This paper gives a unified description of quantum theory and relativity as readings of different axes of a single triadic structure (Δ, Λ, Ψ), presupposing the preceding paper "The Triadic Structure (Δ, Λ, Ψ): Foundations in Orientation, Relation, and Dimension" (DOI: 10.5281/zenodo.20986811). The unification claimed is a unification of description, not of dynamics: no new dynamics are introduced, and "unification" here does not refer to gauge grand unification (GUT). The central new result: the light-cone coordinates of the 1+1-dimensional Minkowski plane form the standard two-dimensional triad as they stand (Proposition 1.1). The two poles V₊, V₋ are the two null directions; parity is P = Ψ; time reversal is T = −Ψ; the total spacetime inversion is PT = −id = Δ²; the boost of rapidity s is exp(sσ) = diag(eˢ, e⁻ˢ); and a boost composed with parity is the scaled exchange Ψ_λ (λ = e⁻ˢ). Under boosts the interval x₊x₋ is invariant while only the ratio x₊/x₋ moves (Proposition 2.1): the invariance of the interval is "a fixed product with a free ratio" — the same structure as the uncertainty relation (the fixed product ℏ with a free ratio). On the quantum side the identifications of the preceding paper are inherited: the complex unit i as the generator Δ (in the sense of right multiplication), the spinor's 2π as Δ², and the KMS antiperiodicity as a conditional reading. A dictionary maps the basic objects of both theories onto the elements and axes of one triad, and the junction identity −id = Δ² = PT ties them to one and the same half-period residue (Theorem 4.1). Gravity is placed as follows: by axiom T2 of the preceding paper, the scale cannot be fixed from inside the structure; the metric field is read as the field that supplies this externally required calibration at each point. The Einstein equations, the gauge content, and the values of the constants are external inputs and are not derived. A demarcation table states the status of every claim (theorem / citation / identification / reading / conditional / unconstructed), and criteria of attainment (U1–U3) for a unification of dynamics are defined, with a self-assessment of this paper against them. Files: Triadic_Unified_Description.pdf (the paper, Draft v2); Triadic_Unified_Description_verification_scripts.zip (three verification scripts — verify_lightcone_triad.py 12/12, verify_interval_product.py 8/8, verify_reversal_T.py 7/7 — with README; dependencies: numpy, scipy). Related: the foundations paper (DOI: 10.5281/zenodo.20986811); the axiomatic proposal for a non-orientable spacetime (DOI: 10.5281/zenodo.20678390).
This paper gives a unified description of quantum theory and relativity as readings of different axes of a single triadic structure (Δ, Λ, Ψ), presupposing the preceding paper "The Triadic Structure (Δ, Λ, Ψ): Foundations in Orientation, Relation, and Dimension" (DOI: 10.5281/zenodo.20986811). The unification claimed is a unification of description, not of dynamics: no new dynamics are introduced, and "unification" here does not refer to gauge grand unification (GUT). The central new result: the light-cone coordinates of the 1+1-dimensional Minkowski plane form the standard two-dimensional triad as they stand (Proposition 1.1). The two poles V₊, V₋ are the two null directions; parity is P = Ψ; time reversal is T = −Ψ; the total spacetime inversion is PT = −id = Δ²; the boost of rapidity s is exp(sσ) = diag(eˢ, e⁻ˢ); and a boost composed with parity is the scaled exchange Ψ_λ (λ = e⁻ˢ). Under boosts the interval x₊x₋ is invariant while only the ratio x₊/x₋ moves (Proposition 2.1): the invariance of the interval is "a fixed product with a free ratio" — the same structure as the uncertainty relation (the fixed product ℏ with a free ratio). On the quantum side the identifications of the preceding paper are inherited: the complex unit i as the generator Δ (in the sense of right multiplication), the spinor's 2π as Δ², and the KMS antiperiodicity as a conditional reading. A dictionary maps the basic objects of both theories onto the elements and axes of one triad, and the junction identity −id = Δ² = PT ties them to one and the same half-period residue (Theorem 4.1). Gravity is placed as follows: by axiom T2 of the preceding paper, the scale cannot be fixed from inside the structure; the metric field is read as the field that supplies this externally required calibration at each point. The Einstein equations, the gauge content, and the values of the constants are external inputs and are not derived. A demarcation table states the status of every claim (theorem / citation / identification / reading / conditional / unconstructed), and criteria of attainment (U1–U3) for a unification of dynamics are defined, with a self-assessment of this paper against them. Files: Triadic_Unified_Description.pdf (the paper, Draft v2); Triadic_Unified_Description_verification_scripts.zip (three verification scripts — verify_lightcone_triad.py 12/12, verify_interval_product.py 8/8, verify_reversal_T.py 7/7 — with README; dependencies: numpy, scipy). Related: the foundations paper (DOI: 10.5281/zenodo.20986811); the axiomatic proposal for a non-orientable spacetime (DOI: 10.5281/zenodo.20678390).
