We identify a class of finite quantum systems — acyclic systems whose transition graph is a directed acyclic graph (DAG) — for which the Born series collapses into an exact algebraic identity with finitely many terms and strictly zero truncation error. The sufficient condition is the nilpotency of the transfer operator T = G₀(E)V. If T^(m+1) = 0, then the exact solution of the Lippmann–Schwinger equation is the finite sum |ψ⟩ = Σ Tᵏ|ϕ⟩ (k = 0 to m), with no condition on ‖T‖. We prove that the acyclicity of the transition graph implies the nilpotency of T (Theorem 19), and that the nilpotency index coincides with the maximal path length of the graph (Proposition 21). The main result (Theorem 23) concerns the four-level quantum system with diamond-graph structure. In this case, the transition amplitude toward the final state is A₄ = t₄₂t₂₁ + t₄₃t₃₁, an exact algebraic identity encoding constructive interference, exact destructive interference (dark state formation), and partial interference. The first-order Born approximation predicts identically zero amplitude in all regimes, thereby failing quantitatively in 100% of the cases. The Born–SON framework additionally provides the exact full resolvent, the exact T-matrix, explicit error control in the quasi-nilpotent regime, and a scalar structural metric — the Born–SON depth — quantifying the intrinsic complexity of an acyclic quantum system. The algebraic core of the framework has been formally verified in Lean 4 with Mathlib (https://doi.org/10.5281/zenodo.20113599). MSC 2020: Primary 81U05. Secondary 47A10, 05C20, 81Q10, 15A16.
We identify a class of finite quantum systems — acyclic systems whose transition graph is a directed acyclic graph (DAG) — for which the Born series collapses into an exact algebraic identity with finitely many terms and strictly zero truncation error. The sufficient condition is the nilpotency of the transfer operator T = G₀(E)V. If T^(m+1) = 0, then the exact solution of the Lippmann–Schwinger equation is the finite sum |ψ⟩ = Σ Tᵏ|ϕ⟩ (k = 0 to m), with no condition on ‖T‖. We prove that the acyclicity of the transition graph implies the nilpotency of T (Theorem 19), and that the nilpotency index coincides with the maximal path length of the graph (Proposition 21). The main result (Theorem 23) concerns the four-level quantum system with diamond-graph structure. In this case, the transition amplitude toward the final state is A₄ = t₄₂t₂₁ + t₄₃t₃₁, an exact algebraic identity encoding constructive interference, exact destructive interference (dark state formation), and partial interference. The first-order Born approximation predicts identically zero amplitude in all regimes, thereby failing quantitatively in 100% of the cases. The Born–SON framework additionally provides the exact full resolvent, the exact T-matrix, explicit error control in the quasi-nilpotent regime, and a scalar structural metric — the Born–SON depth — quantifying the intrinsic complexity of an acyclic quantum system. The algebraic core of the framework has been formally verified in Lean 4 with Mathlib (https://doi.org/10.5281/zenodo.20113599). MSC 2020: Primary 81U05. Secondary 47A10, 05C20, 81Q10, 15A16.
We identify a class of finite quantum systems, namely, acyclic systems whose transition graph is a directed acyclic graph (DAG), for which the Born series collapses into an exact algebraic identity with finitely many terms and strictly zero truncation error. The sufficient condition is the nilpotency of the transfer operator T = G_0(E)V. If T^{m+1} = 0, then the exact solution of the Lippmann-Schwinger equation is the finite sum |psi> = sum_{k=0}^{m} T^k |phi>, with no condition on ||T||. We prove that the acyclicity of the transition graph implies the nilpotency of T (Theorem 19), and that the nilpotency index coincides with the maximal path length of the graph (Proposition 21). The main result (Theorem 23) concerns the four-level quantum system with diamond-graph structure. In this case, the transition amplitude toward the final state is A_4 = t_{42}t_{21} + t_{43}t_{31}, an exact algebraic identity encoding constructive interference, exact destructive interference (dark state formation), and partial interference. The first-order Born approximation predicts identically zero amplitude in all regimes, thereby failing quantitatively in 100% of the cases. The Born-SON framework additionally provides the exact full resolvent, the exact T-matrix, explicit error control in the quasi-nilpotent regime, and a scalar structural metric, the Born-SON depth, quantifying the intrinsic complexity of an acyclic quantum system.
