# GPT-Pro physics consult — actionable synthesis (2026-06-21)
Full answer pasted by user (in session). This file = the load-bearing extraction + the prescribed experiment.

## The decisive reframe: the no-go is REAL but NARROW
- **Same-dynamics, settled-state-only EP computes `J⁻¹ℓ`; the true implicit gradient needs `J⁻ᵀℓ`.**
  True adjoint `λ = −J⁻ᵀℓ` (ℓ=∂L/∂z*); EP's infinitesimal nudge contrast solves `J s = −ℓ` ⇒ `s = −J⁻¹ℓ`.
  Exactness for all ℓ ⇒ requires `J = Jᵀ` (self-adjoint in the physical metric P: `PJ=JᵀP`). **That is the precise no-go.**
- **But the BROADER no-go is FALSE.** `ρ(S⁻¹A)<1` is ONLY the convergence condition of the *first-order* (Neumann) AEP
  expansion — NOT a universal impossibility. Exact non-conservative learning EXISTS if you alter the learning dynamics:
  - **AsymEP** (Scurria 2026): add a local antisymmetric correction in the nudged phase so the modified nudged Jacobian = `Jᵀ`.
  - **Dyadic EP** (Scurria 2026): double the state space; the auxiliary variable becomes a physical error signal ∝ cost gradient. EXACT for arbitrary non-conservative stationary systems.
- ⇒ **The obstruction MOVES from "gradient estimator impossible" to "can the HARDWARE embody the adjoint (transpose couplings / doubled state) without becoming a hidden digital backprop engine?"** Math exists; physical realizability is the open problem.
- Exact error operator (better than the ρ proxy): `E = J⁻¹Jᵀ − I = −2 J⁻¹A`. `‖S⁻¹A‖` is a diagnostic only when S is well-conditioned.

## THE PRESCRIBED PRIORITY EXPERIMENT (GPT-Pro's stated #1: "oracle adjoint physics ablation")
At the SAME checkpoints, implement THREE gradient estimators + sweep an asymmetry knob ε (scale of the antisymmetric attention component):
1. **vanilla/our EP** (`g_EP`) — gives `J⁻¹ℓ`-flavored.
2. **exact transpose relaxation** (`g_transpose`): solve `Jᵀ λ = −ℓ` (GMRES; Jᵀ via vjp), then `g = (∂F/∂θ)ᵀ λ`. = the EXACT implicit gradient.
3. **Dyadic/AsymEP** with exact local `A` from autograd as a simulator ORACLE.
Reference: `g_BPTT` (finite-T).
**DECISION:** if `g_transpose` (and/or dyadic) stays aligned + correctly scaled vs `g_BPTT` even at `ρ(S⁻¹A)>1` (closing our measured 84% gap) ⇒ **the collapse is NOT a mathematical no-go — it's a hardware-realization / convergence problem** ⇒ the research problem becomes "build a local transpose/dyadic substrate." If `g_transpose` ALSO fails ⇒ culprit is finite residual / nonlinearity / saddle instability / loss of contraction, NOT the conservativity assumption.
- MUST run at a HEALTHY tight fixed point (the BPTT good solution, or a converged specnorm ckpt) — at the divergence edge (no fixed point) the solve is ill-conditioned (we saw it: MINRES resid 0.2).

## Q2 (damping) — our suspicion CONFIRMED as plausible
`‖S⁻¹A‖ ≤ ‖A_G‖ / ((1+c) − ‖S_G‖)`. Two routes to high ρ: numerator `‖A_G‖` grows, OR contraction margin `(1+c)−‖S_G‖` collapses (S→singular). Weak-contraction good solutions fit the SECOND. Raising `c` buys EP-validity ONLY for *gratuitous* asymmetry; if you rescale attn/FFN by `(1+c)` to preserve the map, A_G & S_G scale too and ρ may not improve. **Damping sweep is high-value.** Decompose ρ into `‖A‖`, log-norm margin `μ_P(J)`, transient gain `sup_t‖e^{Jt}‖`.

