218 exact constructor checks in package; no empirical data or independent review claimed. Conditional support and observation contract. Physical promotion 0. Current session checkpoint.Markdown source · LaTeX source · Complete package1. Result and status The current finite-chain papers already specify what their amplitudes mean mathematically: real coefficients in an orthonormal oriented-simplex basis. They do not identify those coefficients with geometric densities, integrated fields, or electrical measurements. Shape alone does not supply that interpretation. This note proves a compatibility constraint and gives a conditional extension of the observation contract. Keeping the existing generator and a diagonal, additive, conserved support forces equal weights on each connected Hasse component. A shape-sensitive diagonal support therefore requires a changed generator if it is to remain conserved. For a declared fixed positive diagonal weight matrix, the corresponding local current and positive-ray receipt can be written exactly. Status: internal mathematical development; elementary algebra, no independent novelty claim, no independent human review. No canonical geometric weighting is selected. No physical promotion, spacetime metric, time unit, bench result, or retriangulation equivalence is asserted. The frozen LC acquisition and scoring contracts are unaffected. 2. Baseline and notation Let KK be a finite oriented simplicial complex, using nonaugmented real chains. Stack its simplex coordinates by degree into cc. Let BB be the total degree-lowering boundary matrix, so B2=0B^2=0. For a codimension-one incidence σ⊂η\sigma\subset\eta, Bση=±1B_{\sigma\eta}=\pm1. The existing source uses the dimensionless evolution parameter ss and ċ=A0c,A0=BT−B,S0=cTc. \dot c=A_0c,\qquad A_0=B^T-B,\qquad S_0=c^Tc. Its per-simplex stocks and signed incidence currents are qσ0=cσ2,Jση0=2Bσηcσcη. q^0_\sigma=c_\sigma^2,\qquad J^0_{\sigma\eta}=2B_{\sigma\eta}c_\sigma c_\eta. A positive current transfers stock from the lower-degree coordinate to the upper-degree coordinate. This is an algebraic convention, not a physical identification. The Hasse graph has one vertex per simplex coordinate and an undirected edge per nonzero boundary entry. Connectedness below refers to this graph, not the graph of currently nonzero amplitudes. The latter can have more components when amplitudes vanish. 3. Fixed-source diagonal-support theorem Proposition 1. Let H=diag(hσ)H=\operatorname{diag}(h_\sigma) be fixed, with hσ>0h_\sigma>0. Then SH=cTHcS_H=c^THc is conserved by ċ=A0c\dot c=A_0c for every initial state if and only if hσ=hηh_\sigma=h_\eta on every Hasse incidence. Consequently, HH is a positive scalar multiple of the identity on each connected Hasse component. Fixing a reference weight to one in each component gives H=IH=I. Proof. Conservation for every initial state is equivalent to A0TH+HA0=0. A_0^TH+HA_0=0. Indeed the derivative is the associated symmetric quadratic form, whose vanishing for all real vectors implies the symmetric matrix is zero. Since A0T=−A0A_0^T=-A_0, the matrix is HA0−A0HHA_0-A_0H. Its (σ,η)(\sigma,\eta) entry is (hσ−hη)(A0)ση. (h_\sigma-h_\eta)(A_0)_{\sigma\eta}. Every incidence has nonzero (A0)ση(A_0)_{\sigma\eta}; nonincidences impose no condition. Equality propagates along paths. Conversely, constant weights on components commute with A0A_0. ▫\square This theorem does not force regular tetrahedra. It says that an unchanged incidence-only generator cannot conserve arbitrary relative geometric reweightings of its existing