Paper XXIV of the Six-Throat Directed Geometry program extends the fixed-particle-number multipartite quantum theory of the preceding paper to a finite-mode quantum-field architecture. The construction begins from the previously established directed many-particle Hilbert sectors and completes them through a direct sum over particle number, producing a field space in which particle number is spectrally resolved while the directed branch and permutation-statistics labels remain distinct. Normalized insertion and removal between adjacent particle-number sectors generate the symmetric and antisymmetric ladder structures. These operators are then lifted to quantum-field operators on the antiperiodic global modes inherited from the closed Six-Throat spacetime. The resulting finite-mode field algebra is controlled by a reproducing kernel, while the zero-particle sector remains a nonzero Hilbert state. Retaining the directed structure produces two orthogonal zero-particle states that are exchanged under branch reversal. Free field evolution is represented by a unitary geometric propagator preserving antiperiodicity, particle number, and the equal-time field algebra. The first interacting layer is constructed from a self-adjoint local number-changing vertex. It couples adjacent particle-number sectors, produces an exact continuity law in number space, and preserves total Hilbert probability while permitting particle number to vary. Multiple local vertices combine coherently into higher-order interaction networks. At each interaction order, the ordered amplitudes retain an exact directed branch dependence. Starting from the zero-particle sector, branch reversal changes the phase structure of odd particle-number sectors while leaving the corresponding particle-number probabilities equal between the two directed branches. The accompanying reproducibility package verifies 187 declared conditions across eight derivational blocks. The scope established here is the finite-mode directed quantum-field architecture: Fock-space composition, ladder algebra, global field modes, the directed zero-particle doublet, free propagation, particle-number-changing interactions, coherent multi-vertex interference, and higher-order directed interaction networks. Continuum completion, general multi-field species coupling, full-spacetime interaction-density completion, and renormalized continuum observables are outside the scope of the present paper.
Paper XXIV of the Six-Throat Directed Geometry program extends the fixed-particle-number multipartite quantum theory of the preceding paper to a finite-mode quantum-field architecture. The construction begins from the previously established directed many-particle Hilbert sectors and completes them through a direct sum over particle number, producing a field space in which particle number is spectrally resolved while the directed branch and permutation-statistics labels remain distinct. Normalized insertion and removal between adjacent particle-number sectors generate the symmetric and antisymmetric ladder structures. These operators are then lifted to quantum-field operators on the antiperiodic global modes inherited from the closed Six-Throat spacetime. The resulting finite-mode field algebra is controlled by a reproducing kernel, while the zero-particle sector remains a nonzero Hilbert state. Retaining the directed structure produces two orthogonal zero-particle states that are exchanged under branch reversal. Free field evolution is represented by a unitary geometric propagator preserving antiperiodicity, particle number, and the equal-time field algebra. The first interacting layer is constructed from a self-adjoint local number-changing vertex. It couples adjacent particle-number sectors, produces an exact continuity law in number space, and preserves total Hilbert probability while permitting particle number to vary. Multiple local vertices combine coherently into higher-order interaction networks. At each interaction order, the ordered amplitudes retain an exact directed branch dependence. Starting from the zero-particle sector, branch reversal changes the phase structure of odd particle-number sectors while leaving the corresponding particle-number probabilities equal between the two directed branches. The accompanying reproducibility package verifies 187 declared conditions across eight derivational blocks. The scope established here is the finite-mode directed quantum-field architecture: Fock-space composition, ladder algebra, global field modes, the directed zero-particle doublet, free propagation, particle-number-changing interactions, coherent multi-vertex interference, and higher-order directed interaction networks. Continuum completion, general multi-field species coupling, full-spacetime interaction-density completion, and renormalized continuum observables are outside the scope of the present paper.
Paper XXIII of the Six-Throat Directed Geometry program extends the one-particle quantum structure developed in the preceding paper to composite and multipartite systems. Starting from the conserved positive Hilbert product and unitary one-particle dynamics of the directed spacetime, the construction derives the composite Hilbert tensor product, Schmidt decomposition, reduced density operators, local expectation values, purity, entanglement entropy, correlation tensors, normalized joint probabilities, and the CHSH structure. For identical subsystems, the swap operator produces orthogonal symmetric and antisymmetric permutation sectors together with their dimensions, invariant dynamics, exclusion rule, and occupation counting. The central directed result is that the exchange condition retains two directed branches rather than collapsing them into a single undirected sign. Directed branch and permutation parity therefore remain distinct structural labels throughout the multipartite construction. The resulting two-particle and N-particle laws establish a consistent directed permutation representation. Reversal of the directed branch changes every odd directed exchange while leaving even permutations, branch-blind reduced states, permutation sectors, exclusion, and occupation counting unchanged. The complete numerical closure audit contains 74 checks across nine derivational blocks, with all 74 conditions satisfied. Representative numerical residuals range from exact zero to approximately machine precision. The resulting fixed-particle-number theory contains tensor composition, entanglement, correlations, identical-particle statistics, dual directed exchange branches, and a consistent multipartite permutation structure. Field-level Fock construction, creation and annihilation operators, variable particle number, vacuum structure, and interacting quantum fields are intentionally left to the subsequent paper.