## Q3 (physical adjoint) — buildable, not free
- Transpose relaxation `λ̇ = Jᵀλ + ℓ` = recurrent-backprop/adjoint relaxation (NOT BPTT-through-time), physically allowed IF substrate has transpose couplings. Reciprocal crossbar: `Wᵀ` = swap drive/sense ports. Nonreciprocal attention: transpose includes state-dependent transposes of LN/softmax/QK/value/output ⇒ a "transpose twin" = duplicate routing + calibration + derivative circuits + extra phase.
- **Hamiltonian Echo (HEB/HEL)**: physical adjoint via time-reversal — but does NOT directly extend to DAMPED dissipative dynamics (dissipation destroys time-reversal info). Need bath/ancilla embedding or dyadic doubling. (Pourcel 2026: Lagrangian EP ⊇ HEL, forward-only, passes-independent-of-size, local.)
- Passive RC/resistive networks are energy-based (Kirchhoff ⇒ energy) — that's WHY local EP works there; they will NOT natively give `Jᵀ` for directed attention.
- Falsify: simulate transpose mismatch `Jᵀ+Δ` (sparsity/calibration/quantization error) → measure cos/norm-error/stability vs `‖Δ‖/‖J‖` = the HARDWARE TOLERANCE SPEC.

## Q5/Q6 — non-normality & conservative cost
- Non-normality is NOT purely bad (nnRNN uses it for expressivity w/ stable spectra). Right objective: bound transient gain + adjoint mismatch, not kill A. Physical regularizer = contraction in a metric (`μ_P(J)<−m`), NOT Euclidean spectral radius.
- Conservative-attention's 0.15–0.2 CE gap is likely **parameterization/efficiency cost, NOT an information-theoretic wall** (conservative systems can encode directed maps via latent/paired compartments). Practical path = partially-conservative `F = F_energy + ε F_skew`, controlled ε per head/layer, keep `ρ<1` if using vanilla EP.

## Q7 — make-or-break (staged, not monolithic)
Train 4× (10–50M, same corpus): (a) free non-conservative attn + BPTT [oracle]; (b) conservative Energy/Hopfield/CET + exact digital grad [architecture upper bound]; (c) same conservative + EP at strict residual [EP-estimator test]; (d) partially-conservative ε-skew + exact transpose/Dyadic EP. The result names the WALL: architecture (a bad) / EP-estimator (b good, c bad) / physical-adjoint (transpose works, local impl fails) / non-equilibrium (only finite-T BPTT readable).

## Refs surfaced (pursue)
Dyadic EP + AsymEP = Scurria 2026 (arXiv:2602.03670, the EXACT non-conservative method). Lagrangian EP / Hamiltonian Echo Learning = Pourcel 2026 (arXiv:2506.06248) + RHEL (arXiv:2506.05259, forward-only BPTT proxy). Energy Transformer = Hoover/Krotov (arXiv:2302.07253); Convergent Energy Transformer (EP-trainable, OpenReview Qrfml76eWJ — already in our HW collab docs). Monotone-operator equ. nets = Winston-Kolter (arXiv:2006.08591). Resistive-net universal approx = arXiv:2312.15063. nnRNN = arXiv:1905.12080.

## NEXT (our plan)
1. BUILD the oracle-adjoint probe (extend asym_probe: add `g_transpose` via GMRES solve of `Jᵀλ=−ℓ` + `(∂F/∂θ)ᵀλ`; + synthetic ε-asymmetry sweep on the J operator). [codex]
2. RUN at the BPTT good solution (watcher pings) + the ε-sweep on a current converged ckpt. DECIDES fundamental-vs-hardware.
3. Secondary: ρ(S⁻¹A) decomposition (‖A‖ / μ_P margin / transient gain) at BPTT ckpts; damping-c continuation.