Interference of Two Particle-like States with a Spread Phase: Feature Points of Relative Configuration and a Structural Correspondence with Statistics (Observational Note)
Observational note (thought-experiment style; not a proof or claim paper). A sequel to the preceding note, which defined a single particle-like state as a phase-bearing local mode on the subjective geometry S^4(R_U) with spread W_a = 2 R_a and deferred the multi-particle case. Here we treat the minimal case: two particle-like states with the same spread phase, on a closed one-dimensional relational setting (period-1 phase line, a single great circle of S^4). Representing each particle as a rectangular standing wave (odd harmonics only), the n-th harmonic amplitude of the superposition is A_n = (2/n) cos(n pi f), where f is the relative position in fundamental wavelengths, and the total squared norm (energy-like quantity) is the triangle wave E(f) = (pi^2/4)(1-2f) for 0 <= f <= 1/2 (the autocorrelation of a square wave). Within the continuous f, discrete feature points appear: fusion (f=0), orthogonality / independent coexistence (f=1/4), and complete cancellation (f=1/2). Depending on whether the superposition is symmetrized or antisymmetrized, complete overlap (f=0) is allowed (boson-like) or makes the field vanish, i.e. exclusive (fermion-like). For unequal frequencies, interference arises only from shared odd harmonics (the irreducible ratio r/s with both r,s odd, i.e. matching 2-adic valuation); without sharing, interference vanishes and placement is free. All results are presented as observations of structural correspondence (with standing-wave quantization and the exchange symmetry of quantum statistics); no derivation of quantum statistics or the spin-statistics theorem is claimed. Notably the existence of the three feature points is universal to any half-wave-antisymmetric (odd-harmonic) profile, while only the linear (triangle-wave) envelope is specific to the square-wave (1/n) choice. This record contains Japanese and English versions (Markdown, LaTeX, PDF) and one figure.
Observational note (thought-experiment style; not a proof or claim paper). A sequel to the preceding note, which defined a single particle-like state as a phase-bearing local mode on the subjective geometry S^4(R_U) with spread W_a = 2 R_a and deferred the multi-particle case. Here we treat the minimal case: two particle-like states with the same spread phase, on a closed one-dimensional relational setting (period-1 phase line, a single great circle of S^4). Representing each particle as a rectangular standing wave (odd harmonics only), the n-th harmonic amplitude of the superposition is A_n = (2/n) cos(n pi f), where f is the relative position in fundamental wavelengths, and the total squared norm (energy-like quantity) is the triangle wave E(f) = (pi^2/4)(1-2f) for 0 <= f <= 1/2 (the autocorrelation of a square wave). Within the continuous f, discrete feature points appear: fusion (f=0), orthogonality / independent coexistence (f=1/4), and complete cancellation (f=1/2). Depending on whether the superposition is symmetrized or antisymmetrized, complete overlap (f=0) is allowed (boson-like) or makes the field vanish, i.e. exclusive (fermion-like). For unequal frequencies, interference arises only from shared odd harmonics (the irreducible ratio r/s with both r,s odd, i.e. matching 2-adic valuation); without sharing, interference vanishes and placement is free. All results are presented as observations of structural correspondence (with standing-wave quantization and the exchange symmetry of quantum statistics); no derivation of quantum statistics or the spin-statistics theorem is claimed. Notably the existence of the three feature points is universal to any half-wave-antisymmetric (odd-harmonic) profile, while only the linear (triangle-wave) envelope is specific to the square-wave (1/n) choice. This record contains Japanese and English versions (Markdown, LaTeX, PDF) and one figure.