We study hypergeometric functions of nilpotent operators in finite-dimensional settings, motivated by the algebraic structure of exceptional points in non-Hermitian quantum mechanics. Our starting point is the following exact result: if N is a nilpotent operator of index m+1 in an associative algebra over C, then every generalized hypergeometric function pFq evaluated at N reduces to a finite polynomial in N of degree at most m, without any analytic convergence requirement. This "functional collapse" is distinct from the classical parameter-termination mechanism and arises purely from the nilpotent structure of the argument. The main result is a "nilpotent depth criterion" (Theorem 2): if the first non-constant coefficient of a formal series F appears in degree r >= 1, then the nilpotent part F(N) - F(0)I has nilpotency index bounded above by ceil((m+1)/r). We apply this criterion to Hamiltonians at exceptional points, where H = lambda I + N with N^{m+1} = 0. Theorem 3 establishes that a function F analytic at lambda reduces the Jordan depth of the exceptional point from m+1 to at most ceil((m+1)/r), where r is the contact order of F at lambda. As consequences: the time evolution operator e^{tH} preserves the full Jordan depth for all t != 0; a function with a zero of order m+1 at lambda annihilates the entire Jordan structure; and the order of the pole of the modified resolvent is reduced from m+1 to at most m+1-r. Results are illustrated with explicit 3x3 Jordan block computations for 1F1, 2F1, and the time evolution operator, confirming sharpness of the bounds.
Recursive formulations in theoretical physics often generate very large or formally infinite families of alternative histories, even when the internal state space controlling their generation is finite. This work develops a sector-resolved resummation framework based on propagation, interaction, closed-path monodromy or holonomy, and recursive closure (PIHR). Propagation, interaction, and holonomy are treated as primitive dynamical data, whereas recursion is treated as a closure operation acting on a finite return kernel. For systems admitting a unique first-return factorization, an exact resummation theorem is established: all repeated histories are generated by powers of one return operator and summed by its resolvent whenever the history series converges. The framework is mapped onto Dyson resummation, Lippmann–Schwinger scattering, the Bethe–Salpeter equation, Foldy–Lax multiple scattering, Fabry–Pérot interference, double-barrier tunneling, and Aharonov–Bohm transport. A three-arm flux-threaded quantum network then provides a quantitative benchmark. With complete junction mixing, the number of arm-resolved histories grows exponentially with recursive depth, while the exact coherent amplitude remains encoded in a fixed three-dimensional return operator. Through eight internal returns the cumulative path count already exceeds one hundred forty-five million, although the all-orders solution still requires only a three-dimensional resolvent. Using standard linear-systems realization theory, we additionally give a controllability-and-observability criterion under which this three-dimensional representation is minimal among linear realizations of the recurrence-generating function. The resolvent identity and the individual physical models are established results; the proposed contribution is the PIHR factorization, its cross-domain dictionary, and the quantitative separation of path-growth complexity from finite realization dimension.
