We develop the quantum layer of the finite relational algebraic substrate: observation as the interaction of a Subject and an Object, phase-coherent cyclic subsystems of a common finite Carrier shell. Four identifications carry the construction, each a derivation rather than an interpretation. The wave function is phase: pure states are windings of the Object's cycle, selected by stationarity under the drive, with \( E=hf \) an identity. The probability amplitude is Object-to-Subject projection onto a forced orthogonal basis, so that completeness, the selection rules, and the Lüders update are finite character theory rather than postulates. Probability is a coincidence count in the Subject's ledger, the cyclotomic ring \( \mathbb Z[i] \) of the shared quarter-turn core. This derives the Born rule within the coherence horizon, with the quarter-turn core supplying the canonical complex sector of the amplitudes. Composition is conservation: the single drive conserves the relative offset of a pair, synchronised orbits are the Bell states, and the framework saturates the Tsirelson bound exactly, \( S=2(\zeta_8+\zeta_8^{-1})=2\sqrt2 \), while reproducing the network behaviour of the real-versus-complex discrimination. The same conservation makes gravity an entangling channel. Its distinctive parameter-free signature is equal force but unequal entanglement: a source gravitates by its full mass yet entangles only through its coherent fraction, and entanglement forms at the Newtonian rate under the complementarity \( V^2+C^2=1 \). Every claim is https://github.com/gamayos/frc-numerics/tree/main/22-quantum, verified by exact finite arithmetic in three worked configurations, and the experimental confrontation closes in a single falsifiability ledger on one committed carrier scale.
We develop the quantum layer of the finite relational algebraic substrate: observation as the interaction of a Subject and an Object, phase-coherent cyclic subsystems of a common finite Carrier shell. Four identifications carry the construction, each a derivation rather than an interpretation. The wave function is phase: pure states are windings of the Object's cycle, selected by stationarity under the drive, with ( E=hf ) an identity. The probability amplitude is Object-to-Subject projection onto a forced orthogonal basis, so that completeness, the selection rules, and the Lüders update are finite character theory rather than postulates. Probability is a coincidence count in the Subject's ledger, the cyclotomic ring ( \mathbb Z[i] ) of the shared quarter-turn core. This derives the Born rule within the coherence horizon, with the quarter-turn core supplying the canonical complex sector of the amplitudes. Composition is conservation: the single drive conserves the relative offset of a pair, synchronised orbits are the Bell states, and the framework saturates the Tsirelson bound exactly, ( S=2(\zeta_8+\zeta_8^{-1})=2\sqrt2 ), while reproducing the network behaviour of the real-versus-complex discrimination. The same conservation makes gravity an entangling channel. Its distinctive parameter-free signature is equal force but unequal entanglement: a source gravitates by its full mass yet entangles only through its coherent fraction, and entanglement forms at the Newtonian rate under the complementarity ( V^2+C^2=1 ). Every claim is https://github.com/gamayos/frc-numerics/tree/main/22-quantum, verified by exact finite arithmetic in three worked configurations, and the experimental confrontation closes in a single falsifiability ledger on one committed carrier scale.
We develop the quantum layer of the finite ring cosmology framework: observation as the interaction of a Subject and an Object, phase-coherent cyclic subsystems of a common finite Carrier shell. Four identifications carry the construction, each a derivation rather than an interpretation. The wave function is phase: pure states are windings of the Object's cycle, selected by stationarity under the global drive, with E=hf an identity. The probability amplitude is Object-to-Subject projection onto a forced orthogonal basis, so that completeness, the selection rules, and the Lüders update are finite character theory rather than postulates. Probability is a coincidence count in the Subject's ledger - the group ring Z[i] of the shared quarter-turn core - which derives the Born rule below the coherence horizon and forces complex amplitudes. Composition is conservation: the single drive conserves the relative offset of a pair, synchronised orbits are the Bell states, and the framework saturates the Tsirelson bound exactly, S=2(ζ8+ζ8−1)=22, while reproducing the network behaviour of the real-versus-complex discrimination. The same conservation makes gravity an entangling channel whose distinctive parameter-free signature is equal force but unequal entanglement - a source gravitates by its full mass yet entangles only through its coherent fraction --- with entanglement forming at the Newtonian rate under the complementarity V2+C2=1. Every claim is verified by exact arithmetic in three worked configurations (eleven validation suites, eighty-nine checks), and the experimental confrontation closes in a single falsifiability ledger on one committed carrier scale.
