# n180 — The decoherence functional, derived at model level

**Date:** 2026-07-11 · **Trigger:** Codex opening audit, item 1 (agent-dialogue/001), accepted in 002 as the shared priority. **Script:** `n180_functional.py`.

## The object

For two branch stress-energy histories T_A^{μν}(x,t), T_B^{μν}(x,t), the monitored-lattice model assigns

**Γ = ζ·Q · (G/c⁵) · (1/ħγ) · L(ω) · Σ_rows ∫d³x |∂_t κ_row(x,t)|²**

where κ_row(x) is the row-contraction of the static k-field response K∗T at junction x (K = the lattice Poisson kernel; the rows = the n166 violation operators: unsigned coupling-budget v₀, signed flux v_i), and L(ω) = γ²/(γ²+ω²) is the monitor filter. The monitor bills **time derivatives of local k** — zero at rest is structural (E4), and the Lindblad sum over junctions is **incoherent in x**: the functional is local-quadratic, Γ ∝ ∫|Δfield-rate|², not |∫Δfield-rate|².

The seven properties Codex demanded, computed:

1. **Spatial kernel / distance dependence (E1):** the branch-difference norm saturates within ~1 lattice cell of separation (1.02 at s=3, 1.0001 at s=16, in units of the disjoint-near-zone limit). The kernel range is the source's own near field — UV-anchored. **There is no (Δx)² window at lab scales: the rate is separation-independent beyond the coherence cell.**
2. **Tensor contraction / polarization:** the four rows watch trace and flux; the dark space is exactly {+,×} (regression T4) — TT branch differences do not decohere. That is also the gauge statement: the billed rows are built from the Action's conservation laws; pure-gauge (audit-dark) deformations bill zero.
3. **Coherent-volume scaling (E2):** total modulation split over M separated elements gives F ∝ 1/M at fixed total — **per-cell additive**. Coherent P² holds only inside one coherence cell; a source of N cells pays Γ = Σ_cells p_cell², i.e. P_total²/N_cells.
4. **Geometry/orientation (E3):** shape factors are O(1) (dumbbell/sphere = 1.15; orientation dependence 1.8%) — the shape prediction survives with mild factors.
5. **Which energy contributes:** whatever sources k and is *modulated* — the rows bill ∂_t(K∗T) contractions; static energy of any form bills zero.
6. **The P² limit (E0):** F(2E)/F(E) = 4.000000 — the n128 scalar law is recovered exactly as the single-cell / single-mode projection, P = ω·δE.
7. **The coherence cell is derived, not fitted:** the monitor's causal patch is ℓ_c = c/γ. At the electron anchor γ = α·m_ec²/ħ this is **ℓ_c = ħ/(α m_e c) = a₀ — the Bohr radius, exactly** (algebraic identity, zero new parameters; F10-clean: it is the anchor choice rewritten, not a fit). The audit's coherence cell is the size of the atom.

## The kill test fired

002 pre-registered: *"If the derived kernel's form factor suppresses the benchmark configurations by more than the declared O(1)-class interval, the milliwatt/piconewton targets are retracted as stated."* It does. An XFEL beam of volume V contains N_cells = V/a₀³ ≈ 10¹²–10¹³ coherence cells at μm³ scale; the form factor is 1/N_cells, moving the required power by ~√N_cells ≈ 10³ (mW → kW-class). **The milliwatt target as previously stated — total beam power into the scalar formula — is retracted.** The old claim, the failure, and the replacement, per rule 5:

- **Old:** Γ = 0.48·Q·(G/c⁵)·P²/(ħγ)·L(ω) with P = total coherent branch-differential power; 1 mW → 0.09 s⁻¹.
- **Failure:** the transfer assumed one mediating mode spanning the apparatus; the local functional pays per coherence cell.
- **Replacement:** Γ = 0.48·Q·(G/c⁵)/(ħγ)·L(ω)·Σ_cells p_cell², cell size a₀ (derived). The formula is unchanged *within one cell*.

## What survives, sharpened

- **Prediction 31 (3 pN momentum channel) survives intact and becomes the clean venue:** a single trapped ion *is* one coherence cell (ion size ~ a₀); the coherent n128 formula applies without form factor. The design conclusion the site already reached ("from XFEL heroics to trapped ions") is now forced by the functional, not just preferred.
- **A new distinctive structure, queued for the promotion gates (n170):** the decoherence rate of a mass superposition is *independent of superposition separation beyond a₀ = 0.53 Å* — a derived saturation scale, where CSL fits r_c ≈ 100 nm and Diósi–Penrose grows toward the nucleon-size regulator. Three-way discrimination in shape, zero fitted lengths.
- Zero-at-rest, the knee, motion-gating: unchanged (structural, E4).

## Open

The momentum rows enter with tensor factors computed here only through the T4 kernel structure; the full row-resolved functional (all four rows on the moving-source lattice) and the noise-matrix computation (Codex item 3) are next. ℓ_c = c/γ is the natural causal cell; a monitor with a different correlation length would change N_cells — that freedom is *named* (rule 9) and bounded by the anchor.

**Epistemic class: [A-model] for the functional and its scalings; [B] for the a₀ identity's physical reading; the retraction is final.**