downQuarkSignature
plain-language theorem explainer
Packages the down-quark residual signature: positive B_pow sign, SDGT rung pair (6, 8), and free coupling κ. Closure targets and all-sector tests cite it when freezing global coefficients against quark residuals. Pure structure constructor; positivity of the steps is discharged by decide.
Claim. For any real coupling $\kappa$, the down-quark residual signature is the positive-$B_{\mathrm{pow}}$ sector whose SDGT rung spacings are $6$ (gen $1\to 2$) and $8$ (gen $2\to 3$), with coupling equal to $\kappa$.
background
Item 8 is the open quark sub-leading mass correction in the Recognition mass ladder. This module builds the smallest precise target that would close it: a sign-split ratio family whose global coefficients, once frozen on quark residuals, become out-of-sample tests on leptons and other sectors.
A residual signature is the minimal structural package for one sector's sub-leading law: a $B_{\mathrm{pow}}$ sign class, two positive natural SDGT rung spacings (cube-cell counts from the $Q_3$ decomposition) for the gen $1\to 2$ and $2\to 3$ corrections, and one real coupling scalar. The sign selects which coefficient ($c_{\mathrm{neg}}$ or $c_{\mathrm{pos}}$) is active; the ordered step pair weights the correction via a step fraction.
Down quarks sit in the positive-$B_{\mathrm{pow}}$ class with the derived SDGT pair $(6,8)$, dual to the up-quark and lepton signatures that share the negative class.
proof idea
Definitional constructor, not a proof. Fills the residual-signature fields with sign positive, step12 = 6, step23 = 8, and coupling = κ. The two positivity obligations $0 < 6$ and $0 < 8$ are closed by decide.
why it matters
This is the down-quark half of every Item 8 closure proposition in the module. The basic target asks for unique global coefficients whose predicted residuals match exact up and down pairs under the two quark signatures. The refined and sign-class variants reuse the same down-quark package against refined or per-sign-class coefficient sets.
All-sector tests go further: coefficients frozen on both quark signatures (including this one, typically at $\alpha_s$) must also reproduce lepton residuals under the lepton signature. That is the genuine out-of-sample check the module is built to make falsifiable.
In the broader RS mass story, these signatures encode how sub-leading generation corrections sit on the phi-ladder once the leading rung formula is fixed. Closing Item 8 would freeze the two global correction coefficients and turn later lepton, genetic, and theta instantiations into predictions rather than fits.
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