Two particle product type wavefunctions may be created so as to be symmetric (bosons) or antisymmetric (fermions) on then interchange of the identical particles. As a result, spatial density at a point does not change upon interchange. This approach, however, appears to be mathematical in nature. In Part I of this note, we argued that exp(ipx) free particle momentum wavefunctions within a bound wavefunction W(x) interfere and that full wavefunctions Wn(x) in a linear combination also do the same. As a result, probability interference is ultimately linked to the physical mechanism associated with the periodic exp(ipx) wave behind the Pauli rule that fermions may not occupy the same quantum state. In this note, we try to extend this idea and argue that one does not simply exchange identical particles in the wavefunction as a mathematical procedure, but that actual exchange may occur in the probabilistic quantum mechanical case. We note that a time-independent treatment of scattering from a target already includes the idea of “negative” or out of phase interference within a wavefunction W(x)= exp(ikz) + f(theta, phi) exp(ikr)/r so as to remove some incoming flux as outgoing scattered flux now exists. We suggest that the “minus” sign used in the antisymmetric fermion wavefunction behaves in a similar manner. In (1), we argued that W(x)W(x) in a bound state includes all possible a(p1)exp(i p1 x) a(p2)exp(i p2 x) combinations and represented a product probability. In other words, a particle at x with p1 P(p1/x) may be knocked into p2 at x. Similarly, we argue that cross terms in a two particle fermion wavefunction represent flux W*b(x1)Wa(x1)W*a(x2)Wb(x2) flowing from out of states as an interchange occurs. Main terms such as W*a(x1)Wa(x1)W*b(x2)Wb(x2) represent the presence of particles in the a, b states. If a=b, the presence and flow probabilities cancel locally and there is a vanishing wavefunction i.e. the two fermions may not occupy the same state. We argue that one should be able to describe the Pauli exclusion rule in this manner and note that Feynman (2) has used probability arguments in boson scattering to argue the boson enhancement factor. In other words, the mathematical argument of a symmetric wavefunction is not used, but a more physical argument is given.
Interference is an important observed feature in quantum mechanics. It is often seen in terms of peaks and troughs in two slit interference or in the profile of bound state wavefunction (e.g. harmonic oscillator). In this note, we argue that interference also plays a central role in determining the two types of quantum particles which exist i.e. fermions and bosons. Whereas a wavefunction is subject to a global constraint (its integral over x is 1), a local spatial probability constraint is imposed (at each x) in the case of at least two particles built from a separable wavefunction Wa(x1)Wb(x2). Here there are two indistinguishable particles, two allowable energy levels a and b and one wishes to know the probability of finding a particle at x1 and one at x2 regardless of the energy level. Let us call this P(x1,x2). The modulus squared of the wavefunction yields probability P, so a constructed wavefunction may have the property that the interchange of x1 and x2 yields a phase factor. This argument is presented often although (1) suggests it should be modified, yet the conclusion that there are only fermions and bosons remains. In this note, we consider the existence of two types of interference as the physical basis of this result. From the Schrodinger equation or from P(p/x)= a(p)exp(ipx) / W one may see that W→-W and a(p)-->-a(p) is still a solution. In quantum mechanics not only do exp(ipx) terms interfere, but eigenfunctions Wn may also interfere. Given that a bound eigenfunction may be represented by Wn or -Wn, there exists two kinds of interference. Even in the case in which there is only one particle, one may already create two possibilities: Wa(x)+Wb(x) and Wa(x)-Wb(x). If one sets a=b, the second becomes 0 at each x. This we argue already points to the idea of fermions even though only one particle is considered. This result is due strictly to alternate phase (1, -1) interference). The idea of alternate phase interference Wa-Wb also seems to be linked to a difference in wavefunction i.e. a “kind” of wavefunction representing the change in moving from energy level a to b. Thus we try to argue that there is interference beyond the usual exp(ipx) namely an alternate phase interference which applies to eigenfunctions Wn(x) and leads to the existence of fermions. Bosons occur with no alternate phase interference. These two types of particles emerge from the constraint that P(x1,x2)=P(x2,x1) i.e a local spatial probability constraint.
What quantum mechanics calls "observation" has lumped together three interactions of different character: one that switches the discrete state of the partner, one that leaves the partner's state unswitched and shifts only the internal phases, and one that severs the mutual phase relation. This paper places the substance of observation in the second, the readout of the interference of a phase difference, and adopts only three premises. (P1) The distance between two parties is set by the ratio of a shared viewing angle to the partner's size; only ratios are readable and the absolute scale is not. (P2) An interaction is a single relation in which emission and absorption hold simultaneously; the speed of light appears on the readout side as a conversion factor between local clocks. (P3) The phase of a relation carries a harmonic order n that depends on the kind of wave. From these, the fact that a probability is the product of an amplitude and its conjugate (the form of the Born rule), the two-party correlation E = ∓cos nθ, the upper bound 2√2 of the CHSH quantity (the Tsirelson bound), and no-signaling follow in a few lines. The normalization that turns this into a probability is the division that discards the size ratio, i.e. the choice of the η = 0 section. As a consequence, the bound is 2√2 independently of the kind of wave, and the detector angle giving the maximal violation shrinks in proportion to 1/n (45° for spin-½, 22.5° for photons, 11.25° for gravitational waves). For a localized relation that is a sum of harmonics the correlation is the autocorrelation of the pulse shape; the bound holds for any weights, equality is attained only for a single harmonic, and a rectangular pulse reduces to a local hidden-variable model with S ≤ 2. Japanese original (authoritative) and English translation, with TeX sources, PDFs, five figures (Japanese and English labels) and the figure-generating program.
