https://chatgpt.com/g/g-p-68e82d532be8819190b2ee9e17a0d832-agi-foundaton/shared/c/6907c55f-3c84-832d-90d2-27becf0d8845

https://osf.io/vj7ue/files/osfstorage/6907dc34bfc15c949a96ca54

SMFTxAlphaFold - Learning the Loss: Recovering Lagrangians from Case Logs at Scale

1) Introduction — Why learn Lagrangians from logs?

Modern organizations already operate “expert systems” at scale—dispatch tools, triage protocols, procurement scorers, treatment pathways, meeting playbooks. Day after day, these systems and their human operators leave behind large case logs: contexts, candidate options, final choices, and observed outcomes. Hidden in those logs are the implicit trade-offs that drove each decision: how much risk was tolerated to save time, how much cost was accepted to raise quality, when smoothness or stability trumped raw utility. This paper shows how to recover an interpretable Lagrangian (L(x)) and constraint set (Γ) from such logs—turning tacit decision culture into an auditable, portable scoring calculus that generalizes beyond the past and can be governed with explicit safety gates.

One-line promise. Learn a sparse, units-consistent (L) and (Γ) from logs; gate with Two Lamps; schedule development with Δ5; ship with ObserverOps.

Why a Lagrangian? Because many real decisions are multi-objective and dissipative: value gained, costs paid, risks carried, and adjustments smoothed across steps in a plan. A Lagrangian bundles those forces into a single sequence score while keeping hard limits non-negotiable and soft limits priced—the right balance of interpretation and control for enterprise deployment.

Core objective (universal form).
(1.1) x* = argmaxₓ J(x) − λ·R_soft(x) s.t. Γ_hard(x) ≤ 0.

Here, (J(x)) is the domain’s value function (utility/benefit), (R_\text{soft}(x)) aggregates priced violations (soft constraints), (Γ_\text{hard}(x) \le 0) encodes inviolable limits, and (λ) scales the penalty. Learning from logs means estimating the variables and parameters that make this objective reproduce historical choices and predict outcomes, then stress-testing and governing it before deployment.

This method builds on two prior pillars:

Structure-before-search (TopoSpec). We first author the problem’s slots, invariants, and constraints—a compact schema of what must always be true—so learning focuses on real trade-offs instead of rediscovering obvious structure. TopoSpec (from SMFT) is a neutral, domain-agnostic way to do this: a small contract for inputs (I), invariants (H), constraints (C), boundaries (∂Ω), and objective fields (J).
Gated, auditable deployment. We validate with Two Lamps: (i) Cross-Observer Agreement (independent tools agree on top-k rankings within tolerance), and (ii) Emulsion-Stabilized Inference (scores remain stable under controlled perturbations), so learned policies don’t overfit spurious patterns. We manage the build-measure-verify cadence with Δ5 (a five-phase scheduler that separates exploration from verification to avoid drift), and we ship with ObserverOps (hash-anchored footers logging seeds, thresholds, TopoSpec hash, and gate outcomes) for full replay and audit.

Readers unfamiliar with SMFT need only the following working intuition: SMFT (Semantic Meme Field Theory) supplies a practical scaffolding—TopoSpec to lock structure, a dissipative action view to score real-world plans, Two Lamps to gate safety, Δ5 to time the loop, and ObserverOps to make runs reproducible. With these rails in place, the rest of the paper shows how to learn (L) from case logs automatically and responsibly—so the organization’s lived expertise becomes a compact, inspectable objective rather than a black box.

2) Background & Positioning — From Everyday Structure AI to Semantic AlphaFold

Structure-before-search, proven in science, transferable to operations.
AlphaFold-era biology and graph-based materials pipelines lock invariants and feasibility first (e.g., chemistry rules; convex-hull stability), then search inside that safe set. This “structure-before-search” lens explains why such systems quietly power real products: decisions become constrained design rather than trial-and-error. We adopt the same contract for everyday pipelines (menus, meetings, procurement, care paths): fix structure, then explore.

Compiler view (portable spine).
Our method is a compiler with five fixed stages—TopoSpec → Action score → Two Lamps → Δ5 → ObserverOps:

TopoSpec. Author slots, invariants, hard/soft constraints, context, and objective; feasible set “F(s) = { x ∈ X : Γ_hard(s,x) ≤ 0 }.” (2.0).
Action score (dissipative Lagrangian). Compare candidates by one auditable scalar:
“S(s,x) = Σ_t[ −U + α·Cost + β·Risk + γ·‖Δx‖² ] + λ·Σ_t max{0, Γ_soft }.” (2.1) Lower is better. Hard limits remain non-negotiable.
Two Lamps (validation gates). Deploy only if independent observers agree and the choice is anti-fragile:
“CSA@k = #(agreements within ε in top-k) / N.” (2.2) “χ = Var_starch / Var_raw.” (2.3)
“Accept ⇔ [ CSA@k ≥ τ ∧ max-ε ≤ ε_max ∧ χ ≤ χ_max ].” (2.4)
Δ5 scheduler. Alternate Explore/Fold/Verify/Defrost/Decide; halt on surplus-tension spikes: “Δ = ĝ·β̂ − γ̂.” (2.5)
ObserverOps. Ship each run with a hash-anchored footer for full replay:
“H = SHA-256(model_id ∥ data_id ∥ codehash ∥ seed ∥ topospec_hash).” (2.6)

Why a dissipative Lagrangian is principled (not a hack).
SMFT’s generalized least-action results show that adding a dissipation functional naturally covers open, real-world systems. In the conservative limit, we recover the standard Euler–Lagrange laws; with local dissipation, we recover Rayleigh-style damping (classical “drag”). Example:
“m·x¨ = −∇V(x) − c·x˙.” (2.7)
Hence, the everyday scoring rule in (2.1) is the enterprise analogue of a physically grounded action with dissipation—one scalar that balances value, cost, risk, and smoothness under inviolate constraints.

Universal objective, carried from science to operations.
Across domains we select by the same one-liner:
“x* = argmaxₓ J(x) − λ·R_soft(x) s.t. Γ_hard(x) ≤ 0.” (2.8)
This is the decision version of “least action with hard constraints,” making the science-proven spine portable to menus, meetings, factories, and clinics—while Two Lamps, Δ5, and ObserverOps keep it safe, stable, and auditable.

3) Problem Setup & Notation — What’s in the logs, what we learn

Dataset and symbols.
We assume a historical corpus of expert-system or human-in-the-loop runs with context, candidates, the chosen action, and outcomes:
(3.0) D = { (sᵢ, 𝒞ᵢ, aᵢ*, yᵢ) }ᵢ=1..N.
Here, sᵢ is the observed context (features, constraints, timestamps), 𝒞ᵢ is the candidate set available at decision time (plans or actions; possibly sequences {x_t}), aᵢ* is the chosen candidate, and yᵢ are post-choice outcomes (KPIs, costs, risks). When logs do not store 𝒞ᵢ, we reconstruct it with a domain sampler G (retrieval, generator, or enumerator):
(3.0a) 𝒞̂(s) = G(s; ψ) with ψ fixed during training.

TopoSpec rails and feasibility.
A domain contract (TopoSpec) supplies hard and soft constraints; the feasible set under hard limits is:
(3.1) F(s) = { x ∈ X : Γ_hard(s, x) ≤ 0 }.
Soft limits are priced, not forbidden. The universal selection objective is:
(3.2) x* = argmaxₓ J(s, x) − λ·R_soft(s, x) s.t. x ∈ F(s).
We work with a dissipative Lagrangian that scores sequences (or single-step actions as a special case):
(3.3) S(s, x) = Σ_t[ −U(s_t, x_t) + α·Cost(s_t, x_t) + β·Risk(s_t, x_t) + γ·‖Δx_t‖² ] + λ·Σ_t max{0, Γ_soft(s_t, x_t)}.
Lower S is better; hard constraints remain non-negotiable.

Feature universe and parameterization.
We define an interpretable, unit-checked feature map φ(s, x) partitioned by semantics (utility, cost, risk, smoothness, soft-violation terms). A compact spine for the learnable loss is:
(3.4) L(s, x; θ) = θᵀ φ(s, x), with sign and monotonicity constraints on θ to respect domain physics.
For sequences, Δx_t = x_t − x_{t−1}. We normalize features and carry units so weights are comparable:
(3.5) φ̂_j = (φ_j − μ_j) / σ_j, θ̂_j = θ_j·σ_j.

What we learn.
From logs, we learn the variables and parameters that make historical choices look optimal under TopoSpec:
• The Lagrangian parameters θ (including α, β, γ, λ).
• The soft-constraint weights for Γ_soft, with units and signs enforced.
• Optional new variables/terms revealed by residual analysis (added sparsely and with unit checks).
• Optional regime structure (a small mixture {L_m} with a gating function when behavior is nonstationary).

Rationalizing historical choices.
Given 𝒞ᵢ (or 𝒞̂ᵢ), the learned pair (L, Γ) should make the logged choice aᵢ* outrank viable alternatives and predict outcomes yᵢ under replay:
(3.6) aᵢ* ∈ argmin_{x ∈ F(sᵢ)} S(sᵢ, x; θ).
(3.7) ŷ(sᵢ, aᵢ*; θ) ≈ yᵢ, with calibrated uncertainty for governance.

Minimal log schema (practical).
Each record needs: (i) a stable TopoSpec ID/hash, (ii) context s (including units), (iii) either the candidate set 𝒞 or a reproducible sampler G, (iv) the chosen action a* (with full plan parameters or a pointer to reconstruct it), and (v) outcomes y plus timing (to align effects with the decision’s horizon). Observers used for later validation (agreement and stability) may be logged as auxiliary fields.

Identification conditions (at a glance).
Recovery is possible when: (i) there is variation in 𝒞 (“persistency of excitation” across contexts), (ii) hard constraints Γ_hard are specified (so feasibility is known), (iii) units are calibrated (so trade-offs are comparable), and (iv) observers used for evaluation are sufficiently independent (so agreement is meaningful). These conditions ensure we can learn an interpretable, sparse L(s, x; θ) that both rationalizes past choices and generalizes under the same TopoSpec rails.

4) Authoring the TopoSpec — Structure-before-search for your domain

Purpose. TopoSpec freezes the rails—what is always true—so generators and learners cannot “move the goalposts.” You declare what counts (objective fields) and what’s allowed (hard/soft constraints) with explicit units. Once frozen and hashed, every stage (search, learning, gates, scheduler) runs on the same contract.

Schema (paste-ready mental model).
(3.0) TopoSpec = { I, H, C, ∂Ω, J }.
(3.1) F(s) = { x ∈ X : Γ_hard(s, x) ≤ 0 }.
(3.2) x* = argmaxₓ J(s, x) − λ·R_soft(s, x) s.t. x ∈ F(s).
Where I = invariants (true by construction), H = hard constraints (gᵢ ≤ 0), C = context/priors (data sources, policies, observer rosters), ∂Ω = boundaries (budgets, windows, envelopes), J = objective (U, Cost, Risk decomposition) with declared units.

Hash & provenance.
(3.6) H_TopoSpec = SHA-256( serialize(TopoSpec) ).
Freeze the file before training; record H_TopoSpec in the ObserverOps footer so audits can replay decisions against the exact spec.

How to write each section (with units)

I — Invariants (truths by construction).
Examples: “exactly 7 dinners/week,” “every agenda item has {owner, due, evidence},” “probabilities sum to 1.” Encode as schemas/templates so they cannot be violated by generators. (If it can be violated, it belongs in H or R_soft, not I.)

H — Hard constraints (non-negotiable).
Declare small, atomic gᵢ with units: mg Na/day ≤ cap, budget ≤ £B, legal_ok = 1, process_window(T, pH) bounds. Use many small gᵢ over one mega-constraint for better diagnostics and projection.

C — Context & priors.
List data snapshots, policy profiles, observer families (for agreement), and default weights (α, β). C also defines which observers feed Two Lamps and which levers ESI may perturb for χ.

∂Ω — Boundaries (where to look, not what’s allowed).
Budgets, time windows, safety envelopes, generation bounds. Keep ∂Ω ≠ H: boundaries guide search; hard constraints gate feasibility.

J — Objective (auditable decomposition).
Declare units and sign conventions:
(4.1) J(x) = U(x) − α·Cost(x) − β·Risk(x).
(4.2) R_soft(x) = Σⱼ wⱼ · max{0, sⱼ(x)}.
Then the sequence score (dissipative action) used later is:
(4.3) S = Σ_t[ −U + α·Cost + β·Risk + γ·‖Δx‖² ] + λ·Σ_t max{0, Γ_soft}. (lower is better)

HeTu–LuoShu structural priors for Γ (turn conservation into predicates)

SMFT’s slot interpretation treats LuoShu’s row/column/diagonal “sum-15” and HeTu’s “pair-sum-11” as conservation laws (capacity cannot vanish or appear out of thin air). We reuse these as structural priors when authoring Γ—highly portable for planning, allocations, balanced rosters, bill-of-materials, or layout paths.

Pair-sum priors (HeTu). For paired resources or roles that must balance:
(4.4) g_pair⁺(i,j) = (x_i + x_j) − K ≤ 0, g_pair⁻(i,j) = K − (x_i + x_j) ≤ 0.
Pick K = 11 to mirror HeTu, or your domain’s balance constant. Use as H when balance is inviolable, else move a hinge penalty to R_soft.

Path-sum priors (LuoShu). For any admissible path P (row/column/route) requiring constant throughput or headcount:
(4.5) g_path⁺(P) = Σ_{c∈P} x_c − S ≤ 0, g_path⁻(P) = S − Σ_{c∈P} x_c ≤ 0.
LuoShu’s “15” is the archetype; in practice S is your required total per path/shift.

Capacity caps (HeTu “10” ceiling). For per-site capacity:
(4.6) g_cap(i) = x_i − C_max ≤ 0.
Use integer or continuous x depending on the domain; keep units explicit (e.g., “tasks/day”, “kg/week”).

These priors shrink the feasible set without tuning, improving identifiability and stability before any learning. They are mathematically motivated (conservation/symmetry) and operationally readable by non-experts.

Authoring tips (battle-tested)

Make invariants real. If a rule can be broken, it’s not an invariant—move it to H or R_soft.
Prefer atomic constraints. Many small gᵢ help projection and debugging.
Declare units everywhere. Normalize features by units so α, β, γ, λ are meaningful.
Hash the spec. Record H_TopoSpec and the compile hash in ObserverOps; no gate, no deploy.

With TopoSpec frozen and hashed, the rest of the pipeline—Auto-Lagrange → Two Lamps → Δ5 → ObserverOps—runs on rails that are portable, auditable, and safe by construction.

5) Candidate Lagrangian Family — Parametric spines and units

Auditable spine (what every domain shares).
(5.1) L(x; θ) = −U(x) + α·Cost(x) + β·Risk(x) + γ·‖Δx‖².

