Representation Theory · SU(2)

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VISUAL SYSTEM · MINI AI LABS

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Record: CITY-LABS-VISUAL-SYSTEM-20260915.

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MATHEMATICAL CITY · REPRESENTATION LAYER

The finite carrier and its exact decomposition: why 125 states contain the selected 47-dimensional invariant sector.

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CANONICAL DECOMPOSITION

$V_2^{\otimes3}=V_0\oplus3V_1\oplus5V_2\oplus4V_3\oplus3V_4\oplus2V_5\oplus V_6$

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SELECTED SECTORS

$E_{47}=5V_2\oplus2V_5$ · dimension $25+22=47$.

Continue to Spectral Theory

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Proof-Born Citizenship Rule

Representation-theoretic citizens may parent new City Proofs, but each descendant must establish a distinct theorem, invariant, construction, algorithm, or executable result with its own archive packet and evidence class. Parent validation is premise-level substrate only. Duplicate formulations reconcile to the canonical citizen. Existing registry and archive surfaces remain authoritative.

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Representation Theory district entrance — $SU(2)$ carrier, tensor-power architecture, and the route to the exact $125$-dimensional space.

District purpose: explain why the 125-dimensional carrier and 47-dimensional invariant sector exist.

Main theorem road

$V_2^{\otimes3}=V_0\oplus3V_1\oplus5V_2\oplus4V_3\oplus3V_4\oplus2V_5\oplus V_6.$