non_circular
plain-language theorem explainer
Packages the non-circularity certificate for the six-quark forward mass pipeline: sector yardsticks come only from cube-geometry integers, rungs only from generation torsion, and Z only from the charge-band map. Auditors of the quark sector cite it to certify that no PDG mass enters any right-hand side. The body is a structure assembly of already-proved geometric, torsion, and charge equalities.
Claim. The forward quark-mass pipeline is non-circular. Up-sector yardstick exponents satisfy $B_{\mathrm{pow}}(\mathrm{up})=-1$ and $r_0(\mathrm{up})=35$; down-sector exponents satisfy $B_{\mathrm{pow}}(\mathrm{down})=23$ and $r_0(\mathrm{down})=-5$. The six integer rungs are $4,15,21$ for both up-type and down-type generations. Charge-band values are $Z(\mathrm{up},2/3)=276$ and $Z(\mathrm{down},-1/3)=24$. None of these inputs depends on a measured quark mass.
background
The module builds a single Convention-A forward pipeline for all six quark masses. Sector yardsticks are $A_s=2^{B_{\mathrm{pow}}(s)},E_{\mathrm{coh}},\varphi^{r_0(s)}$ from cube geometry; integer rungs are baseline plus generation torsion; the band correction is $\mathrm{gap}(Z_i)=\log_\varphi(1+Z_i/\varphi)$; predicted mass at the anchor is $m_i(\mu^*)=A_s,\varphi^{r_i-8+\mathrm{gap}(Z_i)}$. Outputs are dimensionless ratios $m_q/m_e$, seam-free and free of absolute calibration.
Non-circularity is the design claim that no measured PDG quark mass appears anywhere on the right-hand side. The certificate structure records three concrete equalities: yardstick exponents fixed by counting-layer geometry, rungs fixed by torsion integers, and $Z$ fixed by the charge-band map. Upstream anchors supply the geometric $B_{\mathrm{pow}}$ and $r_0$ values; the torsion and charge lemmas in this module close the other two legs. $\varphi$ itself is the T5/T6 fixed point from the forcing chain.
proof idea
Definitional structure construction, not a tactic proof. The three fields are filled by named equalities already established upstream or earlier in the module: the four geometric identities for up/down $B_{\mathrm{pow}}$ and $r_0$; the six rung identities from generation torsion; and the two charge-band $Z$ identities. No new algebra is performed; the certificate is the packaged conjunction of those facts under the non-circularity structure type.
why it matters
Closes the dual-coordinate worry for quarks: Convention A alone supplies a complete forward pipeline, so Convention B is a derived coordinate transform rather than a second theory. Downstream, the quark-sector audit structure records non-circularity as a Boolean field of the reconciliation certificate, alongside a unified six-quark formula and a precision target comparable to the lepton chain.
In framework terms this sits on the mass ladder $m=\mathrm{yardstick},\varphi^{r-8+\mathrm{gap}(Z)}$ with $\varphi$ from T5/T6 and counting-layer integers only. The module summary is explicit: all six masses share one forward formula, every input is a counting integer or $\varphi$ or $\alpha$, and equal-$Z$ ratios collapse to pure $\varphi$-powers. Matching PDG numbers is deliberately out of scope here; that is SM RG bookkeeping in Paper IV.
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