Exact action–phase scale on the single filled-K₄ active Hodge sector, without physical promotion. UD Native Hodge Action–Phase Specific Scale Theorem v0.1 Status: EXACT INTERNAL / AUTO_LOCK Carrier: single filled K4K_4 active Hodge sector Depends on POC-1 response selector: NO Physical promotion: 0 1. Exact carrier The current single-cell Hodge carrier supplies P2=I−PH0=−𝒜24,Ω=(−𝒜2)1/2=2P2, P_2=I-P_{H_0}=-\frac{\mathcal A^2}{4}, \qquad \Omega=(-\mathcal A^2)^{1/2}=2P_2, and on the active sector J=𝒜Ω+=𝒜2,J2=−I. J=\mathcal A\Omega^+=\frac{\mathcal A}{2}, \qquad J^2=-I. For active state x=P2xx=P_2x, define a:=∥x∥2. a:=\|x\|^2. The exact lifted Hodge action is ℐHodge=12aΔϕ̃. \boxed{\mathcal I_{\mathrm Hodge}=\frac12a\,\Delta\widetilde\phi.} The native action-dimensional unit is 𝒜0=B0a02τ0. \boxed{\mathcal A_0=\frac{B_0a_0^2}{\tau_0}.} Define 𝒮native:=𝒜0ℐHodge. \mathcal S_{\mathrm native}:=\mathcal A_0\mathcal I_{\mathrm Hodge}. No physical ℏ\hbar enters these definitions. 2. Specific native action–phase scale For a>0a>0 and Δϕ̃≠0\Delta\widetilde\phi\neq0, define α𝒜:=𝒮nativeaΔϕ̃. \alpha_{\mathcal A}:= \frac{\mathcal S_{\mathrm native}}{a\,\Delta\widetilde\phi}. Then α𝒜=𝒜0(12aΔϕ̃)aΔϕ̃=𝒜02. \alpha_{\mathcal A} = \frac{\mathcal A_0(\frac12a\Delta\widetilde\phi)}{a\Delta\widetilde\phi} = \boxed{\frac{\mathcal A_0}{2}}. Hence α𝒜=12B0a02τ0. \boxed{\alpha_{\mathcal A}=\frac12\frac{B_0a_0^2}{\tau_0}.} This is exact, amplitude-independent, and action-dimensional. It is not yet physical ℏ\hbar. 3. Closed-cycle cross-check The single-cell active frequency is 22 in native parameter ss. One recurrence has s=π,Δϕ̃=2π. s=\pi, \qquad \Delta\widetilde\phi=2\pi. For winding n∈ℤn\in\mathbb Z, Δϕ̃n=2πn,ℐn=πna,𝒮native,n=πna𝒜0. \Delta\widetilde\phi_n=2\pi n, \qquad \mathcal I_n=\pi na, \qquad \mathcal S_{{\mathrm native},n}=\pi na\mathcal A_0. Therefore for every n≠0n\neq0, 𝒮native,naΔϕ̃n=𝒜02. \boxed{ \frac{\mathcal S_{{\mathrm native},n}}{a\Delta\widetilde\phi_n} = \frac{\mathcal A_0}{2}. } Clock arithmetic is important here: 2πn≡0(mod2π) 2\pi n\equiv0\pmod{2\pi} while 𝒮native,n\mathcal S_{{\mathrm native},n} is generally nonzero. Endpoint phase alone therefore cannot recover accumulated action. 4. Independent energy/frequency cross-check The canonical spectral Hamiltonian is HHodge(x)=12⟨x,Ωx⟩. H_{\mathrm Hodge}(x)=\frac12\langle x,\Omega x\rangle. On E2E_2, Ωx=2x\Omega x=2x, so HHodge=a. \boxed{H_{\mathrm Hodge}=a.} With native dimensional energy Enative:=𝒜0τ0a E_{\mathrm native}:=\frac{\mathcal A_0}{\tau_0}a and native dimensional time t=τ0st=\tau_0s, the Hodge phase rate is ϕ̃̇=2τ0. \dot{\widetilde\phi}=\frac2{\tau_0}. Thus Enativeaϕ̃̇=𝒜02. \boxed{ \frac{E_{\mathrm native}}{a\dot{\widetilde\phi}} = \frac{\mathcal A_0}{2}. } The action-integral and energy/frequency derivations therefore agree exactly. 5. Normalized-ray corollary For x̂=xa,∥x̂∥2=1, \widehat x=\frac{x}{\sqrt a}, \qquad \|\widehat x\|^2=1, 𝒮native[x̂]=𝒜02Δϕ̃. \boxed{ \mathcal S_{\mathrm native}[\widehat x] = \frac{\mathcal A_0}{2}\Delta\widetilde\phi. } One recurrence gives 𝒮spinor,recnative=π𝒜0, \boxed{\mathcal S_{\mathrm spinor,rec}^{\mathrm native}=\pi\mathcal A_0,} matching the existing native recurrence anchor. This is an explicit normalized representative of a projective ray; it does not prove physical support quantization. 6. Earlier amplitude no-go preserved The prior source result discrete receipts⇏a=1 \text{discrete receipts}\not\Rightarrow a=1 remains valid. The repair is not to assert a=1a=1, but to derive the intensive invariant 𝒮nativeaΔϕ̃=𝒜02. \boxed{ \frac{\mathcal S_{\mathrm native}}{a\Delta\widetilde\phi}=\frac{\mathcal A_0}{2}. } Therefore: EXACT INTERNAL: α𝒜=𝒜0/2\alpha_{\mathcal A}=\mathcal A_0/2; OPEN PHYSICAL BRIDGE: whether a normalized physical quantum-state correspondence identifies this scale with ℏ\hbar. 7. Scope This theorem is exact on the single-cell K4K_4 active sector. It does not automatically universalize to glued carriers with multiple frequencies, and it does not establish quantum identity, Born probabilities, physical time/energy, a continuum limit, or physical ℏ\hbar. Physical promotion=0. \boxed{\text{Physical promotion}=0.}