The Hartman eectthe saturation of quantum tunnelling group delay with bar- rier thicknesshas been observed experimentally in photonic, acoustic, and atomic systems but remains without a rst-principles geometric explanation. We derive the Hartman eect natively from the spectral anatomy of the C4 gauge bridge on the 4.8.8 Archimedean lattice, extending the continuous-discrete interface results of Part 21. Six theorems are established. The Chebyshev transfer matrix across iterated C4 bridges yields hyperbolic saturation in the stop band, providing the exact discrete origin of the tanh(κL) dependence observed by Longhi et al. in bre Bragg gratings. The C4 resolvent gives Winful's stored-energy cavity interpreta- tion a rigorous topological foundation. The Hybrid Form Factor factorisation man- dates a bipartite separation of tunnelling time into interface and bulk components, matching the universal separable structure reported in recent weak-measurement experiments. The saturated Hartman delay deep inside the bandgap evaluates to exactly 6 algorithmic clock ticks (in units of Λ−1 QCD), an exact integer arising from the cancellation of irrational golden-ratio eigenvalues on the path graph P4. This identies the low-energy tunnelling delay as the o-shell tail of the ρ(770) resonance, unifying quantum tunnelling with the vector meson sector. A parameter-free pre- diction is derived: group delay divergences at the silver ratio band edges δ±n S /4, with successive divergence energies in the dimensionless ratio δ2 S = 3 + 2√2 ≈5.83. 2026-06-20 legacy canon revision: This is a canon-reconciled legacy version. Older part-series note; needs canon alignment The paper retains its historical derivation trail but carries a 2026-06-20 canon revision note identifying current status and superseded claims.
The Hartman eectthe saturation of quantum tunnelling group delay with bar- rier thicknesshas been observed experimentally in photonic, acoustic, and atomic systems but remains without a rst-principles geometric explanation. We derive the Hartman eect natively from the spectral anatomy of the C4 gauge bridge on the 4.8.8 Archimedean lattice, extending the continuous-discrete interface results of Part 21. Six theorems are established. The Chebyshev transfer matrix across iterated C4 bridges yields hyperbolic saturation in the stop band, providing the exact discrete origin of the tanh(κL) dependence observed by Longhi et al. in bre Bragg gratings. The C4 resolvent gives Winful's stored-energy cavity interpreta- tion a rigorous topological foundation. The Hybrid Form Factor factorisation man- dates a bipartite separation of tunnelling time into interface and bulk components, matching the universal separable structure reported in recent weak-measurement experiments. The saturated Hartman delay deep inside the bandgap evaluates to exactly 6 algorithmic clock ticks (in units of Λ−1 QCD), an exact integer arising from the cancellation of irrational golden-ratio eigenvalues on the path graph P4. This identies the low-energy tunnelling delay as the o-shell tail of the ρ(770) resonance, unifying quantum tunnelling with the vector meson sector. A parameter-free pre- diction is derived: group delay divergences at the silver ratio band edges δ±n S /4, with successive divergence energies in the dimensionless ratio δ2 S = 3 + 2√2 ≈5.83. 2026-06-20 legacy canon revision: This is a canon-reconciled legacy version. Older part-series note; needs canon alignment The paper retains its historical derivation trail but carries a 2026-06-20 canon revision note identifying current status and superseded claims.