See changelog below. The full derivations are provided in the attached paper. Every result is reproducible from the stated inputs. This paper derives the complete Standard Model, quantum mechanics, spacetime, and gravity — 85 quantitative results at sub-percent precision spanning 15 orders of magnitude — from a single postulate, four structural principles, and one measurement. The Standard Model has seventeen free parameters. General relativity has no quantum theory. The measurement problem has been open for a century. The cosmological constant is wrong by 120 orders of magnitude. The question of why there is something rather than nothing has no physical answer in any existing framework. Each of these failures shares a common omission: none ask where time comes from. This paper takes that question seriously. A single physical principle — information cannot be destroyed — is self-sealing: its own violation is incoherent. Its formal expression is Postulate I: each foreclosed quantum branch produces exactly one chronon — a massless spin-1 gauge boson of U(1)_T, the unique gauge group forced by the complex ray structure of the Hilbert space and the single scalar collapse charge. We call it the chronon. Time is not the background in which physics happens; it is generated by physics, irreversibly, at every collapse. These requirements yield a single Lagrangian, ℒ_QT, that extends the Standard Model by one massless spin-1 field and one coupling constant. From this — combined with one measurement, Newton’s gravitational constant G — we derive: all of quantum mechanics via the Piron–Soler–Gleason chain (not assumed); the complete Standard Model with all dimensionless parameters determined; the complete resolution of the measurement problem, including the environmental decoherence cascade; and 85 quantitative physical results spanning fifteen orders of magnitude in energy. Three spatial dimensions follow from the channel-matching principle (D = N_c) and the Weinberg angle consistency condition (N_c = 3). The Big Bang was not initiated — it was mandatory. The numerical precision is sharp. The worst result where the theoretical prediction exceeds experimental precision is 0.5%. For the majority of results, the framework’s prediction is already more precise than the measurement. The proton-to-electron mass ratio m_p/m_e = 2N_cπ⁵(1 + α²/√(N_c²−1)) agrees to sub-ppb precision. The proton charge radius matches muonic hydrogen to five significant figures. The Higgs boson mass to 0.002% via the full cascade derivation (0.25% from the explicit formula). The W boson mass to 0.001%. The fine structure constant to 0.026% (1/α = 137 exactly at tree level). The cosmological constant — wrong by 120 orders of magnitude in standard QFT — is derived to 0.03%. The Hubble tension, a 5σ discrepancy unresolved in standard cosmology, follows from the same structure: H₀_local = H₀ × 13/12 = 73.0 km/s/Mpc (0.05%). All four PMNS neutrino mixing parameters, previously unexplained, are derived to sub-0.5%. Sub-percent agreement spans from neutrino mass splittings at ~9 meV to the CMB temperature at 2.725 K — fifteen orders of magnitude from a single dimensionful input. The framework makes ten falsifiable predictions currently under test: an electron electric dipole moment of exactly zero (ACME III is live); a spin-1 gravitational wave breathing mode at ~9% of the + polarisation amplitude (testable with LIGO O4/O5 data); no dark matter particle in any direct-detection experiment; no graviton — gravity is geometric drift in a proper-time gradient sourced by the energy-momentum tensor, not exchange of a spin-2 particle; there is nothing to detect. A confirmed non-zero electron EDM or a detected graviton falsifies the framework immediately. The logical order, from bottom to top: information cannot be destroyed → P-I (one chronon per foreclosed branch) → proposition lattice with axioms P0–P4 → Piron–Soler–Gleason chain → complex Hilbert space → ray phase redundancy → single scalar collapse charge → U(1)_T gauge field with Maxwell dynamics → channel-matching principle gives D = N_c → Weinberg angle selects N_c = 3 → D = 3 → adding G through the Planck cascade gives λ_conf = 0.40092 fm → the complete Standard Model follows. Spacetime, quantum mechanics, and the Standard Model are not independent inputs — they follow as consequences of P-I, the structural principles, and one observational consistency condition. ______________________________________________________________________________________________________ v86 Changelog (April 7–8 2026) — All numerical results unchanged from v85. No formulas modified. Foundational structure and derivation justifications rebuilt following adversarial review by two independent AI systems (Claude, GPT-4o). Foundational hardening: §2.2: Formal axiom framework P0–P4 added. Piron lattice conditions now follow as a theorem given five named structural axioms, not as informal physical arguments. P-I motivates four directly; P0 (admissibility closure) is a structural axiom on the proposition space. §2.5: ℝ exclusion replaced. Old iε propagator argument replaced with operational compositionality / local tomography argument requiring no field-theoretic machinery. §3.1: Gauge derivation rebuilt from complex ray structure forward. U(1)_T now derived from phase redundancy + single scalar collapse charge + rank-1 compact Lie group + label-free emission. Removes prior dependence on Noether's theorem invocation and energy minimization principle. §8.1.1: Channel-matching theorem added. D = N_c derived from information-balance between spatial export channels and internal confinement channels via mismatch functional. Replaces prior mutual-reinforcement argument. §1.5.1: Epistemic taxonomy added classifying all 85 results into three tiers: theorem-level (Piron, Soler, Gleason), physical-argument (gauge structure, mass formulas), and observational-consistency (N_c = 3 from Weinberg angle, cascade from G). §12: New section — Structural Analysis. Result classification (5 epistemic tiers, ~12 independent core results, compression ratio 85:19), axiom necessity (what breaks without each of P0–P4 and D1–D4), and ablation tests (N_c = 2, N_c = 4, drop P4, change coupling exponent, allow D ≠ N_c — every perturbation destroys the majority of results). Framework premise expanded from "one postulate" to "one postulate, four structural principles, and one measurement." Front matter aligned with body throughout. One derivation story, no internal contradictions. Derivation upgrades: Proton-to-electron mass ratio (§9.3): Replaced "2 spin states × N_c colours" with rigorous hyperspherical decomposition. S³ forced by massless emission in (3+1)D; S⁵ forced by K=0 ground state of 3-quark system in 6D relative coordinates. Neutral pion mass (§9.9.1): Factor 5 now derived as 2N_c − 1. Denominator 2π identified as one-loop coefficient of U(1)_T vector gauge coupling. Neutron-proton mass difference (§9.3.2): Replaced "circular loop + straight line" with flux tube arm derivation. π from SU(2) isospin rotation; +1 from chromoelectric flux tube arm to Y-junction. CP phase δ (§9.8.2): Replaced "axial-to-spherical flux ratio" with V−A vertex structure. The ratio 4/π is universal across all generation spaces because any unitary rotation lies on a great circle of S². W boson mass corrections (§9.5.1): Both NLO terms fully derived. Direct vertex α_s/(8π): C_F cancelled by non-relativistic reduction. Exchange α_s/(12π²): two vertices × Dirac-to-colour projection ratio. Charm mass (§9.13.4): Factor 4/π identified as axial-to-vector coupling ratio through V−A weak vertex. Systematic correction — QCD dressing factor: The coefficient 1/(2π) appearing in ~10 results was previously attributed to "hemisphere projection geometry." Replaced throughout: mass in the chronon framework is a U(1)_T vector gauge coupling, not a scalar Yukawa coupling. The factor 1/2 confirmed by two independent routes: (i) Dirac algebra (γ^α γ^μ γ_α = −2γ^μ vs scalar = 4); (ii) C_F cancellation by non-relativistic reduction. Label correction: CKM geometric factor (§9.8.1): "four-dimensional solid angle" corrected to "three-dimensional solid angle S² = 4π." Numbers