Summary

• Ticks off one of the unchecked items in Documentation pages #771's docs-pages todo list (Housing Benefit).
• New page at docs/book/programs/gov/dwp/housing-benefit.ipynb covering:
  • the legacy status of HB and its overlap with the UC housing element,
  • eligibility (housing_benefit_eligible gate plus the would_claim_housing_benefit take-up input),
  • the four-stage entitlement pipeline — eligible rent (with LHA cap) -> applicable amount -> means test taper -> benefit cap reduction — with pointers to each underlying variable,
  • a chart of weekly personal allowances by composition (single / lone parent / couple) and age band,
  • statutory references (SSCBA 1992 s.130, SI 2006/213, SI 2006/214) plus the DWP HB caseload statistics page used for calibration.

Test plan

• Notebook is valid JSON and parameter paths used in the allowances chart all resolve via system.parameters.gov.dwp.housing_benefit.allowances.
• Docs build job renders the new page.

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