# PDE C1 — Mori–Zwanzig Energy-Budget Diagnostic (pre-registration) > Mechanism-lane (B), post framing-first. **Purpose:** ground the *known* > Mori–Zwanzig / closure mechanism in this specific cell and **explain the > measured `D_witness` boundary layer** — NOT to claim a closure discovery > (the recon, [`PDE_C1_MECHANISM_RECON.md`](PDE_C1_MECHANISM_RECON.md), found > the phenomenon is MZ/AIM folklore). Status: pre-registered, not built, not > run. Finite-Galerkin, sampled-support, numerical. ## 1. What we measure and why The control-sufficiency result (kNN POSITIVE + paired fiber-constancy) says the safety decision is `Phi_K`-measurable a.e. This diagnostic asks the *mechanistic* question behind it: **how much of the low-band energy tendency is determined by the low modes alone, vs. carried by the unresolved high modes?** If the unresolved coupling `R` is a minority of the budget and its lookahead integral exceeds the decision margin only on a small `mu`-set, that both grounds the mechanism here and explains the ~3.7% `D_witness` residual as the coupling boundary layer. ## 2. The exact decomposition Integrator (from `step()`): `d ω̂(k)/dt = −N̂(k) − ν|k|² ω̂(k) + f̂(k)`, with `N̂ = nonlinear_hat` (advection `u·∇ω`). Differentiating the harness low-band energy `E_low = Σ_low |ω̂(k)/scale|²/|k|²` along the flow: ``` dE_low/dt = D_low + F_low + T_low D_low = −2ν Σ_low |ω̂(k)|²/scale² (low dissipation, low-determined) F_low = Σ_low (2/|k|²) Re[ω̂*(k) f̂(k)]/scale² (forcing input, low-determined) T_low = −Σ_low (2/|k|²) Re[ω̂*(k) N̂(k)]/scale² (nonlinear transfer in) ``` Mori–Zwanzig split of the only unresolved-dependent term, by re-evaluating the nonlinear term on the **low-passed** field `ω̂_L` (all modes outside the 9 signature modes + conjugates zeroed): ``` T_LLL = −Σ_low (2/|k|²) Re[ω̂*(k) N̂(ω̂_L)(k)]/scale² (band-closed; Φ_K only) R = T_low − T_LLL (≥1 high mode; the MZ coupling) g(Φ_K) = D_low + F_low + T_LLL (resolved / Markovian part) dE_low/dt = g(Φ_K) + R ``` `g` depends only on the low modes; `R` is the orthogonal-dynamics coupling. Two nonlinear evaluations per state (full, low-passed). **No closure model is fit** — `R` is measured, not modeled. ## 3. Validation (smoke gates, must pass before the headline) 1. **Budget closure.** `g + R` matches the finite-difference tendency `(E_low(step(ω̂)) − E_low(ω̂))/dt` to `O(dt)` (semi-implicit Euler, so a small discretization residual is expected and bounded, not zero). 2. **`R → 0` on a low state.** For a low-passed input `ω̂_L`, `|R| / |dE_low/dt|` is at machine-noise level (the full and band-closed transfers coincide when there are no high modes). 3. **Sign sanity.** `F_low >= 0` (forcing injects into the band), `D_low <= 0` (dissipation removes). Failing any gate blocks interpretation — the decomposition would be mis-wired. ## 4. Metrics **Level 1 — instantaneous coupling (cheap; primary).** At each of the `50000` adjudication samples (on `supp mu`), decompose `dE_low/dt = g + R`. **Headline (amended 2026-05-29 after validation): population RMS ratio.** The per-sample `|R| / |dE_low/dt|` is ill-conditioned where `dE_low/dt -> 0` (near-laminar states; the validation smoke showed it blowing up to ~10^3). So the robust primary is the population-level ``` rms(R) / rms(dE_low/dt) over the 50000 samples (also rms(R)/rms(g), rms(R)/rms(T_low)) ``` plus the bounded per-sample `rho = |R|/(|g|+|R|)` (median / p90 / mean) and the fraction of samples with `|R| > |g|` as secondaries. *Reading:* a small `rms(R)/rms(dE_low/dt)` (a minority) means the low-band energy tendency is mostly low-determined — the mechanism, measured here. Caveat: instantaneous, not yet decision-tied. Structural validation gate carried into the run: `D_low_max <= 0` (else `MZ_BUDGET_INVALID_DISSIPATION_SIGN`). **Level 2 — lookahead-integrated coupling (run only if Level 1 encouraging; ~2x cost).** Along each `tau`-step lookahead, accumulate `I_R = ∫ R dt` and the realized excursion. *Reading:* the decision flips on `R` only where `|I_R|` exceeds the margin to `E_max`; that set should be ~`D_witness` measure and concentrated near the decision boundary — explaining the residual. Pair-level sharpening (do disagreeing witness pairs have larger `|ΔI_R|`?) is the strongest tie and is the registered follow-up within Level 2. ## 5. Pre-registered interpretation (no goalpost-moving) - This **measures and grounds a known mechanism** (cite Mori–Zwanzig / closure) in this cell and **explains** the `D_witness` boundary layer. It is **not** a closure result and not novel mechanism — the novelty is the observability framing (separation statement §7). - **No pass/fail verdict on C1.