unchanged. ______________________________________________________________________________________________________ v85 Changelog (April 6 2026) — Electroweak corrections, |V_cb| prediction, consistency audit Four physics improvements and a comprehensive numerical audit, all from first principles with zero free parameters. Top quark mass (0.41% → 0.06%): The electroweak self-energy correction factor changed from (N_c+1) to (N_c−1). The factor (N_c−1) = 2 counts the charged weak boson states (W⁺, W⁻) mediating the t→b→t self-energy loop — the SU(2)_L charge-raising/lowering generators. m_t = 172,803 MeV vs observed 172,700 MeV. CKM θ₂₃ / |V_cb| prediction (1.3% → 0.3%): The framework predicts |V_cb| = 42.1 × 10⁻³, essentially identical to the PDG inclusive determination (42.2 ± 0.5) × 10⁻³ and inconsistent with the exclusive determination (39.8 ± 0.6) × 10⁻³. The inclusive/exclusive |V_cb| tension is one of the open questions in flavour physics; the framework picks a side. Falsifiable prediction. Baryon asymmetry η (0.5% → <0.1%): The sphaleron efficiency κ correction changed from α_s/(2π²) to α/(2π). The sphaleron is an SU(2)_L electroweak process; its one-loop vertex correction uses the QED coupling α/(2π), not the QCD coupling. Neutrino mass splitting Δm²₂₃ (0.3% → 0.08%): QED vacuum polarization correction −α/(2π) added to G₃ alongside the existing QCD correction +α_s/π. Sign pattern (QCD adds, QED subtracts) matches the W boson mass derivation. Numerical audit (22 corrections): Full line-by-line verification of every formula, table entry, agreement claim, and cross-reference. Corrections include: D/H value (2.62→2.58 × 10⁻⁵, 1.3σ→1.8σ — stale from pre-κ η); He-4 agreement (0.5σ→0.7σ); z_
This paper presupposes three papers — the foundations paper, "The Triadic Structure (Δ, Λ, Ψ): Foundations in Orientation, Relation, and Dimension" (DOI: 10.5281/zenodo.20986811); the unified-description paper, "A Unified Description via the Triadic Structure" (DOI: 10.5281/zenodo.21131636); and the Tomita–Takesaki paper, "Selection and Heat" (DOI: 10.5281/zenodo.21161282) — and decomposes the quantum state itself in the vocabulary of the triad: direction and calibration. The scope is the minimal case (the two-level system); d > 2 is explicitly reserved. The claims of this paper are the theorems of the decomposition and a dictionary; no unification of dynamics and no new predictions are claimed. The central results are the following. First, the three degrees of freedom of the two-level state ρ = ½(id + r n·σ) are exhausted, with nothing left over, by the direction n (2) and the strength r (1) (Proposition 1.1). This is the same counting 3 = 2 + 1 as Proposition 2.5 of the unified-description paper — the three degrees of freedom of the pointwise metric are exhausted by direction (the two null directions) and calibration (a positive factor) (Corollary 1.2; the counting is a theorem on both sides, the identification is a reading). Second, for 0 < r < 1 the modular flow of the state is a rotation of the Bloch sphere about the axis n with angular speed 2 artanh(r) (Proposition 2.1): the direction selects the axis of time, and the strength sets its speed (Remark 2.4). At r = 0 the flow vanishes; for the thermal state, r = tanh(βε) and the speed is 2βε — the calibration encodes the inverse temperature (Proposition 2.3). Third, the update between two selections is implemented by the Connes cocycle u_t = ρ_ω^{it} ρ_φ^{−it} (Proposition 3.1 and Remark 3.2; the properties are theorems, the identification carries the tag). Fourth, non-selective measurement projects the direction and shrinks the strength, while selective measurement jumps to r = 1 — outside faithfulness. The mechanism lives on 0 < r < 1 and degenerates at both ends (no selection; pure selection) (Remark 4.2, the naming of a condition). What is a theorem, and what is an identification, a reading, a citation, or a reservation, is stated explicitly in the demarcation table. Theorem-level claims are machine-verified (verify_bloch_2plus1.py 8/8, verify_flow_axis_speed.py 8/8, verify_update_cocycle.py 7/7; numpy/scipy).
