https://osf.io/w6be2/files/osfstorage/68f3d5d48a8dd1325519ff88
https://chatgpt.com/share/68f3e15e-ac10-8010-8a31-10bb19776f3e

From Psychoanalytic Constructs to Closed-Loop Control: A Rigorous Mathematical Recast of Freud via Observer-Centric Collapse

1) Introduction: Why Recasting Freud Now

Problem. Classical psychoanalytic ideas—drive, repression, defense, transference—help clinicians think, but they are hard to test, compare, or standardize across cases and schools. We propose a closed-loop, observer-centric mathematical recast that treats therapy as a feedback process: each interpretation is an observation that writes to an internal record (a trace), and that record in turn changes what happens next. This framing gives us falsifiable indicators, reproducible workflows, and lightweight tooling a clinician can actually use.

Core thesis. The act of “making sense” is not neutral measurement; it’s an observer-centric collapse: once something is written into the patient’s lived record, subsequent meanings evolve conditioned on that write. Formally, we keep only a few moving parts—state, observer readout, and a growing trace:

• 𝒯ₜ = [e₁, e₂, …, eₜ] is the list of events the observer has “made real” so far. (1.1)
• yₜ = Ω̂[xₜ] is the observer’s readout (what we deem salient at time t). (1.2)
• xₜ₊₁ = F(xₜ, yₜ, 𝒯ₜ) is closed-loop evolution: what’s next depends on what we saw and what we wrote. (1.3)

Two testable indicators. This paper builds everything around a pair of plain-English, clinic-friendly metrics:

• Δ (stability discriminant). We compress “how hard the framing pulls,” “how fast associations snowball,” and “how much buffer exists” into one number:
Δ := g · β − γ. (1.4)
Here g = macro-guidance gain (strength of therapist frame), β = micro-amplification (branching speed of associations), and γ = damping/buffer (pace control, pauses, grounding). Large positive Δ warns of loop lock-in (e.g., rumination/repetition); negative Δ predicts settling.

• CSA (cross-observer agreement). We estimate objectivity by asking several commuting graders (independent checks whose order shouldn’t matter) to label the same segment; CSA is their order-invariant agreement rate:
CSA := (1/M) Σₘ 1{ graderₘ agrees with others under order-swap }. (1.5)

What “observer-centric collapse” looks like in a room.
Therapist offers a reframe (“You sounded abandoned then, not now”). Patient nods and repeats it later. That moment is an event write eₖ into 𝒯: future choices and memories will be conditioned on “I felt abandoned then,” not “I am abandoned now.” In our terms, yₜ changed, 𝒯 advanced, and (1.3) pushes the trajectory toward a calmer basin—if Δ turned negative (g didn’t overshoot, γ was strong enough).

Every symbol comes with an everyday analogue.
• 𝒯 is a journal: once written, you cannot “unhappen” the entry in your own timeline.
• yₜ is a highlighter: it decides what jumps off the page.
• g is the volume of the therapist’s speaker; β is how quickly the room starts echoing; γ is acoustic panels that absorb echo.
• CSA is “three thermometers agree even if you read them in a different order.”
These analogies run beside each formula throughout the paper to keep the math intuitive.

Contributions (four).

Minimal formalism—only (1.1)–(1.5) and a few operator knobs (introduced later).
Freud→operators mapping—Id as drive potential, Superego as constraint operator, Ego as the observer-controller (Ω̂).
Testable indicators—Δ for stability; CSA and a “CWA certificate” for when averaging across cases is legal.
End-to-end workflow—from transcript segments to a Δ-dashboard, plus SOPs and small datasets any team can reproduce.

Pedagogy promise.
All math is MathJax-free, single-line, Unicode; each object appears with a plain-language example and, where helpful, a tiny toy scenario. When a needed result can be stated from first principles in a few lines, we include it inline; otherwise we add a short appendix sketch and keep the main text readable.