Can a Born-type probability structure arise on a substrate that is only a finite stochastic process? A stochastic program built from six primitive operations supplies the final matrix; a specified layered architecture constructs, from its outputs, a record layer, a two-dimensional real carrier and a quadratic law, and supports an adopted Born calculus. The six operations alone do not give this structure, and the square-root encoding and the projection are construction choices. Our main contribution is the architecture: a directed acyclic graph whose arrows are each marked as derived, chosen, adopted or standard mathematics, with the kind of check behind each. On registered populations, a snapshot predictor whose rank-two case is the law predicts the substrate's sampled label frequencies with lower aggregate loss than named competitors, and, in the construction's canonical basis, its cross terms carry its advantage over the diagonal rule. In each of 2,122 census cases the encoded state is not a function of the anonymous initial record description: two point preparations sharing all three labels differ by at least 1/25 in the trace probability of raw label 0, a statement about exact data, not operational access. This bound is retrospective: the registered bound 3/20 fails in 117 cases of the full census. On the original worked case the updates by the final matrix and its square do not lift to the carrier. The measurement rules are adopted: Lüders tests, all densities and effects, and additivity; on them the trace rule is unique, and finite ontic models with additive responses, and measurable preparation-noncontextual representations preserving the specified mixtures, are excluded. We do not claim that the substrate produces the measurement rule, that the layer forms in the run, or a derivation of quantum theory. The source code of the stochastic program, the constructions, the reproduction pipeline, the verification commands and the Lean 4 proofs are available at https://github.com/ioannist/six-birds-born (release v1.0). AI systems were used throughout this work, as the manuscript describes; the author directed the research, made the scientific decisions and is responsible for the content.
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
The Phase Relational Field Framework (PRFF) presents a unified process ontology intended to connect quantum measurement, gravitational source accounting, and cosmological response as scale-dependent manifestations of a single, continuously maintained process. It does not replace the predictive formalisms of quantum mechanics or general relativity. In a checked constitutive specialization, the interaction-supporting carrier is represented as a continuous, lossless distributed LC medium with phase speed 1/√(L₀C₀) and characteristic phase impedance Z₀ = √(L₀/C₀). The familiar value Z₀ ≈ 377 Ω follows only when L₀ and C₀ are identified with the measured electromagnetic-vacuum response; Z₀ alone does not constitute an absolute load or capacity bound. The central ontological proposition is that a localized entity is maintained through a reaction surface and that the same maintaining process continues beyond its localization boundary as an Omnidirectional Spherical Process-Load Domain (OSPLD). Its Gauss-surface process-load invariant is unsigned and additive. Microscopic quantum interactions may return signed, phase-, polarity-, and orientation-sensitive relational projections, whereas gravitational source accounting retains the unsigned cumulative magnitude. The signed-interaction/unsigned-gravity distinction therefore describes two coupling channels of one continuous process rather than separate quantum and gravitational substrates. The dynamical carrier-to-metric map remains to be derived; Einsteinian curvature, geodesics, and geodesic deviation are retained as macroscopic representation targets. The framework identifies three linked research programmes: Quantum Statistics and Entanglement: PRFF proposes that topological repartition does not erase the relational phase organization of the parent process. Bell-type correlations may consequently arise from contextual, spatial-geometric sampling of an extended joint OSPLD rather than from action at a distance. A complete realization must derive a non-factorizable joint-response kernel and recover the Born rule, singlet correlations, the maximal CHSH value, and no-signalling under a setting-independent sampling measure. Horizon Dynamics and Coalescence: Event horizons are treated as candidate null reaction-surface limits at which local admission, phase relation, and carrier loading become extremal. Bilateral impedance matching and the condition ΓH = 0 provide a conditional, mode-resolved description of reflectionless horizon coupling, while the symmetric Z₀/2 + Z₀/2 partition remains a constitutive ansatz. Binary coalescence is interpreted as a reciprocal phase-commutation transition between complete OSPLDs. The resulting carrier-stress transient is proposed as the source-level process whose transverse-traceless projection is represented macroscopically by the + and × gravitational-wave polarizations. Cosmological Background and the Hubble Tension: Under an Olbers-type shell accumulation model, an unsigned common-mode matter contribution may diverge with increasing sampling radius unless a separate nonlinear carrier-saturation law intervenes. Its large-scale asymptote could be represented gravitationally as an effective cosmological term rather than as a fundamental local repulsive force. Because cosmic dilution changes the cumulative load distribution, the associated saturation radius and effective cosmological response may drift with the scale factor. This supplies a concrete ontological mechanism to investigate as a possible contribution to the Hubble tension, rather than claiming its resolution in advance. By separating derived relations, constitutive interpretations, working hypotheses, and theorem targets, PRFF establishes an explicit and falsifiable research architecture. Its principal open requirements are the derivation of a covariant carrier and boundary action, the carrier-to-metric map, the nonlinear saturation law, the joint quantum-response kernel, and quantitative confrontation with gravitational-wave and cosmological observations.
