Wall-3 theorem addendum: arXiv:2603.26969 (PAR) hardens the J^T obstruction; price list + Demo-M reframe

Co-Authored-By: Claude Fable 5 <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_014FAPDWQ49M5Ye3NpTndTpn

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@@ -96,3 +96,39 @@ The feasibility question: an EP run does ~tens-of-thousands of update STEPS; wit
 - **Device-nudge insight**: an EP/Coupled-Learning *nudge* changes resistance far less than a full state write, so per-nudge endurance is plausibly >> rated full-write endurance (needs empirical confirmation, but favorable). 
 - **ECRAM (electrochemical RAM)** = the symmetric/linear-analog-update + high-endurance technology specifically aimed at in-situ training: "open-loop analog programmable electrochemical memory array" (Nature Comms 2023, s41467-023-41958-4) — but **research-only** (not commercially available; lab/foundry fab). It's the device-physics frontier Wenjuan Zhu / Wong-type collaborators work in.
 - **VERDICT: endurance is survivable** with HfOx-class RRAM (10^10) + threshold-program; ECRAM is the better-but-research-only upgrade. The make-or-break risk is NOT endurance — it's **update linearity/symmetry + device variation** (the asymmetric-nonlinear-update problem), which the digital-accumulate scheme + compensation (stochastic rounding) mitigates. [from earlier run + this]
+
+---
+
+## 2026-07-07 addendum — the J^T wall is now a THEOREM (arXiv:2603.26969), and the price list
+
+**Stern, Frim, Candás, Liu, Balasubramanian, "Contrastive learning in tunable dynamical systems"
+(UPenn physical-learning group — the same lineage as our Demo-M blueprint):** for dynamics that break
+time-reversal symmetry (= non-reciprocal = our non-conservative operators), *exact* gradient descent
+via local rules + a causal, scalable supervisor is impossible **in principle** — the gradient needs the
+adjoint (backward-in-time / transposed) propagator, which the forward physics of a non-reciprocal
+system cannot supply. Their constructive answer: **PAR (Probably Approximately Right)** — a causal
+"forward supervisor" whose update kernel is NOT the gradient but is often positively correlated with
+it; learning succeeds iff the average projection is positive.
+
+**What this hardens:** our "Wall 3" (exact AsymEP correction needs J^T v) is physics, not engineering
+— *for single-ended, forward-dynamics-only hardware*. Our plain-nudge measurement (cos 0.30 at C512)
+is their impossibility measured; our corrected estimator's cos 0.88–0.94 is their PAR correlation
+parameter at LM scale.
+
+**The price list (escape routes, ranked):**
+1. **Bidirectional crossbars** — the transpose of every LINEAR element is physically free by
+   reverse-driving the same array (row-drive vs column-drive; classic analog-BP practice); diagonal
+   nonlinear gains are self-transpose (sample-and-hold the operating point). Cost: ~2x terminal
+   circuitry (drivers/ADCs both sides). This is the Phase-3 chip spec item (Shanbhag conversation).
+2. **Conservative corner (CET energy attention)** — J = J^T by construction: the forward system IS its
+   own adjoint; plain EP exact; the theorem's exemption zone. The expressivity tax (no causal masking,
+   per our Onsager argument) is now weighed against *exact physical trainability* — CET's stock rises.
+3. **PAR acceptance** — run plain contrastive, eat the bias. Price being measured NOW at C512
+   (runs/par_baseline: warm s2000 + plain nudge + stability guard; the corrected twin reached 1.75).
+4. **Mixed-signal (Demo S, current plan)** — correction computed digitally; the theorem says this
+   digital slice cannot be eliminated for free, only bought via (1) or avoided via (2)/(3).
+
+**Demo-M reframe (upgrade):** the OTA board's local contrastive rule on non-reciprocal coupling *is* a
+PAR process — position the demo as "PAR learning + our spectral/residual stability control on
+non-reciprocal analog hardware": complementary to (and citing) their paper, covering exactly the half
+they don't (forward stability), on their home turf.