IT is well known from experiment, as well as from classical theory, that strictly monochromatic light waves originating from different sources do interfere.' An example in which the waves are nearly monochromatic is found in the case of radio waves, and it is known that radio waves from two stations that have the same frequency show an interference effect. We may describe the phenomenon quantum mechanically if we represent the light quanta emitted by both sources by waves in Schroedinger configuration space. So long as we consider the quanta emitted by one source as different from those emitted by the other, the total wave function will be the product of the wave functions corresponding to the quanta emanating from both sources, and there will be no interference effect. In the present paper, we will see how things are changed when we symmetrize or antisymmetrize the wave function, i.e. assume either Bose or Fermi statistics for the particles. In the simple quantum mechanical picture that we will use here, we will assume a definite number of photons coming out from each source, and this implies that the phase relations will be undetermined.2 As a result, even the symmetrized wave functions must be expected to give, at any point in space, the same density of quanta that classical theory determines, when the intensity is averaged over the phase differences between the two sources. This implies that no interference is observed between the waves from the two sources. If we want to look for an interference effect when dealing with a quantum picture in which we have a definite number of photons, we must look for a phenomenon whose classical description should be independent of the phase. We shall make use of the fact that, in classical theory,
In this paper, we verify the existence conditions and stability of composite states (clusters consisting of many cells) in the reciprocal dual model by explicit construction of interference patterns. First, consolidation of the logic wave hypothesis: the per-axis value of the counting condition, |ki| + 1/2, is an odd-harmonic sequence with the zero point 1/2 as its base, and a spectrum containing only odd harmonics is the fingerprint of a binary symmetric signal (square wave, logic wave). That the phase wave carries no amplitude is not an approximation but a coherence condition demanded both by the theorem layer (the integer partition structure) and by curvature consistency. Information resides in edge positions, i.e., in the relative phases of the harmonic ladder, and a parallel translation of a square pulse is exactly equivalent to a linear phase multiplication on the coefficients (position = phase). Second, kinematic stabilization by half-wavelength censorship: the maximum of the curvature-induced anharmonic shift of the ladder is exactly 1/2 (the lowest mode, verified in rational arithmetic), and the censorship condition "shift ≤ half the resolution" is satisfied exactly at the limit. Without censorship, if the shift is taken as real dynamics, a composite dephases and collapses within 3–6 fundamental periods; under censorship, the continuous decay channel is kinematically closed, and the only remaining decays are the discrete channels (Paper 7). The stability of composite particles is not a dynamical accident but a quantization of the deformation space. As a by-product, we observe that the maximum dimension in which this protection holds is d = 4 (the fixing of normalization is deferred to Paper 11). Third, interference closure tests and the existence ceiling: a single cell can be constructed exactly from odd harmonics alone (width/period = 1/2 is the unique filling fraction at which the even harmonics vanish identically). The condition for odd-harmonic ladders to nest is that the scale ratio be odd (the odd nesting theorem — a wave-level rederivation of the odd-k rule of the genealogy). Computing, by exact Fourier analysis over the entire odd-s ladder, whether an occupancy structure can be certified by its own waves alone (the eye opening in the fundamental-wave band), we have established that the stable species s = 1, 3, 5 certify themselves with strong margins (≥ +0.22), the decay threshold s = 7 cannot be certified, s ≥ 9 is critical, and for s ≥ 25 certification at the composite scale disappears, while for s ≥ 49 certification fails in every band of the odd sector (verified down to band depth 31; the margin deteriorates monotonically with s). A control experiment shows that allowing even harmonics restores certification, so the closure is intrinsic to the logic wave (odd, binary) structure. As a consequence, a large content S cannot exist as a flat single entity, but only as a nesting of certifiable small entities — hierarchization is not a choice but a compulsion. Bilingual edition (Japanese and English): Markdown, LaTeX, and PDF for each language, plus three figures (PNG).