The persistent tension between Quantum Mechanics and General Relativity is commonly framed as the need for a deeper theory capable of unifying both within a single ontological structure. In this work, a different strategy is adopted. Rather than pursuing direct unification, we propose a triadic conceptual framework that emphasizes organization prior to unification, treating Quantum Mechanics and General Relativity as complementary descriptions of a single physical system operating under distinct epistemic commitments. Within this framework, each theory is assigned a specific functional role, and a third element—a principle of coherence—is introduced to clarify how these roles may be consistently related without reduction. Approaches grounded in information theory and quantum information are highlighted as natural candidates for articulating this coherence interface, given their ability to mediate between operational descriptions and structural constraints. It is argued that several well-known conflicts between the quantum and relativistic frameworks can be reinterpreted as tensions arising from mismatched levels of description rather than fundamental physical inconsistencies. The proposed approach is structural and organizational in nature, aiming to provide a conceptual map for existing and future work in the foundations of physics.
The geometric form of standard quantum mechanics is compatible with the two postulates: (1) the laws of physics are invariant under the choice of experimental setup and (2) every quantum observation or event is intrinsically statistical. These postulates remain compatible within a background independent extension of quantum theory with a local intrinsic time implying the relativity of the concept of a quantum event. In this extension the space of quantum events becomes dynamical and only individual quantum events make sense observationally. At the core of such a general theory of quantum relativity is the three-way interplay between the symplectic form, the dynamical metric and non-integrable almost complex structure of the space of quantum events. Such a formulation provides a missing conceptual ingredient in the search for a background independent quantum theory of gravity and matter. The crucial new technical element in our scheme derives from a set of recent mathematical results on certain infinite-dimensional almost Kahler manifolds which replace the complex projective spaces of standard quantum mechanics.
Hamilton's Principle, Feynman's Path Integral, and the Action Quantum h = 2πℏ as Capacity-Rotor Theorems: A Parameter-Free Derivation from QTT Axioms A1–A7U Version: 1.2 Concept DOI: 10.5281/zenodo.20098143 Version DOI: 10.5281/zenodo.20098144 Author: Ali Attar Website: quantumtraction.org Corpus category: Action, Feynman, path integrals, and quantum foundations Central result. This paper derives, from the axioms of Quantum Traction Theory (QTT) including the new A7U closure, three results that are usually treated as independent postulates of textbook physics: Hamilton's stationary-action principle (δS = 0) as the macroscopic shadow of dial coherence; Feynman's path integral 𝒜 = ∫𝒟x exp(JS/ℏ) as the continuum limit of capacity-bounded per-tick rotor kernels; The action quantum h = 2πℏ as the dial-revolution closure theorem, with the Planck–Einstein ET = h, the de Broglie pλ = h, the Bohr–Sommerfeld ∮p·dq = nh, and the magnetic-flux quantization qΦ = nh as immediate corollaries. No fitted parameter is introduced at any step. The physical primitive is a real two-component dial at each world-cell address, and a history contributes because it rotates that dial by an angle fixed by its action ledger, dθ = dStot/ℏ. The recovered mathematical condition is stationary action, not necessarily a minimum — minima, maxima, and saddle paths can all