** Unlike the cell-set adjudicators, this is explanatory, not promotion-bearing. It cannot make C1 stronger or weaker as a *claim*; it can only explain the existing measured numbers. Honest outcomes: (a) small `rho` + boundary-concentrated `I_R` → mechanism cleanly grounded; (b) large `rho` but control still holds → the sufficiency comes from cancellation/averaging over `tau`, a subtler and still-interesting story; (c) decomposition fails validation → fix wiring, do not interpret. - C1 promotion status is **unchanged** by this diagnostic. ## 6. Build + run plan - Add an `mz-budget` diagnostic to the harness: the decomposition (§2) + the three validation gates (§3) as a `--self-test` path, Level-1 accumulation at sample points, Level-2 accumulation along the lookahead behind a flag. Deterministic; reuses `nonlinear_hat` / `low_energy`. - Smoke + validation gates first. Then: ``` # Level 1 (primary, ~22 min) python scripts/pde_c1_kolmogorov_cell.py --preset lock_v5 --adjudicator mz-budget --out results/proof/c1-mz-budget-g200 python scripts/pde_c1_kolmogorov_cell.py --preset lock_v7_g300 --adjudicator mz-budget --out results/proof/c1-mz-budget-g300 # Level 2 (only if Level 1 encouraging, ~44 min) — same presets, --mz-lookahead ``` ## 8. Level-1 v1 result + the energy-conservation finding (2026-05-29) The first Level-1 run (G=200, `results/proof/c1-mz-budget-g200/`) returned a result that **verification caught before interpretation**: the band-closed transfer `T_LLL` is **identically zero** (`8e-17`, machine precision), not small. This is a conservation law, not a bug — the budget-closure and `R->0` validation gates all passed. The self-advection of the low-passed field conserves the low-passed field's energy, so its net transfer summed over the band is exactly zero: ``` T_LLL = -Σ_low (2/(scale²k²)) Re[ω̂_L*·N̂(ω̂_L)] = -(2/scale²) Σ_all (1/k²) Re[ω̂_L*·N̂(ω̂_L)] = 0. ``` **Physical finding (real, worth keeping):** *the low band cannot self-determine its own energy* — **all** net nonlinear transfer into it is out-of-band (high-mode) mediated. So `g = D_low + F_low` (dissipation + forcing only) and `R = T_low` (the entire inter-scale transfer). Measured at G=200: `rms(R)/rms(dE/dt) ≈ 14`, `rms(R)/rms(g) ≈ 1`, `corr(g,R) ≈ -0.69`, `rms(dE/dt)/rms(g) ≈ 0.24` — i.e. a large transfer `R` quasi-balanced against dissipation+forcing, leaving a small net tendency. **Consequence:** "is `R` small?" is the wrong question — `R` is the whole transfer, by conservation. Control-sufficiency cannot mean `R` is small; it must mean `R` is **predictable from `Φ_K`**. This reframes Level 1. ## 9. Level-1 v2 (redesign): is `R` predictable from `Φ_K`? (held-out R²) The corrected mechanistic test — does the signature pin the net high-mode coupling `R = T_low`? **First estimator tried and rejected (kNN conditional variance).** The natural disintegration `eta_R = E_i[Var_local(R)]/Var(R)` over kNN signature- neighbours was implemented and **failed its own calibration**: the positive control `eta_g` (for `g`, an *exact* function of `Φ_K`) came out `≈ 0.42`, not `≈ 0`. Cause: kNN conditional variance is confounded by (i) the target's steepness (`g` is quadratic in `Φ_K`, so it varies a lot inside any ball) and (ii) 18-dim neighbourhood width (curse of dimensionality — kNN balls are never truly local). Both inflate the estimate independent of predictability. Recorded, not rescued. **Estimator used (held-out regression R²).** The statistically correct, steepness-agnostic, unbiased test of "is `R` a function of `Φ_K`": ``` slaving_index = R²( R predicted from Φ_K ), held out on a block split (first 70% train / last 30% test — no temporal leakage) ``` A flexible regressor (`HistGradientBoostingRegressor`) is fit on the train block and scored on the held-out block. `R² -> 1` means `R` is (an approximate) function of `Φ_K` — **the high modes' net energy-transfer into the band is signature-determined even though the high modes themselves roam (twin states)**: slaving of the *relevant functional*, not the state. Unifies with the kNN action result (`Var(action|Φ_K)->0`). **Built-in calibration controls (known-answer, same data, same regressor):** - `R²(g)` — **positive control.