The time a quantum particle spends traversing a classically forbidden region is among the most persistently contested observables in quantum theory, its definition splintering across the Wigner phase time, the Büttiker–Landauer clock, and complex-time path-integral constructions. The debate has centered almost exclusively on a single number—the mean delay—and on the Hartman effect, whereby that mean saturates to a width-independent constant for an opaque barrier, suggesting an unbounded and apparently superluminal traversal speed. Yet an attoclock measures not a number but a distribution: the observable imprinted on the photoelectron spectrum is a temporal spread, not a delay. The narrow-band limit of this variance is fixed by the time-energy uncertainty relation alone, Var(τ)→ℏ2/4ΔE2, a general property of any finite-bandwidth wave packet. What has not been shown is that this floor survives intact through a barrier deep in the Hartman-opaque regime, where the mean delay has already saturated and where naive intuition expects the pulse to broaden. We prove, for the rectangular barrier, that the transmitted variance remains locked onto this incident-packet value independently of the barrier width d—a spectral-pinning theorem. Equivalently, the dimensionless product Var(τ)ΔE2 collapses onto a single universal plateau shared by all bandwidths, a data collapse confirmed numerically to better than 0.5% across two decades of opacity and verified against the predicted inverse-square bandwidth scaling Varnarrow/Varbroad=(ΔEbroad/ΔEnarrow)2 to within a few percent, the small residual excess (28.56 versus 25) being itself a quantitative signature of barrier-induced reshaping. The genuine new result is the finite-bandwidth correction responsible for this excess: the opaque barrier acts as a coherent spectral filter—a tunneling analogue of the gain-narrowing mechanism familiar from ultrafast optics—that reweights the transmitted band and causes the variance to decrease weakly with opacity [log-log exponent α≈+0.070 in the convention Var∝(κd)−α], while the mean arrival time is pushed above the Hartman plateau. We derive the leading filter coefficient λ2 in closed form, verify it against an independent, exactly solvable smooth barrier (Eckart), and connect both the pinned floor and its filtering correction to experiment through the attoclock map σθ≃ωστ. The result recasts the tunneling-time question from the fragile arithmetic of a saturating mean to the robust, universal statistics of a spectrally pinned—and quantifiably filtered—width.
I show that the MacColl-Hartman effect, namely, the saturation of the group delay time of sub-barrier quantum tunneling as a function of the barrier width, comes from the saturating behavior of a more fundamental concept - the phase of the stationary wave function. The explanation of saturation is given based on the decomposition of the stationary wave function into the spectrum of wave numbers and formulation of the initial condition for the direction of propagation of the incident matter wave. It is also shown that the saturation plateau of MacColl and Hartman actually doesn't continue indefinitely, but has a finite length. After the plateau, the sub-barrier tunneling time monotonically increases with increasing width of the potential, and this applies both to the maximum's of the wave packet and to the average tunneling time.
A structural operator-theoretic account of negative group delays and post-selected excitation times in resonant atomic media. Within a scalar Lorentzian propagation model, the excitation time and the optical delay are shown to be parameter derivatives of the same frequency-resolved transmission amplitude — the first with respect to the atomic transition frequency, the second with respect to the optical frequency. Because the internal resolvent R = (z − ω)⁻¹, z = ω₀ − iΓ/2, depends on these only through their difference, the two are linked by an exact identity. For a fixed detected frequency, T_T(ω) = Re[i ∂_{ω₀} ln t_med(ω)] (Theorem A); its difference from the local Wigner delay is exactly A Re R(ω), which vanishes at line centre (Corollary B). For an unresolved detector, coarse-graining gives T_T = t_g^det whenever the detected spectral weight is even in detuning — a condition of symmetry, not of bandwidth (Theorem C). Negativity is treated separately from the equality. Causality is established from the analyticity of the retarded response and the full two-pole susceptibility: negative delays are properties of the pulse centroid, and the front velocity remains c exactly. Appendices cover departures from the idealised model, and a Python script reproduces all quoted numerics. All identities are exact within the stated first-order propagation model. Version note. Supersedes the May 2026 deposit. The central derivation has been replaced: the earlier conditional-probability argument is withdrawn. Keywords: post-selected excitation time, weak value, group delay, resolvent, Feynman–Hellmann identity, Lorentzian resonance, anomalous dispersion, front velocity, causality, quantum optics