Paper XXII of the Six-Throat Directed Geometry program derives the mathematical structure of one-particle quantum mechanics from the directed spinor dynamics established on the regular closed six-throat spacetime. Starting from the conserved curved-spacetime Dirac current, the construction produces a positive and slice-independent inner product and its Hilbert-space completion. The corresponding geometric propagator is unitary and satisfies composition, inverse, and adjoint closure. Real physical observables are represented by self-adjoint operators with real spectra and orthogonal spectral resolutions. The compact global geometry yields an exact periodic commutator algebra, while the standard canonical position-momentum relation is recovered on regular local charts. The Robertson and Schrödinger-Robertson uncertainty relations then follow from the same Hilbert-space structure. The tetrad Clifford algebra generates the intrinsic spin-one-half representation, including the expected spin magnitude, the spinor sign reversal under a full two-pi rotation, and the rotational multiplets obtained by coupling orbital and intrinsic angular momentum. Detector-surface current is promoted from the direct Born-flux construction of Paper XXI to a general positive measurement structure described by a non-projective POVM and the trace probability rule. Density operators, purity, von Neumann entropy, conditional access states, and Kraus dephasing channels arise within the same measurement framework. The positive-energy Dirac branch reduces at low energy to Pauli dynamics and, when the spin-covariant terms commute appropriately, to the Schrödinger Hamiltonian. The same unitary framework yields the Liouville-von Neumann equation, the Heisenberg operator equation, and the Ehrenfest relations in the local directed potential. A forty-three-condition numerical closure audit satisfies all 43 declared conditions across the derived one-particle structure. Representative numerical results include Hilbert-product and unitary residuals at approximately machine precision, exact spin-one-half algebra to numerical precision, POVM completeness at machine precision, low-energy wave-packet infidelity of 3.16 × 10⁻¹⁰ for the stated audit configuration, and Schrödinger-Heisenberg expectation agreement at 5.66 × 10⁻¹⁷. The result is a single geometric chain from conserved spacetime flux to one-particle state space, unitary dynamics, observables, commutators, uncertainty, spin, measurement operators, statistical states, and the low-energy quantum equations. Composite systems, entanglement, exchange symmetry, quantum statistics, and variable-particle-number field theory remain subsequent structural layers and are not claimed as part of the present one-particle closure.
Paper XXI completes the microscopic extension of Six-Throat Directed Geometry by introducing a directed spinor dynamics on the regular six-throat spacetime and carrying that dynamics through the exact periodic closure. The construction separates persistent physical identity, directed state, and observer access, so a state may pass through a direction-neutral condition without annihilation while observational localization remains an access restriction rather than a second global evolution law. The microscopic dynamics is formulated with a curved-space Dirac equation containing a directed scalar mass law. At directional zero, the effective mass returns smoothly to its reference value and the spinor evolution remains regular. The associated Dirac current is positive and conserved, providing an invariant microscopic measure through directed reversal. The closed geometry is equipped with an antiperiodic spin structure over the 15.6-length geometric cycle, while bilinear observables remain periodic after one complete spatial circuit. The global construction yields a regular zero passage, a resolved directed spectral splitting, dynamical conservation of the spinor norm, channel-resolved decoherence, and a geometric detector measure. Observer localization is represented by projection onto the observer-accessible subspace while the global physical state remains intact. For flux-orthogonal detector channels, the conserved current partitions according to the squared channel amplitudes. The resulting normalized detector measure reproduces the standard Born weights as a current-flux theorem rather than as a primitive probability postulate. An explicit six-channel detector-surface calculation gives a maximum Born-weight residual of approximately 1.43 × 10^-14, while the integrated cross-channel surface flux remains below approximately 2.22 × 10^-16. The accompanying reproducibility package includes the global kinetic spectrum, parametric mass spectrum, directed spectral splitting, zero-passage dynamics, dynamic channel flux, detector-surface flux, final consistency audit, and the inherited geometric generators required by the calculation chain. The complete numerical outputs are distributed with the package. The result is a microscopic closure in which existence, direction, zero passage, geometric emergence, coherence, decoherence, observer access, and detector statistics are represented within one regular geometric framework.