independent stocks. A common overall weight may still depend on externally fixed geometry; it contributes no relative shape weighting within a component. Scope check: non-diagonal metrics are not excluded. Put ℒ=−A02=A0TA0\mathcal L=-A_0^2=A_0^TA_0. Then H×=I+ℒH_\times=I+\mathcal L is positive definite and commutes with A0A_0, hence is conserved by the same generator. For a filled tetrahedron it has nonzero off-diagonal entries. Such a quadratic form has cross terms and does not retain the original independent stocks hσcσ2h_\sigma c_\sigma^2. Proposition 1 is not a uniqueness theorem for all conserved quadratic forms. 4. Declared weighted dynamics and local continuity For a fixed positive diagonal HH, retain the same boundary matrix and define its weighted adjoint and a candidate generator: B†H=H−1BTH,AH=B†H−B. B^{\dagger_H}=H^{-1}B^TH,\qquad A_H=B^{\dagger_H}-B. Then AHTH+HAH=0A_H^TH+HA_H=0. Set qσ=hσcσ2,SH=∑σqσ,JσηH=2Bσηhσcσcη. q_\sigma=h_\sigma c_\sigma^2,\qquad S_H=\sum_\sigma q_\sigma,\qquad \boxed{J^H_{\sigma\eta}=2B_{\sigma\eta}h_\sigma c_\sigma c_\eta.} The weight in this expression belongs to the lower-degree simplex. Proposition 2. Under ċ=AHc\dot c=A_Hc, q̇σ=∑ξ:ξ⊂σJξσH−∑η:σ⊂ηJσηH, \dot q_\sigma= \sum_{\xi:\,\xi\subset\sigma}J^H_{\xi\sigma} -\sum_{\eta:\,\sigma\subset\eta}J^H_{\sigma\eta}, where both sums are over codimension-one incidences. Proof. An upper neighbor contributes −Bσηcη-B_{\sigma\eta}c_\eta to ċσ\dot c_\sigma; multiplication by 2hσcσ2h_\sigma c_\sigma gives −JσηH-J^H_{\sigma\eta}. A lower neighbor contributes hσ−1Bξσhξcξh_\sigma^{-1}B_{\xi\sigma}h_\xi c_\xi, giving +JξσH+J^H_{\xi\sigma}. Summing over coordinates cancels every incidence pair. Summing over a subset cancels internal incidences and gives the corresponding cut balance. ▫\square In normalized coordinates y=H1/2cy=H^{1/2}c, this is JσηH=2Bσηhσ/hηyσyη. J^H_{\sigma\eta}=2B_{\sigma\eta} \sqrt{h_\sigma/h_\eta}\,y_\sigma y_\eta. The normalized generator H1/2AHH−1/2H^{1/2}A_HH^{-1/2} is skew-symmetric, but its incidence coupling magnitudes are hσ/hη\sqrt{h_\sigma/h_\eta}. It is generally not the old unit-coupling generator. Weighted normalization does not make the dynamics unchanged. 5. A compatible weighted receipt Assume c≠0c\ne0, the positive weights hσh_\sigma are known, and all data refer to the same state. Record pσ=hσcσ2SH,χση=sgn(BσηJσηH) p_\sigma=\frac{h_\sigma c_\sigma^2}{S_H},\qquad \chi_{\sigma\eta}=\operatorname{sgn}(B_{\sigma\eta}J^H_{\sigma\eta}) on a deterministic spanning forest of the nonzero support graph, plus one root sign per component. Include explicit zero stocks and the simplex, orientation, weight, forest, and normalization conventions. Proposition 3. Propagate signs from the roots along the forest. If the resulting sign at σ\sigma is ϵσ\epsilon_\sigma, then ŷσ=ϵσpσ=hσcσSH,ĉσ=ϵσpσ/hσ=cσSH. \widehat y_\sigma=\epsilon_\sigma\sqrt{p_\sigma} =\frac{\sqrt{h_\sigma}c_\sigma}{\sqrt{S_H}},\qquad \boxed{\widehat c_\sigma=\epsilon_\sigma\sqrt{p_\sigma/h_\sigma} =\frac{c_\sigma}{\sqrt{S_H}}.} Zeros reconstruct as zero. This recovers the positive ray; absolute amplitude additionally requires SHS_H. Proof. Positive weights do not change the relative sign: χση=sgn(cσcη)\chi_{\sigma\eta}=\operatorname{sgn}(c_\sigma c_\eta). Tree induction therefore recovers coefficient signs. The recorded stocks give the displayed squared magnitudes. ▫\square A root sign is required for each connected nonzero support component, including isolated nonzero coordinates. Neither a zero state nor an uncertain forest-current sign can be normalized