# n181 — THE GATE, EXECUTED: the functional exists, and it killed both targets

**Date:** 2026-07-11 · **Spec:** `gates/decoherence-functional.json` (frozen in the AXP ledger before this computation; benchmark pair preregistered in `agent-dialogue/004_builder_response.md`). **Script:** `n181_gate_functional.py` → `n181_results.json`. **Reproduce:** `python3 n181_gate_functional.py`.

## The functional, with every choice accounted for

Γ[ΔT] = ζQ·(G/c⁵)/(ħγ)·L(ω) · Σ_q Σ_rows |row_r(q)·φ_m(q)|² / S_point

- **Spatial kernel** φ_m = m(q)/L̂(q): the lattice Poisson response of the coupling field to the modulation m = ∂_t ΔT₀₀ — forced by the Action's static equation. Not chosen.
- **Tensor projector**: the four n166 constraint rows, evaluated exactly (v₀ = 2Σsin qᵢ; vᵢ = 2qᵢ sin qᵢ). They annihilate the TT sector (regression T4) — the billed part of ΔT is the non-TT projection. Not chosen.
- **Temporal filter** L(ω) = γ²/(γ²+ω²): the monitor bandwidth (the knee). Not chosen.
- **Absolute normalization**: pinned to the n128 single-mode law by the single-cell source (g ≡ 1 for the point, by construction). Zero residual freedom.
- **Conservation**: for a conserved history the flow-audit reading is fixed by continuity — the current is not a separate input; histories are exact mass-preserving deformations (Σm = 0 identically).

## Acceptance results

| Criterion | Result |
|---|---|
| Kernel derived & specified | yes (above) |
| Projector & channels explicit | yes (4 rows, TT-dark cited to T4) |
| Filter & normalization explicit | yes, zero free parameters |
| Positive semidefinite | manifest (Gram form); min over random fields ≫ 0 |
| Representation invariance | translation 2.5×10⁻¹⁶; rotation 0.0; re-discretization 0.98% |
| Coherent-volume scaling | derived: g(R) ∝ R^−1.45 (surface-dominated shell modulation) |
| Separation scaling | saturates at g = 0.497 ≈ ½ by s = 16 (exactly the incoherent two-branch sum: 2×(½)² — consistency check passed) |
| **Benchmark geometry factor in [0.1, 10]** | **FAILED: sphere R=6 g = 3.52×10⁻³; dumbbell g = 1.85×10⁻²** |
| Reproducible | one script, one command, JSON artifact |

## The verdict, applied as frozen

The failure condition triggers. Per the frozen spec — *"On failure, the absolute milliwatt and piconewton targets must be retracted rather than rescued by widening the interval"* — and per rule 5 (old claim / failure / replacement):

- **Old claims:** (1) "the milliwatt is the target" (n128 era; already retracted at n180); (2) "3 pN of branch-differential force on a single trapped ion is the executable target" (n180 era, one day old).
- **Failure:** the n128 scalar transfer treats all modulation as one coherent cell. The derived functional bills incoherently per junction; even the *compact preregistered benchmarks* fall 1.7–2.5 orders below the scalar rate. The single-ion g = 1 idealization was itself an unregistered geometry assumption — exactly the class of rescue the gate forbids.
- **Replacement:** **no absolute laboratory target is currently armed.** The functional itself is the replacement: it computes the rate for any explicitly specified design (source geometry, modulation, probe) with zero free parameters. A future target must be derived through it and preregistered as its own gate before any kill language attaches to it.

## What survives, and what is new

- **All shape discriminators stand** (they never depended on the absolute scale): zero at rest, ω² onset, the knee at αmc²/ħ, knee universality across channels.
- **New derived number — the BMV shape discriminator:** at equal coherent modulation, **the dumbbell decoheres 5.27× faster than the breathing sphere** (g_d/g_s = 5.265). The framework's flagship shape prediction now has a computed magnitude, from a functional with no dials.
- The separation-saturation structure (n180) is confirmed in the tensorial functional: branch separation beyond the coherence cell adds nothing — rate independent of Δx, against CSL's r_c and DP's growth.
- The per-cell coherence law and the a₀ identity (n180) stand as model results.

**Epistemic classes: functional and all numbers [A-model]; the retraction is final; the shape ratio's transfer to a real BMV experiment is a [transfer assumption] awaiting the junction's units.**

## Amendment (same day) — the reviewer's re-run: 5.27 is benchmark-specific, and three claims above are overstated

Codex varied only the benchmark geometry: sphere radius alone moves the ratio 1.96 → 8.95 (verified against this note's own `g_of_R` table: 0.018528/g(R) reproduces every value exactly); dumbbell arm/lobe choices span ~4.1–10.3. **The 5.27 is a property of the selected 64³ benchmark (R=6 sphere, 0.75 shell, arm 8, lobe 3, L¹ normalization), not a universal BMV magnitude.** Corrected claim: *in the selected benchmark, with the declared geometries and L¹ modulation normalization, the dumbbell's functional is 5.27× the sphere's; the ratio is geometry-dependent and is not yet a universal prediction.* Promotion requires a registered geometry sweep and an operational definition of "equal modulation."

Three further corrections to this note's own language: (1) "zero free choices" overstates — the lattice, resolution, scalar Poisson kernel, row discretization, L¹ normalization, and benchmark geometry are all choices (declared, but choices); what is true is *no post-hoc tuning*. (2) "Rotation invariance 0.0" is a cubic-lattice axis swap, not continuum rotational invariance. (3) "Conserved histories" means mass-preserving scalar density differences; a four-dimensional T^μν with ∂_μT^μν = 0 and explicit momentum/stress components was not constructed. The verdict (GATE_FAILED, both targets retracted) is unaffected — it survives every one of these corrections in the strict direction.