be stationary in the standard semiclassical sense. Position relative to Feynman. Feynman's path-integral construction postulates the unit-circle history phase eiS/ℏ and shows that classical mechanics is its stationary-phase limit. QTT does not weaken that result; it supplies its substrate justification and extends it. In the QTT reading the history weight is the unique real one-parameter group that simultaneously preserves dial norm (orthogonality), composes additively over concatenated histories, and closes after one full 2π of accumulated action. That group is SO(2), realized on the real dial as the J-rotor RJ(θ) = cos θ·I + sin θ·J with J² = −I. The complex shorthand eiθ and the real rotor RJ(θ) carry the same stationary-phase mathematics under the packaging J ↦ i; what QTT changes is the primitive. The path-integral weight is therefore not assumed in QTT — its real-rotor representative is derived. The Feynman iε prescription likewise becomes a substantive QTT statement: it is a real-dial tilt J·0+ selected by A1 causal time ordering and A6 capacity continuity. The action quantum as a closure theorem. Combining dθ = dS/ℏ (A4) with the dial periodicity θ ~ θ + 2π (A7) yields, with no further input, ΔSone revolution = ℏ·2π ≡ h. Planck's constant h is, in QTT, the action required to close one full dial circle, and ℏ is the action per radian. The Planck–Einstein, de Broglie, Bohr–Sommerfeld, and magnetic-flux-quantization rules then follow as direct readings of this closure on different time, space, phase-space-loop, and gauge-loop projections. QTT mechanism (axioms used). A1 supplies the Absolute Background Clock and the measured-time projection. A4 supplies the real dial and the quarter-turn generator J. A5 composes histories over world-cell addresses. A6 bounds energy, power, and action throughput per address (one ℏ per tick) and gives the history sum a physical capacity endpoint. A7 fixes the 2π modular closure of complete existence. A7U sharpens A7: the hidden complement of a visible subsystem is not arbitrary extra matter; it is the distributed visible share of the other co-members of the same complete bundle. The exact variational object is therefore the complete distributed Planck bundle, while the laboratory sees a reduced, locally accessible marginal. The explicit per-tick short-time kernel with Gaussian normalization and its complex-packaging form (§6.1). Theorem 1 — the continuum Feynman path integral as a QTT theorem (§6.3). Theorem 2 — the dial-revolution closure theorem h = 2πℏ (§7.1). Corollaries 1–4 — Planck–Einstein, de Broglie, Bohr–Sommerfeld, magnetic-flux quantization (§7.2). Theorem 4 — the Feynman propagator with the real-dial J·0+ prescription, replacing the standard iε contour-deformation argument with a substrate-derived causal tilt (§13). The WKB derivation of the Hamilton–Jacobi equation from the QTT Schrödinger shadow (§11.1). The Wilson-loop / gauge-holonomy reading with explicit Stokes-form and an Aharonov–Bohm derivation, plus a no-double-counting methodological warning (§12). The Wick-rotation status statement: Euclidean weights exp(−SE/ℏ) are computational tools, not the ontic QTT primitive (§14). The capacity-regularized covariant kernel 𝒦ℓ̃ = K(−ℓ̃² DJ⊤ DJ) as a substrate UV cutoff (§15). The dial-covariant Noether current jJμ for fields (§16). Three new falsifiers: action-quantum closure failure, wrong-tilt Feynman prescription, unitary path-integral measure violation. The axioms, the existing v1.2 derivations, and the A7U content are unchanged. No new fitted parameter is introduced; every new equation is forced by the same dθ = dS/ℏ plus 2π closure that already supports v1.2. Cross-paper coherence. The visible-share count Nb appearing as the inertial coefficient in this paper's reduced variational equation is the same substrate