# Structural Failure Coincidence — Cut 2 C2-B Resolution: `pen(q)` and the admissible `q_a` range Pre-registration: [`README.md`](README.md) Run spec: [`P2_RUN_SPEC.md`](P2_RUN_SPEC.md) Admission check: [`P2_SPEC_ADMISSION.md`](P2_SPEC_ADMISSION.md) Parent condition: [`P2_CUT2_C2_NUISANCE_AND_BRIDGE.md`](P2_CUT2_C2_NUISANCE_AND_BRIDGE.md) (C2-B) Consumers unblocked: [`P2_CUT2_C3_DECOY_TERM_AND_TEMPTATION.md`](P2_CUT2_C3_DECOY_TERM_AND_TEMPTATION.md) (C3-T, C3-B(ii), C3-D) · [`P2_CUT2_C4_DERIVED_AUDIT.md`](P2_CUT2_C4_DERIVED_AUDIT.md) (C4-C / D1) Filed: **2026-05-16 (PT)**. Status: **C2-B FILED FOR AUDIT — HOLD FOR EXECUTION**. Resolves the free-`q_a` degeneracy by design only. Cut-2 execution remains **HELD** on the C2-B calibration blocker below, the still-open C2-A/C/D, C3-A/B/C/D, C4-A/B/C/D, and C5, and a fresh P2-spec admission re-check. No harness written; nothing run. ## Purpose C2-B is the cascade hub. The C2 route ridge `I_route(q;bundle) = exp(−[ f_par_obs − R22/cos(q_h) − q_a ]² / 2σ²)` with `q_a` **unconstrained** has a continuum of exact maxima: for any `q_h`, set `q_a = f_par_obs − R22/cos(q_h)` ⇒ bracket `= 0` ⇒ `I_route = 1`. So `argmax_q I_route` is not unique, `π_route` is undefined, C2 P-A is undefined, and — via C4-C — D1 has no construction-level object to audit. This resolution makes the route optimum unique and well-posed, which unblocks C3-T's `π_route` baseline, C3-B(ii), C3-D, and C4-C/D1. ## 1. The resolution — receipt-grounded convex anchor-prior penalty The reason `q_a` exists is to let the controller hypothesise the unknown anchor error `ε`. C2 already declares that error **bounded and zero-centred**: `ε ~ U[−A,+A]`, `A = ρ·R22`. So: - **Admissible range:** `q_a ∈ [−A, +A]`, `A = ρ·R22`. This is the receipt-grounded anchor-error support. `A` is a constant times the scale-lock `R22`; **no `h`** (A1-safe). - **Penalty:** a strictly convex, zero-centred prior on the anchor correction (Occam: the smallest anchor correction consistent with the signal): ``` pen(q) = λ · (q_a / A)² , λ > 0 (pre-registered, A3 tolerance) O(q) = I_route(q;bundle) − pen(q) ← what the controller climbs ``` ### Unique argmax (degeneracy broken) On the zero-bracket manifold `q_a*(q_h) = f_par_obs − R22/cos(q_h)`, `O = 1 − λ(q_a*(q_h)/A)²`, uniquely maximised where `|q_a*|` is **minimised**, i.e. `q_a* = 0` ⟺ `R22/cos(q_h) = f_par_obs` ⟺ `q_h = arccos(R22/f_par_obs)`. Off the manifold `I_route < 1` and `pen ≥ 0`, so `O < 1`. Hence the **unique global maximum** is ``` (q_h*, q_a*) = ( arccos(R22 / f_par_obs) , 0 ) , O* = 1 ``` `q_h*` is exactly `q_naive(h,ε) = arccos(R22/f_par_obs)` — the **biased naive inverse**, not true `h` (bias = the C2 anchor-noise bias, which blows up at the L1 low-leverage band). Therefore: - **`π_route` is well-defined** (climb `O`; converges to `q_naive`). - **C2 P-A holds and is now computable:** `argmax_q I_route(−pen) = q_naive ≠ h` on the must-differ band — the route construction is an honest-but-anchor-biased inverse, never a readout of the hidden cause. - **P-B unaffected:** `O` is smooth; climbability is the separate conditioning question pinned numerically with C2-A. ## 2. C2-B load-bearing calibration window (surfaced adversarially) `λ` must thread a window — the now-standard pattern (cf. C3-B, C4-B): - **`λ` too small ⇒ degeneracy not broken in practice.** The along-manifold curvature of `O` near `q_h*` scales like `2λ·(R22 sin q_h*/cos²q_h* / A)²`; as `λ→0` it →0, `argmax` is numerically non-unique, `π_route` non-reproducible. - **`λ` too large ⇒ it moves the optimum.