# CSM_RH Paper 09 v0.1

Campaign 08 expands the canonical Heath-Brown identity at K=3 before taking absolute values.

Main results:

- the K=3 identity is split exactly into Q0, Q1, Q2, Q3 by the number of non-unit Möbius factors;
- Q0 has the exact divisor-polynomial form `log(n) * [3 - 3 d2(n)/2 + d3(n)/3]`;
- the alternating unit algebra cancels rough semiprimes exactly but preserves primes and prime squares with the correct von Mangoldt weights;
- more generally, for every fixed K, the all-unit sector equals Lambda on every U-rough integer;
- every prime p>U therefore survives in the all-unit sector with coefficient log p;
- increasing fixed K cannot algebraically eliminate the rough-prime channel;
- Campaign 09 is prepared to audit growing K against binomial and block-complexity costs.

State version advances to CSM_RH v1.0 as a closure-framework maturity milestone only. RH remains open.

No proof or disproof of RH is claimed.
No live GLM provider run is claimed.