** `g = D_low + F_low` is an exact function of `Φ_K`, so `R²(g) ≈ 1` (sets the regressor's achievable ceiling). Validated inline at **0.997**. - `R²(permuted R)` — **negative control.** Permuted `R` has no signature dependence, so `R² ≈ 0` (negative = worse than the mean). Validated inline at **−0.20** (confirms no block-split leakage). **Pre-registered reading + validation gates:** - Validation: `R²(g) > 0.90` AND `R²(perm) < 0.10` AND `D_low_max <= 0`; else `MZ_COUPLING_ESTIMATOR_INVALID`, do not interpret. - `slaving_index >= 0.70` → `COUPLING_SIGNATURE_SLAVED` (`R` is `Φ_K`-pinned). - `0.30 <= slaving_index < 0.70` → `COUPLING_PARTIALLY_SLAVED`. - `slaving_index < 0.30` → `COUPLING_NOT_SLAVED` (sufficiency is integrated/averaged → Level 2). - Still **explanatory, non-promotion**; C1 status unchanged either way. **Inline pre-run validation (G=200, 4000-sample probe):** `R²(g)=0.997`, `R²(perm)=-0.20` (controls pass), and the measurement `R²(R|Φ_K)=0.998` — `R` is predictable from `Φ_K` *at the exact-function ceiling*. The full 50000-sample headline runs confirm this on the registered samples. Re-runs: `results/proof/c1-mz-coupling-g200/`, `…-g300/` (new dirs; the v1 `c1-mz-budget-*` dirs stand as the conservation-finding record). ## 10. Level-1 v2 result (2026-05-29) — COUPLING_SIGNATURE_SLAVED, both regimes The corrected diagnostic ran on the full registered 50000-sample sets and returned `COUPLING_SIGNATURE_SLAVED` at both Grashof regimes, with both calibration controls clean. | quantity | G=200 (`lock_v5`) | G=300 (`lock_v7_g300`) | | --- | --- | --- | | verdict | `COUPLING_SIGNATURE_SLAVED` | `COUPLING_SIGNATURE_SLAVED` | | **`slaving_index = R²(R\|Φ_K)`** | **0.998** | **0.990** | | `R²(g)` positive control (ceiling) | 0.999 | 0.980 | | `R²(permuted R)` negative control | −0.001 | −0.0005 | | `corr(g,R)` | −0.85 | −0.93 | | `rms(R)/rms(dE/dt)` | 14.1 | 2.9 | | `T_LLL` check `rms(R)/rms(T_low)` | 1.0 | 1.0 | | train / test (held out) | 35000 / 15000 | 35000 / 15000 | Results: `results/proof/c1-mz-coupling-g200/`, `…-g300/`. **Reading.** The net high-mode energy-transfer `R = T_low` into the low band is **~99% predictable from the signature `Φ_K`** at both regimes — *at the exact-function ceiling* set by the `g`-control — while the negative control (permuted `R`) sits at ~0 on 15000 held-out points, ruling out block-split leakage. So `R` is, empirically, an approximate **function of `Φ_K`**, even though the high modes themselves are certified **not** reconstructable from `Φ_K` (twin-state non-injectivity). Equivalently the low-band energy *tendency* is ~99% signature-determined (`R` and `dE/dt` differ only by the exact function `g`). **The mechanism, measured.** The state roams; its decision-relevant aggregate is slaved to the signature. This is the local, approximate **energy-budget closure** `R ≈ R(Φ_K)` — the Mori–Zwanzig orthogonal/memory part is only ~1% here — and it holds *precisely where state reconstruction fails*. That is why the safety decision is `Φ_K`-measurable (control-sufficiency): not because the coupling is small (it is the whole transfer, by conservation), but because it is signature-determined. Closes the loop with the observability framing (separation statement §7): functional observability of the decision works because the energy-budget closure survives the unresolved state. **Scope / honesty.** Explanatory, not promotion-bearing — it explains *why* the measured control-sufficiency holds; it does not change C1's status. Instantaneous-budget predictability (the lookahead-integrated tie, Level 2, remains a registered option but is now less necessary — the instantaneous coupling is already ~99% slaved). Finite-Galerkin, sampled-support, two Grashof points at `k_f=2`. The `c1-mz-budget-*` dirs (superseded magnitude diagnostic) stand as the `T_LLL≡0` conservation-finding record. **C1 stays PROVISIONAL, UNPROMOTED.** ## 7. Cross-references - [`PDE_C1_MECHANISM_RECON.md`](PDE_C1_MECHANISM_RECON.md) — why this is grounding, not discovery. - [`PDE_C1_SEPARATION_STATEMENT.md`](PDE_C1_SEPARATION_STATEMENT.md) §7 — the observability framing the mechanism supports. - [`PDE_C1_PAIRED_FIBER_CONSTANCY.md`](PDE_C1_PAIRED_FIBER_CONSTANCY.md) — the `D_witness` boundary layer this explains. - Mori–Zwanzig closure: Parish–Duraisamy, Phys. Rev. Fluids 2017 (arXiv 1611.03311).