Prism Derive S State

Step N: propose

Read working_dir/derivations/P1-T3_s_state_correction.tex so far. Identify the next unresolved goal from the outline. Compose a LaTeX snippet (≤15 lines inline math) with an English justification tied to either:

• A prior line of this derivation (cite line number)
• A postulate in prism_formal_spec.md (cite postulate id)
• A standard QM/QFT theorem (cite source)

Write the proposal to chat, not yet to disk:

### Proposed step N
**Goal:** <one sentence>
**Justification:** <one or two sentences, cite sources>

$$
<LaTeX snippet>
$$

**Output:** <what this step produces for the next step to consume>

Step N: review

Ask the user: ACCEPT / REJECT / REVISE?

• ACCEPT → append the step to .tex, update the symbolic form in .json, continue to step N+1.
• REJECT → record the rejection in derivation_log.md with the user's reason, do not append, re-plan.
• REVISE: <user edit> → apply the edit, re-present for confirmation before appending.

Step N: persist

On ACCEPT:

1. Append the snippet to working_dir/derivations/P1-T3_s_state_correction.tex.
2. Update working_dir/derivations/P1-T3_s_state_correction.json with:

   {"step": N, "symbolic_form": "<sympy-parseable>", "variables": {...}, "postulates_used": [...]}
   

3. Append one line to working_dir/derivations/derivation_log.md.

Success criteria (from spec §8, P1-T3)

The completed derivation must deliver all three:

1. Integral expression for the s-state self-energy analog in the native spiral basis.
2. Integral is finite and well-defined (no renormalization handwaves).
3. Explicit function of n with no free parameters beyond α.

When all three are met, the derivation is complete. Hand off to P1-T4 (prism-computer) for numerical evaluation.

Failure modes (from spec §8, P1-T3)

• integral_diverges: PRISM needs an explicit regularization scheme. Escalate to user.
• expression_requires_tuning: PRISM is not parameter-free at current specification. Document plainly and continue — but flag in the phase-1 verdict.

Both are legitimate outcomes. Do not paper over either.

Outputs required at task completion

• working_dir/derivations/P1-T3_s_state_correction.tex
• working_dir/derivations/P1-T3_s_state_correction.json
• working_dir/derivations/derivation_log.md (appended)
• working_dir/derivations/P1-T3_s_state_correction.tex.meta.yaml per handoff contract

Estimated: 80K-150K tokens, 2-3 weeks wall time including user review cycles.