As the final paper of the series, this paper constructs the minimal system in which two composites coexist — the twin minimal space model — and examines how much of the behavior that looks quantum mechanical is derived as a reading of hierarchy interfaces in a discrete relational model that assumes neither amplitudes nor continuous time. External face of the stable species: The handles (hairs) by which a composite is observed from outside divide into three ledgers: the face fundamental wave ν0 = 1/(2√s) (the dominant line of the self-container spectrum), the parent-record line νobs = |k|, and the energy-like norm R' = √s. The exact identity ν0R' = 1/2 (the zero point) links the first two, and R' never appears as a line — it is readable from outside as the spacing 1/R' of the odd comb (the particle-level realization of the record theorem). The external face is a minimal-uncertainty quasi-rectangular wave packet that carries a built-in odd comb of spacing 1/R' and is localized within an extent 2R' (half-width × fundamental wave = 1/2). Construction of the twin system: The initial value Ω0² = 2ν1² = 18 is an even label and, by the classification theorem (Paper 5), cannot exist as a single state — being born as a pair is forced by parity (the first event requires no dynamics). The allowed lattice separations of the twins (each s = 9) are quantized to |Δ|² ∈ {2, …, 20}, and |Δ|² ≡ 1 (mod 4) is entirely forbidden. The relational classes (21 under B4 × swap) are completely separated once the branch (cos/sin) channels are read in addition to the fringe intensity data (intensity-only fringe data degenerates to 18 kinds as ordered data, and to 11 kinds under the operationally available unordered reading) — the scale marks of space are born as chord sets of equal-frequency shells. The exact form of measurement: Inspecting the two conditions for measurement (affecting the system; detecting the change), the detection of change requires knowledge of the prior state and regresses infinitely; in this model, however, the record of the prior state is absent by censorship, and the λ ledger is append-only (the monotonic increase theorem, Paper 6), so the regress is structurally cut. Measurement = an append to the record (absence → presence); "collapse" is the appearance of a record, and its irreversibility reduces to the same single bit as the arrow of time. Computing the first decay of the twins explicitly, the appearance of the reference wave (s = 1) promotes the information class of the record from interferometric to holographic — the measuring apparatus is generated by the event itself. We also show that the final-state statistics depend on the choice of measure (one-shot equal weight over configurations 60/61 versus sequential conditional counting 396/403) — measure bifurcation, with an exact difference of +0.00098. Layer resolution of entanglement: The self-spectrum of a single fragment lies on all-components-even vectors and is therefore identically invisible in the logical-wave (odd) sector — only counts and relations can be read (the relational-only readout theorem). The decay of one twin instantaneously rewrites the branching statistics of the other (total variation 0.774), but this shift does not depend on the twin separation at all, and the influence does not propagate through the space of the child layer — instantaneity is not a speed but a mismatch of levels. Since order and attribution do not exist in the final record, operational no-signalling follows. An entangled pair, being an even label, cannot become a single particle at the layer above (parity protection). The same map applies to two-path interference of a single wave packet. Vertex deficits and the two hidden axes: An exhaustive inspection of the record continuity rule across an event (that the parent's identity line persists in the cross spectrum of the daughter system) derives, as the persistence condition, a conservation law of position vectors v5 ± v3 = ±v9 (a momentum analogue), which holds for parents in the (2,1,0,0) orbit but fails for parents in the (1,1,1,1) orbit by exactly one unit (one axis / one quantum). Furthermore, in the bookkeeping of the excitation degree ε = (−1)Σ|k| there exists a second one-bit deficit independent of this one (specific orbits at s = 7, 9). The visible four-axis ledgers demand two invisible directions — R on the curvature side and Q on the bare-parity side — and intermediate displacements along the invisible directions, being unrecorded, are permitted (the licensing principle for virtual displacements). This licence virtually unlocks two-body splittings that are forbidden in the visible ledgers, and decomposes every decay into sequences of cubic vertices (multiple paths to the same final state). Finally, we organize the correspondence between the obtained structure and standard theory as a transport principle (a theorem is transported whenever the counterparts of its premises are verified), and state explicitly the candidate answers this model gives to questions that remain postulates on the standard-theory side (the measurement problem, the location of the cut, the problem of time, dimensionality, the measure), together with falsifiability (gravitationally mediated entanglement experiments, etc.). Bilingual edition (Japanese and English): Markdown, LaTeX, and PDF for each language, plus four figures (PNG). v0.3 (2026-06-12): Corrects Sec. 3.2 and the abstract: the twin pairs decompose into 21 relational classes under B4 × swap (not 18); intensity-only fringe data degenerates to 18 kinds (ordered) / 11 kinds (unordered, operational), and complete separation is recovered by the branch-channel readout. All other results are unchanged.