Paper 640 Strategy & Next Steps

What we have now

Paper 640: Full outline + §1-2 prose ✓

  • §1 Introduction (1,200 words): pedagogical disconnect → hidden duality → why now → three examples → roadmap
  • §2 Category theory (2,500 words): limits/colimits for chemists → adjoints → Grassmannian → Spec ⊣ Lan precise statement → ISA tiers → β* snap → why category theory

Status of each section:

Section Status Pages Experiments
§1 Introduction ✓ Draft complete 3–4 None
§2 Category theory ✓ Draft complete 4–5 None
§3 H₂ worked example Outline ready 2–3 x640a
§4 N₂ dissociation Outline ready 3–4 x640b
§5 Fe-SCO Outline ready 4–5 x640c-e
§6 QEC sidebar Brief outline 1–2 None
§7 Comparison table Outline ready 1–2 —
§8 Implications Outline ready 2–3 —
§9 Discussion Outline ready 2–3 —
§10 Conclusion Template 1–2 —
Total ~60% done 12–15 pages 5 experiments

Immediate next step: prose or experiments?

Recommendation: BOTH IN PARALLEL

Path A (Prose next, experiments follow)

  1. Write §3 (H₂ example) this week — pure theory, no computation
  2. Parallelize:
    • Continue writing §4-5 while x640a runs
    • Run all 5 experiments while revising early sections

Advantage: Prose draft gets done faster; experiments inform revisions

Path B (Experiments first, prose after)

  1. Design and run x640a-e this week
  2. Write §3-5 next, using real data from experiments
  3. Prose is richer, validated

Advantage: Experiments might reveal gaps in theory; prose is stronger

My recommendation: Path A (prose + experiments parallel)

  • Reason: §3 (H₂) is self-contained and needs no computation; write it while setting up x640a
  • §4-5 can reference “results from x640b” while x640b is running

Decision: Experiments priority order

Start immediately:

  1. x640a (H₂ Rayleigh-Ritz): 1–2 days. Simple, clean, no dependencies. Output: plots of E(θ) on Gr(1,2).
  2. x640c (Fe-SCO CASSCF): 5–7 days. More involved but standard CASSCF. Output: LS and HS energies, structures.

Start after above succeed:

  1. x640b (N₂ dissociation): 10–14 days. Needs CASSCF at multiple R values + Berry phase. Output: dissociation curve, phase calculation.

Final pass (after main results):

  1. x640d (Fe-SCO + SOC): 3–5 days. Add spin-orbit to x640c results.
  2. x640e (Fe-SCO thermal): 2–3 days. Boltzmann distribution at varying T.

Total computation time: ~25–30 days if sequential. Estimated calendar time with parallelization: 3–4 weeks.

Paper 641 & 642 strategy

Paper 641 (QEC duality):

  • Start detailed outline after Paper 640 §1-2 are finalized
  • Plan to write § while Paper 640 is in revision cycle
  • Should not delay 640 submission; can be companion paper

Paper 642 (FeMoco):

  • Only attempt if x640a-e validate the theory cleanly
  • Defer to Q4 2026 / Q1 2027 if we want more confidence
  • Alternative: publish 640-641, then decide on 642

Success criteria for Paper 640

Prose quality:

  • Is the adjunction explanation clear to someone with no category theory training?
  • Do chemists find the tier structure useful (not just abstract)?
  • Does the H₂ → N₂ → Fe-SCO progression build intuition?

Experimental validation:

  • Does E(θ) on Gr(1,2) match HF/FCI for H₂? (expect yes, trivial)
  • Does N₂ avoided crossing line up with β* snap prediction? (expect yes, non-trivial)
  • Do Fe-SCO LS-HS results match experimental thermodynamics? (harder, but doable with literature data)

Universality claim:

  • Is the QEC sidebar credible even to QC non-experts?
  • Do readers see the connection as real, not just superficial?

Impact:

  • Would chemists adopt Spec ⊣ Lan language in active-space selection?
  • Would QC community recognize fault-tolerance as β* snap?

Risks and mitigations

Risk 1: §1-2 is too abstract for chemists

  • Mitigation: Test on 2–3 chemistry colleagues after §3 written. Get feedback.

Risk 2: Experiments show theory is wrong

  • Mitigation: Modest expectations for x640a-e. They validate narrative, not prove it.
  • If x640b (N₂) fails to show Berry phase at avoided crossing, it’s still interesting (means our theory is incomplete).

Risk 3: FeMoco turns out to be necessary for the paper to be “convincing”

  • Mitigation: Paper 640 does NOT promise to handle H². It’s about H⁰ → H¹ transition. FeMoco is Paper 642, separate claim.

Risk 4: Paper is too long (15+ pages)

  • Mitigation: Trim QEC sidebar to 1 page. Combine §7-8. Keep §9-10 brief.

Authorship & timeline

Week 1-2: Prose §3-5 + start x640a Week 3-4: Experiments x640b-e in parallel with prose revisions Week 5-6: Integrate experiments into §3-5, write §6-8 Week 7-8: Draft §9-10, full revision pass Week 9: Final proof, figures, submit to JCP

Total: 9 weeks

Decision framework (go/no-go after Week 4)

After x640a-b results are in (4 weeks from now):

Ask:

  1. Is Spec ⊣ Lan narrative holding up?
    • Yes → Continue to §6-10
    • No → Revise theory or reconsider scope
  2. Do x640a-b validate the tier structure?
    • Yes (β* snap appears where predicted) → Proceed to Paper 641 planning
    • Partial (some evidence but not clean) → Rewrite §6-8 to hedge; still publishable
    • No (results contradict theory) → Consider abort or radical rethink
  3. Is the time budget realistic?
    • Yes (running on schedule) → Stay the course
    • Slipping (experiments taking longer) → Drop x640d-e, focus on x640a-c

What you should do now

  1. Read §1-2 draft — does the pedagogical narrative resonate?
  2. Decide: prose or experiments first?
  3. Greenlight x640a — trivial H₂ calculation, should start ASAP
  4. Outline x640b — N₂ dissociation at multiple R, CASSCF settings, Berry phase calculation method

Then we can start writing §3 in earnest.