---
name: project-pending-riemann-cliff-map-alpha-dial-foldability
description: "DONE (STEP 1169, 2026-05-27) — Riemann difficulty-cliff map via α-dial × foldability; near-vertical wall finding + rigid/Poisson controls + honest scope (suggester not RH proof)"
metadata: 
  node_type: memory
  type: project
  originSessionId: 4d9333d7-e932-4180-904a-8c123eea02b6
---

**✅ DONE** — implemented 2026-05-27 as **STEP 1169** (藤本さん requested it 2026-05-27 "帰宅後"; done same day on return). Files: `src/aios/qft/riemann-cliff-map.ts` + `test/step1169-riemann-cliff-map-test.ts` (**23/23 PASS**) + `scripts/generate-riemann-cliff-map.ts` → `data/riemann-cliff-map/latest.json` (backend-only, no site). CLAUDE.md STEP-table + qft/index.ts export + package.json (`test:step1169`, `riemann-cliff:refresh`) updated. Regression step1164/1165/1167/1168 all PASS / 0 breaking.

## ★ Key empirical finding (differed from the naive design — honest)
- The **spacing-based dial does NOT produce a cliff** (first few hundred zeros: raw spacings look like GUE noise to LZ at every α, fold≈0.04–0.10 flat). The predictable smooth structure lives in the **cumulative positions (counting function N̄)**, not local spacings. So the cliff dial is built on POSITIONS.
- **Smooth-removal dial (final design)**: unfolded positions u_n=θ(t_n)/π → least-squares linear trend L_n (b≈1) → fluctuation φ_n=u_n−L_n → `x_n(ρ)=φ_n+ρ·(L_n−L_0)` (ρ=remaining-trend fraction, α=1−ρ), swept on a **log-spaced ρ grid** {1,.5,.1,.03,.01,.003,.001,.0003,.0001,0} to resolve the wall. Each ρ → `sequenceFoldability` (STEP 1168).
- **The cliff is a NEAR-VERTICAL WALL at α*≈0.997**: foldability 0.93 (α=0, N̄ intact, A/TRUE) plateau → sharp collapse around ρ≈0.01–0.003 → 0.05 (α=1, bare fluctuation, **B/INFINITY**). Trend rise (~M) dwarfs fluctuation std (~0.24, ratio ~1000) ⇒ you must strip ~99.7% of the smooth model before irreducibility surfaces. Interpretation: the smooth counting function carries ~ALL the foldable structure; the fluctuation carries ~ALL the difficulty; **no partial-credit middle** (= conservation of difficulty made visible).
- **Controls (share the same linear trend)**: rigid (φ≡0) → **NO cliff** (flat A/TRUE, floor 0.97) ⇒ the cliff is the signature of irreducible fluctuation, not an artifact of unfolding. Poisson (uncorrelated → cumulative random walk) → cliff present but **floor=NEITHER (0.40)**, distinct from Riemann's **INFINITY (0.05)**. Since LZ symbolize is quantile/rank-based (scale-free), this floor-axis difference is a genuine SHAPE difference (rigid GUE noise vs diffusive random-walk drift), a heuristic rigidity proxy.

## ★ HONEST SCOPE (locked, as designed)
Suggester / cliff MAP, **NOT a proof of RH and NOT progress toward it** — only locates where knowable smooth structure (N̄) ends and irreducible fluctuation (where RH's difficulty lives) begins. Measuring the cliff does not reduce it (conservation of difficulty; re-encoding can't manufacture structure — Paper 155 efficiency_le_one/SemanticDPI). Rigorous GUE-vs-Poisson discrimination stays with long-range spectral statistics (⟨r⟩/Σ²(L), STEP 1162-1165) — complementary. D-FUMT₈ projection heuristic. Pattern 5 self-aware: extends STEP 1162-1165 (Riemann↔GUE) + 1167 (α-dial) + 1168 (foldability).

## API added (for future reuse)
`riemann-cliff-map.ts`: `unfoldedPositions(zeros)`, `linearTrend(u)→{a,b,L}`, `partiallyDetrendedPositions(u,ρ)`, `buildRiemannCliffMap(M)→CliffMap`, `buildControlCliffMaps(M)→{rigid,poisson}`, `analyzeRiemannCliff(M)→{riemann,rigid,poisson,verdict,honestNote}`. CliffMap has rows[{rho,alpha,foldability,regime,dfumt8Axis}], foldabilityIntact/Floor, floorAxis, alphaStar, steepestDrop, hasCliff.

Session lineage: [[project-session-2026-05-27-paper157-publish-and-maintenance]]. Base lenses: STEP 1162-1165 (`riemann-gue.ts`/`spectral-lens.ts`), 1167 (`ab-transition-observer.ts`), 1168 (`problem-foldability.ts`).