CALPHAD inverse-discovery

Recover Redlich-Kister binary interaction parameters (L0, L1, L2) from a phase-diagram target using the governed inverse-PINN engine — and refuse the answer when the problem is ill-posed.

Headline number: 1 POST → L0/L1/L2 recovered within ~0.7% of ground truth, GREEN, with a signed Khipu receipt.

CALPHAD (CALculation of PHAse Diagrams) models the excess Gibbs energy of a binary solution as a Redlich-Kister polynomial. Inverse-discovery is the materials-by-design move: given measured thermochemistry, recover the interaction parameters. The SZL inverse-PINN does this with a self-doubt gate — it labels a non-identifiable fit RED/REFUSE instead of asserting a number.

Honest scope. Recovered values are labelled MODELED (fit to data, not measured). The identifiability gate is real (Fisher information / FIM condition number). Λ is Conjecture 1 (advisory, ≤ 0.99). PDD-fingerprint injectivity for the materials registry is Conjecture 2 — OPEN, NOT proven. This recipe runs against the live a11oy Space.

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Prerequisites

python3 -m pip install httpx

Live base: https://a-11-oy.com. Check the organ first:

curl -s https://a-11-oy.com/api/a11oy/v1/pinn/health | jq '{calphad_available, supported_systems}'
# => {"calphad_available": true, "supported_systems": ["duffing","redlich_kister"]}

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Quickstart (live, verified)

import httpx

BASE = "https://a-11-oy.com"

# GREEN case: well-posed CALPHAD target → recover L0,L1,L2
r = httpx.post(f"{BASE}/api/a11oy/v1/pinn/identify", json={"demo": "calphad"}, timeout=120).json()
print("convergence:", r["convergence"]["label"])           # => GREEN
for p in r["discovered"]:
    print(p["name"], "=", round(p["value"], 1), p["units"],
          "ground_truth", p["ground_truth"], "label", p["label"])
# => L0 = -40262.9 J/mol ground_truth -40000.0 label MODELED
# => L1 =   7880.4 J/mol ground_truth   8000.0 label MODELED
# => L2 = ...                                        label MODELED
print("lambda_advisory:", r["lambda_advisory"]["value"], r["lambda_advisory"]["status"])
print("receipt organ:", r["receipt"]["payload"]["organ"])  # => a11oy-pinn

Now the ill-posed case — too few points, too much noise — must REFUSE:

r = httpx.post(f"{BASE}/api/a11oy/v1/pinn/identify",
               json={"demo": "calphad", "case": "ill_posed"}, timeout=120).json()
print(r["convergence"]["label"])         # => RED
print(r["convergence"]["criteria"]["min_fisher"])   # below floor ⇒ unidentifiable
# The engine reports RED and does NOT assert the parameters as trustworthy.

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Full walkthrough

Step 1 — The model

The excess Gibbs energy of a binary A–B solution:

[ G^{xs} = x_A x_B \sum_{k=0}^{n} L_k ,(x_A - x_B)^k = x_A x_B \big[L_0 + L_1 (x_A - x_B) + L_2 (x_A - x_B)^2 + \dots\big]. ]

The unknowns are the Redlich-Kister coefficients (L_0, L_1, L_2) (units J/mol).

Step 2 — Why governed inverse, not a black-box fit

A least-squares fit will always return some numbers. The governance question is: are those numbers identifiable from the data you actually have? The engine computes the Fisher information matrix and its condition number κ(FIM). If min_fisher < 1e-08 or κ(FIM) ≥ 1e+08, the parameter is UNIDENTIFIABLE and the convergence label flips to RED.

Criterion	GREEN	RED
κ(FIM)	< 1e+06 (IDENT)	≥ 1e+08
min_fisher	above 1e-08 floor	below floor ⇒ UNIDENTIFIABLE
normalised_data_rms	< 0.05	—

Step 3 — Read the recovery

Each discovered parameter carries value, a 95% CI (analytic + bootstrap), std, fisher_information, ground_truth, recovery_abs_err, and an F19 Bekenstein plausibility check (the inequality is locked-proven; the application is MODELED with SAMPLE R, E). The GREEN demo recovers L0 ≈ −40 263 J/mol against a −40 000 ground truth (~0.7% error).

Step 4 — Keep the receipt

The receipt.payload is a szl.lake.receipt/v1 envelope (organ a11oy-pinn, kind inverse_pinn_identify) hash-chained into szl-lake. Verify the chain head against the lake:

h = httpx.get(f"{BASE}/api/lake/v1/health", timeout=30).json()
print(h["organs"]["a11oy-pinn"]["chain_head"])

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Honest scope table

Claim	Status
Redlich-Kister recovery	MODELED (fit to data; not measured)
Identifiability gate (Fisher / κ(FIM))	REAL, deterministic
F19 Bekenstein bound	inequality PROVEN (locked-8 @ c7c0ba17); application MODELED
Λ advisory	Conjecture 1 — advisory, ≤ 0.99, never a proof
PDD-fingerprint injectivity	Conjecture 2 — OPEN, Lutar/Materials/PDDInjective.lean
Receipt	SHA3-256 chained; signed or honest DSSE_PLACEHOLDER

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See also

• 19 — Inverse-PINN physics discovery — the Duffing demo, same engine.
• 18 — Governed materials prediction — novelty + PAC-Bayes certify.
• 09 — PAC-Bayes confidence margin — the certified-bound primitive.

Cite this recipe

@misc{szl_cookbook_calphad_inverse_2026,
  title        = {CALPHAD inverse-discovery (SZL Cookbook recipe 16)},
  author       = {{SZL Holdings}},
  year         = {2026},
  howpublished = {\url{https://github.com/szl-holdings/szl-cookbook/blob/main/recipes/16-calphad-inverse-discovery.md}},
  note         = {Redlich-Kister L0/L1/L2 recovery; values MODELED; identifiability gate REAL. Λ = Conjecture 1.}
}

References: Redlich & Kister 1948, Ind. Eng. Chem. 40:345; Lukas, Fries & Sundman 2007, Computational Thermodynamics: The Calphad Method (CUP). Bekenstein 1981, Phys. Rev. D 23:287 (F19).

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Doctrine v11 LOCKED — 749/14/163 — kernel c7c0ba17 · Λ = Conjecture 1 · SLSA L1 (honest)