Derives the internal Hodge–Dirac action geometry and exact 2π ratio; physical normalization and a numerical xi-hbar value remain open. Governed release package · 0.1Xi-Hbar and Hodge–Dirac Action RecoveryDerives the internal Hodge–Dirac action geometry and exact 2π ratio; physical normalization and a numerical xi-hbar value remain open.StatusPARTIAL PASS + EXACT INTERNALPhysical promotion0Package files12Hash manifestVerified · 11 filesDownload immutable release ZIP ↓Package compositioncode1governance1manuscript1no_go1package root1report1results1theorem1verification4Show all 12 preserved pathsHASH_MANIFEST.jsoncode/verify_hd_action.pygovernance/UD_XI_HBAR_PRETARGET_EXCLUSION_RECEIPT_v0.1.jsonmanuscript/UD_JOP_v2.40_HD_ACTION_UNIT_ADDENDUM_v0.1.mdno_go/UD_XI_HBAR_PHYSICAL_NORMALIZATION_NO_GO_v0.1.mdreport/UD_XI_HBAR_HD_ACTION_RECOVERY_REPORT_v0.1.mdresults/UD_XI_HBAR_HD_ACTION_RESULT_v0.1.jsontheorem/UD_HD_RECURRENCE_SYMPLECTIC_ACTION_THEOREM_v0.1.mdverification/INDEPENDENT_NUMERIC_REPLAY.txtverification/PRIMARY_EXACT_REPLAY.txtverification/UD_XI_HBAR_HD_ACTION_VERIFICATION_v0.1.jsonverification/independent_numeric_replay.pyreport/UD_XI_HBAR_HD_ACTION_RECOVERY_REPORT_v0.1.mdUD ξℏ\xi_\hbar Source Recovery and Hodge–Dirac Action Audit v0.1 Result: PARTIAL_PASS / NEW EXACT INTERNAL THEOREM / PHYSICAL NORMALIZATION OPEN Physical promotion: 0 Objective Recover or derive the strongest current source-complete action-unit law from the existing finite Hodge–Dirac/receipt framework, explicitly excluding measured hh, mpm_p, GG, and α\alpha from derivation and candidate selection. Source audit The current ledger still classifies G06 ξℏ\xi_\hbar as SOURCE_STRATUM / OPEN because the historical operational PASS receipt is not reproducibly recovered. The current textbook also records reciprocal historical conventions for ξℏ\xi_\hbar, and states that native-to-physical unit mapping remains downstream of the Be/ξℏB_e/\xi_\hbar audit. The single-tetrahedron finite carrier, however, is source-complete enough to derive a native action geometry: 𝒜T=−𝒜\mathcal A^T=-\mathcal A; 𝒜2=−4(I−PH0)\mathcal A^2=-4(I-P_{H_0}); one harmonic zero mode and 14 nonharmonic modes; exact nonzero frequency magnitude 22; exact recurrence parameter π\pi; conserved positive chain support Sμ=∥c∥2S_\mu=\|c\|^2. New exact result On the nonharmonic carrier, JHD=𝒜/2 J_{\mathrm HD}=\mathcal A/2 is an orthogonal complex structure. It induces a canonical symplectic form and a recurrence circulation ℭHD(c)=π∥(I−PH0)c∥2. \boxed{ \mathfrak C_{\mathrm HD}(c)=\pi\,\|(I-P_{H_0})c\|^2. } The corresponding action variable is IHD(c)=12∥(I−PH0)c∥2. \boxed{ I_{\mathrm HD}(c)=\frac12\,\|(I-P_{H_0})c\|^2. } Hence ℭHD=2πIHD. \boxed{ \mathfrak C_{\mathrm HD}=2\pi I_{\mathrm HD}. } For unit nonharmonic support: ℭHD=π,IHD=1/2. \boxed{ \mathfrak C_{\mathrm HD}=\pi, \qquad I_{\mathrm HD}=1/2. } Governance ruling AUTO_LOCK — EXACT_INTERNAL orthogonal complex structure JHD=𝒜/2J_{\mathrm HD}=\mathcal A/2 on the 14-dimensional nonharmonic carrier; canonical symplectic form induced by JHDJ_{\mathrm HD} and the