Why now.
Modern practice needs common measures that respect depth and enable cumulative evidence. By casting interpretation as controlled observation-with-trace, we earn simple levers (turn g down, raise γ, shape β) and clear guardrails (raise CSA before committing to a case-level claim). This paper lays the self-contained backbone; Part II (a separate paper) maps the same symbols onto EEG/MEG/fMRI so the clinic and the lab can finally speak the same language.

2) Reader’s Roadmap & Style Conventions

2.1 Style (how to read formulas, symbols, and boxes)

Unicode Journal Style. All formulas are single-line, MathJax-free, tagged like “(2.1)”.
Example tags we’ll reuse later:
Δ := g · β − γ. (2.1) CSA := (1/M) ∑ₘ 1{ graders agree under order-swap }. (2.2)
Symbols.
• States/signals: (x_t, y_t) as plain Unicode (subscripts only).
• Operators: hats, e.g., Ω̂ (observer/decoder), Ŝ (constraint), R_θ (frame rotation).
• Trace: 𝒯ₜ = [e₁,…,eₜ] as a growing list.
• Scalars: g (guidance), β (amplification), γ (damping), Δ (stability discriminant).
• Kernels/fields: K(Δτ) (memory kernel), A_θ (direction/“pull” field).
• Default time base: discrete steps t = 1,2,… unless stated otherwise.
AMS-style blocks, zero heavy prerequisites. Short, stand-alone lemmas appear inline; any proof longer than ~8 lines is sketched in Appendix B with plain language.
Pedagogy boxes used throughout.
Analogy (plain-English mental model), Estimator (how to compute a number), Guardrail (what not to do), Clinic Note (what a therapist actually says).

2.2 Roadmap at a glance (who should read what first)

Clinician-first path (practical): §3 → §5 → §6 → §7 → §8 → §9 → skim §10.
Methodologist-first path (formal): §3 → §4 → §8 → §9 → §10.
Everyone: §1–§2 for setup; §11 for limits/ethics; §12 for the short bridge to neuroscience.

§3 Minimal Mathematical Toolkit. Self-contained definitions (states, operators, traces, closed-loop). Read this if you prefer first principles with examples.
§4 Observer-Centric Collapse. The two postulates (write-to-trace; agreement via commuting checks), notation, and why these give falsifiable claims.
§5 Recasting Freud’s Tripartite Model. Id/Superego/Ego → drive potential V, constraint Ŝ, observer-controller Ω̂; one-line update law.
§6 Defense Mechanisms as Operators. Repression, isolation, projection, sublimation → knobs on V, Ŝ, Ω̂, plus frame rotations and channel decoupling.
§7 Dreams, Transference, Repetition. Condense/shift operators, direction field A_θ, and the repetition attractor (Δ-based).
§8 Agreement & Certificates. CSA (how we quantify objectivity) and the CWA certificate (when group averaging is legal).
§9 Clinical Workflow. From transcript segments to a Δ-dashboard with early-warning “hitting-time” checks.
§10 Case Mini-Studies. Three short N=1 narratives with pre-registered falsification gates.
§11–§12 Limits & Bridge. Failure modes, ethics, and a preview of the neural mapping we develop in the companion paper.