into a valid exact receipt. The exact current consistency test becomes (JσηH)2=4hσhηqσqη,(JσηHSH)2=4hσhηpσpη. \boxed{(J^H_{\sigma\eta})^2 =4\frac{h_\sigma}{h_\eta}q_\sigma q_\eta,}\qquad \left(\frac{J^H_{\sigma\eta}}{S_H}\right)^2 =4\frac{h_\sigma}{h_\eta}p_\sigma p_\eta. Signs must still have product one around every cycle of the nonzero support graph. Given positive stocks, compatible signs, root signs, and known weights, reconstruction realizes currents with these magnitudes and signs. At zero endpoints the current must be zero. The legacy unit-weight test cannot be retained unchanged where endpoint weights differ and both stocks are nonzero. These formulas are a conditional extension of the existing PSR contract, not evidence of a measured device. The weight matrix is supplied metadata; no inverse problem establishing its recovery from observations is solved here. One could instead report rescaled currents J̃ση=hη/hσJσηH\widetilde J_{\sigma\eta}=\sqrt{h_\eta/h_\sigma}J^H_{\sigma\eta}, which satisfy the legacy product identity for yy. Those rescaled quantities are not the currents in the local continuity law above. 6. Explicit nonregular counterexample Use the previous disphenoid family p0=(1,1,r),p1=(1,−1,−r),p2=(−1,1,−r),p3=(−1,−1,r). p_0=(1,1,r),\quad p_1=(1,-1,-r),\quad p_2=(-1,1,-r),\quad p_3=(-1,-1,r). At r=2r=2, the previous squared-volume-ratio candidate (Rule A) gives h0=1h_0=1, h01=5/2h_{01}=5/2. Set c0=2c_0=2, c01=−3c_{01}=-3, all other coordinates zero, with B0,01=−1B_{0,01}=-1. Then q0=4,q01=45/2,SH=53/2,J0,01H=12. q_0=4,\quad q_{01}=45/2,\quad S_H=53/2,\quad J^H_{0,01}=12. The correct squared current is 144144, while the legacy expression 4q0q014q_0q_{01} gives 360360. Multiplication by h0/h01=2/5h_0/h_{01}=2/5 restores 144144. Under the old generator A0A_0, this same weighted support has instantaneous derivative 1818; under AHA_H, its derivative is zero. This is an exact algebraic witness that geometry-dependent weights cannot simply be inserted into the old support and receipt formulas. 7. What an amplitude interpretation would add The existing source definition is complete for its abstract algebra. A geometric interpretation is additional structure. Three distinct contracts should be distinguished: Interpretation of a coefficient Declared norm Geometric implication Existing abstract chain amplitude ∑cσ2\sum c_\sigma^2 Unit weights; dihedral angles can accompany the state as geometry metadata Constant scalar density on a cell of measure VV ∫ρ2dV=ρ2V\int\rho^2\,dV=\rho^2V Relative weight V/VrefV/V_{\mathrm{ref}} after reference normalization Integrated scalar quantity Q=ρVQ=\rho V on that cell ∫ρ2dV=Q2/V\int\rho^2\,dV=Q^2/V Relative weight Vref/VV_{\mathrm{ref}}/V after reference normalization The last two rows are conditional derivations for constant scalar densities, with separately chosen reference units for each interpretation. They do not establish a norm for all simplex degrees at once. Mixing dimensions requires explicit reference measures and amplitude units. In particular, the two squared-volume candidates in the previous interface note are not selected by either scalar-density calculation. For differential forms, simplex integrals are cochain degrees of freedom. A Whitney reconstruction ω=∑iaiWi\omega=\sum_i a_iW_i has ∥ω∥L22=aTMa,Mij=⟨Wi,Wj⟩L2. \|\omega\|_{L^2}^2=a^TMa,\qquad M_{ij}=\langle W_i,W_j\rangle_{L^2}. Overlapping basis forms generally produce off-diagonal terms. An inverse-volume diagonal rule is not automatically the Whitney mass matrix. This chain/cochain distinction is standard in finite element exterior calculus [3]; adopting that route requires a specified interpolation and inner product, beyond the present diagonal-stock contract. 