Fee Options Paper: Fixed, Capped or Phased

Act as a pricing adviser to an independent consultant. I want fee structures I can explain to the client. Don't supply market rates or say what I should charge.

Scope summary: {{SCOPE_SUMMARY}}
My estimate of effort in days, by phase: {{EFFORT_BY_PHASE}}
My day rate (the figure I have decided): {{DAY_RATE}}
What the outcome is worth to the client, in their words and figures: {{CLIENT_VALUE}}
Main uncertainties that could change effort: {{UNCERTAINTIES}}

Produce:
1. Three fee structures: fixed fee, time-and-materials with a cap, and phased fee with a gate after discovery. For each, show the arithmetic from my days and day rate, step by step, and state the total.
2. For each, who carries the risk, when the client pays, and what behaviour it encourages.
3. A contingency in days tied to my uncertainties, shown separately from the rate.
4. A plain-English paragraph per option for the proposal.
5. Questions about budget and approval to ask the client before I present.

Rules: use only my numbers; show every calculation and check it twice. Mark assumptions [INFERRED]. Don't value the outcome beyond the client's own figures. Output: a comparison table, then the paragraphs and questions.