2.3 Terminology crosswalk (classical → operator/metric → plain-English knob)

Classical term	This paper (symbol)	What it does (1-line, clinic knob)
Id (drive)	Drive potential V_id(x)	Pulls content toward urges; raise/lower its “slope” to modulate pressure.
Superego (rules/guilt)	Constraint operator Ŝ	Fences off taboo regions; “tighten/loosen” boundary.
Ego (reality testing)	Observer-controller Ω̂	Chooses what to highlight/ask next; sets pacing and focus.
Repression	Tighten constraint Ŝ_tight	Exclude content; watch for rebound when loosened.
Isolation	Channel decoupler U_⊥	Prevents narratives from interacting; useful short-term, risky long-term.
Projection	Frame rotation R_θ	Reassigns internal content outward; detect via stance shift.
Sublimation	Path redirection (edit V)	Same energy, safer basin (work/art/service).
Displacement	Topic shift Shift_θ	Moves affect to a nearby object; reversible by re-anchoring.
Denial	Hard gate Gate_τ	Drops evidence at intake; soften gate by paced exposure.
Rationalization	Mask M_logic	Polishes a fixed outcome; test with order-swap checks.
Free association	Amplifier β	Branching rate of links; tame by slowing and chunking.
Dream work	Condense/shift Condense, Shift_θ	Many-to-one compression + angle change; integrate by gentle unpacking.
Transference	Direction field A_θ	Bias toward past templates; lower by labeling and time-indexing.
Counter-transference	Therapist Ω̂ drift	Calibrate using CSA and supervision.
Repetition compulsion	Stability Δ := g·β − γ	Δ↑ → loop risk; act by ↓g and ↑γ.
Reality testing	Agreement CSA	Multiple commuting checks converge; raise CSA before committing.
Working through	Damping γ	Pauses, grounding, homework; increases “recovery to baseline.”
Cathexis	Basin depth in V	Investment level; rebalance by reshaping V.

(We reuse these names consistently so the paper reads like a bilingual dictionary: classical term ⇄ operator knob.)

2.4 What “independence” means for sections 3–10

Each technical section begins with a boxed recap of any symbols it needs from §3–§4.
Any formula introduced is paired with one toy example and one 30-second clinic vignette.
If a claim needs heavier math, we give a one-paragraph plain-language proof idea and push details to Appendix B.

2.5 Cheat-sheet card (2 pages, printed at the end)

Page A: Symbols & Knobs. One-liners for Ω̂, Ŝ, V, Δ, g, β, γ, CSA, CWA; plus how to change them in practice.
Page B: Mini-SOP. 6-step workflow from transcript → segments → Δ/CSA estimates → guardrails → intervention templates.
Copy-paste equations (Unicode):
𝒯ₜ = [e₁,…,eₜ]. (2.3) yₜ = Ω̂[xₜ]. (2.4) xₜ₊₁ = F(xₜ, yₜ, 𝒯ₜ). (2.5)
Δ := g · β − γ. (2.6) CSA := (1/M) ∑ₘ 1{ agree under order-swap }. (2.7)

Pedagogy reminder. Throughout the paper we (i) re-introduce any needed math from first principles when it fits in a short box, (ii) keep every equation single-line Unicode with a tag, and (iii) sit each symbol next to a live example so non-PhD readers and busy clinicians can follow without extra references.

3) Minimal Mathematical Toolkit (Self-Contained)

Promise: no advanced prerequisites, no MathJax. All formulas are single-line “Unicode Journal Style” with numbered tags like (3.1). Each concept comes with a clinic-floor example.

3.1 Signals, States, and Simple Operators

State. A state is just a list of numbers you track at time t (e.g., mood, arousal, sleep, alliance):
x_t ∈ ℝ^n (3.1)

Signal. A signal is a time-stamped list of observations:
s_{1:T} = [(t_1, y_1), (t_2, y_2), …, (t_T, y_T)] (3.2)

Operator (a “recipe”). An operator takes a vector and returns a transformed vector. Think: rotate, select, threshold.
A: ℝ^n → ℝ^m, (A ∘ B)(x) = A(B(x)) (3.3)

Examples you’ll actually use.
• Selection (a “mask” that keeps only components in index set S): P_S(x)_i = x_i if i ∈ S else 0. (3.4)
• Projector (apply selection twice = once): P_S ∘ P_S = P_S. (3.5)
• Threshold label (e.g., “inhibition high?”): Θ_τ(x) = 1 if uᵀx ≥ τ else 0. (3.6)

Therapy-room example. If x_t = [PHQ-9, GAD-7, sleep_hours], then P_{PHQ}(x_t) extracts just depression severity; Θ_τ flags “risk_high” when a weighted sum crosses a threshold.