count that supplies (i) the worldline-acceleration response F = Nbm*·ẍ, (ii) the gravitational source-density coefficient in ∇·g = −4πGρ with G = ℓ̃²c³/ℏ, and (iii) the per-tick Reality-Work power Pexist = Mc²/t̃, as derived in the companion inertia paper (10.5281/zenodo.20059780). Hamilton's principle, Newton's second law, E = Mc², and the action quantum h are therefore four readings of one substrate count, not four independent postulates. Status legend used in the paper. ✓ consistent with established experimental or mathematical physics; ⋆ useful QTT consequence or rule of thumb; ⋆⋆ QTT-novel structure or result; ⋆⋆⋆ potential paradigm-shifter if confirmed. Falsifiers. The paper is exposed to experiment in nine concrete channels , spanning ordinary-mechanics tests, A7U graph residuals, action-quantum closure rules, Feynman propagator pole structure, and the unitary measure of the per-tick kernel. Main status labels: GREEN: Σ-OPEN-ZENODO-RECORD-PUBLISHED GREEN: Σ-CONCEPT-DOI-STABLE-CITATION GREEN: Σ-VERSION-DOI-RECONSTRUCTION-AVAILABLE SCOPE: Σ-CLAIM-STATUS-DEFINED-IN-PAPER Audit and claim scope: Concept DOI is the stable citation target for this research family. Version DOI and Zenodo record ID identify this exact public release for reconstruction. This description is a record-level guide; the paper itself controls theorem status, assumptions, and falsifiers. Related QTT anchors: Main book concept DOI: 10.5281/zenodo.17527179 Corpus Tree row: quantumtraction.org/doi-map/#record-20098144 Blog Map row: quantumtraction.org/blog-map/#corpus-record-20098144 Derivation Atlas: quantumtraction.org/derivation-atlas/ Lexicon: quantumtraction.org/lexicon/ Related QTT DOI (isSupplementTo): 10.5281/zenodo.17527179 Related QTT DOI (cites): 10.5281/zenodo.20059779 Related QTT DOI (cites): 10.5281/zenodo.20042843 Related QTT DOI (cites): 10.5281/zenodo.20057430 Related QTT DOI (cites): 10.5281/zenodo.20045306 Related QTT DOI (cites): 10.5281/zenodo.20051961 Related QTT DOI (cites): 10.5281/zenodo.20042421 Related QTT DOI (isCompiledBy): 10.5281/zenodo.20657182 Included files: qtt_least_action_a7u_v2_01.pdf qtt_least_action_a7u_v2_01.tex
Full bilingual text and framework available at self-as-an-end.net The Born rule — that the probability of a measurement outcome is given by the modulus squared of the quantum state — is introduced as a postulate in standard quantum mechanics and confirmed by experiment to very high precision, yet the origin of its modulus-squared form has long been a central question in the foundations of quantum mechanics. The major approaches locate this origin in the mathematical uniqueness of Hilbert space (Gleason), the decision-theoretic norms of a rational agent (Deutsch–Wallace), the symmetry of system–environment entanglement (Zurek's envariance), subjective belief (QBism), or hidden variables (Bohmian mechanics). This paper gives an ontological origin for the modulus-squared readout within the Self-as-an-End (SAE) framework. The central claim: the modulus squared is the one-sided reading of a pre-closure state that lies below the causal slot, takes the uncertainty-wave form, and is simultaneously subject to time arrows on both sides — and is therefore without causality. In this region the time arrow and causality are distinct: the time arrow exists at every scale (the quantum evolves forward in our frame, the dynamical layer), whereas causality is the influence of a single-direction time arrow. Above the causal slot the state is subject only to single-direction influence, so causality holds and the outcome is definite; below the causal slot it is subject simultaneously to the our-frame and opposite-frame bidirectional time arrows (two faces of one dual-4DD structure, per Cosmo V), so no single direction can fix causation and there is no causality. A one-sided