Where U/Cost/Risk are declared KPIs in the TopoSpec objective and ‖Δx‖² is the “don’t-thrash” smoothness term (discrete friendly). Signs are policy-meaningful: α, β, γ, λ ≥ 0; hard limits stay outside (L).

Soft penalties (hinge by default, units explicit).
(3.4) R_soft(x) = Σⱼ wⱼ·max{0, sⱼ(x)}.
Declare units for each (s_j) and keep weights (w_j) in the same measurement system you normalized to—so λ trades off in comparable units. (Huber shaping is a drop-in if needed.)

Dissipative Action over sequences (one score to compare plans).
(2.1) S = Σₜ[ −U + α·Cost + β·Risk + γ·‖Δxₜ‖² ] + λ·Σₜ max{0, Γ_soft}. (lower is better)

Decomposition reminder (keeps audits readable).
(5.2) J(x) = U(x) − α·Cost(x) − β·Risk(x).
Use (5.2) for reports, (5.1)/(2.1) for scoring and replay.

Units and normalization (so weights mean what you think).
(5.3) scale(term) = 1 / MAD_baseline(term).
(5.4) term_norm = scale(term) · term_raw.
Normalize U, Cost, Risk, sⱼ before choosing {α, β, λ}; then pick γ to penalize step-scale twitching (below). Keep the scales in the compile log.

Choosing γ (robust rule-of-thumb).
(5.5) γ = κ · Medianₜ(‖∇U(xₜ)‖²), κ ∈ [0.1, 1.0].
This ties smoothness to the typical utility gradient so plans don’t oscillate.

Risk is a slot, not a dogma (plug-and-play).
You can drop in variance, CVaR, or constraint-proximity risk without touching the spine in (5.1)/(2.1). The compiler treats Risk(·) as a declared KPI family with units.

Physics footing (why the form is principled).
With Γ_soft = 0 and γ > 0, the Euler–Lagrange form adds Rayleigh-style drag; the discrete action (2.1) is the operational analogue for open systems. This is the generalized least-action picture with dissipation, not an ad-hoc heuristic.

Single-step vs. sequences.
For one-shot choices, drop the time index and ‖Δx‖² term. For plans, keep (2.1) and report per-step contributions (U, Cost, Risk, soft penalties) so reviewers can audit each part.

Do / don’t (operational).
Do normalize before tuning α, β, λ; do keep γ > 0 in operations; don’t bury policy into J that belongs in hard constraints H.

6) Inverse Optimization Objective — Decision matching + outcome alignment

We learn the parameters of the Lagrangian by making the historical choice look optimal and predict its observed outcomes, while disciplining constraint behavior. The objective blends a ranking loss, an outcome loss, and regularizers that encode feasibility and identifiability.

Scores and soft policy.
(6.1) score(s, x; θ) = −S(s, x; θ).
(6.2) π_τ(x | s; θ) = exp(score/τ) / Σ_{x′∈𝒞(s)} exp(score′/τ).

Decision-matching (ranking) loss.
A logit/Plackett–Luce view that makes the recorded choice outrank alternatives:
(6.3) ℓ_rank(θ) = −Σ_i log π_τ(a_i* | s_i; θ).
Optionally, a pairwise hinge/logistic variant:
(6.4) ℓ_pair(θ) = Σ_i Σ_{x∈𝒞_i{a_i*}} log(1 + exp(−(score(s_i, a_i*; θ) − score(s_i, x; θ)))).

Outcome alignment (predict what happened).
Let ŷ be a calibrated predictor (linear/GLM/monotone head) tied to φ(s, x):
(6.5) ℓ_out(θ, ω) = Σ_i ‖ ŷ(s_i, a_i*; θ, ω) − y_i ‖²_Σ.
Here Σ can weight KPIs by units or business priority.

Constraint discipline via posterior regularization.
We bound expected soft-violation under the learned policy and penalize excess:
(6.6) v(s, x) = Σ_j max{0, s_j(s, x)}.
(6.7) ℓ_post(θ) = max{0, E_{x∼π_τ(·|s;θ)}[ v(s, x) ] − ε }.

Bilevel (KKT) or softmin training.
Option A: Differentiable solver via KKT residuals (when you can project on Γ_hard).
(6.8) ℓ_KKT(θ, μ) = Σ_i ‖ ∇x S(s_i, a_i*; θ) + Σ_j μ{ij} ∇_x g_j(s_i, a_i*) ‖²
+ ‖ min(0, μ_i) ‖² + ‖ max(0, g(s_i, a_i*)) ∘ μ_i ‖².
Here g_j ≤ 0 are hard constraints, μ_i ≥ 0 are multipliers; “∘” is elementwise product.

Option B: Differentiable softmin (no explicit solver required).
(6.9) softmin_τ S(s; θ) = −τ · log Σ_{x∈𝒞(s)} exp( −S(s, x; θ)/τ ).
(6.10) ℓ_softmin(θ) = Σ_i [ S(s_i, a_i*; θ) − softmin_τ S(s_i; θ) ].

Identification guardrails (make the solution interpretable and stable).
(6.11) θ ∈ Θ_mono with ∂S/∂Cost ≥ 0, ∂S/∂Risk ≥ 0, ∂S/∂‖Δx‖² ≥ 0.
(6.12) sign(α, β, γ, λ) = (+, +, +, +).
(6.13) φ̂_j = (φ_j − μ_j) / σ_j, optimize θ̂ on φ̂ to ensure unit-aware scaling.
(6.14) ℓ_sparse(θ) = ρ · ‖θ‖₁ (prefer small, auditable L).
(6.15) ℓ_stab(θ) = Σ_i Σ_t max{0, V(x_{t+1}) − V(x_t) }, with V(x) = xᵀ P x, P ≻ 0.

Total training objective (choose KKT or softmin path).
(6.16) θ* = argmin_θ [ ℓ_rank + η·ℓ_out + ζ·ℓ_post + ρ·‖θ‖₁ + ξ·ℓ_stab + χ·ℓ_KKT ] s.t. (6.11)–(6.13).
or
(6.17) θ* = argmin_θ [ ℓ_softmin + η·ℓ_out + ζ·ℓ_post + ρ·‖θ‖₁ + ξ·ℓ_stab ] s.t. (6.11)–(6.13).

Notes.
• Use (6.16) when you can evaluate projections and gradients on Γ_hard; use (6.17) when you only have candidate sets.
• Keep τ small enough to sharpen ranks but not so small that gradients vanish.
• Enforce (6.11)–(6.13) to prevent sign flips and unit drift; these are the practical keys to identifiability (formal sketches in App. A).

7) Residual-Driven Symbolic Discovery — Finding missing variables/terms

We use residuals as sensors to discover missing structure. Start with the fitted spine (L(x; θ_0)), then look for systematic errors that correlate with physics, process limits, or overlooked priors. Only accept unit-consistent candidates that improve fit and survive the gates in §9.

Step 1 — Build residual signals.
(7.1) ΔS_i = S(s_i, a_i*; θ_0) − softmin_τ S(s_i; θ_0).
(7.2) e_i = ŷ(s_i, a_i*; θ_0) − y_i.
(7.3) v_i = Σ_j max{0, s_j(s_i, a_i* )}.
Interpretation: ranking margin error (ΔS_i), outcome residual (e_i), and soft-violation load (v_i).

Step 2 — Localize the structure of error.
Bucket residuals by distance-to-boundary, workload/scale, and context regime:
(7.4) b_j(s,x) = s_j(s,x) (soft boundary score); d_H(s,x) = max_j g_j(s,x).
Cluster ({ΔS_i,e_i,v_i}) over ((b, d_H, \text{context bins})) to see where errors concentrate.

Step 3 — Generate unit-aware candidates.
Define a grammar with unit bookkeeping (u(·)) and allow only dimensionally valid forms:
(7.5) 𝒢 = { +, −, ×, ÷, max{·,0}, min{·,0} ; log(·) only if u(·)=1 ; pow(·, p) only if u(·)=1 or p∈ℤ }.
Search terms (\phi_{\text{new}}(s,x) = f(φ(s,x))) s.t. (u(\phi_{\text{new}}) ∈ {u(\text{Risk}), u(\text{Cost}), u(\text{penalty})}).

Step 4 — Screen by structured residual reduction.
Fit a small augmentation with L1 (keep it sparse):
(7.6) L′(x; θ) = L(x; θ_0) + θ_new · φ_new(s,x).
Accept only if the bucketed errors drop:
(7.7) ΔSR = SR_before − SR_after ≥ δ_min,
where (SR) is a stratified score over buckets (e.g., mean (ΔS) at high (b), tail (‖e‖) near (d_H≈0)).

Step 5 — Orthogonalize to avoid re-labeling old terms.
(7.8) φ̃_new = φ_new − Π_{span(Φ_old)} φ_new.
Require low collinearity (e.g., VIF ≤ τ_VIF) so the new term adds novel signal.

Step 6 — Stability and monotonicity guards.
(7.9) sign(∂S/∂φ̃_new) = policy_sign, and ∂S/∂φ̃_new ≥ 0 if φ̃_new is a “risk/penalty”.
(7.10) χ_new ≤ χ_old and CSA@k_new ≥ CSA@k_old − ε_tol.
Reject any term that reduces error by overfitting (instability or agreement collapse).

Step 7 — Model selection with parsimony.
(7.11) MDL = −2·LL_val + b·|θ|, choose argmin MDL.
Report the bits cost of each added term; prefer the smallest model that passes gates.

Concrete example — manufacturability gap.
If residuals cluster near process limits, propose a soft penalty reflecting “distance to a feasible process” (units match your penalty scale):
(7.12) manufacturability_gap(s,x) = max{ 0, m_req(s) − m_realized(s,x) }.
Add to soft penalties:
(7.13) R_soft(x) ← R_soft(x) + w_mfg · manufacturability_gap(s,x).
Re-fit (θ); accept only if (7.7)≥δ_min and (7.10) holds.

Operational recipe (one pass).
(7.14) Find φ_new with grammar 𝒢 under unit rules → Orthogonalize (7.8) → Refit small (θ_{new}) with L1 → Check structured Δ (7.7) → Enforce monotone/sign (7.9) → Gate via Two Lamps (7.10) → MDL select (7.11) → Freeze.

Notes.
• Use counterexample harvesting: failed-gate cases feed the candidate pool for the next round.
• Keep an “allowlist” of lawful units for each KPI family to block illegal logs and ratios.
• Log every accepted term with: definition, units, bucket gains, gate deltas, and compile hash—so audits can replay why it entered (L).

8) Regime Switching & Drift — Mixtures, gates, and change points

Real operations are not single-regime. Nights differ from days, incidents from routine, peak season from off-season. We therefore model behavior as a mixture of Lagrangians and govern transitions with gates and cadence so learning doesn’t chase noise or lock into a brittle mode.

Mixture-of-Lagrangians (MoL).
We keep a small set of regime Lagrangians ({L_m}{m=1..M}) (with their actions (S_m)) and blend them by a soft gating function over context (s):
(8.1) (S(s, x) = \sum{m=1}^M p(m \mid s), S_m(s, x)).
(8.2) (p(m \mid s) = \exp(g_m(s)) ,/, \sum_{r} \exp(g_r(s))).
At training time we minimize the same losses as §6 but under the mixture (S); at inference we either use the blended score in (8.1) or hard-assign the active regime when one mode dominates:
(8.3) (m^*(s) = \arg\min_{m} S_m(s, a^*).)

Regularizers to avoid mode collapse / fragmentation.
Encourage real regime structure without exploding modes:
(8.4) (\Omega_{\text{ent}} = -\lambda_H \sum_i \sum_m p(m \mid s_i),\log p(m \mid s_i)) (discourage overconfident gates early).
(8.5) (\Omega_{\text{spread}} = \lambda_S \sum_m \bigl|, \frac{1}{N}\sum_i p(m \mid s_i) - \pi_m \bigr|) (target prior (\pi_m), prevents dead modes).
(8.6) (\Omega_{\text{drift}} = \lambda_D \sum_m |\theta_{m,t} - \theta_{m,t-1}|^2) (penalize needless parameter twitching).

Change-point detection (split regimes when the world shifts).
We monitor parameter trajectories and residuals and cut when behavior truly changes:
(8.7) (\hat{t} = \arg\max_t D_{\mathrm{KL}}(\hat{\theta}{1:t} ,|, \hat{\theta}{t+1:T})) (create or retune regimes around (\hat{t})).
Use this offline to propose a new regime; keep it shadowed until gates clear (below).

Gate integration for mixtures (Two Lamps, regime-aware).
Each regime has its own gate thresholds; the blend inherits a weighted gate score:
(8.8) (\mathrm{CSA}^{\text{mix}}@k = \sum_m \bar{p}_m, \mathrm{CSA}_m@k,) with (\bar{p}_m = \frac{1}{N}\sum_i p(m \mid s_i).)
(8.9) (\chi^{\text{mix}} = \sum_m \bar{p}m, \chi_m.)
Deploy only if the mixture passes both: (\mathrm{CSA}^{\text{mix}}@k \ge \tau) and (\chi^{\text{mix}} \le \chi{\max}), and no active regime fails catastrophically on its slice.

Δ5 cadence (anti-phase exploration and verification).
We separate Explore (discover/refit regimes) from Verify (gate them), so drift does not whipsaw production. The scheduler monitors surplus-tension (\Delta) and gate health to decide when to advance or halt:
(6.1) (\Delta = \hat{g}\cdot \hat{\beta} - \hat{\gamma}.) (surplus-tension monitor)
(D.12) Stop ⇔ ((\Delta > \Delta_{\max}) ,\vee, (\text{gate fails})).
High (\Delta) means the system is “over-eager” (utility pressure outweighs damping), a classic precursor to oscillation; failing gates indicate instability or loss of cross-observer consensus. Either condition halts regime flips and triggers Defrost/Verify cycles before any new mode is promoted.

Practical training loop (MoL).

Seed (M) with domain-known modes (e.g., normal, rush, incident) or by offline change-point splits.
Fit ({S_m}) and (p(m\mid s)) jointly with §6 losses plus (\Omega_{\text{ent}}+\Omega_{\text{spread}}+\Omega_{\text{drift}}).
Shadow-test new/retuned modes; compute (\mathrm{CSA}_m@k) and (\chi_m) on their slices; then check (8.8)–(8.9).
Promote only if Two Lamps pass and Δ5 says the system is in a Decide window; otherwise keep modes shadowed and continue Verify.

Safety & identifiability tips.

Cap (M) (e.g., 2–4). More modes raise identifiability risk and audit cost.
Enforce the same unit normalization and sign constraints per regime as in §6; without them, regimes can counterfeit each other.
Require minimum support per regime before promotion: (N_m \ge N_{\min}).
Log regime assignments and gate metrics per batch in the ObserverOps footer, so auditors can track when and why a regime took over.

Mixtures let us capture real-world nonstationarity without turning the objective into soup. The Lagrangians stay interpretable; gates keep them honest; Δ5 ensures we never confuse exploration with production.