Euclidean chain norm; native recurrence circulation ℭHD=πS⊥\mathfrak C_{\mathrm HD}=\pi S_\perp; normalized action variable IHD=S⊥/2I_{\mathrm HD}=S_\perp/2; exact internal ratio ℭHD/IHD=2π\mathfrak C_{\mathrm HD}/I_{\mathrm HD}=2\pi. AUTO_FREEZE — unsupported stronger claims ξℏ=π\xi_\hbar=\pi, 1/π1/\pi, 1/21/2, 22, or any other numerical value from convention choice alone; identification of ℭHD\mathfrak C_{\mathrm HD} or IHDI_{\mathrm HD} with physical hh or ℏ\hbar; identification of this local recurrence action with G05/Q136 coarse obstruction action; reuse of the single-cell period π\pi as a universal physical clock. Main conclusion The historical ξℏ\xi_\hbar numerical receipt was not recovered. A new source-complete internal action theorem was derived instead. This closes the missing finite action geometry but proves that the physical ξℏ\xi_\hbar value remains underdetermined until a physical unit/correspondence map is independently supplied. manuscript/UD_JOP_v2.40_HD_ACTION_UNIT_ADDENDUM_v0.1.mdManuscript Addendum — Native Hodge–Dirac Recurrence Action Geometry The single filled tetrahedron admits an additional exact internal structure relevant to the action-normalization frontier. Let PH0P_{H_0} denote the projector onto the unique harmonic vertex mode. Since 𝒜2=−4(I−PH0), \mathcal A^2=-4(I-P_{H_0}), the restriction JHD=𝒜/2J_{\mathrm HD}=\mathcal A/2 to the 14-dimensional nonharmonic carrier is an orthogonal complex structure. Together with the Euclidean chain norm it defines the symplectic form ωHD(u,v)=⟨u,JHDv⟩. \omega_{\mathrm HD}(u,v)=\langle u,J_{\mathrm HD}v\rangle. For the closed flow ċ=𝒜c\dot c=\mathcal A c, the support functional ℋHD=∥c∥2\mathcal H_{\mathrm HD}=\|c\|^2 generates the flow under this symplectic form. The exact recurrence parameter is π\pi, and the associated closed-orbit symplectic circulation is ℭHD(c)=π∥(I−PH0)c∥2. \boxed{ \mathfrak C_{\mathrm HD}(c) = \pi\,\|(I-P_{H_0})c\|^2. } The conventional action variable obtained by division by 2π2\pi is IHD(c)=12∥(I−PH0)c∥2, \boxed{ I_{\mathrm HD}(c) = \frac12\,\|(I-P_{H_0})c\|^2, } so ℭHD=2πIHD\mathfrak C_{\mathrm HD}=2\pi I_{\mathrm HD} exactly. These quantities are dimensionless internal finite-carrier invariants. They do not by themselves identify a physical quantum of action and do not resolve the current ξℏ\xi_\hbar provenance gap. In particular, the theorem does not choose between the historical reciprocal conventions ℏ=ξℏ𝒜0\hbar=\xi_\hbar\mathcal A_0 and 𝒜0=ξℏℏ\mathcal A_0=\xi_\hbar\hbar, nor does it supply physical values for the burden, length, and time units entering 𝒜0=B0a02/τ0\mathcal A_0=B_0a_0^2/\tau_0. The action-unit bridge remains open while the native recurrence-action geometry is now exact. theorem/UD_HD_RECURRENCE_SYMPLECTIC_ACTION_THEOREM_v0.1.mdUD Hodge–Dirac Recurrence Symplectic-Action Theorem v0.1 Status: EXACT_INTERNAL / AUTO_LOCK candidate for append-only merge Physical promotion: 0 Source anchors For the filled tetrahedron chain carrier 𝒞=C0⊕C1⊕C2⊕C3,dim𝒞=15, \mathcal C=C_0\oplus C_1\oplus C_2\oplus C_3, \qquad \dim\mathcal C=15, the current