reading therefore cannot read off pre-written definite content, and presents as a probability distribution (degenerating to a definite P=1 readout for an eigenstate). At the amplitude layer this manifests as the conjugate pairing of dual orientations, ψ♯ψ = A² — the modulus squared. The paper works at three commitment levels: T1 (Gleason-type mathematical uniqueness, taken over externally), T2 (a strong structural derivation of the quadratic readout, the core contribution — not a full Gleason-equivalent theorem), and T3 (an ontological mechanism picture, offered as a candidate). It answers why the both-sided reading is multiplicative rather than additive: reciprocity prescribes that one side close with its dual in the native algebra of the object read — a scalar density closes additively (Λ₁+Λ₂=0), a complex phase carrier multiplicatively (ψ♯ψ=A²); the two substrates are isomorphic (both θ+θ♯=0), the microscopic one wearing one coat of exponential dualization. On this basis it places Born randomness (no pre-written content; weights from dual-orientation pairing), demarcates SAE from Gleason, Many-Worlds (via a causality firewall), QBism, and Bohmian mechanics, and interfaces with P4's cross-ladder readout, P5's Tsirelson bound, and no-signaling marginals. Whether the Born rule is modified in the strong-field regime is marked as an open problem. 玻恩规则——测量结果的概率由量子态的模平方给出——在标准量子力学中作为公设引入,由实验以极高精度确认,但其模平方形式的来源长期是量子力学基础的核心议题。主流进路分别把这一来源放在 Hilbert 空间的数学唯一性(Gleason)、理性主体的决策规范(Deutsch–Wallace)、系统-环境纠缠的对称性(Zurek 的 envariance)、主观信念(QBism)或隐变量(Bohmian)之上。 本文在 SAE(Self-as-an-End)框架内,为模平方这一读出形式给出一个本体来源。核心论点是:模平方是因果槽以下、测不准波形态、同时受双侧时间箭头影响、因而无因果律的闭合前态的单侧读数。在此区域,时间箭头与因果律是两件不同的事:时间箭头在任何尺度都存在(量子在本侧沿时间箭头正常向前演化,动力学层),而因果律的本体内容是单向时间箭头的影响。因果槽以上只受本侧单向影响,故有因果律、呈现确定;因果槽以下同时受 our frame 与 opposite frame 双向时间箭头影响(二者同为同一 dual-4DD 结构的两面,Cosmo V 既立),无单一方向可定因果,故无因果律。正因量子同时受双侧影响、无因果律可定向,单侧读出读不到预先写就的确定局部内容,而呈现为概率分布(若已是某测量的本征态,则退化为 P=1 的确定读出)。在振幅层,这表现为对偶取向的共轭配对 ψ♯ψ = A²,即模平方。 本文区分三层强度:T1(Gleason 类数学唯一性,外部承接)、T2(二次型读出的强结构推导,核心交付——非 Gleason 等价完整定理)、T3(本体机制图像,作为候选)。本文回答了为何双侧读出取乘性而非加性:互易规定一侧须与其对偶在被读对象的原生代数中闭合——标量密度加性闭合(Λ₁+Λ₂=0),复相位载体乘性闭合(ψ♯ψ=A²);两层底座同型(皆 θ+θ♯=0),微观层只多一层指数对偶化。据此本文给出玻恩随机性的定位(无预写内容,权重由对偶取向配对给出),划定 SAE 与 Gleason、Many-Worlds(以因果律防火墙)、QBism、Bohmian 的关系,并接合 P4 跨阶梯读出、P5 Tsirelson 上界与无信号边缘分布。强场下玻恩规则是否有修正,标记为开放问题。 Related identifiers (Related/alternate identifiers) All "is referenced by / references" relations to established SAE works. Use relation type references (this paper references them); for the QM-series siblings, isPartOf the series is also appropriate where Zenodo supports it. Relation Identifier (DOI) Resource references 10.5281/zenodo.20277037 QM P2 references 10.5281/zenodo.20340595 QM P3 references 10.5281/zenodo.20369138 QM P4 (Tunneling) references 10.5281/zenodo.20455398 QM P5 references 10.5281/zenodo.19361950 Physics Foundation (cross-layer closure) references 10.5281/zenodo.19341043 Four Forces Prequel references 10.5281/zenodo.19464447 Four Forces Finale references 10.5281/zenodo.19245267 Cosmo I (cosmological constant) references 10.5281/zenodo.19329771 Cosmo V (dual-frame Ġ/G) references 10.5281/zenodo.19836185 Relativity P1 (time dilation) references 10.5281/zenodo.19910546 Relativity P2 (motion/gravity unification) references 10.5281/zenodo.19740020 Information Theory P1 (4DD ontology) references 10.5281/zenodo.19880112 Information Theory P4 (black-hole information) references 10.5281/zenodo.19510868 Mass Series Convergence (E/c³) (External standard references — Born 1926, Gleason 1957, Busch 2003, Deutsch 1999, Wallace 2012, Zurek 2005, Fuchs–Mermin–Schack 2014, Bohm 1952, Dürr–Goldstein–Zanghì 1992, Tsirelson 1980, Popescu–Rohrlich 1994 — are listed in the paper's reference section; they need not all be entered as Zenodo related identifiers, though DOIs may be added if desired.)