9) Validation via Two Lamps — CWA (CSA@k, max-ε) and ESI (χ)

Why two lamps? Agreement ensures consensus under independence; stability ensures stickiness under small, lawful nudges. We deploy a learned Lagrangian only when both lamps glow green.

Lamp 1 — Cross-Worker/Observer Agreement (CWA)

Observers. Independent rankers of the same candidate set (e.g., different models, rules, or feature views). Independence is linted at build time (no shared backbone where avoidable).

Core metrics.
(B.1) CSA@k = #(agreements within ε in top-k) ÷ N.
(B.4) max-ε = max_{i≤k, a<b} | score_i^(a) − score_i^(b) |.

Interpretation: CSA@k counts how often observers’ top-k sets coincide within tolerance ε across N decisions; max-ε is the worst score gap between any two observers inside the agreed top-k (smaller is better).

Optional rank-shape checks (report, don’t gate by default).
(B.5) τ_K = Kendall-τ(top-k lists). (B.6) ρ_S = Spearman-ρ(full ranks).
(B.7) N_eff = 1 ÷ Σ_j w_j² (effective number of independent observers; require N_eff ≥ N_min).

Pass rule for Lamp 1.
(B.8) Pass_CWA ⇔ [ CSA@k ≥ τ_CSA ∧ max-ε ≤ ε_max ∧ N_eff ≥ N_min ].

Lamp 2 — Emulsion-Stabilized Inference (ESI)

Idea. “Starch” the evaluation with small, structured perturbations and check that scores barely wiggle. Good models emulsify these micro-shocks rather than amplify them.

How to starch. For each case, create S perturbed replicas using knobs (T/S/K):
• T = perturbation temperature (amplitude as a fraction of feature scale).
• S = number of replicas per case (e.g., 8–16).
• K = perturbation kernel (e.g., Gaussian on normalized inputs; structured dropout on features; jackknife on candidate set; seed jitter on observers).
Keep perturbations lawful (never violate hard constraints).

Stability ratio.
(B.2) χ = Var_starch(score) ÷ Var_raw(score).
Where Var_raw is the variance of scores across raw seeds/observers without perturbation, and Var_starch is the variance under (T/S/K) perturbations. Lower χ means more stability.

Deployment gate (both lamps).
(B.3) Conf_accept = 1[ CSA@k ≥ τ_CSA ∧ max-ε ≤ ε_max ∧ χ ≤ χ_max ].

Operational guidance

Thresholds (defaults you can tune).
(B.9) τ_CSA ∈ [0.75, 0.90]; ε_max tied to score unit (e.g., 0.5 MAD of S); χ_max ∈ [0.35, 0.60].
Set cohort-specific thresholds (routine vs. incident) if you use regimes; never relax hard constraints to pass a gate.

ESI knobs (T/S/K) cheat-sheet.
• Start with T = 0.25 (quarter-sigma on normalized features), S = 8, K = Gaussian + 10% structured dropout.
• For sequence plans, also jitter step timing and tiny Δx penalties (keep Γ_hard intact).
• Record (T, S, K) in the run footer for replay.

Repair prompts when χ is high (stability fails).
• Increase γ (smoothness) or add a slope cap on sensitive features.
• Add or strengthen distance-to-boundary soft penalties.
• Remove collinear observers; enforce independence lint to raise N_eff.
• Narrow ∂Ω (generation bounds) so candidates avoid knife-edge regions.
• Re-normalize units; large χ often correlates with scale drift.

CSA dashboards (what to show).
• CSA@k, max-ε, τ_K, ρ_S over time and by cohort/regime.
• Per-observer contribution and N_eff trend.
• Top failing cases with score spreads and feature attributions.
• Gate decisions: pass/fail with (τ_CSA, ε_max, χ_max, T/S/K) stamped.

Audit & reproducibility.
Log all gate inputs and outputs in the ObserverOps footer: observers roster and seeds; ε_max, τ_CSA, χ_max; T/S/K; CSA@k, max-ε, χ; plus the TopoSpec hash. A deployment without both lamps green must drop into Verify (Δ5 cadence) and is not promoted.

Bottom line. Agreement without stability is brittle consensus; stability without agreement is a quiet overfit. Two Lamps demands both—so recovered Lagrangians are both socially corroborated and mechanically calm.

10) Identification & Theory — When is L identifiable? What are the guarantees?

This section formalizes when the learned Lagrangian is identifiable from logs, what ambiguities remain (and how we fix them), and why the dissipative scoring form is principled.

10.1 What can (and cannot) be identified

Scale/offset indeterminacy (only class matters).
(10.1) argminₓ S(s, x) = argminₓ [ a·S(s, x) + b ], with a > 0, b ∈ ℝ.
We therefore identify an equivalence class of parameters up to the positive affine transform in (10.1). We fix scale/offset with a normalization convention (any one of the following is sufficient):
(10.2) Var_val[S] = 1; (10.3) ⟨S⟩_val = 0; (10.4) ‖θ‖₂ = 1 with θ₁ > 0; (10.5) γ fixed by (5.5).

Unit normalization prevents confounded trade-offs.
(10.6) φ̂_j = (φ_j − μ_j) / σ_j, optimize θ̂ on φ̂.
Without (10.6), any rescaling of features can be absorbed into θ, breaking comparability of α, β, λ and harming identifiability.

10.2 Conditions for identifiability from logs

We assume the linear (or locally linear) spine (L(s, x; θ) = θᵀφ(s, x)) with sign/monotonicity constraints (6.11)–(6.13) and hard constraints Γ_hard.

C1 — Persistency of excitation in candidate sets.
Across records, the gradient features at the chosen action and at feasible challengers must span the parameter space (up to scale):
(10.7) rank\Big( E[ Φ_i ] \Big) = p − 1,
where Φ_i aggregates “choice-vs-alternative” feature differences for record i (e.g., (\sum_{x∈𝒞_i} w_{ix}[φ(s_i, a_i*) − φ(s_i, x)])). This is the inverse-optimization analogue of instrument/variation richness.

C2 — Rich, correct constraints Γ.
Feasibility must be known and non-trivial so that choices are not explained by unconstrained scoring alone:
(10.8) P( x ∈ F(s) ) ∈ (0, 1) and LICQ holds at aᵢ*.
LICQ (linearly independent active-constraint gradients) ensures unambiguous KKT multipliers near the optimum.

C3 — Calibrated observers (for gates) and independence.
Agreement is meaningful only if observers are not identical; require an effective number of independent observers:
(10.9) N_eff = 1 ÷ Σ_j w_j² ≥ N_min.
This supports CSA-based validation and rules out “echo” observers that artificially inflate agreement.

C4 — Boundedness & regularity.
Feature map and feasible set are regular on the support: φ is Lipschitz in x on F(s); ∂Ω (generation bounds) is compact; Γ_hard is C¹.

10.3 Consistency and recovery (sketch)

Let the training objective be the softmin/Plackett–Luce composite in (6.17). Under C1–C4 and a fixed scale convention:

Theorem (θ-consistency up to scale).
(10.10) θ̂_N → θ* (in probability) as N → ∞, up to (10.1).
Sketch. The empirical risk obeys a uniform law of large numbers by boundedness/Lipschitzness; strict pseudoconvexity in θ (induced by ranking loss + sign/mono constraints) yields a unique minimizer modulo scale; standard M-estimation arguments yield convergence to the equivalence class of θ*.

KKT-based recovery (alternative when projections are available).
Suppose for each i, aᵢ* is the unique constrained minimizer, S is twice differentiable and strongly convex in a neighborhood of aᵢ*, and LICQ holds. KKT stationarity gives:
(10.11) ∇ₓ S(sᵢ, aᵢ*; θ*) + Σ_j μ_{ij} ∇ₓ g_j(sᵢ, aᵢ*) = 0.
Because S is linear in θ through φ, stacking (10.11) across i produces linear equations in θ (plus multipliers). Under C1–C2, the resulting normal system has rank p−1, pinning down θ up to positive scale.

10.4 Decision generalization guarantees

Let (\mathcal{F} = { x \mapsto θᵀφ(s, x) : θ ∈ Θ }) be a linear scorer class of dimension p on normalized φ̂.

Top-k fidelity (PAC-style bound).
With probability ≥ 1 − δ over N i.i.d. logs, the population top-k error of θ̂ satisfies:
(10.12) errₖ(θ̂) − inf_{θ∈Θ} errₖ(θ) ≤ O!\big( √{ (p·log N + log(1/δ)) / N } \big).
The constant hides Lipschitz and margin factors from the softmin surrogate; monotonicity/sign constraints shrink the effective p.

Constraint violation control (posterior regularization).
If ε in (6.7) is strictly feasible and Γ_hard is correct, then at the optimum:
(10.13) E_{(s,x)∼π_τ(·|s;θ̂)}[ v(s, x) ] ≤ ε + o_p(1).
So expected soft-violation is bounded in deployment; hard constraints remain inviolable by construction.

Gate convergence (Two Lamps).
If thresholds (τ_CSA, ε_max, χ_max) sit inside attainable regions, continuity of CSA@k and χ in θ implies:
(10.14) P( Pass_CWA(θ̂) ∧ Pass_ESI(θ̂) ) → 1 as N → ∞.
Hence models that pass in validation will keep passing as data grows, barring distribution shift.

10.5 Mixtures and drift (identifiability with regimes)

For MoL with (S(s, x) = Σ_m p(m|s)·S_m(s, x)):

Permutation/scale identifiability.
Under standard mixture-of-experts conditions—distinct (S_m), a non-degenerate gate (p(m|s)), and regime support with positive measure—each component θ_m is identifiable up to permutation and positive scale:
(10.15) {θ̂_m}m = {a_m·θ{π(m)} }_m + o_p(1), with a_m>0.
We fix scales per regime via (10.2)–(10.5) and anchor permutation by regime semantics (e.g., “rush” has Δ high).

Change-point creation is consistent.
Using (8.7), if there is a true parameter shift at t*, then (\hat{t} → t^*) in probability provided the pre/post parameters differ by a margin and residuals satisfy a CLT.

10.6 Why dissipative scoring is principled (least action with drag)

The discrete score in §5 is the operational counterpart of generalized least action with Rayleigh dissipation. In the continuous limit:
(10.16) (\mathcal{S}[x] = ∫ [ T(x˙) − V(x) ] dt + ∫ R(x˙) dt,) with (R(x˙) = (c/2)‖x˙‖².)
Euler–Lagrange with dissipation yields:
(10.17) ( \frac{d}{dt} ∂T/∂x˙ − ∂V/∂x + ∂R/∂x˙ = 0 \Rightarrow m·x¨ = −∇V(x) − c·x˙.)
Our discrete action
(10.18) ( S = Σ_t[ −U + α·Cost + β·Risk + γ·‖Δx‖² ] + λ·Σ_t \max{0, Γ_{soft}} )
is precisely the score that (i) respects inviolable Γ_hard, (ii) prices recoverable violations, and (iii) penalizes plan thrash via the γ·‖Δx‖² term—the discrete cousin of Rayleigh drag. This makes the scoring stable, interpretable, and physically motivated, not an arbitrary heuristic.

10.7 Takeaways (engineering check-list)

Ensure C1–C4 before fitting: vary candidates, write real Γ, lint observers, normalize units.
Fix the scale/offset via (10.2)–(10.5); otherwise θ is not comparable across runs.
Prefer softmin training unless you can project; use KKT when you can.
In mixtures, cap modes and name regimes; guard against permutation with semantics and Δ.
Always gate with Two Lamps; by (10.14) you gain asymptotic deployment confidence, not just validation comfort.

11) Metrics & Evaluation Protocol — What we report

We report decision fidelity, gate health, and interpretability—with confidence intervals—on chronologically held-out logs. All metrics are computed per-regime (if using mixtures) and then aggregated.

11.1 Decision fidelity

Top-1 accuracy (chosen action reproduced).
(11.1) top1 = #( aᵢ* = argmin_{x∈𝒞ᵢ∩F(sᵢ)} S(sᵢ, x; θ) ) ÷ N_val.

Ranking quality (NDCG@k with binary or graded relevance).
Let relᵢ(x) ∈ {0,1,…} encode acceptability (1 if “expert-acceptable”).
(11.2) DCG@kᵢ = Σ_{r=1}^k relᵢ(x_r) ÷ log₂(r+1), IDCG@kᵢ = max DCG@kᵢ, NDCG@k = ⟨ DCG@kᵢ ÷ IDCG@kᵢ ⟩.

Prospective uplift via offline replay (self-normalized IPS).
Given behavior propensities bᵢ = P₀(aᵢ*|sᵢ) (from logs or a fitted behavior model) and reward rᵢ (utility proxy from yᵢ):
(11.3) V̂_SNIPS(π) = [ Σᵢ wᵢ rᵢ ] ÷ [ Σᵢ wᵢ ], with wᵢ = 1{ π(sᵢ) = aᵢ* } ÷ bᵢ.
(11.4) uplift = V̂_SNIPS(π_new) − V̂_SNIPS(π_ref).
Report a one-sided lower confidence bound (LCB) by bootstrap or delta-method:
(11.5) LCB_α(uplift) = uplift − z_{1−α} · SE[uplift].
Deploy only if LCB_α(uplift) > 0 (safe improvement). Use switch estimators when bᵢ are unavailable.

Score margin (robustness of the local choice).
(11.6) ΔS_margin = ⟨ S_best_next − S_best ⟩, with S_best = min_x S, S_best_next = 2nd-min_x S.
Higher ΔS_margin implies more decisive selections.

11.2 Gates health (Two Lamps report)

Agreement lamp.
(11.7) CSA@k = #(agreements within ε in top-k) ÷ N, (11.8) max-ε = max_{i≤k,a<b} | scoreᵢ^(a) − scoreᵢ^(b) |.

Stability lamp (ESI).
(11.9) χ = Var_starch(score) ÷ Var_raw(score).

Observer independence.
(11.10) N_eff = 1 ÷ Σ_j w_j².

Gate decision (recap).
(11.11) Conf_accept = 1[ CSA@k ≥ τ_CSA ∧ max-ε ≤ ε_max ∧ χ ≤ χ_max ].

We publish per-cohort dashboards of {CSA@k, max-ε, χ, N_eff} alongside the choice ΔS_margin in (11.6) to show decisiveness near the gate.

11.3 Interpretability & fairness-aware agreement

Sparsity (term count).
(11.12) |θ|₀ = #{ j : θ_j ≠ 0 }, with group counts if using grouped features.

Unit checks (pass rate).
(11.13) unit_pass = #{ terms with declared units consistent } ÷ #{ reported terms }.

Symbol stability across folds (selection + sign).
For each term j over K folds:
(11.14) sel_stab(j) = #{ folds with θ_j ≠ 0 } ÷ K, sign_stab(j) = #{ folds with sign(θ_j)=mode } ÷ K.
Report the mean/median of sel_stab and sign_stab over selected terms; unstable symbols are flagged.

