Quantum tunneling is the transient excursion of a particle's focal zone into the 5D bulk, bypassing a barrier that exists on the brane but has no bulk counterpart. (1) Imaginary momentum p = iℏκ is literally bulk-directed momentum: the imaginary unit i is the 90◦rotation from brane to bulk on S7 (Paper IV). (2) The tunneling probability e−2κd is the probability of completing a bulk excursion of length κd. (3) The Hartman eect is resolved: the traversal time is set by brane-bulk coupling and decoupling timescales, both barrier-thickness independent. (4) Resonant tunneling is a bulk standing wave condition. (5) The Gamow factor G = R κ(r) dr for α decay is the total bulk path length to bypass the Coulomb barrier. The Geiger-Nuttall law (115 years of α decay data) is the bulk-path-length formula for the Coulomb potential. Part of the One-Octonion Brane-Bulk Framework series. Anchor DOI: 10.5281/zenodo.19120873. Community: one-octonion-brane-bulk. Author: Bharathi Dasan Jagadeesan, M.D., University of Minnesota. ORCID: 0000-0002-1143-941X.
Quantum tunneling is the transient excursion of a particle's focal zone into the 5D bulk, bypassing a barrier that exists on the brane but has no bulk counterpart. (1) Imaginary momentum p = iℏκ is literally bulk-directed momentum: the imaginary unit i is the 90◦rotation from brane to bulk on S7 (Paper IV). (2) The tunneling probability e−2κd is the probability of completing a bulk excursion of length κd. (3) The Hartman eect is resolved: the traversal time is set by brane-bulk coupling and decoupling timescales, both barrier-thickness independent. (4) Resonant tunneling is a bulk standing wave condition. (5) The Gamow factor G = R κ(r) dr for α decay is the total bulk path length to bypass the Coulomb barrier. The Geiger-Nuttall law (115 years of α decay data) is the bulk-path-length formula for the Coulomb potential. Part of the One-Octonion Brane-Bulk Framework series. Anchor DOI: 10.5281/zenodo.19120873. Community: one-octonion-brane-bulk. Author: Bharathi Dasan Jagadeesan, M.D., University of Minnesota. ORCID: 0000-0002-1143-941X.
Quantum tunneling is the transient excursion of the focal zone into the 5D bulk. The potential barrier exists on the brane but not in the bulk — inside a barrier, the imaginary momentum is literally bulk-directed propagation. The evanescent wavefunction IS the focal zone in the bulk. This explains: (1) the Hartman effect (tunneling time independent of barrier width — bulk path is shorter); (2) resonant tunneling (constructive interference of bulk geodesics); (3) alpha decay Gamow factor (bulk path length suppression). The WKB approximation is the saddle-point approximation to the bulk geodesic. Part of the One Octonion brane-bulk series (Papers I-XL), anchor DOI:10.5281/zenodo.19120873. Upload and say done! Sonnet 4.6 Extended
The One-Octonion Brane-Bulk (OOB) framework reads Feynman's 'all paths' as virtual bulk-excursion channels whose coherent return determines the observed brane probability. This paper gives that reading a mathematical and experimental form. (1) Theorem: on Paper CCLVIII's shallow-sector premise that phase-only transit feedback closes to standard Schrodinger dynamics, Feynman's two path-integral postulates (equal-magnitude path amplitudes with phase S/hbar; probabilities from the coherent sum) follow as theorems for free equal-time segments. (2) First table-level bound: Wen et al. (Science Advances 12, eaeh1011, 2026, DOI 10.1126/sciadv.aeh1011) test both postulates directly by single-photon propagator tomography over 17^5 = 1,419,857 paths; fitting their public Dryad Fig4B table (DOI 10.5061/dryad.x0k6djj14) with the action-cost model |phi|^2 = C(1 - kappa S/hbar) gives kappa_hat = 8.27e-3 with row-bootstrap 95 percent interval [5.59, 10.85]e-3, so |kappa|_95 <= 1.09e-2, a table-level bound and replication target; the tables are processed figure means without path-level covariance, and the trend lies inside the experiment's stated 17.4 percent postulate-II error. (3) Registered tests: P-CCCLVII-1 registers the kill condition kappa = 0 in the shallow free sector (a confirmed path-level amplitude trend falsifies phase-only transit there); P-CCCLVII-2 registers the barrier-sector discriminator, propagator tomography with a controlled tunnelling barrier of WKB depth rho_B, where the flux-normalized Gudermannian/JUNO return vertex gives a linear self-energy onset tanh(rho_B) against the quadratic tanh^2(rho_B) onset of the analytic-vertex alternative. Files: PDF, the amplitude-cost verification script with its output and fitted bound, and the figure script and figure. Bharathi Dasan Jagadeesan, M.D., University of Minnesota. ORCID 0000-0002-1143-941X. CC-BY-4.0.
