Wave-Nature Unification Theory: Foundational Document and Derivation Notes (English Edition)

[v7] Relativistic completion of the gravity sector. The Foundational Overview is updated to v2.3, and a new Derivation Note v4 "Relativistic Completion of the Gravity Sector — The Tensor Sector, the 1PN Lagrangian, and Observational Constraints" is added. (1) The signs of Eqs. (3.1) and (3.2) in Chapter 3 are corrected — under the convention Φ>0 the previous pair gave a repulsion; the correct pair is ∇²Φ = -(4πG/c²)ρ and F = +mc²∇Φ. (2) The status of the two-body expression Φ_AB = Φ₀ - (τ_A+τ_B)/r² is corrected — Eq. (3.1) is linear and its two-point-source solution is of 1/r form in the distances from each body. The 1/r² term is reassigned as a 1PN correction rather than a Newtonian solution, and τ_A = (2β_PPN-1)G²m_A/(2c²) is fixed; the additive form is found to reproduce exactly the structure of the static term of the 1PN two-body problem. (3) The tensor sector is introduced — the previous framework gave the effect of mass only to the clock rate, implying the PPN parameter γ=0, i.e. half the observed deflection of light (0.876″ vs 1.752″, excluded at 4.3×10⁴σ by Cassini). Solving ledger item (16) of the previous version, "sharpening the source term", shows that the source is T_μν rather than ρ, and hence that the field must be a rank-2 tensor h_μν; this promotion yields γ=1 with no adjustable parameter. In the language of refractive index, a clocks-only description gives n = 1+Φ while one that also modulates space gives n = 1+2Φ — the missing factor of two is the spatial sector. (4) The verdict on aberration is changed from [Open] to [conditionally resolved] (Carlip 2000) — Laplace's argument tacitly assumes a delayed central force and does not apply to a Lorentz-covariant field equation. The theory's greatest theoretical challenge moves from "resolving the aberration problem" to "constructing a Lorentz-invariant action from the provenance account". (5) The bound on the non-minimal coupling ξ is updated from 10⁶ to 10⁻⁴, about ten orders of magnitude. (6) It is stated explicitly that the field equation is precisely that of linearised general relativity [C]; the theory is not "an alternative to general relativity" but "a theory that attempts to explain why the field general relativity describes is there at all". All results are verified by symbolic computation and numerical integration. See Appendix C (Change History) inside the document and Derivation Note v4. [v6] The Foundational Overview is updated to v2.2. (1) Refinements from the literature check of Chapter 4 — the lineage note is sharpened (Kelvin (1867) identified vortices with atoms, not charge; the closest precedent for "a conserved quantum number as soliton winding" is Skyrme, baryon number as topological winding), and Sec. 4.5(1) now records that particle-vortex duality is a theorem of 2+1 dimensions, the 3+1-dimensional dual of a vortex string being a string coupled to a two-form gauge field. (2) Resolution of ledger item (21) — the charging problem of the dark vortex strings is resolved by the two kinds of winding (wavefront phase = charge; arrest-field phase = dark strings), restricting the scope of identification A to the wavefront phase, with the falsifiable corollary (dark strings interact through tension alone) agreeing with Chapter 7. See Appendix C inside the document. [v5] The Foundational Overview is updated to v2.1. Main changes: (1) a new Chapter 4, "The Electromagnetic Force — Where Does Sign Come From?" — the two-sector structure (gravity = the scalar sector of arrest density; electromagnetism = the signed sector of winding number); from the identification charge = winding there follow charge quantization, charge conservation (= a rediscovery of the existing pair-creation prohibition), the identification of the annihilation channel, and the emergence of sign structure; subsequent chapters are renumbered and ledger items (18)-(21) added. (2) The zero-extinction refinement in Chapter 3, Sec. 3.4 — the transparency requirement is extended from zero absorption to zero extinction (absorption plus scattering); by the optical theorem, drag and heating