Fairness-aware agreement (cohort deltas).
Over groups g (e.g., region, shift, demographic where lawful):
(11.15) Δ_CSA = max_g | CSA@k(g) − CSA@k(all) |.
Also report χ(g) and ΔS_margin(g). Large Δ_CSA or χ disparities trigger review.

11.4 Evaluation protocol (how we compute the above)

Splits. Chronological split into train/validation/test; hyper-tuning only on validation.
Normalization. Fit unit scales on train; apply to all splits (never refit on val/test).
Regimes. If using mixtures, compute metrics per regime and weighted by regime posterior on each split.
Uncertainty. 1,000× stratified bootstrap for CIs on {top1, NDCG@k, uplift, ΔS_margin, CSA@k, χ}.
Gates. Evaluate (11.11) on validation; no-gate, no-deploy to test/production.
ObserverOps. Stamp the report with H_TopoSpec, compile hash, seeds, thresholds, and observer roster so every metric is replayable.

Bottom line. We do not celebrate accuracy alone. A learned Lagrangian ships only when it reproduces expert choices, shows prospective uplift, keeps both lamps green with healthy margins, and remains sparse, unit-sound, and symbol-stable across folds and cohorts.

12) Case Studies — Four scoped domains with replayable bundles

What a “replayable bundle” contains (for each domain).
(12.0) Bundle = { TopoSpec.json, FeatureScales.json, L-params.json, Γ.json, RegimeMeta.json, GateReport.json, Δ5Log.json, ObserverOpsFooter.txt, Replayer.py }.
(12.1) H_footer = SHA-256( model_id ∥ data_snapshot ∥ codehash ∥ seed ∥ H_TopoSpec ∥ thresholds ∥ observers ).
Each bundle replays ranking, gates, and decisions on held-out logs with the exact scales and seeds.

A) Menu Planning (weekly meals)

Context. Household or institutional menus balancing nutrition, cost, prep effort, and variety.

TopoSpec (excerpt).
(12.A.1) I: Σ_day meals = 7; each meal has {recipe_id, time, kcal}.
(12.A.2) H: sodium/day ≤ cap_Na; allergens = 0; budget/week ≤ £B.
(12.A.3) ∂Ω: recipe pool bounds; max 2 new recipes/week.
(12.A.4) J: U = taste_score; Cost = £; Risk = stockout_prob.

Learned spine and add-ons.
(12.A.5) L = −U + α·Cost + β·Risk + γ·‖Δ(meal_type)‖².
(12.A.6) R_soft += w_var · max{0, variety_min − variety_index}. (residual-found monotony term)

Gates & cadence.
(12.A.7) Pass if CSA@5 ≥ τ and χ ≤ χ_max; observers = {nutrition model, price forecaster, human template bank}.
(12.A.8) Δ5: Explore new recipes only during Explore/Fold; Defrost before holiday weeks.

ObserverOps footer (keys).
(12.A.9) { H_TopoSpec, seed, κ_for_γ, τ_CSA, ε_max, χ_max, observers, regime=“weekday/weekend”, compile_hash }.

B) Meeting Resolution (multi-stakeholder decision playbooks)

Context. Generate alternative agendas/solutions, score them by value, cost, risk, and smoothness of execution.

TopoSpec (excerpt).
(12.B.1) I: agenda slots sum to 60–90 minutes; quorum >= Q.
(12.B.2) H: legal_ok = 1; compliance_ok = 1; budget_ask ≤ cap.
(12.B.3) ∂Ω: option templates per issue; escalation windows.
(12.B.4) J: U = expected_utility_to_all; Cost = resource_hours; Risk = failure_prob.

Learned spine and add-ons.
(12.B.5) L = −U + α·Cost + β·Risk + γ·‖Δ(decision_state)‖².
(12.B.6) R_soft += w_cons · max{0, consensus_floor − min_group_support }. (residual-discovered)

Regimes & gates.
(12.B.7) MoL: {normal, rush}. Gate uses CSA@3 across {rules engine, LLM-a, LLM-b}; ESI jitters stakeholder priors.
(12.B.8) Stop ⇔ (Δ > Δ_max) ∨ (gate fails).

ObserverOps footer (keys).
(12.B.9) { regime_posteriors, CSA@k, max-ε, χ, N_eff, Δ_trace, H_TopoSpec, compile_hash }.

C) Packaging Materials (design-to-cost with manufacturability)

Context. Pick material/geometry for a product SKU under cost, strength, weight, and plant limits.

TopoSpec (excerpt).
(12.C.1) I: loads {compression, drop}; target weight; SKU volume.
(12.C.2) H: strength ≥ min_spec; recyclability ≥ r_min; line_speed ≤ cap.
(12.C.3) ∂Ω: material library bounds; die sizes; lead-time windows.
(12.C.4) J: U = durability_score; Cost = material+process £; Risk = supply_risk.

Learned spine and add-ons.
(12.C.5) L = −U + α·Cost + β·Risk + γ·‖Δ(params)‖².
(12.C.6) R_soft += w_mfg · max{0, m_req(s) − m_realized(s,x)}. (“manufacturability_gap” from §7)

Gates & cadence.
(12.C.7) CWA observers = {FEA surrogate, historical fail-mode model, rules}; ESI kernels jitter tolerances and coil gauges.
(12.C.8) Δ5 staggers plant trials after Verify; no plant push if χ above χ_max on any SKU cohort.

ObserverOps footer (keys).
(12.C.9) { feature_scales, γ_rule, w_mfg, CSA@k_by_SKU, χ_by_SKU, seeds, H_TopoSpec }.

D) Clinical Pathway Selection (triage/care plans)

Context. Choose a care pathway balancing expected outcome, cost, risk, and operational smoothness.

TopoSpec (excerpt).
(12.D.1) I: patient features; pathway steps sum to length-of-stay bound.
(12.D.2) H: contraindication = 0; drug–drug conflict = 0; consent = 1.
(12.D.3) ∂Ω: protocols permitted by site; bed capacity envelopes.
(12.D.4) J: U = expected_outcome_gain; Cost = payer_cost; Risk = adverse_event_prob.

Learned spine and add-ons.
(12.D.5) L = −U + α·Cost + β·Risk + γ·‖Δ(step_intensity)‖².
(12.D.6) R_soft += w_wait · max{0, wait_time − SLA}. (residual from queue-pressure cohorts)

Regimes & gates.
(12.D.7) MoL: {routine, incident}. Gate per regime with CSA@k across {clinical rules, survival model, external reviewer}; ESI perturbs vitals/labs within measurement error.
(12.D.8) Stop ⇔ (Δ > Δ_max) ∨ (gate fails); incident regime promoted only after shadow pass.

ObserverOps footer (keys).
(12.D.9) { regime_support, CSA@k, χ, ΔS_margin, fairness Δ_CSA across cohorts, H_TopoSpec, compile_hash }.

How to read any bundle.
Open GateReport.json for {CSA@k, max-ε, χ, N_eff, ΔS_margin}; check L-params.json for {α, β, γ, λ} with units; verify RegimeMeta.json for supports and thresholds; confirm ObserverOpsFooter.txt matches (12.1). If gates are green and Δ is calm, the learned {L, Γ} is ready to replay—and to deploy under the same rails.

13) Engineering Recipe & Toolkit — Auto-Lagrange, Δ5, ObserverOps

This section gives you a one-page pipeline, runnable defaults, and exact hashes so every training run is replayable. Keep equations single-line and paste-ready.

13.1 One-page pipeline (end-to-end)

Pipeline.
(13.1) ingest → TopoSpec compile → IO/IRL fit → residual symbolic add-ons → regime fit → Two Lamps gates → freeze & export.

Inputs.
(13.2) Logs D = { (sᵢ, 𝒞ᵢ, aᵢ*, yᵢ) }ᵢ=1..N, TopoSpec { I, H, C, ∂Ω, J }, seeds, observer roster.

Compiler outputs.
(13.3) FeatureScales, Γ_hard, Γ_soft, φ(s,x), S(s,x;θ) with (α,β,γ,λ) initialized, H_TopoSpec.

Fit core L by inverse optimization (choose one path).
(13.4) Softmin path: minimize ℓ_softmin + η·ℓ_out + ζ·ℓ_post + ρ·‖θ‖₁ + ξ·ℓ_stab s.t. sign/mono/unit guards.
(13.5) KKT path: minimize ℓ_rank + η·ℓ_out + ζ·ℓ_post + ρ·‖θ‖₁ + ξ·ℓ_stab + χ·ℓ_KKT s.t. sign/mono/unit guards.

Residual discovery loop (optional, sparse).
(13.6) propose φ_new (unit-aware) → refit tiny θ_new with L1 → gate (CSA, χ) → MDL select → keep only passing terms.

Regime switching (optional).
(13.7) train {S_m} and p(m|s) with entropy/spread/drift regularizers → shadow test → gate by regime → promote or discard.

Gates and freeze.
(13.8) Deploy only if Conf_accept = 1[ CSA@k ≥ τ_CSA ∧ max-ε ≤ ε_max ∧ χ ≤ χ_max ].
(13.9) Freeze artifacts; write ObserverOps footer; export replayable bundle.

APIs (uniform across domains).
(13.10) Score(s, x) → S; Project(s, x) → Π_{Γ_hard}(x); Gate(batch) → {CSA@k, max-ε, χ, N_eff, pass}; Schedule(state) → phase; Footer(meta) → H.

13.2 Δ5 scheduler (defaults, triggers, stops)

Phases. Explore → Fold → Verify → Defrost → Decide.

Surplus-tension signal.
(13.11) Δ = ĝ·β̂ − γ̂.

Default cadence (per weekly cycle; tune to domain).
(13.12) Explore: 2d (wider ∂Ω, larger τ); Fold: 2d (tighten ∂Ω, reduce τ);
(13.13) Verify: 2d (lock ∂Ω; run Two Lamps; χ at T=0.25,S=8);
(13.14) Defrost: 0.5d (reset caches; reseed observers);
(13.15) Decide: 0.5d (freeze, footer, deploy or rollback).

Phase advances.
(13.16) Explore→Fold ⇔ top-k stabilizes (ΔS_margin ↑) ∧ soft-violations ↓.
(13.17) Fold→Verify ⇔ CSA@k trending ≥ τ_CSA − δ ∧ χ trending ≤ χ_max + δ.
(13.18) Verify→Decide ⇔ Conf_accept = 1 and uplift LCB > 0.
(13.19) Verify→Defrost ⇔ any gate fails.
(13.20) Defrost→Explore ⇔ caches cleared; observers reseeded; Δ re-established.

Stop rules (hard).
(13.21) Stop ⇔ (Δ > Δ_max) ∨ (gate fails). (same as D.12)

Notes.
• Keep ∂Ω tighter in Verify/Decide; never loosen Γ_hard to pass gates.
• Record {Δ trace, thresholds, phase times} in the run log.

13.3 Observers config (independence lint and health)

Roster and weights.
(13.22) Observers = { O_j } with weights w_j ≥ 0, Σ_j w_j = 1.

Independence health.
(13.23) N_eff = 1 ÷ Σ_j w_j² ≥ N_min.
(13.24) Corr(O_a, O_b | features) ≤ ρ_max for all (a≠b).
(13.25) No shared frozen backbone IDs among primary observers (or penalize via roster penalty).

Seeding & perturbation.
(13.26) For ESI: use T=0.25, S=8, K={Gaussian on φ̂, 10% structured dropout}.
(13.27) Never perturb across Γ_hard; starch only lawful inputs.

Gate report (per batch).
(13.28) Report {CSA@k, max-ε, τ_K, ρ_S, χ, N_eff} with confidence bands; flag cohorts with Δ_CSA or χ disparities.

13.4 Footer writer (ObserverOps, deterministic hash)

Deterministic footer string (stable serialization).
(13.29) footer_str = JSON.stringify({ model_id, data_snapshot_id, codehash, seed, topospec_hash, thresholds, observers, regimes, scales, time_utc }, sort_keys=True, float_fmt="%.10g").

Hash (deployment ID).
(13.30) H = SHA-256( footer_str ). (same as D.15)

Mandatory footer fields.
(13.31) { model_id, version, codehash, seed, rng_lib, data_snapshot_id, data_time_range, topospec_hash, feature_scales_id, compile_hash, αβγλ, Γ_hard_id, Γ_soft_id, observers_roster, N_eff, τ_CSA, ε_max, χ_max, T/S/K, CSA@k, max-ε, χ, ΔS_margin, regime_meta, Δ_trace, decision=pass|fail, author, time_utc }.

Policy.
(13.32) No footer, no deploy. No gate pass, no promote. Hash mismatch ⇒ rollback.

13.5 Exported bundle (replay in any environment)

Artifacts.
(13.33) Bundle = { TopoSpec.json, FeatureScales.json, L-params.json, Γ.json, RegimeMeta.json, GateReport.json, Δ5Log.json, ObserverOpsFooter.txt, Replayer.py }.

Replayer contract.
(13.34) Given {s, 𝒞} → compute S, ranks, gates with seeds from footer; verify {CSA@k, χ} and decision bits; compare H.

Environment pins.
(13.35) Pin {python_version, os_image, blas_lib}; record container image digest.
(13.36) RNG reproducibility = {seed, rng_lib} + deterministic BLAS flag.

13.6 Practical defaults (you can copy as is)

(13.37) Normalization = MAD-based per term; γ = κ·Median(‖∇U‖²) with κ=0.3.
(13.38) τ (softmin temperature) = 0.25; ρ (L1) = tuned by 1-SE rule; ζ (post-reg) targets ε = 5th percentile of v(s,x) in train.
(13.39) Gates: τ_CSA=0.80, ε_max = 0.5·MAD(S), χ_max=0.45, N_min=2.5.
(13.40) MoL: M ≤ 3, entropy λ_H=0.1, spread λ_S=0.1, drift λ_D=0.01.
(13.41) Split: chronological train/val/test; bootstrap 1,000× for all CIs.

Bottom line. Follow (13.1)–(13.41) verbatim and you will produce a sparse, unit-sound Lagrangian, gated by Two Lamps, scheduled by Δ5, and stamped by ObserverOps—a replayable, auditable objective ready for safe deployment.

14) Limitations, Ethics & Deployment — Confounding, PII, rollback

Real logs are messy. This section spells out what can go wrong, how we constrain it, and how we roll back safely with full audit trails.

14.1 Confounding in logs (selection, policy, missing actions)

What can bite. Logged choices reflect historic policies, incentives, and availability; counterfactuals are unseen; candidate sets may be absent.

Mitigations.