** A steep `q_a`-penalty makes accepting a small `r≠0` (exp cost `~r²/2σ²`) worthwhile to shrink `q_a`, pulling `q_h*` off `arccos(R22/f_par_obs)` — P-A would then be a λ-artifact, not the biased naive inverse. C2-B is not closed until a **pre-run numeric demonstration** shows the frozen `λ` satisfies **both**: - **C2-B(i):** the post-penalty argmax is unique with along-manifold conditioning ≥ a pre-registered floor on the eligible/biased band (degeneracy genuinely broken; `π_route` reproducible). - **C2-B(ii):** the located `q_h*` deviates from `arccos(R22/f_par_obs)` by < a pre-registered negligible tolerance over the C2 bias grid (the penalty breaks the tie **without moving the optimum** — P-A stays exactly the biased naive inverse). ## 3. Honest couplings and cross-condition findings (recorded, not papered over) - **C2-B(ii) ↔ C2-A.** The (ii) demonstration is computed on the C2 bias grid; `λ`, the conditioning floor, and the P-A tolerance **fold into the C2-A numeric freeze** (one set of frozen numbers). - **Exposes C2-D, does not absorb it.** When `f_par_obs < R22` there is **no** zero-bracket manifold (`R22/cos(q_h) ≥ R22 > f_par_obs` ∀ `q_h∈[0,80]`); the unique `O`-max then sits at the penalty-best-fit domain edge. C2-B makes this geometry explicit; classifying it (abstain / invalid) is **C2-D's** job and is deferred there, not silently resolved here. - **C4-C / D1 clarification (genuine tension, flagged for the C4 reviewer).** Once C2-B makes `argmax I_route(−pen) ≡ arccos(R22/f_par_obs)` *by construction*, C4-C's stated D1 — "compare `argmax I_route` against `arccos(R22/f_par_obs)`" — becomes **degenerate** (the two are identical, so a literal reading makes D1 trivially fail). D1's anti-Cut-1 intent must therefore be the **P-A form**: `argmax I_route (= q_naive) ≠ **true h** on the must-differ low-leverage band` — the route construction is not a clean readout of the *hidden cause* (the actual Cut-1 sin), not "differs from its own closed form." This is raised for the C4 reviewer in the `P2_SPEC_ADMISSION` filing log; the frozen C4 body is **not** edited here. ## 4. Cut-2 C2-B binding rules 1. `q_a` is hard-clamped to `[−A,+A]`, `A = ρ·R22`; an unbounded `q_a` in any Cut-2 run is **void**. 2. `pen(q) = λ(q_a/A)²` reads only `q_a` and the constant `A`; any `h`-dependence in `pen` or the `q_a` range ⇒ run **VOID** (A1). 3. `λ`, the C2-B(i) conditioning floor, and the C2-B(ii) tolerance are pre-registered with the C2-A freeze and **never** edited post-results (A3). Immutable geometry/receipt boundaries unchanged. 4. The C2-B(i)/(ii) artifacts are produced and frozen **before** any controller instantiation. ## Explicit non-bindings (cannot satisfy C2-B) - An unbounded or `h`-dependent `q_a` / `pen` (degeneracy or A1 leak). - A non-convex / non-zero-centred `pen` that does not single out the minimal anchor correction (degeneracy not provably broken). - A `λ` outside the C2-B(i)/(ii) calibrated window. - Quietly handling `f_par_obs < R22` here instead of in C2-D. - Tuning `λ`/floor/tolerance after seeing any controller result. ## Open items C2-B files the `pen(q)` form, the admissible `q_a` range, the unique- argmax derivation, and the C2-B(i)/(ii) calibration obligation for audit. Still open before any Cut-2 run: - **C2-A** absorbs the C2-B numerics (`λ`, conditioning floor, P-A tolerance) alongside `ρ, A, σ, seed`, the grid and `q_h/q_a` domains. - **C2-C** (leverage-confidence function) and **C2-D** (`f_par_obs