whose stationary sector coincides with the empirically identified classical variational sector is forced into a unique Euler--Lagrange equivalence class. Hence the universal form of an action is fixed by descriptional structure alone, while the specific microscopic Lagrangian and couplings remain system-dependent semantic input. Second, independently of microscopic stochasticity, finite prefix-free languages exhibit exponential redundancy: many distinct programs encode the same coarse history, and this redundancy induces a universal exponential multiplicity weight on histories. Requiring this weight to be real and bounded below selects a real Euclidean representative for stable local bosonic systems, yielding the standard Euclidean path-integral form. When Osterwalder--Schrader reflection positivity holds, the Euclidean measure reconstructs a unitary Lorentzian amplitude.
A hybrid classical-quantum dynamics whose decoherence, geometry noise, and causal-order action are all driven by one record-export current faces a sharp question the moment it is written down non-relativistically: is the single laboratory time in its intensities and its equal-time noise a removable scaffold, or a load-bearing preferred simultaneity? This paper audits that question for the record-history kernel P_ker adopted earlier in this series, in a fully declared formal setting and with a pre-committed either-way verdict. The scope is fixed before anything is scored: a fixed globally hyperbolic background (Minkowski benchmark) under a minimal dressing assumption D0, at leading order in the export couplings — never background independence, never Planckian physics. The setting types record exports as non-selective instruments implemented in the Fewster-Verch measurement-scheme sense; lifts the kernel's adaptedness clause to a causal-set-indexed filtration (intensities measurable with respect to their own causal past); and requires export-sourced noise to admit a representation as a retarded kernel integrated against an adapted martingale measure, separating retarded response (M3a) from screened common-past correlations (M3b). Four covariant lifts of the export intensity are scored on eight criteria, including a local-implementation witness and covariant trade-off data. The hypersurface candidate fails as a primitive — the live example of the barred move, "notation-covariantization" — while the world-tube candidate passes, with the density and diamond forms as its aggregates. For the surviving class the paper constructs, rather than asserts, its core: an explicit measurement-scheme witness model (a pointer coupling to a fragment carrier field, compactly supported in the event diamond and switched by the apparatus's own trigger field) for the self-adjoint dephasing subclass of the kernel's operator family — a declared narrowing carried as an explicit scope condition; an explicit admissible-noise construction (compensated-Poisson impulses along the tube, retarded profiles), whose covariance computation recovers the kernel's equal-time form as its non-relativistic cell limit and whose screening property is proved, not assumed; an export stress tensor constructed pathwise — the difference of one closed, internally switched action's on-shell conserved stress tensor along two coupled realizations differing only in the jump innovation at the event diamond — so that conservation is a pathwise identity, causal support follows by finite propagation speed, and the monopole vanishes by global conservation, with the trigger's locality carried as a displayed hypothesis; a typing theorem showing the kernel's three type declarations make every witnessed export a causally admissible local operation (with a two-step counterexample showing each declaration is necessary — the Sorkin impossible-measurement threat is blocked by exactly those declarations); a leading-order foliation-independence proposition (spacelike-separated update maps commute; adapted increments compose; the multi-event kernel is independent of the causal linear extension); and the tube-wise decoherence-diffusion floor with the Minkowski benchmark numbers recovered unchanged. The verdict is scoped, not discharged. What is certified is a constructed microcausality core at leading order for the exhibited, witnessed class: after this paper, the registered sentence "nothing relativistic certified" no longer stands as written. What remains open is named, not hidden: the general-frame operator derivation of the trade-off floor for boosted extended populations; the multi-event composition law beyond leading order, where geometry back-reaction couples updates within a light cone; background independence — the dressed-source seam assigned to the next paper; and measurement-scheme witnesses for the kernel's operator family beyond the self-adjoint dephasing subclass. Because those conjuncts are open, the covariant-extension debt as a whole is not paid, and the honest verdict under the grouping's shared discharge type is scoped, with the kernel keeping its declared non-relativistic restriction and the series' U3-level claims keeping their published ceiling until the completion closes. Within the series this opens the thirteenth grouping — the Debt-Discharge Program (Papers 45-47) — whose charter, naming rule, and verdict type are defined here and reused by the dressed-source and species/ultraviolet panels. Nothing is adopted, calibrated, or scored; no band moves; the substantive preferred-foliation rival is declined, not refuted.