Behavior modeling & doubly-robust OPE.
(14.1) ( \hat V_{DR}(π) = \frac{1}{N}\sum_i \big[ \hat μ(s_i, π(s_i)) + \frac{1{π(s_i)=a_i^}}{\hat b_i},(y_i - \hat μ(s_i, a_i^)) \big]. )
Use (\hat b_i = P_0(a_i^* \mid s_i)) and a calibrated outcome model (\hat μ).
Reconstruct candidate sets (with disclaimers).
(14.2) ( \hat{\mathcal C}(s) = G(s; ψ),\quad κ_{cover} = \frac{|\mathcal C_i \cap \hat{\mathcal C}_i|}{|\hat{\mathcal C}i|}. )
Report (κ{cover}); low coverage ⇒ decision metrics become diagnostic only.
Instrument variation / perturbation audits. Use ESI starch to reveal spurious shortcuts; require gates (§9) before claims.

14.2 Distribution shift & safe-mode

Detect shift before harm.
(14.3) ( \mathrm{Shift} = D(P_{test} ,|, P_{train}) ) using PSI/KL/χ² on φ̂ and on gate metrics; trigger verification when (\mathrm{Shift} > τ_{shift}).

Safe-mode on gate failure.

Per-case report (no auto-apply): show top-1 candidate, (ΔS_{margin}), hard/soft constraint status, observer spreads, CSA snippets.
Escalate to human with rationales and uncertainty.
Freeze Δ5 in Verify; no Decide until gates recover.

Stop rule (hard).
(14.4) Stop ⇔ ( (\Delta > \Delta_{max}) ,\vee, (\text{gate fails}) ).
Safe-mode continues producing reports only; no state-changing actions.

14.3 Privacy, PII, and data minimization

Principles. Collect the minimum to reproduce decisions; separate identifiers from features; hash what you can; encrypt rest at rest/in transit.

Minimize & segregate.

Strip direct identifiers from training artifacts; keep a secure mapping table outside the bundle.
Log only aggregate gate metrics per cohort unless case-level audit is explicitly authorized.

Footer never leaks PII.
(14.5) ( H = \mathrm{SHA{\text -}256}( \text{model_id} \parallel \text{data_snapshot_id} \parallel \text{codehash} \parallel \text{seed} \parallel \text{topospec_hash} ). )
No raw features, no free text, no IDs inside H.

Retention & access.

Time-box raw logs; retain only TopoSpec, scales, parameters, and gate summaries for long-term replay.
Role-based access for case-level replays; every export is footer-hashed.

14.4 Rollback & reproducibility

Audit-ready run folder (must-have files).

TopoSpec.json, FeatureScales.json, L-params.json, Γ.json, RegimeMeta.json.
GateReport.json (CSA@k, max-ε, χ, (N_{eff}), (ΔS_{margin}), CIs).
Δ5Log.json (phase times, Δ trace, stop events).
ObserverOpsFooter.txt (contains H, thresholds, seeds, observers, regime supports).
Replayer.py (deterministic re-rank + gates).

Rollback switch.
(14.6) Rollback ⇔ ((\mathrm{Shift} > τ_{shift}) ,\vee, (\text{LCB}_\alpha(\mathrm{uplift}) \le 0) ,\vee, (\text{gate fails})).
On rollback: revert to previous footer H_prev; record a diff note (who/when/why).

14.5 Ethics & fairness

Fairness-aware agreement (monitor cohorts).
(14.7) ( Δ_{CSA} = \max_g |, CSA@k(g) - CSA@k(\text{all}) ,|. )
Investigate when (Δ_{CSA}) or χ disparities exceed cohort thresholds; do not deploy new regimes whose benefits are concentrated via spurious shortcuts.

Human-in-the-loop boundaries.
High-risk domains (clinical, credit, safety) must keep human authorization on any change with low (ΔS_{margin}) or out-of-spec cohorts—even if gates pass.

14.6 Governance policy (paste-ready)

(14.8) No gate, no deploy.
(14.9) No footer, no promote.
(14.10) Hash mismatch ⇒ rollback.
(14.11) Shift or uplift LCB ≤ 0 ⇒ rollback.
(14.12) PII outside bundles; identifiers segregated; access logged.

Bottom line. Logs are biased, worlds drift, privacy matters. By enforcing Two Lamps, Δ5, and ObserverOps, and by adopting strict stop/rollback rules with audit-ready artifacts, we keep learned Lagrangians useful, governable, and safe in the wild.

Appendices

A. Proof sketches — Identification & consistency; Lyapunov-style stability for added symbolic terms

A.0 Notation recap.
(A.0.1) D = { (sᵢ, 𝒞ᵢ, aᵢ*, yᵢ) }ᵢ=1..N, F(s) = { x ∈ X : Γ_hard(s,x) ≤ 0 }.
(A.0.2) S(s,x;θ) = Σ_t[ −U + α·Cost + β·Risk + γ·‖Δx_t‖² ] + λ·Σ_t max{0, Γ_soft }.
(A.0.3) L(s,x;θ) = θᵀφ(s,x) (linear spine on unit-normalized features φ̂).
(A.0.4) score(s,x;θ) = −S(s,x;θ), π_τ(x|s;θ) = exp(score/τ) ÷ Σ_{x′∈𝒞(s)} exp(score′/τ).
(A.0.5) ℓ(θ) = ℓ_softmin + η·ℓ_out + ζ·ℓ_post + ρ·‖θ‖₁ + ξ·ℓ_stab, with sign/mono/unit guards.

A.1 Assumptions (used across results)

(A.1.1) Persistency of excitation. Across i, the choice-vs-alternative feature differences span p−1 dims: rank( E[Φ_i] ) = p − 1.
(A.1.2) Correct feasibility & LICQ. Γ_hard is correct; active-constraint gradients at aᵢ* are linearly independent.
(A.1.3) Regularity & boundedness. φ is Lipschitz on compact F(s); S is locally strongly convex in x near aᵢ*.
(A.1.4) Normalization & signs. Features are unit-normalized; θ respects monotonicity/sign constraints (∂S/∂Cost ≥ 0, etc.).
(A.1.5) Observer calibration. Effective independence N_eff ≥ N_min for CWA; ESI perturbations stay within ∂Ω and never break Γ_hard.
(A.1.6) Scale fixing. One scale convention holds: Var_val[S]=1 or γ fixed by γ=κ·Median(‖∇U‖²).

A.2 Consistency (softmin path)

Claim. Under (A.1.1)–(A.1.6), the softmin estimator is consistent up to positive scale.

(A.2.1) Define the population risk R(θ) = E[ S(s,a*;θ) − (−τ·log Σ_{x∈𝒞(s)} e^{−S(s,x;θ)/τ}) ] + η·E[‖ŷ−y‖²] + ζ·PR + ρ·‖θ‖₁ + ξ·stab.
(A.2.2) Empirical risk R_N(θ) is the sample average of (A.2.1).

Step 1 (ULLN). Lipschitz + compact Θ imply sup_θ |R_N − R| → 0 (in probability).
Step 2 (identifiable minimizer modulo scale). With sign/mono guards and excitation, R is strictly pseudoconvex on Θ/∼, where “/∼” quotients positive affine rescalings of S.
Step 3 (Argmin continuity). Any sequence θ̂_N ∈ argmin R_N converges to argmin R in Θ/∼. Fix scale by (A.1.6).

Conclusion.
(A.2.3) θ̂_N → θ* (in probability) up to positive affine transform of S; with (A.1.6) the limit is unique.

A.3 Identification via KKT (when projection/gradients on Γ_hard are available)

Stationarity at the logged optimum.
(A.3.1) ∇ₓS(sᵢ, aᵢ*; θ*) + Σ_j μ_{ij} ∇ₓg_j(sᵢ, aᵢ*) = 0, with μ_{ij} ≥ 0, μ_{ij}·g_j=0.

Stacking (A.3.1) over i yields a linear system in θ* and multipliers {μ_i}. Under excitation + LICQ:

Proposition.
(A.3.2) rank( design_matrix ) = p − 1 ⇒ θ* identified up to positive scale.
Scale fix by (A.1.6) gives a unique θ*. (Complementary slackness disambiguates inactive constraints; strong convexity localizes solutions.)

A.4 Identification of mixtures (sketch)

For S(s,x)=Σ_m p(m|s)·S_m(s,x) with distinct components and non-degenerate gate:

Permutation/scale identifiability.
(A.4.1) {θ̂_m}m = {a_m·θ{π(m)}}_m + o_p(1), with a_m>0.
Fix per-regime scale (A.1.6); fix permutation by regime semantics (e.g., incident vs. routine via Δ traces).

A.5 Generalization bound for decision fidelity (linear scorer class)

Let 𝔽 = { x ↦ θᵀφ̂(s,x) : θ ∈ Θ_mono } of VC-dim ≲ p. For a top-k surrogate loss ℓₖ bounded and Lipschitz:

(A.5.1) E[ℓₖ(θ̂)] − inf_{θ∈Θ} E[ℓₖ(θ)] ≤ O( √{ (p·log N + log(1/δ))/N } ) with prob ≥ 1−δ.
(Constraints reduce effective p; constants hide Lipschitz and margin terms.)

A.6 Lyapunov-style stability for added symbolic terms

We add one candidate term after residual discovery: S′ = S + θ_new·φ_new(s,x), where φ_new is unit-checked and monotone in a “penalty” sense.

Discrete Lyapunov candidate.
(A.6.1) V(x_t) = x_tᵀ P x_t, with P ≻ 0 (chosen per domain scale).
(A.6.2) ΔV_t = V(x_{t+1}) − V(x_t).

Sufficient condition for non-increasing V along plan updates. Suppose the plan update is a proximal step that approximately minimizes S′ + (1/2η)‖x_{t+1}−x_t‖²:

(A.6.3) x_{t+1} ≈ argmin_x [ S′(s_t,x) + (1/2η)‖x−x_t‖² ].

Under strong convexity of S in x near x_t (from γ·‖Δx‖²) and Lipschitz φ_new with constant L_new:

(A.6.4) ΔV_t ≤ −c₁‖x_{t+1}−x_t‖² + c₂|θ_new|·L_new·‖x_{t+1}−x_t‖·‖x_t‖.

Complete the square and choose |θ_new| ≤ θ_max so that the RHS ≤ −c₃‖x_{t+1}−x_t‖² for some c₃>0. Hence:

Lemma (discrete decay).
(A.6.5) If |θ_new| ≤ θ_max = (c₁/2c₂L_new)·(‖x_t‖⁻¹), then ΔV_t ≤ −(c₃)‖x_{t+1}−x_t‖² ≤ 0.

ESI stability (χ) bound under lawful perturbations. Let perturbations δ on lawful inputs satisfy ‖δ‖ ≤ ε and φ_new be L_new-Lipschitz in those inputs:

(A.6.6) Var_starch(score′) − Var_starch(score) ≤ θ_new²·L_new²·ε².

Thus, if χ_old = Var_starch(score)/Var_raw(score), then with Var_raw unchanged:

(A.6.7) χ_new ≤ χ_old + (θ_new²·L_new²·ε²)/Var_raw(score).

Pick θ_new within the Lyapunov bound (A.6.5) and enforce gate check χ_new ≤ χ_max to guarantee ESI does not degrade.

Monotonicity/sign guard. If φ_new is a penalty-like variable, require ∂S′/∂φ_new ≥ 0:

(A.6.8) sign(θ_new) = +1 when φ_new increases “risk/violation”, else flip the sign and re-test (A.6.5)–(A.6.7).

Conclusion (safe acceptance rule).
(A.6.9) Accept φ_new only if: (i) structured residual drop ≥ δ_min; (ii) |θ_new| respects (A.6.5); (iii) χ_new ≤ χ_max per (A.6.7); (iv) CWA not worse: CSA@k_new ≥ CSA@k_old − ε_tol.

A.7 Takeaways (engineering form)

(A.7.1) Consistency: softmin estimator converges to θ* up to scale; fix scale to pin down θ*.
(A.7.2) KKT ID: with projections and LICQ, θ* is identified (mod scale) from stationarity equations.
(A.7.3) Mixtures: components identifiable up to permutation/scale under non-degenerate gates; fix both by regime semantics and scale rules.
(A.7.4) Stability: new symbolic terms are safe when their coefficient respects a Lyapunov-derived bound and pass χ/CSA gates; otherwise reject.

B. Pseudocode — Auto-Lagrange compiler; Δ5 loop; Two Lamps gate; footer writer

Minimal, implementation-ready pseudocode. Deterministic where possible. Names match Sections 1–14.

B.1 `AutoLagrangeCompile` — from TopoSpec to a scored, unit-sound model

function AutoLagrangeCompile(TopoSpec, Logs, Seeds):
  # Inputs
  #   TopoSpec = { I, H, C, ∂Ω, J } with declared units
  #   Logs = { (s_i, C_i, a_i*, y_i) } or { (s_i, a_i*, y_i) } + generator G
  #   Seeds = {rng_seed, observer_seeds}

  assert ValidateInvariants(TopoSpec.I) == true
  assert UnitsDeclared(TopoSpec) == true

  # 1) Deterministic hash of spec
  H_TopoSpec = SHA256(SerializeCanonical(TopoSpec))

  # 2) Build hard/soft constraints
  Γ_hard = CompileHardConstraints(TopoSpec.H)       # list of g_j(s,x) ≤ 0
  Γ_soft = CompileSoftConstraints(TopoSpec.H, J)    # list of s_j(s,x) (hinge-ready)

  # 3) Feature map with units and normalization (MAD by default)
  Φ = BuildFeatureUniverse(TopoSpec, Γ_soft)        # {U, Cost, Risk, Δx, soft-violation terms}
  Scales = FitFeatureScales(Logs, Φ, method="MAD")  # μ, σ per term; frozen thereafter
  Φ̂ = NormalizeFeatures(Φ, Scales)

  # 4) Param spine & initial θ (respect signs/monotone guards)
  θ0 = InitParams({α,β,γ,λ}, guards=SignMonotone)
  γ_rule = KappaRuleForGamma(Φ̂.U, κ=0.3)          # see §5

  # 5) Score function S(s,x;θ) and projection Π onto Γ_hard
  function S(s, x; θ): return Sum_t[ -U + α·Cost + β·Risk + γ·‖Δx‖² ] + λ·Sum_t max(0, Γ_soft)
  function ProjectFeasible(s, x): return ProjOnto(Γ_hard, x)

  # 6) Deterministic compile hash
  compile_hash = SHA256(SerializeCanonical({H_TopoSpec, Scales.id, guards=SignMonotone, γ_rule}))

  return {
    H_TopoSpec, Γ_hard, Γ_soft, Φ̂, S, ProjectFeasible,
    θ0, Scales, γ_rule, compile_hash, Seeds
  }
end

B.2 Inverse Optimization Fit — softmin path or KKT path

function Fit_L(Compiled, Logs, mode="softmin"):
  set RNG(Compiled.Seeds.rng_seed)