Papers I and II derived gravitational geometry and second-order dynamics from finite accessibility without invoking measurement, observation protocols, or collapse as primitive notions. This paper extends the operational program to the measurement sector, deriving the structure of record formation and probabilistic weights from the same finite accessibility and compositional constraints. Finite accessibility restricts admissible operations on a subsystem to a proper operator subalgebra. Because bounded certification excludes operators requiring nonlocal coordination or arbitrarily fine phase resolution, the accessible algebra is strictly smaller than the full operator algebra. When a system becomes correlated with orthogonal environment sectors, off-diagonal operators between those sectors fall outside the accessible algebra. The reduced state is therefore indistinguishable, under all admissible operations, from its diagonal ensemble. Definiteness emerges as an algebraic consequence of sector restriction, not as a collapse postulate. The weights of these sectors are fixed uniquely by compositional consistency. Phase invariance, multiplicativity under independent refinement, and stability under record coarse-graining constrain any admissible assignment of sector weights to satisfy a functional equation whose only continuous, non-contextual solution is quadratic in amplitude. The Born rule therefore arises as the unique invariant measure compatible with finite accessibility. Irreversibility follows from action scaling: reversal requires operators outside the admissible tensor algebra and incurs action cost proportional to the number of correlated subsystems. Since admissible operations are bounded by the action gap, macroscopic record reversal is excluded. Measurement introduces no new physical primitives beyond finite accessibility, locality, and compositional stability. Record formation, classical definiteness, and Born probabilities are structural consequences of algebraic restriction under finite accessibility. **This paper is part of the Finite Accessibility series, which reconstructs gravitational geometry, quantum dynamics, gauge structure, cosmological perturbation theory, and the dark sector from a single operational primitive: a finite certification ceiling ω* governing the maximum rate at which physical relations can be maintained across extended domains.** **Introductory papers** - Geometric Limits to Coherent Timekeeping in Extended Reference Frames — https://doi.org/10.5281/zenodo.18824556 - Accessibility Selection Law (ASL) — https://doi.org/10.5281/zenodo.18824649 - Phase Boundary Gravity — https://doi.org/10.5281/zenodo.18824707 **Pillars (numbered I–IV)** - Paper I: Deriving Gravitational Geometry and Schrödinger Dynamics from Finite Accessibility — https://doi.org/10.5281/zenodo.18601600 - Paper II: Deriving Inertia, Motion, and Wave Propagation from Finite Accessibility — https://doi.org/10.5281/zenodo.18625174 - Paper III: Accessible State Spaces, Record Stability, and the Emergence of Quantum Statistics — https://doi.org/10.5281/zenodo.18625192 - Paper IV: Frame Dissolution, the Certification Ceiling, and Observable Deformations — https://doi.org/10.5281/zenodo.18654458 **Extension papers** - Dark Matter from Finite Accessibility — https://doi.org/10.5281/zenodo.18654433 - Dark Energy from Finite Accessibility — https://doi.org/10.5281/zenodo.18654523 - No-Go Theorems and the Admissibility Corridor — https://doi.org/10.5281/zenodo.18654493 - Gauge Structure from Finite Accessibility — https://doi.org/10.5281/zenodo.18928152 - Perturbation Theory from Finite Accessibility — https://doi.org/10.5281/zenodo.18928213 **Experimental paper** - Experimental Probes of Finite Accessibility: GW Dispersion and Vacuum Path Independence — https://doi.org/10.5281/zenodo.18931126 **Keywords:** Born rule derivation, measurement problem, quantum statistics, finite accessibility, operator subalgebra, record formation, definiteness emergence, wave function collapse, compositional consistency, accessible state spaces, action gap, irreversibility, sector restriction, quantum foundations, algebraic quantum mechanics, decoherence
A finitist, relational axiomatization of physics: quantum theory and a relativity sector derived from four axioms about group-valued relations and closure. Contributions to the foundations of quantum mechanics (Born rule derivation, quantum reconstruction, local tomography, complex field selection), the emergence of time and causal structure in discrete networks, discrete Noether theorems, lattice gauge structure, and the Schrödinger equation derived by two independent routes. Every claim is graded in a Claim Ledger; every numerical claim is machine-verified with archived scripts; negative results are reported at full strength. What this paper does The universe is taken to be constituted by relations rather than objects. Stable relations form closures (self-consistent cycles) whose holonomies carry the physical content. Four axioms — primacy of relations (group-valued, edge-borne), closure stability, chirality from order, and