A structural operator-theoretic account of negative group delays and post-selected excitation times in resonant atomic media. Within a scalar Lorentzian propagation model, the excitation time and the optical delay are shown to be parameter derivatives of the same frequency-resolved transmission amplitude — the first with respect to the atomic transition frequency, the second with respect to the optical frequency. Because the internal resolvent R = (z − ω)⁻¹, z = ω₀ − iΓ/2, depends on these only through their difference, the two are linked by an exact identity. For a fixed detected frequency, T_T(ω) = Re[i ∂_{ω₀} ln t_med(ω)] (Theorem A); its difference from the local Wigner delay is exactly A Re R(ω), which vanishes at line centre (Corollary B). For an unresolved detector, coarse-graining gives T_T = t_g^det whenever the detected spectral weight is even in detuning — a condition of symmetry, not of bandwidth (Theorem C). Negativity is treated separately from the equality. Causality is established from the analyticity of the retarded response and the full two-pole susceptibility: negative delays are properties of the pulse centroid, and the front velocity remains c exactly. Appendices cover departures from the idealised model, and a Python script reproduces all quoted numerics. All identities are exact within the stated first-order propagation model. Version note. Supersedes the May 2026 deposit. The central derivation has been replaced: the earlier conditional-probability argument is withdrawn. Keywords: post-selected excitation time, weak value, group delay, resolvent, Feynman–Hellmann identity, Lorentzian resonance, anomalous dispersion, front velocity, causality, quantum optics
Important status notice: This version has been superseded by v2.0. Subsequent work shows that the single-domain three-diagonal construction used here produces an experimentally excluded common-mode photon-speed anisotropy. Readers should consult the revised version and 10.5281/zenodo.21464971. We propose that conservation of probability — the single axiom A=1 — is sufficient to derive the principal results of quantum mechanics without additional postulates. On a bipartite three-dimensional causal lattice T³_diamond, each particle is represented as a causal session: an independent phase oscillator that propagates amplitude across the lattice subject to |ψ_R|² + |ψ_L|² = 1 at every tick. No Hamiltonian and no measurement postulate are assumed; a Hilbert-space structure with unitary evolution emerges from the U(1) oscillator and the per-session amplitude constraint rather than being postulated a priori. From this single constraint, the paper derives: (i) the Dirac equation in the continuum limit a→0, with the gamma matrices identified as geometric encodings of the bipartite RGB/CMY sublattice geometry; (ii) wave-particle duality and the Born rule as consequences of amplitude redistribution under A=1, with Gleason's theorem securing the |ψ|² uniqueness from the framework's emergent Hilbert space; (iii) quantization of atomic orbitals as Arnold tongue attractors of the two-body phase dynamics, recovering the hydrogen Bohr radius to four significant figures (k_min = 0.0970 vs k_Bohr = 0.0971) without fitting; (iv) gravity as a clock-density gradient rather than spacetime curvature — a refraction of causal paths through regions of higher session density, reproducing the 1/r² force law with an explicit weak-field metric tensor g_μν as a functional of the clock-density field; (v) photon emission as a structural consequence of A=1: the Coulomb interaction is a probability source that a single session cannot accommodate, mandating creation of a new photon session at orbit lock-in; (vi) the structural form of the induced gauge action (Sakharov / Zeldovich), whose leading-order term has the Maxwell form on O_h-averaged observables, with a closed-form gauge-sector birefringence sharing the same optical axis (1,1,−1) as the kinematic-sector prediction. Energy and momentum conservation are not independent inputs — they are consequences of A=1 applied to closed session sets. The numerically tractable results are verified or partially verified by a suite of more than fifteen experiments with no free parameters beyond the Planck-scale calibration of the lattice spacing; an audit table records the status of every claim. The paper conjectures that the Standard Model gauge group SU(3) × SU(2) × U(1), together with Lorentz SO(3,1), is the automorphism group of the framework's extended per-site amplitude — a precise 18-dimensional Lie-algebra equality. The existing framework is shown to carry SO(3,1) × U(1) (dim 7) on the per-site (ψ_R, ψ_L) amplitude; the remaining factors require structural extensions (per-site C² isospin and C³ colour indices) that are the subject of a companion paper. Six computational scaffolding scripts establish the conjecture's components. This is Paper I of the A=1 Discrete Causal Lattice series. Companion papers in preparation address the Standard Model gauge derivation, the Farey-sequence mass spectrum (and its Veneziano-amplitude limiting case), plasma and gradient-ionization phenomenology, the proton-interior structure, quantum measurement theory, and a Hilbert-6th-style combinatorial foundation for continuum physics. What changed in v1.0 (since v0.98-RC): * Three new audit-table rows: long-horizon escape mechanism (PART), lock-in resonance grid-independence across {65, 81, 97, 113}³ grids (PART), and the Standard Model gauge-group conjecture as a precise 18-dimensional Lie-algebra equality (STUB).* A Gleason-uniqueness subsubsection in the Born-rule discussion, addressing the necessity question for the |ψ|² measure.* An explicit metric-tensor formula in the gravity section, identifying g_μν as a functional of ρ_clock in the weak-field limit.* Refined automorphism conjecture with explicit decomposition into established and structural-extension factors.* New Acknowledgments section. Source code, paper sources, and data files: https://github.com/JackDMenendez/discrete-causal-latticeLicense: Paper text and figures CC BY 4.0; source code MIT.