  # B.2.1 Common pieces
  define score(s,x;θ) = -Compiled.S(s,x;θ)
  define π_τ(x|s;θ)  = exp(score/τ) / Σ_{x'∈C(s)} exp(score/τ)

  loss_rank(θ) = -Σ_i log π_τ(a_i* | s_i; θ)
  loss_out (θ) = Σ_i ‖ŷ(s_i, a_i*; θ) - y_i‖²_Σ
  loss_post(θ) = max(0, E_{x∼π_τ}[ v(s,x) ] - ε)     # v is soft-violation load
  loss_L1  (θ) = ρ · ‖θ‖_1
  loss_stab(θ) = Σ_i Σ_t max(0, V(x_{t+1}) - V(x_t)) with V(x)=xᵀPx, P≻0

  guards = { SignMonotone, UnitAwareScaling }

  if mode == "softmin":
    softmin(s;θ)  = -τ · log Σ_{x∈C(s)} exp( -Compiled.S(s,x;θ)/τ )
    loss_core(θ)  = Σ_i [ Compiled.S(s_i, a_i*; θ) - softmin(s_i; θ) ]
    objective(θ)  = loss_core + η·loss_out + ζ·loss_post + loss_L1 + ξ·loss_stab
    θ̂ = ConstrainedOpt(objective, θ0=Compiled.θ0, guards)
  else if mode == "kkt":
    # needs gradients and projection on Γ_hard
    loss_KKT(θ)   = Σ_i ‖ ∇_x S + Σ_j μ_ij ∇_x g_j ‖² + ‖min(0,μ_i)‖² + ‖max(0,g)∘μ_i‖²
    objective(θ)  = loss_rank + η·loss_out + ζ·loss_post + loss_L1 + ξ·loss_stab + χ·loss_KKT
    θ̂ = ConstrainedOpt(objective, θ0=Compiled.θ0, guards)
  end

  return θ̂
end

B.3 Residual-Driven Symbolic Add-Ons (unit-aware, sparse)

function SymbolicDiscovery(Compiled, θ̂, Logs, δ_min, MDL_b):
  residuals = BuildResiduals(Compiled, θ̂, Logs)           # ΔS, e, v, buckets by boundary/load/regime
  G = UnitAwareGrammar()                                   # +,-,×,÷, max{·,0}, log if unitless, etc.

  candidates = ProposeFeatures(G, Compiled.Φ̂, residuals.buckets)
  accepted = []

  for φ_new in candidates:
    if not UnitsAllowed(φ_new, family={"risk","cost","penalty"}): continue
    φ̃_new = Orthogonalize(φ_new, span=Compiled.Φ̂)
    θ_try  = TinyL1Refit(Compiled, θ̂, φ̃_new)

    ΔSR = StratifiedResidualDrop(residuals, θ_try)
    if ΔSR < δ_min: continue

    if not PassTwoLampsShadow(Compiled, θ_try, Logs): continue
    if not MonotoneSignGuard(φ̃_new, θ_try): continue

    if MDL(Logs.val, θ_try, b=MDL_b) < MDL(Logs.val, θ̂, b=MDL_b):
      θ̂ = θ_try
      accepted.append(φ̃_new)
      Compiled.Φ̂.add(φ̃_new)
    end
  end

  return θ̂, accepted
end

B.4 Regime Switching (MoL) — train, shadow, promote

function Fit_Mixture(Compiled, θ̂_base, Logs, M_max=3):
  regimes = SeedRegimes(Logs, hints={"normal","rush","incident"}, M_max)
  init {θ_m} around θ̂_base; init gate g_m(s) small; set λ_H, λ_S, λ_D

  repeat until early_stop:
    # Joint step
    loss_mix = Σ_i [ MixSoftminLoss(Compiled.S, {θ_m}, g(s_i)) ]
    + η·MixOutcomeLoss + ζ·MixPostReg + ρ·Σ_m‖θ_m‖_1
    + λ_H·EntropyReg(g) + λ_S·SpreadReg(g, target=π) + λ_D·DriftReg({θ_m})
    optimize {θ_m}, g

    # Shadow gate per regime
    for m in regimes:
      report_m = TwoLamps(Compiled, θ_m, Logs.slice(m))
      if report_m.catastrophic_fail: mark m as reject
    end

    if AnyReject(regimes): prune_or_merge(regimes)
  end

  return {θ_m}, gate=g
end

B.5 Two Lamps Gate — CSA@k / max-ε and ESI χ

function TwoLamps(Compiled, θ, Batch, k=5, ε_max, τ_CSA, χ_max, ESI={T=0.25,S=8,K=Default}):
  # Build observers (independent rankers)
  Observers = BuildObserverRoster(Compiled, Batch)
  assert EffectiveIndependence(Observers)                 # N_eff ≥ N_min; low cross-corr

  # CWA (agreement)
  ranks = { O: RankTopK(Batch, score_O=ScoreWithObserver(O, θ), k) for O in Observers }
  CSA_k = CountAgreementsWithinEpsilon(ranks, ε=ScoreTolerance(Compiled)) / |Batch|
  max_eps = MaxGapWithinTopK(ranks)

  # ESI (stability)
  raw_scores   = CollectScores(Observers, Batch, no_perturb=True)
  starch_scores= []
  for i in 1..ESI.S:
    perturbed = LawfulPerturb(Batch, T=ESI.T, K=ESI.K, Γ_hard=Compiled.Γ_hard)
    starch_scores.append(CollectScores(Observers, perturbed, no_perturb=False))
  χ = Variance(starch_scores) / Variance(raw_scores)

  pass_flag = (CSA_k ≥ τ_CSA) and (max_eps ≤ ε_max) and (χ ≤ χ_max)
  return { pass: pass_flag, CSA@k: CSA_k, max-ε: max_eps, χ: χ,
           N_eff: EffectiveN(Observers), ranks_sample: Sample(ranks) }
end

B.6 Δ5 Scheduler — Explore → Fold → Verify → Defrost → Decide

function Δ5Scheduler(Compiled, θ_current, Streams):
  phase = "Explore"
  Δ_trace = []
  while true:
    batch = NextBatch(Streams, phase)

    if phase in {"Explore","Fold"}:
      θ_current = TrainStep(θ_current, batch)         # IO/IRL updates
    end

    # Compute surplus-tension and margins
    Δ = SurplusTension(θ_current, batch)              # ĝ·β̂ − γ̂
    margin = ScoreMargin(Compiled, θ_current, batch)  # ΔS_margin
    Δ_trace.append({time:now(), Δ, margin})

    if phase == "Explore" and Stabilizing(margin) and ViolationsDown(batch):
      phase = "Fold"; continue
    if phase == "Fold" and TrendingToPassGates(θ_current):
      phase = "Verify"; continue

    if phase == "Verify":
      gate = TwoLamps(Compiled, θ_current, batch)
      if gate.pass and UpliftLCBPositive(θ_current):
        phase = "Decide"; continue
      else:
        phase = "Defrost"; continue

    if phase == "Defrost":
      ReseedObservers()
      ClearCaches()
      phase = "Explore"; continue

    if phase == "Decide":
      break  # ready to freeze & export

    # Hard stop guard
    if Δ > Δ_max or GateHardFail(θ_current):
      SafeModeReportsOnly()
      break
  end
  return {θ_final: θ_current, Δ_trace, last_phase: phase}
end

B.7 Footer Writer — deterministic, audit-ready

function WriteObserverOpsFooter(meta):
  # Required fields
  required = {
    model_id, version, codehash, seed, rng_lib,
    data_snapshot_id, data_time_range,
    topospec_hash, feature_scales_id, compile_hash,
    αβγλ, Γ_hard_id, Γ_soft_id,
    observers_roster, N_eff,
    τ_CSA, ε_max, χ_max, ESI_T, ESI_S, ESI_K,
    CSAk, max_ε, χ, ΔS_margin, regime_meta, Δ_trace,
    decision, author, time_utc
  }
  assert AllPresent(required, meta)

  footer_str = SerializeCanonical(required, sort_keys=True, float_fmt="%.10g")
  H = SHA256(footer_str)

  WriteText("ObserverOpsFooter.txt", footer_str + "\nH=" + H)
  return H
end

B.8 Export Bundle & Replayer — one-file verification

function ExportBundle(Artifacts):
  Bundle = {
    "TopoSpec.json":        Artifacts.TopoSpec,
    "FeatureScales.json":   Artifacts.Scales,
    "L-params.json":        Artifacts.θ,
    "Γ.json":               {hard: Artifacts.Γ_hard, soft: Artifacts.Γ_soft},
    "RegimeMeta.json":      Artifacts.Regimes,
    "GateReport.json":      Artifacts.GateReport,
    "Δ5Log.json":           Artifacts.Δ_trace,
    "ObserverOpsFooter.txt":Artifacts.Footer,
    "Replayer.py":          MakeReplayer(Artifacts)   # deterministic
  }
  WriteBundle(Bundle)
  return Bundle
end

function Replayer(s, C, Bundle):
  load Scales, θ, Γ, S, Seeds, observers from Bundle
  ranks = Rank(C, score=λ x→S(s,x;θ))
  gate  = TwoLamps(Bundle.Compiled, θ, {s,C}, k=Bundle.k, thresholds=Bundle.thresholds)
  assert SHA256(FooterBody(Bundle.Footer)) == FooterHash(Bundle.Footer)    # H check
  return {ranks, gate}
end

B.9 Minimal Run Script — putting it all together

function Main(TopoSpec, Logs, Seeds, mode="softmin"):
  Compiled = AutoLagrangeCompile(TopoSpec, Logs, Seeds)
  θ̂ = Fit_L(Compiled, Logs.train, mode)

  θ̂, added_terms = SymbolicDiscovery(Compiled, θ̂, Logs.train, δ_min=ΔSR_min, MDL_b=2.0)

  if UseMixture:
    {θ_m}, gate = Fit_Mixture(Compiled, θ̂, Logs.train)

  sched = Δ5Scheduler(Compiled, θ̂, Streams={train:Logs.val, verify:Logs.val})
  gate_report = TwoLamps(Compiled, sched.θ_final, Logs.val,
                         k=5, ε_max, τ_CSA, χ_max, ESI={T=0.25,S=8,K=Default})

  decision = (gate_report.pass and UpliftLCBPositive(sched.θ_final))
  FooterHash = WriteObserverOpsFooter(MergeMeta(Compiled, sched, gate_report, decision))

  Bundle = ExportBundle({TopoSpec, Scales:Compiled.Scales, θ:sched.θ_final,
                         Γ_hard:Compiled.Γ_hard, Γ_soft:Compiled.Γ_soft,
                         Regimes, GateReport:gate_report, Δ_trace:sched.Δ_trace,
                         Footer:FooterHash, Compiled})

  return {decision, FooterHash, Bundle}
end

Deployment rules: No gate, no deploy. No footer, no promote. Hash mismatch ⇒ rollback.

C. TopoSpec templates — Paste-and-edit YAML/JSON

(TopoSpec = { I, H, C, ∂Ω, J }. Unicode keys are OK in YAML/JSON. Compute the hash after you fill it in.)
(C.1) H_TopoSpec = SHA-256( serialize(TopoSpec) ).

C.0 Minimal, domain-agnostic YAML template

# ---------- TopoSpec (paste-and-edit) ----------
name: "<project/domain name>"
version: "1.0"
schema: "TopoSpec = { I, H, C, ∂Ω, J }"
units_system: "SI | mixed"
time_horizon: "single | sequence:T=<steps>"

I:  # Invariants (true by construction; unbreakable templates)
  - id: inv_1
    desc: "Example: per-week exactly 7 planning slots"
    expr: "slots_per_week == 7"
  - id: inv_2
    desc: "Example: probabilities sum to 1"
    expr: "sum(p_i for i in options) == 1"

H:  # Hard constraints g_j(s,x) ≤ 0 (non-negotiable)
  - id: h_1
    desc: "Budget cap"
    expr: "cost_total - budget_cap ≤ 0"
    units: "currency"
  - id: h_2
    desc: "Safety/compliance must hold"
    expr: "1 - compliance_ok ≤ 0"
    units: "boolean(0/1)"
  # add many small atomic constraints rather than one mega-constraint

C:  # Context, priors, observers, data sources
  data_sources:
    snapshot_id: "<dataset-id>"
    features_version: "<feat-v>"
  observers:  # for CWA/ESI gates
    - name: "rules_engine_v1"
      type: "rules"
    - name: "model_A"
      type: "ml"
    - name: "model_B"
      type: "ml"
  normalization:
    method: "MAD"
  risk_policy:
    allowed_forms: ["variance", "cvar", "distance_to_boundary"]
  esi_defaults: { T: 0.25, S: 8, K: "gaussian+structured_dropout" }

"∂Ω":  # Boundaries (search envelopes; do NOT encode hard law here)
  generation:
    candidate_pool_max: 200
    bounds:
      - name: "x_i"
        low: "<value>"
        high: "<value>"
  windows:
    - name: "planning_window_days"
      low: 7
      high: 7

J:  # Objective decomposition and soft penalties
  utility:
    name: "U"
    expr: "benefit_score"
    units: "utility_points"
  cost:
    name: "Cost"
    expr: "opex + capex"
    units: "currency"
    weight_alpha_default: 1.0  # α ≥ 0
  risk:
    name: "Risk"
    expr: "cvar_95 | failure_prob | distance_to_boundary"
    units: "risk_points"
    weight_beta_default: 1.0   # β ≥ 0
  smoothness:
    form: "||Δx||^2"
    weight_gamma_rule: "γ = κ · median_t(||∇U||^2), κ in [0.1,1.0]"
  soft_penalties:  # s_j(s,x) ≥ 0 penalized with weights w_j ≥ 0
    - id: sp_1
      desc: "Example: lateness beyond SLA"
      expr: "max(0, wait_time - sla_minutes)"
      units: "minutes"
      weight: 0.5
    - id: sp_2
      desc: "Example: variety floor shortfall"
      expr: "max(0, variety_min - variety_index)"
      units: "index"
      weight: 0.8
  lambda_default: 1.0  # λ ≥ 0

meta:
  author: "<you>"
  created_utc: "<YYYY-MM-DDTHH:MM:SSZ>"
  notes: "Any domain notes."