source gives the real skew Hodge–Dirac generator 𝒜\mathcal A, the harmonic projector PH0P_{H_0}, and 𝒜T=−𝒜,𝒜2=−4(I−PH0),rank𝒜=14, \mathcal A^T=-\mathcal A, \qquad \mathcal A^2=-4(I-P_{H_0}), \qquad \operatorname{rank}\mathcal A=14, with exact nonzero frequency magnitude 22 and recurrence parameter srec=π. s_{\mathrm rec}=\pi. No measured hh, ℏ\hbar, mass, gravitational constant, or electromagnetic coupling enters this theorem. Theorem Let 𝒱:=(I−PH0)𝒞 \mathcal V:=(I-P_{H_0})\mathcal C be the 14-dimensional nonharmonic carrier, and define JHD:=12𝒜|𝒱. J_{\mathrm HD}:=\frac12\mathcal A\big|_{\mathcal V}. Then JHD2=−I𝒱,JHDT=−JHD,JHDTJHD=I𝒱. J_{\mathrm HD}^2=-I_{\mathcal V}, \qquad J_{\mathrm HD}^T=-J_{\mathrm HD}, \qquad J_{\mathrm HD}^T J_{\mathrm HD}=I_{\mathcal V}. Therefore JHDJ_{\mathrm HD} is an orthogonal complex structure on 𝒱\mathcal V. Define ωHD(u,v):=⟨u,JHDv⟩. \omega_{\mathrm HD}(u,v):=\langle u,J_{\mathrm HD}v\rangle. Then ωHD\omega_{\mathrm HD} is a nondegenerate skew bilinear form. With the internal Hamiltonian-shaped support functional ℋHD(c):=∥c∥2, \mathcal H_{\mathrm HD}(c):=\|c\|^2, the flow ċ=𝒜c \dot c=\mathcal A c is Hamiltonian on 𝒱\mathcal V in the convention ωHD(Xℋ,v)=dℋ(v), \omega_{\mathrm HD}(X_{\mathcal H},v)=d\mathcal H(v), because ωHD(𝒜c,v)=⟨2JHDc,JHDv⟩=2⟨c,v⟩=dℋ(v). \omega_{\mathrm HD}(\mathcal A c,v) = \langle 2J_{\mathrm HD}c,J_{\mathrm HD}v\rangle = 2\langle c,v\rangle = d\mathcal H(v). Let the radial primitive one-form be λc(v):=12ωHD(c,v),dλ=ωHD. \lambda_c(v):=\frac12\omega_{\mathrm HD}(c,v), \qquad d\lambda=\omega_{\mathrm HD}. For a nonharmonic orbit c(s)=es𝒜c0,0≤s≤π, c(s)=e^{s\mathcal A}c_0, \qquad 0\le s\le\pi, the support ∥c(s)∥2\|c(s)\|^2 is constant and λc(s)(ċ(s))=−∥c0∥2. \lambda_{c(s)}(\dot c(s))=-\|c_0\|^2. Hence the orientation-independent recurrence circulation is ℭHD(c0):=|∮λ|=π∥c0∥2. \boxed{ \mathfrak C_{\mathrm HD}(c_0) := \left|\oint\lambda\right| = \pi\,\|c_0\|^2. } For a general state c=h+c⊥c=h+c_\perp, with h=PH0ch=P_{H_0}c, the harmonic component is static and ℭHD(c)=π∥(I−PH0)c∥2. \boxed{ \mathfrak C_{\mathrm HD}(c) = \pi\,\|(I-P_{H_0})c\|^2. } The associated normalized action variable is IHD(c):=ℭHD(c)2π=12∥(I−PH0)c∥2. \boxed{ I_{\mathrm HD}(c) := \frac{\mathfrak C_{\mathrm HD}(c)}{2\pi} = \frac12\,\|(I-P_{H_0})c\|^2. } Therefore ℭHDIHD=2π \boxed{ \frac{\mathfrak C_{\mathrm HD}}{I_{\mathrm HD}}=2\pi } for every nonzero nonharmonic state. For unit nonharmonic support, ℭHD=π,IHD=12. \boxed{ \mathfrak C_{\mathrm HD}=\pi, \qquad I_{\mathrm HD}=\frac12. } Channel decomposition The nonharmonic carrier splits into the three orthogonal adjacent-degree exchange sectors 𝒱=𝒱01⊕𝒱12⊕𝒱23,dim(𝒱01,𝒱12,𝒱23)=(6,6,2), \mathcal V = \mathcal V_{01}\oplus\mathcal V_{12}\oplus\mathcal V_{23}, \qquad \dim(\mathcal V_{01},\mathcal V_{12},\mathcal V_{23})=(6,6,2), and the recurrence circulation is additive: ℭHD=π(∥c01∥2+∥c12∥2+∥c23∥2). \boxed{ \mathfrak C_{\mathrm HD} = \pi\left( \|c_{01}\|^2+\|c_{12}\|^2+\|c_{23}\|^2 \right). } Equivalently the complex dimensions are 3+3+1=73+3+1=7, matching the 14 real