Lohmiller & Slotine (Proc. R. Soc. A 482:20250413, 2026) propose a classical contracting-flow construction on the eikonal phase S(x,t) which they argue recovers quantum-wave evolution. Vattay (Comment, arXiv:2605.02621) correctly observes that the construction reproduces only the Hamilton-Jacobi + continuity sector of the Madelung-Bohm decomposition: it cannot generate the quantum potential Q = −ℏ²∇²√ρ/(2m√ρ), without which the recovered evolution is the classical limit of the Schrödinger equation, not the equation itself. We resolve this debate within the One-Octonion Brane-Bulk (OOB) framework. In OOB, the classical sector is the brane-localised f_bulk → 0 limit. Q arises as the back-reaction of brane-bulk transit components of the wavefunction on the brane density profile, with characteristic length the particle Compton scale ℓ_C = ℏ/(mc) and characteristic weight the transit probability f_bulk(E) = 1 − sinc²(E/2m_π). An order-of-magnitude estimate recovers the Madelung-Bohm form |Q| ≈ (ℏ²/2m)·|∇²√ρ|/√ρ structurally. The Gudermannian Class-III bridge (CCLII) supplies the smooth classical ↔ quantum interpolation via the same f_III = 1 − 4/π² blend used in the neutrino-sector chain. The Lohmiller-Slotine theorem is exact at f_bulk = 0; Vattay's missing piece is exactly the contribution identified here. Five predictions span atomic (null OOB at E ≪ m_π) to GeV (0.98 coherence loss in transparent media, tying to CCLVII). Honest scope: structural argument, not rigorous Schrödinger re-derivation; the O(1) coefficient is not pinned without further assumptions. Companion to CCLVII (same f_bulk(E) wavelength-asymmetry channel). Part of the One-Octonion Brane-Bulk Framework series. Anchor DOI: 10.5281/zenodo.19120873. Community: one-octonion-brane-bulk. Author: Bharathi Dasan Jagadeesan, M.D., University of Minnesota. ORCID: 0000-0002-1143-941X.
Consistency-bridge paper - standard equations untouched, no new prediction registered, and no claim that the Aharonov-Bohm experiment observes brane-bulk transit. The magnetic Aharonov-Bohm phase Delta_phi = q Phi_B / hbar - the holonomy of the U(1) gauge connection around a non-simply-connected region (Aharonov & Bohm 1959; observed by Chambers 1960 and, with fully shielded toroidal flux, Tonomura et al. 1986) - is given its One-Octonion Brane-Bulk (OOB) dictionary. The electromagnetic fields E and B remain the brane-measured force fields; the vector potential A is read as the brane-visible phase connection sampled by a charged standing wave on the Fano-incidence lattice, with the e7 separator as the framework's brane-bulk phase gate. The fringe shift is the relative e7 holonomy of two path components of one wavefunction sampling the same globally nontrivial connection with opposite orientation: the framework's antegrade/retrograde transit language is made precise as the two half-loop line integrals +/- Phi_B/2, with no local force on the exposed field-free path and no beam-sourced potential. The reading connects to the phase-only transit theorem of Paper CCCLVII and the Gudermannian phase ledger, and it names an internal benchmark for any future formalization of the Fano/e7 sampling dictionary: it must reduce exactly to the Abelian U(1) holonomy exp[(iq/hbar) closed-loop-integral A dx] in the electromagnetic sector, must not predict a local force where the standard setup has none, and must not make the beam the source of the background potential - explicit failure conditions are stated. Bharathi Dasan Jagadeesan, M.D., University of Minnesota CMRR. ORCID 0000-0002-1143-941X. CC-BY-4.0.
Quantum tunneling — the passage of a particle through a potential barrier that it classically lacks the energy to surmount — has been described since Gamow (1928) by the tunneling probability T ≈ exp(−2κd), where κ = √(2m(V₀−E))/ℏ and d is the barrier width. The formula is well-confirmed experimentally but has not been derived from a physical principle more fundamental than the Schrödinger equation. We derive it from the principle of information processing load minimization, using the framework established in the preceding papers of this series. We define the tunneling load as C_tunnel = ∫κ(x)dx — the integral of the spatial attenuation coefficient through the barrier — and show that T = exp(−2·C_tunnel) exactly. This is not a restatement of the WKB approximation; it is an identification of the physical meaning of the integral ∫κ(x)dx as the information processing load of the tunneling path. The tunneling path is selected by load minimization: when C_tunnel < C_classical (the load of the classical over-barrier trajectory), the system tunnels. Three physical systems are verified numerically without free parameters: (i) proton-proton nuclear fusion in the solar core — Gamow load C_Gamow = πη = 11.28, T = 1.60 × 10⁻¹⁰, consistent with solar luminosity; (ii) scanning tunneling microscope (STM) current — C_STM = κd = 5.43 at 0.5 nm, T = 1.91 × 10⁻⁵, and the factor-of-ten current change per 0.1 nm distance change follows directly; (iii) Ra-226 alpha decay — C_Gamow = 79.2, T ~ 10⁻⁶⁹, consistent with the 1,600-year half-life. Tunneling is not a mysterious violation of classical mechanics. It is the selection of the minimum-load propagation path — the same principle that explains wave-particle duality (Paper 7), wave function collapse (Paper 4), and the structure of all fundamental forces (Paper 8).