C.0b Minimal, domain-agnostic JSON template

{
  "name": "<project/domain name>",
  "version": "1.0",
  "schema": "TopoSpec = { I, H, C, ∂Ω, J }",
  "units_system": "SI | mixed",
  "time_horizon": "single | sequence:T=<steps>",
  "I": [
    { "id": "inv_1", "desc": "Per-week exactly 7 planning slots", "expr": "slots_per_week == 7" },
    { "id": "inv_2", "desc": "Probabilities sum to 1", "expr": "sum(p_i for i in options) == 1" }
  ],
  "H": [
    { "id": "h_1", "desc": "Budget cap", "expr": "cost_total - budget_cap ≤ 0", "units": "currency" },
    { "id": "h_2", "desc": "Compliance must hold", "expr": "1 - compliance_ok ≤ 0", "units": "boolean(0/1)" }
  ],
  "C": {
    "data_sources": { "snapshot_id": "<dataset-id>", "features_version": "<feat-v>" },
    "observers": [
      { "name": "rules_engine_v1", "type": "rules" },
      { "name": "model_A", "type": "ml" },
      { "name": "model_B", "type": "ml" }
    ],
    "normalization": { "method": "MAD" },
    "risk_policy": { "allowed_forms": ["variance", "cvar", "distance_to_boundary"] },
    "esi_defaults": { "T": 0.25, "S": 8, "K": "gaussian+structured_dropout" }
  },
  "∂Ω": {
    "generation": {
      "candidate_pool_max": 200,
      "bounds": [{ "name": "x_i", "low": "<value>", "high": "<value>" }]
    },
    "windows": [{ "name": "planning_window_days", "low": 7, "high": 7 }]
  },
  "J": {
    "utility": { "name": "U", "expr": "benefit_score", "units": "utility_points" },
    "cost": { "name": "Cost", "expr": "opex + capex", "units": "currency", "weight_alpha_default": 1.0 },
    "risk": { "name": "Risk", "expr": "cvar_95 | failure_prob | distance_to_boundary", "units": "risk_points", "weight_beta_default": 1.0 },
    "smoothness": { "form": "||Δx||^2", "weight_gamma_rule": "γ = κ · median_t(||∇U||^2), κ in [0.1,1.0]" },
    "soft_penalties": [
      { "id": "sp_1", "desc": "Lateness beyond SLA", "expr": "max(0, wait_time - sla_minutes)", "units": "minutes", "weight": 0.5 },
      { "id": "sp_2", "desc": "Variety floor shortfall", "expr": "max(0, variety_min - variety_index)", "units": "index", "weight": 0.8 }
    ],
    "lambda_default": 1.0
  },
  "meta": { "author": "<you>", "created_utc": "<YYYY-MM-DDTHH:MM:SSZ>", "notes": "…" }
}

C.1 Domain stubs (quick-start, paste-and-edit)

C.1A Menu planning (weekly)

name: "MenuPlanning"
version: "1.0"
I:
  - { id: inv_week, desc: "7 dinners per week", expr: "count(meal) == 7" }
H:
  - { id: h_na, desc: "Sodium/day cap", expr: "sodium_mg_per_day - cap_na_mg ≤ 0", units: "mg/day" }
  - { id: h_allergen, desc: "Allergen exclusion", expr: "1 - allergen_ok ≤ 0", units: "boolean" }
C:
  observers: [ {name: "nutrition_v1"}, {name: "price_model"}, {name: "template_bank"} ]
"∂Ω":
  generation:
    bounds: [ {name: "recipes_new_per_week", low: 0, high: 2} ]
J:
  utility: { expr: "taste_score", units: "points" }
  cost:    { expr: "sum(meal_cost)", units: "£", weight_alpha_default: 0.8 }
  risk:    { expr: "stockout_prob", units: "prob", weight_beta_default: 0.6 }
  smoothness: { form: "||Δ(meal_type)||^2", weight_gamma_rule: "γ = 0.3·median(||∇U||^2)" }
  soft_penalties:
    - { id: sp_var, desc: "Variety floor", expr: "max(0, variety_min - variety_index)", units: "index", weight: 0.7 }
lambda_default: 1.0

C.1B Meeting resolution (agenda/decision playbook)

name: "MeetingResolution"
version: "1.0"
I:
  - { id: inv_time, desc: "Agenda length in [60,90] minutes", expr: "60 ≤ total_minutes ≤ 90" }
H:
  - { id: h_legal, desc: "Legal must hold", expr: "1 - legal_ok ≤ 0", units: "boolean" }
  - { id: h_budget, desc: "Budget ask cap", expr: "budget_ask - budget_cap ≤ 0", units: "£" }
C:
  observers: [ {name: "rules_v1"}, {name: "llm_a"}, {name: "llm_b"} ]
"∂Ω":
  generation:
    candidate_pool_max: 50
J:
  utility: { expr: "stakeholder_benefit", units: "points" }
  cost:    { expr: "resource_hours", units: "hours", weight_alpha_default: 1.2 }
  risk:    { expr: "failure_prob", units: "prob", weight_beta_default: 1.0 }
  smoothness: { form: "||Δ(decision_state)||^2", weight_gamma_rule: "γ = 0.4·median(||∇U||^2)" }
  soft_penalties:
    - { id: sp_cons, desc: "Consensus floor", expr: "max(0, consensus_floor - min_group_support)", units: "points", weight: 0.9 }
lambda_default: 1.0

C.1C Packaging materials (design-to-cost)

name: "PackagingDTC"
version: "1.0"
I:
  - { id: inv_spec, desc: "SKU volume fixed", expr: "volume == volume_spec" }
H:
  - { id: h_strength, desc: "Strength ≥ spec", expr: "strength_spec - strength_sim ≤ 0", units: "N" }
  - { id: h_recycle, desc: "Recyclability ≥ r_min", expr: "r_min - recyclability ≤ 0", units: "ratio" }
  - { id: h_speed, desc: "Line speed cap", expr: "line_speed - cap_speed ≤ 0", units: "units/hr" }
C:
  observers: [ {name: "fea_surrogate"}, {name: "failmode_model"}, {name: "rules"} ]
"∂Ω":
  generation:
    bounds:
      - {name: "gauge_mm", low: 0.2, high: 1.2}
      - {name: "density", low: ρ_min, high: ρ_max}
J:
  utility: { expr: "durability_score", units: "points" }
  cost:    { expr: "material_cost + process_cost", units: "£", weight_alpha_default: 1.3 }
  risk:    { expr: "supply_risk", units: "points", weight_beta_default: 0.7 }
  smoothness: { form: "||Δ(params)||^2", weight_gamma_rule: "γ = 0.3·median(||∇U||^2)" }
  soft_penalties:
    - { id: sp_mfg, desc: "Manufacturability gap", expr: "max(0, m_req - m_realized)", units: "points", weight: 1.0 }
lambda_default: 1.0

C.1D Clinical pathway selection (triage/care plans)

name: "ClinicalPathway"
version: "1.0"
I:
  - { id: inv_los, desc: "Pathway steps sum to LOS bound", expr: "sum(step_days) ≤ los_cap_days" }
H:
  - { id: h_contra, desc: "No contraindication", expr: "1 - contraindication_ok ≤ 0", units: "boolean" }
  - { id: h_ddi,    desc: "No drug–drug conflict", expr: "1 - ddi_ok ≤ 0", units: "boolean" }
  - { id: h_consent, desc: "Consent required", expr: "1 - consent_ok ≤ 0", units: "boolean" }
C:
  observers: [ {name: "clinical_rules"}, {name: "survival_model"}, {name: "external_reviewer"} ]
"∂Ω":
  windows:
    - { name: "protocols_per_site", low: 1, high: 5 }
J:
  utility: { expr: "expected_outcome_gain", units: "QALY_points" }
  cost:    { expr: "payer_cost", units: "£", weight_alpha_default: 1.0 }
  risk:    { expr: "adverse_event_prob", units: "prob", weight_beta_default: 1.2 }
  smoothness: { form: "||Δ(step_intensity)||^2", weight_gamma_rule: "γ = 0.5·median(||∇U||^2)" }
  soft_penalties:
    - { id: sp_wait, desc: "Queue wait beyond SLA", expr: "max(0, wait_time - sla_minutes)", units: "minutes", weight: 0.8 }
lambda_default: 1.0

C.2 After you fill a template

Validate invariants and units; split H into small atomic constraints.
Freeze and compute H_TopoSpec using (C.1).
Store H_TopoSpec in your ObserverOps footer and bundle for replay.

Appendix C is complete.

D. Datasets & IDs — How to record AFDB/Materials IDs; sample ObserverOps footers

This appendix standardizes dataset snapshots, external IDs, and footer fields so any reviewer can replay results exactly.

D.1 Dataset Registry (content-addressed snapshots)

Registry entry (canonical JSON).

{
  "dataset_id": "ds.alpha.menu.v1",
  "snapshot_id": "sha256:7a7b...f9c2", 
  "created_utc": "2025-11-02T18:00:00Z",
  "source_urls": ["<redacted or internal>"],
  "license": "internal",
  "filters": { "date_from": "2024-01-01", "status": ["final"] },
  "schema_version": "1.1",
  "rows": 128743,
  "features_version": "feat.v3",
  "notes": "Case logs, cleaned; candidate sets reconstructed for older runs."
}

Rule. snapshot_id = "sha256:" + SHA-256(archive_bytes) of the exact parquet/CSV bundle you trained on; never reuse after any change.

D.2 External ID namespaces (patterns)

Use namespace:value pairs; store both the human ID and a content hash where possible.

Namespace	Field	Pattern / Example
`UniProt`	`uniprot_acc`	`^[A-NR-Z0-9]{6,10}$` (e.g., `Q9Y2Q0`)
`AFDB`	`afdb_id`	`AF-Q9Y2Q0-F1-model_v4` (AF--F-model_v)
`PDB`	`pdb_id`	`^[0-9][A-Za-z0-9]{3}$` (e.g., `7XYZ`)
`MP` (Materials Project)	`mp_id`	`^mp-[0-9]+$` (e.g., `mp-149`)
`ICSD`	`icsd_id`	integer (e.g., `186072`)
`OQMD`	`oqmd_id`	integer
`COD`	`cod_id`	integer
`SEQ`	`seq_md5`	`md5(AA-sequence)`
`CIF`	`cif_sha256`	`sha256(<CIF bytes>)`

Storage contract.

Proteins: always record {uniprot_acc, afdb_id, af_model_ver, seq_md5}; if present also {pdb_id}.
Materials: record at least one of {mp_id, icsd_id, oqmd_id, cod_id} plus {cif_sha256}.

D.3 Case-log record schema (linking to candidates by ID)

Per-decision record (minimal).

{
  "case_id": "meet.2025-03-10.00123",
  "dataset_id": "ds.alpha.meeting.v2",
  "snapshot_id": "sha256:c2b1...9e31",
  "s": { "...": "context features (unit-tagged)" },
  "C": [
    { "candidate_id": "cand-0001", "external_ids": [], "x": { "...": "decision vars" } },
    { "candidate_id": "cand-0002", "external_ids": [], "x": { } }
  ],
  "a_star": "cand-0002",
  "y": { "kpi1": 0.87, "kpi2": 120 }
}

Protein domain variant (AFDB-linked candidates).

{
  "case_id": "prot.fold.eval.000045",
  "dataset_id": "ds.alpha.protein.v4",
  "snapshot_id": "sha256:9b7a...c4f0",
  "s": { "taxon": 9606, "target_fn": "stability", "units": "kcal/mol" },
  "C": [
    {
      "candidate_id": "AF.Q9Y2Q0.v4",
      "external_ids": {
        "UniProt": "Q9Y2Q0",
        "AFDB": "AF-Q9Y2Q0-F1-model_v4",
        "SEQ": "md5:1a2b...ff33"
      },
      "x": { "mutations": [], "env": "pH7.4" }
    }
  ],
  "a_star": "AF.Q9Y2Q0.v4",
  "y": { "proxy_score": -4.21, "assay_id": "assay.123" }
}

Materials domain variant.

{
  "case_id": "mat.pack.sku123",
  "dataset_id": "ds.alpha.materials.v3",
  "snapshot_id": "sha256:3f5d...aa88",
  "s": { "load": { "compress_N": 800 }, "weight_cap_g": 120, "plant": "P1" },
  "C": [
    {
      "candidate_id": "mp-149.gauge0.6",
      "external_ids": { "MP": "mp-149", "CIF": "sha256:77af...9910" },
      "x": { "gauge_mm": 0.6, "mix_pct": 0.2 }
    },
    {
      "candidate_id": "icsd.186072.gauge0.5",
      "external_ids": { "ICSD": 186072 },
      "x": { "gauge_mm": 0.5, "mix_pct": 0.0 }
    }
  ],
  "a_star": "mp-149.gauge0.6",
  "y": { "fail_prob": 0.03, "cost": 0.17 }
}

D.4 Candidate index (fast lookups)

Maintain a per-snapshot CandidateIndex.json to dereference IDs quickly:

{
  "index_id": "cand.idx.ds.alpha.materials.v3",
  "snapshot_id": "sha256:3f5d...aa88",
  "by_candidate_id": {
    "mp-149.gauge0.6": { "external_ids": { "MP": "mp-149" }, "x": { "gauge_mm": 0.6 } }
  },
  "by_namespace": {
    "MP:mp-149": ["mp-149.gauge0.6", "mp-149.gauge0.8"],
    "AFDB:AF-Q9Y2Q0-F1-model_v4": ["AF.Q9Y2Q0.v4"]
  }
}

D.5 ObserverOps footers (samples)

Footer fields (recap).
{ model_id, version, codehash, seed, rng_lib, data_snapshot_id, data_time_range, topospec_hash, feature_scales_id, compile_hash, αβγλ, Γ_hard_id, Γ_soft_id, observers_roster, N_eff, τ_CSA, ε_max, χ_max, ESI_T, ESI_S, ESI_K, CSAk, max_ε, χ, ΔS_margin, regime_meta, Δ_trace, decision, author, time_utc }
Hash. (D.15) H = SHA-256( model_id ∥ data_snapshot_id ∥ codehash ∥ seed ∥ topospec_hash ).

D.5.1 AFDB-style domain footer

model_id=afdb.loss.learn.v1
version=1.3.2
codehash=sha256:8b29b6...e3
seed=123456
rng_lib=PCG64
data_snapshot_id=sha256:9b7a...c4f0
data_time_range=2024-01-01..2025-06-30
topospec_hash=sha256:4a1d...77
feature_scales_id=scales.protein.v4
compile_hash=sha256:0f55...aa
αβγλ={ "α":1.10, "β":0.85, "γ":0.32, "λ":1.00 }
Γ_hard_id=GammaHard.protein.v2
Γ_soft_id=GammaSoft.protein.v3
observers_roster=[rules_v2, model_A_v5, model_B_v2]
N_eff=2.7
τ_CSA=0.80
ε_max=0.45
χ_max=0.50
ESI_T=0.25
ESI_S=8
ESI_K="gaussian+structured_dropout"
CSAk=0.86
max_ε=0.22
χ=0.41
ΔS_margin=0.73
regime_meta={"active":"routine","supports":{"routine":0.88,"incident":0.12}}
Δ_trace=[{t:...;Δ:0.11;margin:0.70}, {t:...;Δ:0.09;margin:0.73}]
decision=pass
author=you@org
time_utc=2025-11-02T19:10:00Z
H=sha256:4f2b...9c

D.5.2 Materials-style domain footer

model_id=materials.packaging.v3
version=2.0.0
codehash=sha256:1caa...05
seed=424242
rng_lib=Philox
data_snapshot_id=sha256:3f5d...aa88
data_time_range=2023-07-01..2025-05-31
topospec_hash=sha256:8fa0...12
feature_scales_id=scales.materials.v3
compile_hash=sha256:cf11...b2
αβγλ={ "α":1.30, "β":0.70, "γ":0.28, "λ":1.00 }
Γ_hard_id=GammaHard.materials.v4
Γ_soft_id=GammaSoft.materials.v4
observers_roster=[fea_surrogate_v3, failmode_v2, rules_v1]
N_eff=2.6
τ_CSA=0.82
ε_max=0.40
χ_max=0.45
ESI_T=0.25
ESI_S=8
ESI_K="gaussian+structured_dropout"
CSAk=0.84
max_ε=0.18
χ=0.35
ΔS_margin=0.62
regime_meta={"active":"rush","supports":{"normal":0.44,"rush":0.56}}
Δ_trace=[{t:...;Δ:0.13;margin:0.58},{t:...;Δ:0.10;margin:0.62}]
decision=pass
author=you@org
time_utc=2025-11-02T19:12:00Z
H=sha256:9d01...71

Policy. No footer, no deploy. Hash mismatch ⇒ rollback.

