full_anchor_tuple_forces_coefficients_and_offset
plain-language theorem explainer
Matching the three SM anchor band labels 1332 (lepton), 276 (up), and 24 (down) forces the even-quartic coefficients and color offset of the topology-compatible Z-map to (a,b,c)=(1,1,4). Mass-layer forcing theorems cite this uniqueness when bridging topology-derived polynomials to canonical ChargeIndex outputs. The proof unfolds the polynomial definitions into three integer equations and closes by linear arithmetic.
Claim. Let $a,b,c\in\mathbb{Z}$. Write the even degree-$\le 4$ band polynomial $Z=a\tilde{Q}^{2}+b\tilde{Q}^{4}$, with leptons using the bare polynomial and quarks using a constant color offset $c$. If the lepton value equals $1332$, the up-quark value equals $276$, and the down-quark value equals $24$, then necessarily $a=1$, $b=1$, and $c=4$.
background
This module derives the charge-to-band map $Z(\tilde{Q})$ from recognition boundaries on the 3-cube, without mass phenomenology. Stage 1 integerizes SM charges $Q\in{-1,2/3,-1/3}$ by the face count $F=2D=6$ (T8 forces $D=3$), so $\tilde{Q}=FQ$ is integral: lepton $\tilde{Q}=\pm 6$, up $\pm 4$, down $\pm 2$.
Stage 2 requires a gauge-invariant band label: even in $\tilde{Q}$, non-negative, and vanishing at neutrality. The minimal such polynomial is $Z_{\mathrm{poly}}(a,b;\tilde{Q})=a\tilde{Q}^{2}+b\tilde{Q}^{4}$. Stage 3 adds a quark-only color offset $c$, from $2^{D-1}$ edge channels along one spatial direction of the cube, giving $Z_{\mathrm{quark}}=c+Z_{\mathrm{poly}}$.
The three hypotheses pin the symbolic family against the fixed anchor triple $(1332,276,24)$. Sibling definitions package the branches: leptons use the bare polynomial on $\tilde{Q}{\ell}$; up and down use the offset form on $\tilde{Q}{u}$ and $\tilde{Q}_{d}$.
proof idea
Unfold the three branch definitions. The lepton hypothesis becomes the bare polynomial identity $a\cdot 36+b\cdot 1296=1332$ (using $\tilde{Q}_{\ell}^{2}=36$). The up and down hypotheses become $c+(a\cdot 16+b\cdot 256)=276$ and $c+(a\cdot 4+b\cdot 16)=24$.
Subtract the two quark equations to cancel $c$, obtaining $12a+240b=252$. Feed that difference together with the lepton equation into linarith to force $b=1$, then $a=1$. Substitute both into the up equation to force $c=4$. Package the three equalities as a conjunction.
why it matters
This is the algebraic uniqueness engine behind Stage 2–3 of the topological Z-map derivation: once the anchor triple is fixed, the only topology-compatible coefficients are the canonical ones $(a,b,c)=(1,1,4)$, i.e. $Z_{\ell}=\tilde{Q}^{2}+\tilde{Q}^{4}$ and $Z_{q}=4+\tilde{Q}^{2}+\tilde{Q}^{4}$ with color offset $2^{D-1}=4$.
Downstream, canonical_tuple_forced_from_anchor_outputs in the mass-layer bridge rephrases the same claim against ChargeIndex.Z sector outputs, quoting: if a topology-compatible family matches the canonical anchor outputs, then $(a,b,c)$ are forced to $(1,1,4)$. That bridge is how the first-principles cube derivation locks the mass-formula band labels.
Framework landmarks: T8 ($D=3$) supplies both the face-count scale $F=6$ behind the $\tilde{Q}$ values and the edge-count offset $2^{D-1}=4$. The result does not itself derive the anchors; it shows they admit only one integer solution in the allowed polynomial family.
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