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

Related to: https://doi.org/10.5281/zenodo.20657182 (The Artian Lagrangian Framework: A Finite Action Ledger for Quantum Traction Theory, from Completed Address Events to Laboratory Least Action)

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 = dS_tot/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 e^iS/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 R_J(θ) = cos θ·I + sin θ·J with J² = −I. The complex shorthand e^iθ and the real rotor R_J(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, ΔS_{one 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(−S_E/ℏ) are computational tools, not the ontic QTT primitive (§14).
• The capacity-regularized covariant kernel 𝒦_ℓ̃ = K(−ℓ̃² D_J^⊤ D_J) as a substrate UV cutoff (§15).
• The dial-covariant Noether current j_J^μ 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 N_b 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 = N_bm_*·ẍ, (ii) the gravitational source-density coefficient in ∇·g = −4πGρ with G = ℓ̃²c³/ℏ, and (iii) the per-tick Reality-Work power P_exist = 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.