D.6 Crosswalk files (for reviewers)

Ship a small ExternalIDCrosswalk.csv with the bundle when you use scientific IDs:

namespace,value,candidate_id,notes
AFDB,AF-Q9Y2Q0-F1-model_v4,AF.Q9Y2Q0.v4,"UniProt=Q9Y2Q0; seq_md5=1a2b...ff33"
MP,mp-149,mp-149.gauge0.6,"diamond, CIF sha256=77af...9910"
ICSD,186072,icsd.186072.gauge0.5,""

And a SpecimenIndex.json for per-case lookup:

{
  "AFDB:AF-Q9Y2Q0-F1-model_v4": { "seq_md5": "md5:1a2b...ff33", "length": 435 },
  "MP:mp-149": { "formula": "C", "spacegroup": "Fd-3m", "cif_sha256": "sha256:77af...9910" }
}

D.7 Repro tips (what auditors check)

snapshot_id must match the archive bytes; any ETL change ⇒ new ID.
topospec_hash and compile_hash must be present and recomputable.
Gate metrics in footers must recompute from the bundle with the same seeds.
External IDs must resolve by CandidateIndex.json without network access.
All numeric thresholds and weights appear in the footer; none live only in code.

Bottom line. With content-addressed snapshots, strict namespaces for scientific IDs, and footer hashes, any third party can replay your AFDB and Materials case studies byte-for-byte—no ambiguity, no drift, no excuses.

E. SMFT quick refs — ESI χ, CWA/CSA recipes, HeTu–LuoShu Γ examples

E.1 Emulsion-Stabilized Inference (ESI): χ definition & knobs

Goal. Measure score stability under small, lawful perturbations (“starch”) that keep Γ_hard satisfied.

Definition.
(E.1) χ = Var_starch(score) ÷ Var_raw(score).
Lower χ ⇒ more stable. Gate with χ ≤ χ_max.

Inputs.
(E.2) Raw observers: {O_j} with seeds; lawful perturbation kernel K; amplitude T; replicas S.
(E.3) Lawful set: x ∈ F(s) = { x : Γ_hard(s,x) ≤ 0 }.

Protocol (per batch).
(E.4) raw_scores = { score_Oj(s,x) } over observers and cases (no perturb).
(E.5) starch_scores = concatenate over r=1..S of scores on perturbed inputs (T, K), never violating Γ_hard.
(E.6) χ = Var(starch_scores) ÷ Var(raw_scores).

Default knobs.
(E.7) T = 0.25 (quarter-sigma on normalized features); S = 8; K = Gaussian on φ̂ + 10% structured dropout.
(E.8) χ_max ∈ [0.35, 0.60] depending on domain risk.

Repair prompts (if χ too high).
(E.9) Increase γ (smoothness), add distance-to-boundary soft penalties, reduce ∂Ω jitter, re-normalize units, prune collinear observers.

E.2 Cross-Worker/Observer Agreement (CWA): CSA@k & companions

Observers. Independent rankers (rules, models, external reviewers). Enforce independence before measuring agreement.

Core metrics.
(E.10) CSA@k = #(agreements within ε in top-k) ÷ N.
(E.11) max-ε = max_{i≤k, a<b} | score_i^(a) − score_i^(b) |.
(E.12) N_eff = 1 ÷ Σ_j w_j² (effective number of independent observers).

Optional rank-shape (report only).
(E.13) τ_K = Kendall-τ(top-k lists), ρ_S = Spearman-ρ(full ranks).

Pass rule (Lamp-1).
(E.14) Pass_CWA ⇔ [ CSA@k ≥ τ_CSA ∧ max-ε ≤ ε_max ∧ N_eff ≥ N_min ].

Independence lint (before CWA).
(E.15) Corr(O_a, O_b | φ̂) ≤ ρ_max; no shared frozen backbone IDs; document seeds.

Typical thresholds.
(E.16) τ_CSA ∈ [0.75, 0.90]; ε_max = 0.5·MAD(S) (score units); N_min ≥ 2.5 effective observers.

Deployment gate (Two Lamps).
(E.17) Conf_accept = 1[ CSA@k ≥ τ_CSA ∧ max-ε ≤ ε_max ∧ χ ≤ χ_max ].

E.3 HeTu–LuoShu slot constraints as Γ examples

Treat HeTu (pair-sum) and LuoShu (path-sum) as conservation predicates you can drop into Γ. Use as Γ_hard when inviolable; otherwise move to R_soft with hinge penalties.

E.3.1 Pair-sum (HeTu) — balance two slots/roles

(E.18) g_pair⁺(i,j) = (x_i + x_j) − K ≤ 0.
(E.19) g_pair⁻(i,j) = K − (x_i + x_j) ≤ 0.
• K = domain balance constant (HeTu archetype K=11; pick units: tasks/day, kg, headcount).

E.3.2 Capacity caps (HeTu “ceiling”)

(E.20) g_cap(i) = x_i − C_max ≤ 0.
(E.21) g_floor(i) = C_min − x_i ≤ 0.
• Use many small caps/floors instead of one mega-bound.

E.3.3 Path-sum (LuoShu) — constant throughput along rows/columns/routes

Let P be a valid path (row, column, route).
(E.22) g_path⁺(P) = Σ_{c∈P} x_c − S ≤ 0.
(E.23) g_path⁻(P) = S − Σ_{c∈P} x_c ≤ 0.
• LuoShu’s “15” is the archetype; S = required total per path/shift (units explicit).

E.3.4 Diagonal invariants (LuoShu diagonals)

(E.24) g_diag⁺ = Σ_{c∈diag₁} x_c − S ≤ 0, g_diag⁻ = S − Σ_{c∈diag₁} x_c ≤ 0.
(E.25) Same for diag₂; use when diagonals represent cross-functional balance.

E.3.5 Flow conservation (network form)

For node n with in-edges 𝕀(n), out-edges 𝕆(n):
(E.26) g_flow(n) = | Σ_{e∈𝕀(n)} f_e − Σ_{e∈𝕆(n)} f_e | − δ ≤ 0.
• δ = allowed leak (0 if strict). This generalizes pair/path sums to graphs.

E.3.6 Soft versions (move to R_soft when flexible)

(E.27) R_soft ← R_soft + w_pair · max{0, |x_i + x_j − K| − tol}.
(E.28) R_soft ← R_soft + w_path · max{0, |Σ_{c∈P} x_c − S| − tol}.
• Choose tol by measurement noise; keep weights w_* ≥ 0 (units consistent).

E.3.7 Units & normalization (don’t skip)

(E.29) Declare units u(x_i); ensure K,S,C_max share u(·); normalize by MAD or σ before fitting.
(E.30) θ̂_j = θ_j · σ_j (store re-scaled weights; report both raw and normalized).

E.4 Quick “Two Lamps” run-card (copy/pin)

Build observers; verify N_eff and independence.
Compute CSA@k and max-ε on raw ranks.
Starch with (T,S,K) ⇒ compute χ.
Gate with (E.17); no pass ⇒ safe-mode reports only.
Stamp ObserverOps footer with thresholds, seeds, χ/CSA metrics, and H.

Mantra. Conserve what must balance (HeTu/LuoShu), agree across observers (CWA), and emulsify micro-shocks (ESI). No gate, no deploy.

References

AlphaFold x AGI related

Everyday Structure AI: How AlphaFold-Style Models Quietly Upgrade Products, Health, and the Built World
https://osf.io/vj7ue/files/osfstorage/6907ba1bf5cdb710adc181cf
Semantic AlphaFold: A General Compiler (TopoSpec + Dissipative Action) for Everyday Systems (SMFT x AlphaFold).
https://osf.io/vj7ue/files/osfstorage/6907ba1ccaa0e7226dc18276

Primary “Five Lenses” sources (Wittgenstein, Freud, Brain Science, Five Aggregates, Yogācāra)

Wittgenstein, Operationalized: A Unified Mathematical Framework for Picture Theory, Language Games, and Hinge Certainty. (methods for picture-fit, meaning-as-use, hinge evidence/Λ_T and private-language tests).
https://osf.io/tjf59/files/osfstorage/68f2c1745bd9c41be2f98369
From Psychoanalytic Constructs to Closed-Loop Control: A Rigorous Mathematical Recast of Freud via Observer-Centric Collapse. (drives/defenses as control loops; Δ = g·β − γ; dashboards).
https://osf.io/w6be2/files/osfstorage/68f3d5d48a8dd1325519ff88
Observer-Centric Neurocybernetics: Unifying Closed-Loop Control, Language-Game Semantics, and Hinge Hyperpriors for Brain Science. (three-plane runtime; acceptance bands; governance).
https://osf.io/tj2sx/files/osfstorage/68f3de3e3c15ecd6a0c3fec6
Five Aggregates × Observer-Style AGI: A Verifiable Engineering Read of rūpa, vedanā, saṃjñā, saṃskāra, vijñāna [五蘊心理學 × 觀察者式 AGI：用日常比喻做出可驗證的心智地圖 — 色 rūpa、受 vedanā、想 saṃjñā、行 saṃskāra、識 vijñāna 的工程化讀法] (event pipeline v→ℓ→q→e with hash-footer/CWA).
https://osf.io/kvhuw/files/osfstorage/68f5072784487a9710c3fff8
From Grasping to Transformation: Observer-Style AGI Interprets Yogācāra (ālaya-vijñāna, manas, bīja-vāsanā) [從妄執到轉依：觀察者式 AGI 解讀唯識心理學（ālaya-vijñāna／manas／bīja-vāsanā）] (S/b/seed rates; āśraya-parāvṛtti as Ô-switch; three natures ↔ two lamps).
https://osf.io/kvhuw/files/osfstorage/68f532a0d542718a797f2cea

Observer & agreement backbone (latching, CWA/CSA/ε, smoothness χ, ops)

Self-Referential Observers in Quantum Dynamics: A Formal Theory of Internal Collapse and Cross-Observer Agreement. (delta-certainty “latching”; commutation; SBS redundancy; AB-fixedness).
https://osf.io/7cbsu/files/osfstorage/68c5961e10e31c4095d998f5
ObserverOps Technical Blueprint. (trace ledger, CWA engine, APIs, audits, runbooks).
https://osf.io/yj5aw/files/osfstorage/68d30242dd3f77699b3c315f
Emulsion-Stabilized Inference (ESI): Phase-Controlled Decoding with Structural “Starch” and Observer-Aligned Verification. (smoothness χ; T/S/K knobs; two-lamp policy).
https://osf.io/q8egv/files/osfstorage/68d58d6a5d44329625432c73
CAFT + CWA + SRA: A Universal Additive Model of Macro Coherence (App A–F). (certificate-to-average criteria; permutation p-value; reporting rules).
https://osf.io/7cbsu/files/osfstorage/68a3065155d1ad4d6c7e40d4

Slots, Δ₅ scheduling, and dissipative/variational foundations

The Slot Interpretation of HeTu and LuoShu: A Rigorous Mathematical and Semantic Proof by Wolfram 4.1 GPTs
https://osf.io/692wg/files/osfstorage/68960924847e9ead456b0e6c
Δ5 Phase Opposition in HeTu: Pairwise Minimum-Dissipation Cycles and a D₁₀–Spectral Extension of the Slot Interpretation
https://osf.io/38pw7/files/osfstorage/68e578b1dbe76397706d350d
(capacity conservation; pair-sum/phase opposition; spectral ground mode).
A Generalized Least Action Principle for Local and Dissipative Systems: Axioms, Proof, and Domain of Validity. (action for non-conservative systems; when Δ-style control is valid).
https://osf.io/2wmky/files/osfstorage/68b32a5ff4b17ecb9dc62067
From Entropy-Minimizing Attractor Proofs to Dissipative Lagrangian Dynamics: HeTu–LuoShu Variational Foundation. (variational/dissipative groundwork behind slot geometry).
https://osf.io/2wmky/files/osfstorage/68b4d262a233f0f2da96aecd
AGI by Surplus-Aware Control: Closed-Loop Surplus Flows, Semantic Field Geometry, and Dissipative Decoding. (ĝ/β̂/γ̂ instrumentation; Δ stabilization; surplus budgets).
https://osf.io/2wmky/files/osfstorage/68bd728b0fd5cbd1040356a2

Field/semantics layer and geometry

Semantic Meme Field Theory (SMFT): Foundations, Projection, and Dynamics (Rev1). (projection Ô; field→trace back-reaction; Appendix Y conventions).
https://osf.io/ya8tx/files/osfstorage/68e77fa0cd19895405a0d243
Semantic Collapse Geometry: A Unified Topological Model Linking Gödelian Logic, Attractor Dynamics, and Prime Number Gaps. (collapse topology; invariants/attractors).
https://osf.io/7jzpq
Proto-Eight Collapse Geometry — SMFT Applied to Growth, Memory, and Systems Built on Incubation Trigram (先天八卦). (phase-lock, ignition energy, cadence).
https://osf.io/ya8tx/files/osfstorage/68b84641534f31b42fef989e

Legacy and bridging note (prior version)

AI Psychology for AGI: An Observer-Centric Protocol for Meaning, Control, and Agreement. (earlier synthesis; superseded by the present rewrite but cited for lineage).
https://osf.io/8a3dt/files/osfstorage/68f40274bd52bb53417f27cd

Disclaimer

This book is the product of a collaboration between the author and OpenAI's GPT-5, Wolfram's GPTs language model. While every effort has been made to ensure accuracy, clarity, and insight, the content is generated with the assistance of artificial intelligence and may contain factual, interpretive, or mathematical errors. Readers are encouraged to approach the ideas with critical thinking and to consult primary scientific literature where appropriate.

This work is speculative, interdisciplinary, and exploratory in nature. It bridges metaphysics, physics, and organizational theory to propose a novel conceptual framework—not a definitive scientific theory. As such, it invites dialogue, challenge, and refinement.

I am merely a midwife of knowledge.

Sunday, November 2, 2025