How a finite real cycle acquires a complex structure and a rule for composition Quantum theory uses a common imaginary unit when individual systems become a composite. This framework constructs that compatibility from finite real source histories. On the complete contrast sector of a three-cycle, the forward and backward comparisons give J=3P−P2,J2=−I The complex structure appears after the real comparison is specified. At a qualified unmarked boundary, opposite reference shifts select QAB=2I−JA⊗JB The range is the balanced complex tensor product. Its many-source composition is associative. A finite controlled contact supplies a nonfactorizable complex-linear operation, and exact residual bounds quantify departure from the selected sector. A source model that produces the contact A reversible controller compares operation orders on 24 proper signed frames. Its ordered, arrow-insensitive axis detector produces the retained cyclic response. The contact algebra gives [[G,G],[G,G]][G,G]≅C3. The construction prints the controller, scratch cleanup, full contrast probes, admissible operations and refusal states. A triaxial real response tensor realizes the detector and provides an explicit map between material contrasts and event-count contrasts. A record must survive the test of future operations A present tensor can hide a material marker that a later operation reveals. The framework derives independent force-gradient calibration and a delayed odd-field release. With a nonzero release field, energy and vector torque reconstruct the entire marker change: DU=−Δp⋅E,Dτ=Δp×E.Δp=∣E∣2E×Dτ−DUE The test includes shared calibration errors, stray fields, finite release pulses, marker relaxation and a complete operation inventory. It distinguishes detection of a retained record from agreement with its frozen prediction and from discrimination between physical theories. From proved structure to existing experiments The paper adds a conditional probability-correspondence theorem for the balanced composite, examines independent-source quantum-network experiments, and reproduces all eight measured rows of a public torque dataset. These supply composition benchmarks and demonstrated apparatus capabilities. Their published significances are kept attached to the hypotheses they actually test. The calibrated material release law is also a conventional electrostatic prediction; it is not presented as a QTT-exclusive signal. One material coefficient, two readouts A new archival calculation transfers a published same-particle libration calibration to a separate precession measurement. In the qualified rotor regime, the cross-readout relation is Ξ=ωlib2EpωsΩElcosϕl=alibaprec. An unchanged dipole-to-inertia coefficient gives unity. The calculation yields a slope of 2.909 from the libration calibration, compared with the published precession slope of 2.950, a central difference of 1.4135%. This shared electrostatic consistency relation eliminates the separate dipole and inertia; unequal field gains, dynamical corrections and shared uncertainty remain explicit. Reversal and long-term dipole-stability observations identify complementary laboratory capabilities. The complete matched-tensor delayed-release test remains an experimental target. When do two descriptions differ in any future experiment? The finite-reference extension follows a controlled source contact through shared references, fresh references, inverse operations, intermediate measurements and feedback. Its complete order-three response is gn=∑k=02pkζkn,ζ=e2πi/3,g3=1,g2=g1. A finite observable-span algorithm tests two fully specified models across every permitted future word. It either certifies agreement or returns the preparation, operation sequence, terminal outcome and exact probability difference. In the populated compatible-instrument benchmark, the complete quantum representation gives the same records without an additional target fit. A source-selected physical coupling or preparation remains a distinct route to predictive value. The framework includes nine vector figures, the retained source proofs and countermodels, complete reference-history benchmarks, calibration and vector-reconstruction theorems, runnable exact and numerical checks, published-input receipts, a data assessment and an experimental packet specification. The introduction defines the QTT terms needed to follow the construction without prior familiarity with the corpus. The probability-interface theorem, finite-memory bounds and covariance-weighted witness specify which further laboratory comparisons can distinguish physical models. Source anchors QTT Main Book, especially pp. 56, 248-250, 352 and 827-829; Artian's Completed-Event Four-Capacity Theorem; Frame-cycle derivation map, completed-event lexicon, and Artian's Universe and laboratory readout. Stable citation: 10.5281/zenodo.22882084.
This paper develops finite temporal observability in two parts, with a third in a companion[1]. Part I isolates the structure of a single finite-observability conjugate pair (u, v) andpushes it as far as it goes with no specific physical identification assumed. A finite observerresolves real registers whose entire content is a dimensionless ledger of distinctions, capped atλmax and carrying a total λ+ and a split λ−; positing the symplectic pairing dresses these asthe cell area K = δu δv and the slope Luv = δv/δu. A single shared distinguishability ledgerthen makes the reciprocal cell a consequence — split-invariance forces exponential resolutionmaps, the cell boundary is the full-budget locus, and the saturated cell carries the compatiblecomplex structure supplying the bracket’s imaginary unit. From this alone follow noncommutativityat scale K, the complex unit, an implied wave with a Born rule read as a Fubini–Studytransition probability, a causal bound with one unconditional theorem (the causal slope equivalentto non-negative mass-squared), a reversible generator, the forced first-order (Clifford) form,the non-fixability of the scale, and composite objects as differentiated cell bundles. The singleresidue of the wave construction — the Wallstrom single-valuedness condition — is reduced tothe cell’s prequantum integrality, leaving the integral period 2πK as the one named quantuminput rather than a free-standing posit. Part II supplies the physical reading — the conjugatepairexploration. Validated on the regular pairs the wave is Schr¨odinger’s, the bound is therelativistic mass-shell with L = c, and the first-order form is Dirac’s; extended to exotic pairs(horizon angle and area, holonomy and flux, modular flow and entangling area, the cosmologicalconstant and four-volume) a single per-pair criterion — whether the exchange rate Luv vanishes— sorts discrete from continuous spectra, consistent with the careful literature each time; and asurvey of laboratory prospects finds the frontier mapped but no forced, allowed, novel signature.The architectural extensions to gauge fields and gravity — fermionic statistics from stability,the gauge coupling from comparing cells, and the metric with the Lovelock field equations —are developed in the companion [1]. Every claim is graded — theorem, conditional construction,or candidate route. The whole is offered as a constraint grammar of possible universes, not aderivation of ours.