# CSM_RH Paper 47 package

This package advances CSM_RH from state v1.37 to v1.38 and closes Campaign 43.

Main certified results:

- exact unpartitioned Heath-Brown cross-j recombination is a finite convolution binomial identity;
- the binomial remainder has no support below the inherited cutoff, so the full alternating j-sum equals Lambda on the prescribed output range;
- actual dyadic output-product overlap exists across adjacent j-levels, but cancellation is global after multiplicative pushforward rather than cellwise;
- every linear translated-frequency operator commutes with the exact recombination;
- after squaring, the complete cross-j matrix is exactly the Lambda-level norm, so alternating signs alone do not supply a fixed-X-power estimate;
- pre-square recombination removes the separated Type-II interface required by B-RH-014, so that component amplifier cannot simply be applied after recombination;
- in the actual theorem the recombined object is Lambda-Lambda-sharp, not a new component theorem;
- PESC has the exact triangular positive-lag autocorrelation form, giving the next root-observing continuation;
- a separate fixed-power Lambda-sharp-to-root bridge would still be required before a residual theorem could promote PESC/MLEPG.

WL4 closes as `CLOSED_AS_EXACT_BINOMIAL_RECOMBINATION_TO_THE_ORIGINAL_LAMBDA_RESIDUAL_WITHOUT_POWER_GAIN`.

WL5 closes as `CLOSED_AS_ROOT_RECOUPLING_WITHOUT_A_NEW_FIXED_POWER_THEOREM`.

Campaign 43 closes as `CLOSED_AS_COMPONENT_MECHANISMS_EXHAUSTED_AND_EXACTLY_RECOUPLED_TO_ROOT_PRIME_ERROR`.

Next continuation: Campaign 44 / PK1 — `EXACT_PESC_TRIANGULAR_KERNEL_NORMALIZATION`.

No RH proof or disproof is claimed.