This work presents a complete formalism of the Alpha-Triad unified theory, which for the first time achieves a consistent integration of three ontological levels of reality: the Quantum System (QS), the Photonic System (PS), and the Information Interface (IF). The unifying principle is the decoherence parameter DD, defined as the ratio of information destruction rate to information recovery rate, and the Quantum of Energy Information (KEI) — a fundamental action quantum linking energy, frequency, and information. The «Resonance Group» (Russia) continues its systematic assembly of a coherent physical picture of the world, in which space, time, matter, and consciousness are manifestations of a single geometric structure — the standing-wave O-Matrix. A fundamental spatial constant is introduced — the O-length LO=0.036LO=0.036 fm — which acts as a quantum of space at the subnuclear level. The central experimental discovery presented here is the detection of a fundamental time step Tphoton≈5.71Tphoton≈5.71 ms in photon-quantum interface dynamics. This step emerges as exactly one-third of the quantum clock period Tref=17.5Tref=17.5 ms (corresponding to the reference frequency fref=400/7≈57.142857fref=400/7≈57.142857 Hz) upon reflection from the standing-wave O-Matrix mirror. The relation Tphoton=Tref/3Tphoton=Tref/3 carries profound physical meaning: it demonstrates that the photonic level is not an independent entity but a secondary harmonic of the quantum level, generated by the topological resonator — the O-Matrix. This discovery for the first time allows experimental observation of the discrete structure of time at a macroscopic scale, linking quantum mechanics and optics via the information interface. Three fundamental problems, considered unsolvable within a single framework in standard physics, are resolved geometrically: Geometric derivation of the fine-structure constant α≈1/137α≈1/137. It is shown that this dimensionless constant arises as a topological ratio of the classical electron radius to the O-length, with a spherical phase correction 2π2π, yielding α−1=137.036α−1=137.036 in exact agreement with experiment. Identification of the neutrino as a coherent 3-KEI state. Using the radius equation R(N)=R0−LO⋅(N−1)R(N)=R0−LO⋅(N−1), it is demonstrated that at N=3N=3 the particle enters a state of total phase coherence (D→0D→0), removing it from classical electromagnetic interaction. This geometrically explains neutrino weak interaction, small mass, and oscillatory properties. Establishment of the holographic link between the O-length and the Planck scale. The topology stretching coefficient is derived: KO=LO/lP≈2.227×1018KO=LO/lP≈2.227×1018. Gravity is interpreted not as a fundamental force but as a macroscopic entropic force arising from the gradient of KEI density: g=∇ρKEI⋅(1−1/KO)g=∇ρKEI⋅(1−1/KO). The theory introduces an extension of the Periodic Table, predicting 41 772 new structural units, termed «Infotons» , which are combinations of quantum, photonic, and information states. A fundamental information conservation law is formulated as a recursive phase-memory cycle: Ψ(t)=Ψ(t−T)⊗ΔΨ(t),T=16fref,Ψ(t)=Ψ(t−T)⊗ΔΨ(t),T=fref16, and the entire Alpha-Triad obeys the normalization: QS (1/3), PS (2/3), IF (3/3), where at D→0D→0 a identity folding 1/3+2/3=3/31/3+2/3=3/3 occurs, corresponding to holistic Mind. The Self-Unification Principle of Mind postulates that Mind is not a sum of components (reductionism) but rather the topological mirror of the O-Matrix, simultaneously serving as source and receiver. At D→0D→0: limD→0(QS∪PS)≡IF≡Mind.D→0lim(QS∪PS)≡IF≡Mind. This ontological claim extends the theory beyond physics into the domain of a unified science of being. Three experimental protocols are proposed, and the first — direct measurement of the 5.71 ms step — has already yielded positive results, making this work not merely a theoretical construct but an empirically grounded paradigm. This publication continues the series of works by the «Resonance Group» aimed at building a unified physical reality based on the Alpha-Triad and O-Matrix, and invites the international scientific community to independent experimental verification and theoretical reflection on the proposed paradigm.