One Electron. Two Directional Amplitudes. One Photon Field. Could the missing information be hidden in intensity itself? Can a single electron carry two simultaneous directional amplitudes without dividing into two particles? Can a single photon field preserve directional information that ordinary intensity measurements fail to distinguish? And can this hidden information be transformed into a measurable difference between two detector outputs? This research investigates these questions within Six-Throat Directed Geometry, developing a mathematical connection between directed electronic states, quantum transitions, photon-field coherence, and direction-resolved measurement. The central result establishes that one photon field can carry both unsigned intensity and signed directional information without requiring a second photon species. Through an explicitly constructed two-mode representation, directional coherence that remains inaccessible to an intensity-only measurement can be converted into a difference between two output photon counts using a coherent, number-preserving transformation. The analysis begins with the inherited six-throat spacetime metric and its directed electronic operator, preserving the established geometry and global quantum structure. A reproducible spectral investigation tracks electronic states across four Fourier cutoffs and verifies 28 inherited geometric and numerical identities. For the inherited two-route source profiles, the study obtains an exact restricted overlap of 0.9945940085424316, corresponding to a signed coherence magnitude of 0.1038400605330016. This result supplies a quantitative mathematical foundation for examining directional information within a single photon field. The paper further develops a four-class Z2 × Z2 algebraic structure linking fermionic and bosonic source sectors. Its conditional joint-covariance construction demonstrates how directional information can remain encoded in correlations even when individual measurement averages vanish. A separate quantum-transition analysis establishes precise conditions under which a single electronic state can retain simultaneous directional amplitudes while some transition measurements remain insensitive to their signed difference. The experimental significance is equally important: the research specifies an independently testable optical protocol involving coherent mode preparation, mode matching, interferometric transformation, balanced photon-count measurements, and controlled comparisons. It derives measurable predictions, coherence requirements, discrimination limits, and finite-sample precision conditions. The accompanying reproducibility package reports 13,066 successful mathematical and computational checks, together with source code, numerical data, figures, and file-integrity verification. The result is a theoretical and computational framework for investigating a fundamental possibility: directional physical information may remain concealed in total intensity while becoming accessible through coherent measurement of the same photon field. This work advances the study of directed matter–radiation coupling, quantum observability, and photon-information structure, and provides explicit mathematical predictions and experimental procedures for independent scientific examination.
This study asks which structures can be derived from a single finite relational system to which no prior physical meaning, coordinates, or time are assigned. A selector-free relational lift and its minimal self-sustaining orbit generate an intrinsic separator order and grading, providing a directed order axis without an external clock. Boolean-diamond incidence defines transverse locality, and five-K4 incidence produces an exact S3 minimal persistent complex whose canonical continuation yields a four-dimensional associated realization. An explicit finite witness gives a nonzero commutator between longitudinal incidence and transverse locality, ruling out a simple product of fixed space with independent time. A complementary stable-quotient readout contains a refinement branch that closes as a one-dimensional metric-graph continuum. Combined with the S2 fiber of the standard S4 representation, this gives a regular spatial sector locally of the form (0,1) x S2(R). The S2 Laplace spectrum provides a branch-independent nonzero rest gap for every l >= 1; under the stated internal second-order wave representation this gap is read as a rest-mass parameter. Gravity is represented by an exact finite source-to-field map using the Moore-Penrose Green operator, and the regular three-dimensional branch yields the local 1/r^2 acceleration exponent in the weak-static regime. The same relational development also supplies three distinct internal Hilbert representations. Conditional on the full orthogonal-projector event structure and standard noncontextual probability consistency, Gleason's theorem fixes the Born form in the spatial representation. The incidence representation is noncommutative and yields the Robertson uncertainty relation, while procyclic reaction gives exact discrete unitary dynamics by Haar-measure invariance. No canonical identification of these Hilbert spaces, unique continuous Hamiltonian, measurement-update rule, spin-1/2 structure, or Bose/Fermi statistics is claimed. Complete derivations and branch audits are provided in a Supplementary Formula Atlas.
This is Paper 007 in the Anaxiomatic Mechanics / pq-Spacetime stack. It develops the emergence of Fock-like structure within Binary Mechanics and addresses how variable excitation number can be represented without treating particles, Fock space, creation operators, or annihilation operators as primitive assumptions. The paper begins from the binary-mechanical picture in which the realized substrate is a finite resolved graph, while amplitudes arise from admissible finite histories of that graph. Particle-like objects are defined not as fundamental entities, but as stable, repeatable, coarse-grained graph-history patterns whose completion channels remain trackable across admissible continuation structure and effective boundary maps. Within this framework, the vacuum sector is interpreted as average non-excited graph behavior rather than an empty state. One-excitation sectors arise from stable trackable patterns, many-excitation sectors arise when multiple excitations are independently addressable, and variable excitation number requires an effective direct-sum representation over admissible excitation-number sectors. This yields a finite Fock-like space as an emergent representation, not a primitive input. The paper also clarifies how creation- and annihilation-like operators appear as effective boundary maps transferring amplitude support between excitation-number sectors, and how exchange constraints for indistinguishable excitations are tied to admissible exchange-history phases. The result provides a clean finite bridge from binary-mechanical quantum kinematics toward the representation architecture normally associated with quantum field theory.