8. Consequence for dihedral-angle research The prior shape audit showed that the disphenoid’s two interior-dihedral cosine classes determine its dimensionless shape parameter r2r^2, but not its scale. That information is already a legitimate geometric observable. It does not by itself select a meaning for chain amplitudes or a transport law. The next well-defined modelling decision is therefore an amplitude-to-geometry contract: state the represented quantity, its units or reference normalization, its interpolation if applicable, the conserved quadratic form, and the resulting observable currents. This note supplies the exact checks for a diagonal contract. Until such a contract is adopted, preserve the existing orthonormal-amplitude model and report shape observables separately. If geometry varies with ss, the fixed-weight proof no longer suffices: dds(cTH(s)c)=cTḢ(s)c \frac{d}{ds}(c^TH(s)c)=c^T\dot H(s)c when the instantaneous generator is AH(s)A_{H(s)}. An additional transport or exchange law would be needed to remove or interpret this term. No such law is chosen here. The parameter ss remains dimensionless and is not identified with a sink-reversal count or an LC period. 9. Verification and provenance verify_receipts.py uses only Python’s standard library and rational arithmetic. It builds the oriented filled tetrahedron directly, checks the 28-incidence constraint matrix has rank 14, and verifies a non-diagonal conserved-form counterexample. It evaluates both previous geometric rules at r=1/2,1,3/2,2,3r=1/2,1,3/2,2,3, with dense, sparse incident-pair, and disconnected-support states. The script checks weighted skew-adjointness, per-coordinate continuity, global balance, reconstruction of squared magnitudes and signs, forest sizes, weighted current consistency, the explicit legacy-contract failure, and zero-state rejection. Result: 218 exact constructor checks pass, including ten shape/rule cases and three support patterns per case. Finite checks are regression evidence; the propositions above supply the general algebraic arguments. These checks are not exhaustive state-space verification, independent review, or physical data. The source snapshots reproduce the local living-repository files consulted during this session, not newly verified publication editions. The previous shape and interface packages remain separate, unchanged development artifacts. The running LaTeX source accompanies this note; no compiled PDF is claimed. References Current UD finite-chain source, local living repository papers/sources/paper-1.md, especially the finite-chain generator, support, and local-current definitions. Snapshot included as dependencies/PAPER_I_SOURCE.md. Current UD positive-ray receipt source, local living repository papers/sources/paper-2.md, Sections 2–3. Snapshot included as dependencies/PAPER_II_SOURCE.md. D. N. Arnold, R. S. Falk, R. Winther, Finite element exterior calculus: from Hodge theory to numerical stability, Bulletin of the American Mathematical Society 47 (2010), 281–354. DOI: 10.1090/S0273-0979-10-01278-4. Author preprint: https://arxiv.org/abs/0906.4325. Used for the established cochain/finite-element distinction, not as a novelty attribution for Propositions 1–3. Session development packages UD_TETRAHEDRAL_SHAPE_AUDIT_v0.1 and UD_GEOMETRIC_INNER_PRODUCT_INTERFACE_v0.1. Their archive hashes are recorded in this package’s provenance file.