are resolved simultaneously by one condition; the observational bound from the persistence of stellar peculiar velocities is registered as ledger item (22). See Appendix C (Change History) inside the document. [v4] The Foundational Overview is fully revised (document v2). Main changes: separation of the two roles of the arrest parameter (a: degree of arrest / χ: time velocity / Φ: pressure-deficit potential); Chapter 3 restated at the level of a field equation, with a new section answering the classical objections to Le Sage-type gravity (drag, heating, aberration); retraction of the overtone law m_n = n²·m_e and its replacement by the equipartition constraint of the charged-lepton triplet (Koide\u2019s formula, known); consolidation of the MOND attribution onto the coherence-time mechanism; claim labels [A/B/C/Open] applied throughout, with a new Chapter 0 and a change-history Appendix C. See Appendix C inside the document for details. [v3.1 Corrigendum] A corrigendum (corrigendum_v3_1_EN.pdf) concerning the lattice numerical claims of Derivation Note v3, Sec. 7, has been added. The Sec. 7(i) values depend solely on matrix-valued couplings not derivable from the medium model, and Sec. 7(ii) could not be reproduced under pre-registered protocols; the network-level claims of Sec. 7 are therefore withdrawn. The single-link results (plasticity equation, retention law, non-destructive readout) are unaffected. The independent reimplementation code (lattice_reimplementation_code.zip) is included. Japanese-English split edition (English record) of the Wave-Nature Unification Theory (a-theory, Arrest Parameter Framework), containing the Foundational Document (Overview) and Derivation Notes v2 and v3. The Japanese edition is published as a separate record (DOI: 10.5281/zenodo.21850306). Reconstructed from the former combined record (DOI: 10.5281/zenodo.21740126). Contents: Foundational Document (Overview) (Markdown, dated 2026-07-30) / Derivation Note v2 (PDF + LaTeX source) / Derivation Note v3 (PDF + LaTeX source). [v2] Derivation Note v2 "Unification of the Averaging Stiffness θ′ — Dispersion θ′(k), Determination of the Coefficient A, Interpretation of ε₀, and the Lifetime Formula". θ′ is redefined as the averaging stiffness of the field itself, establishing: the effective stiffness θ′_eff(k) = c²(k_g/k + k/2k_g)²; the exact coincidence of its minimum with the arrest ground mode k₁ = π/L_s (a variational re-derivation of L_s; total ground energy = ε₀ = 2m_ec²); the complete determination of the selection-rule coefficient A = 6θ′k_g² (= 6U″(φ₀)); the unification of the two readings of ε₀ via topological pair creation; and the lifetime formula τ_n = 2τ₀/(n(n²−3)) with stability boundary n² = 3 and asymptotics Γ ∝ m^{5/2}. Four falsifiable predictions and five open issues are stated explicitly. [v3] Derivation Note v3 "History Retention in the Interaction Medium — the Plasticity Equation and the Forgetting Action S_rec". Formalizes, using only previously derived results and zero additional parameters, the mechanism by which the interaction medium between arrested configurations retains history (a synapse-like plastic coupling). Main results: (1) the plasticity equation dw/dt=(1/τ₀)⟨Θ(E_loc−ε_th)⟩(1−w)−w/τ_r, with the learning rate set by the theory's unique time scale τ₀, the Hebbian coincidence gate derived from the pair-creation threshold and wave interference, and saturation/weight quantization from the packing rule 2L_s; (2) two independent formulations of the forgetting action S_rec (real-space pinning tunnelling vs. order-parameter phase slip S₁=1.16) agree at the 1.2% level through the barrier identification V_PN=μξ_h=1.70𝓔₀ — a cross-validation of the S₁ calculation; (3) the retention law τ_r(d)=ω_att⁻¹exp[2S₁d/ξ_h], programmable from nanoseconds to cosmological scales by the write separation; the separation for age-of-the-universe non-volatility, 38.2ξ_h, equals the dark-structure survival cut L_q(t₀); (4) numerical experiments on a 26-direction cell lattice demonstrating distributed memory and threshold-protected non-destructive readout. Five falsifiable predictions and five open items are stated explicitly.