observation as bidirectional group exchange — yield, at explicitly graded and conditional levels: Quantum sector Exchange-completeness theorems for finite label groups: abelian at all cycle ranks; non-abelian on the single cycle (Wilson readouts plus class-sum convolutions generate the full matrix algebra); the multi-cycle non-abelian case on the operational-equivalence quotient, with the obstruction completely characterized via class-preserving outer automorphisms — and the quotient's necessity now stratified by the source of word blindness (see New in Version 24.0). The Born rule, proved for finite abelian networks from a strictly weaker relational readout premise (statistics of realizable effects depend only on relational content); non-contextuality has this single source; additivity is basis-relative and declared as such. The Schrödinger equation by two routes of unequal premise grade, maintained side by side: a spectral route — the action operator is a weighted Cayley-graph Laplacian in the Pontryagin dual basis, the derived Gibbs state's unique canonical (modular) flow yields the Schrödinger equation with Hamiltonian equal to the action operator, no continuum limit, zero conjectures consumed, the single non-theorem step one flagged identification; and a Madelung route — physical and mechanism-rich (probability density derived from constraint gaps), valid exactly on the domain of the pseudo-continuity conjecture, with its full premise bill stated (continuity equation, Wallstrom circulation condition, m, V). A dedicated comparison subsection gives the line-by-line premise accounting and the machine-measured agreement region. Complex field selection: ℂ selected over ℝ and ℍ within the GPT comparison class in the covered sectors, with a second independent route from chirality (Frobenius–Schur dichotomy, quaternionic double exclusion); "uniquely selected" expressly not claimed. Positivity as pairing: all five in-text positive-semidefiniteness theorems traced to one mechanism (M = B†B with native factor); the Kraus form and complete positivity derived rather than assumed. The Noether sector on Noether's two theorems exactly: Noether I recovered on the holonomy action (compact Lie realization); Noether II identified as present (Bianchi as horn-filling, Schläfli as its Regge form); selection rules recovered for finite groups; the GNS bridge proved non-vacuously on the full exchange algebra; the sole potential extension — conservation as the kernel of exchange — isolated as a registered conjecture, consumed nowhere. A local–global representation theorem: six generative quantities (closure invariant, flux quantization, Weyl phase, non-abelian Stokes, orientation visibility, terminal normalization) are evaluations of cohomology classes against closed chains — one syntax, six exact machine probes. The gatekeeper no-go theorems registered and discharged: Kochen–Specker (the framework posits no global valuation — outcomes are resolutions of constraint gaps, and basis-relative additivity blocks state-independent variants), Pusey–Barrett–Rudolph (the statistical envelope is over resolutions yet to happen, not values already held; machine witness: identical configuration ensembles with opposite relative phase have opposite statistics), Wallstrom (registered as an explicit input, structurally discharged in the finite-abelian sector), Hegerfeldt (the Lieb–Robinson exponential tail identified as the only claimable front form for the continuum limit), Frauchiger–Renner (cross-agent certainty transfer presupposes kernel-free record transport, exactly what exchange-incompleteness excludes), and no-cloning (inherited in the covered sectors); Conway–Kochen and Colbeck–Renner noted as concordant. Relativity sector Time from finite capacity plus genuine randomness: the staging theorem (Θ(d) dependency depth, unbounded chains via the Moore bound; the quantitative failure law conditionalized on measure-level independence with exact minimal witnesses); the temporal quantum defined combinatorially, no Hamiltonian input. The speed of light grounded in an exact discrete dependency cone (combinatorial, no quantum input), the Lieb–Robinson bound consumed at cross-check grade only; a boost no-go theorem reducing Lorentz invariance to named statistical conditions, with the Collins fine-tuning objection recorded as a gate. The trace-descent theorem group: the growth-history measure forced onto the Mazurkiewicz trace quotient; chain reflection positivity at every depth with completely explicit spectrum. An equation sector of nine verification families (Schrödinger, Dirac, Maxwell/U(1), canonical commutator, Yang–Mills SU(2), Regge calculus, Klein–Gordon, Noether loop, Cartan curvature), where every machine-precision claim is the shadow of a proved backbone lemma — including the factor ½ of H = ½L + V, identified as the Wilson Fourier coefficient fixed by the action theorem. Four experimentally testable predictions, including a discriminating flat-versus-sigmoid visibility protocol. The Standard Model gauge group, ħ, and G remain open, with attack routes registered. How to read Three entry