nonharmonic dimensions as 14=2⋅714=2\cdot7. This is typed chain geometry only; it is not a seven-particle or seven-dimension statement. Scope This theorem derives a dimensionless native recurrence circulation/action geometry. It does not derive a physical action unit, ℏ\hbar, hh, a physical clock, or a physical energy scale. no_go/UD_XI_HBAR_PHYSICAL_NORMALIZATION_NO_GO_v0.1.mdUD ξℏ\xi_\hbar Physical-Normalization No-Go v0.1 Status: LOCKED_NO_GO / AUTO_FREEZE stronger claim Physical promotion: 0 Current source state The governed source retains 𝒜0=B0a02τ0 \mathcal A_0=\frac{B_0a_0^2}{\tau_0} as a well-typed action-dimension object, while historical records use reciprocal conventions ℏ=ξℏ𝒜0 \hbar=\xi_\hbar\mathcal A_0 and 𝒜0=ξℏℏ. \mathcal A_0=\xi_\hbar\hbar. The current reproducible numerical/provenance receipt for ξℏ\xi_\hbar is not recovered. The new Hodge–Dirac theorem independently derives ℭHD=πS⊥,IHD=12S⊥,ℭHD=2πIHD. \mathfrak C_{\mathrm HD}=\pi S_\perp, \qquad I_{\mathrm HD}=\frac12 S_\perp, \qquad \mathfrak C_{\mathrm HD}=2\pi I_{\mathrm HD}. No-go statement The current finite source does not determine a unique physical ξℏ\xi_\hbar. Reason: the internal carrier supplies dimensionless recurrence geometry but does not yet supply all of: a physical burden/energy normalization for B0B_0; a physical length normalization for a0a_0; a physical time normalization for τ0\tau_0; a theorem identifying a specific internal recurrence circulation/action variable with the physical quantum of action; a convention theorem deciding which historical reciprocal ξℏ\xi_\hbar definition is canonical. Thus the implications ℭHD=π⇒ξℏ=π \mathfrak C_{\mathrm HD}=\pi \Rightarrow \xi_\hbar=\pi or IHD=12⇒ξℏ=12 I_{\mathrm HD}=\frac12 \Rightarrow \xi_\hbar=\frac12 are not licensed. They would insert a physical correspondence not derived by the finite theory. Likewise the local Hodge–Dirac recurrence action must not be silently identified with the separate coarse path-obstruction/action lane ΔLhist\Delta L_{\mathrm hist}, gloopnormalizedg_{\mathrm loop}^{\mathrm normalized}, or the source-open master loop weight κloopmaster\kappa_{\mathrm loop}^{\mathrm master}. Those objects have distinct carriers and provenance. Finally, the exact single-tetrahedron recurrence cannot be promoted to a universal action clock because the currently analyzed glued A/B/C complexes have no finite nonzero common Hodge–Dirac recurrence. What is newly closed The internal normalization ambiguity is narrowed to a clean statement: native full recurrence circulation=2π×native action variable. \boxed{ \text{native full recurrence circulation} = 2\pi\times \text{native action variable}. } This exact 2π2\pi relation is source-complete internally. The remaining ξℏ\xi_\hbar problem is now purely a unit/correspondence bridge, not an absence of internal action geometry. Reopen condition Reopen physical ξℏ\xi_\hbar only after an independently derived map supplies physical units and identifies the internal action object to be compared with ℏ\hbar, with the convention frozen before numerical target comparison.