paths are marked in the document. Path A (derivational core): axioms → group structure → Noether sector → least action → Born rule → positivity. Path B (physical mechanisms): constraint propagation → time as determination → wave function as gap envelope → fields. Path C (scope and limits): the Claim Ledger, then Discussion and Open Problems — for the reader who wants to know what is claimed and what is not before reading anything else. The Claim Ledger is the single authoritative register of claim status; wherever body text sounds stronger, the Ledger governs. New in Version 24.0 (Version 23.0 is the immediately preceding public release; this version supersedes the internal V12–V15 and V17–V19 series, whose numbers are not used externally.) Version 24.0 resolves the structure of the multi-cycle non-abelian obstruction — the sector's longest-standing open front — following an external research exchange with three rounds of independent re-verification (fourteen core figures, three commutant instruments implemented independently, exact integer arithmetic on the obstruction side) and two rounds of counter-review that corrected one quantifier on each side: A dichotomy of the multi-cycle obstruction (new remark). Word blindness on multi-cycle networks has two identified sources, jointly covering every tested case, with opposite behaviour as cycle rank grows. Type I (global; graded): a class-preserving outer automorphism yields a commutant element by an argument free of cycle-rank reference; its non-triviality holds at theorem grade for b₁ ≥ d(G) via generating tuples, is monotone in b₁, and the threshold is measured at b_σ = 2 on the tested Wall group (at b₁ = 1 the permutation is the identity — no universal claim is made, and the existence of groups with threshold above 2 is registered open). Type II (local; measured): Out_c trivial but fusion fails on a proper subgroup, the blind pairs sitting in non-generating sectors — the same mechanism that defeats Wilson generation; this blindness heals: on every verified group (A₄, SL(2,3), A₄×ℤ₂) the full commutant collapses to trivial at b₁ = 3 by exact computation on two independently implemented instruments, although word readouts remain blind — the shift primitives separate, above a threshold, and the natural explanation (tuples becoming generating) is refuted by the persistence of blind blocks. Three boundaries at full strength: no exhaustiveness proof (a third source is not excluded); no general healing theorem (the verified groups all heal at 3, cause unidentified); the verified Type II sample is narrow. The quotient proposition restated in stratified form. The operational-equivalence quotient's necessity is now part of the proposition's own statement, in three strata: necessary where Out_c ≠ 1 at every b₁ ≥ b_σ(G), with b_σ ≤ d(G) at theorem grade; one-dimensional at b₁ = 2 on the fusion-type groups (verified cases); trivial at b₁ ≥ 3 on every verified fusion-type group and at every tested b₁ for word-separation-complete groups (a fourteen-group scan of orders ≤ 31 found no third source). Pre-registered validation preceded the restatement: a stratification-exhaustiveness scan (pass), a new-witness search (no new witness in range, downgrade clause applied as pre-registered), and a zero-conflict audit of the three strata against all archived data. The residual open problem is now precisely located. What remains of multi-cycle completeness: the rare Out_c ≠ 1 groups (54 in the exhaustive range through order 96) above their measured thresholds, the low-b₁ fusion stratum, a general healing theorem, and whether a threshold above 2 occurs. The multiplicity-freeness conjecture's mechanism sentence is corrected with the two regimes stated separately (at b₁ = 2 the shifts do not separate — the blind-pair transposition commutes with every primitive, integer-exact — and reduce the commutant to
Why do coherent causal alternatives contribute to a sum over histories, and why are detector outcomes described by quadratic measures? This article addresses these questions separately within Recursive Causal Closure (RCC). On finite directed acyclic networks, recorded causal restitution, additive admission and a specified linear local-response rule give an exact nilpotent resolvent. Its finite expansion enumerates all admissible directed routes, without separately postulating a sum over paths. Exact elimination of internal events preserves the boundary response and is associative under successive coarse-grainings. When a persistent phase is supplied by the companion temporal analysis, each coherent route acquires an action-dependent complex factor. A separate positive bilateral detector comparison yields a strongly positive quantum measure with no irreducible third-order interference. As a continuum benchmark, the real causal-set chain construction reproduces the massive retarded Klein–Gordon kernel in 1+1 dimensions in expectation, with a controlled variance. A universal continuum path measure, the microscopic quantum generator, Born normalisation and the identification of the action scale with Planck's constant are not yet derived.