3.2 Closed-Loop Dynamics (why “the talk” today changes next week)

A closed loop feeds output back into next input:
x_{t+1} = F(x_t, u_t, ξ_t) and y_t = H(x_t) + ε_t. (3.7)

• x_t is the client’s latent condition we don’t fully see.
• u_t is what we do (intervention, homework, medication adjustment).
• ξ_t, ε_t are disturbances and measurement noise.

Analogy. A thermostat reads the temperature (y_t), writes “heat_on” to its log, then acts (u_t = turn furnace on), changing tomorrow’s temperature x_{t+1}. That’s closed loop in one sentence.

3.3 Traces: the “internal record” that therapy writes to

We formalize a running record of consequential events:
e_t = (τ_t, label_t, meta_t). The trace up to time t is 𝒯_t = [e_1, e_2, …, e_t]. (3.8)

• τ_t is when it happened.
• label_t is the categorical outcome or tag we care about (“avoidance_detected”, “grief_accessed”, “insight_A”).
• meta_t can include rater ID, confidence, or sensor context (e.g., EEG bandpower window).

Therapy-room example. After a session you append e_t = (“2025-10-18 15:02”, “exposure_completed”, {SUDS_drop: 3, rater: “Dr. Lin”}). Next week’s plan conditions on this trace.

3.4 What is an Observer (informal → formal)

Plain English. An observer (therapist, client self-monitor, or instrument) (i) measures/labels, (ii) writes to the trace, and (iii) conditions its next step on that updated trace.

Minimal formal version (components and loop).
Observer ℴ is a triple ℴ = (M, W, Π) with:
• M: measurement/labeling rule, M(x_t, 𝒯_{t−1}) → ℓ_t ∈ Labels. (3.9)
• W: write rule, W(𝒯_{t−1}, ℓ_t) → 𝒯_t. (3.10)
• Π: policy (how we act next), Π(𝒯_t) → u_t. (3.11)

Closed-loop with an observer is then:
x_{t+1} = F(x_t, Π(W(𝒯_{t−1}, M(x_t, 𝒯_{t−1}))), ξ_t). (3.12)

Therapy-room example. M = “rate avoidance 0–3 from session notes”, W = “append label to EHR trace”, Π = “if avoidance ≥ 2 for 2 weeks, schedule imaginal exposure first”.

3.5 Observer-Centric Collapse (why “making sense” changes the state)

When an observer assigns a label, the system’s effective state space shrinks to the labeled slice; next dynamics follow that slice. This “collapse” is not quantum mysticism—it’s conditional dynamics given the record you just wrote.

One-line update.
Given ℓ_t = M(x_t, 𝒯_{t−1}), define the collapsed next-step map F_{|ℓ_t}(·) whose parameters or branches are selected by ℓ_t:
x_{t+1} = F_{|ℓ_t}(x_t, u_t, ξ_t). (3.13)

Clinic analogy. Once the team writes “trauma_cue_confirmed”, future sessions route through the exposure protocol branch (F_{|trauma}), not the generic supportive branch—so the label you wrote changes the path.

Pointers to rigor. For a neuroscience-friendly development of observer/observed hierarchies and how labeling re-routes integration across networks (PFC, DMN, claustrum), see the uploaded monograph on observer/observed dynamics; it motivates “collapse” as conditioning on a recorded observation and forecasts empirical tests (EEG/fMRI, TMS) that should show re-routing after labeling.

3.6 Compatibility, Commutation, and CSA Agreement (trust in labels)

Two checks “commute” on x if doing A then B equals doing B then A and they land on the same label.

Commutation on x.
A and B commute on x if A(B(x)) = B(A(x)). (3.14)

Label agreement on x.
Let L_A(x) and L_B(x) be their labels; they agree if L_A(x) = L_B(x). (3.15)

Compatibility set.
Compat(A, B) = {x : A(B(x)) = B(A(x)) and L_A(x) = L_B(x)}. (3.16)

CSA (Cross-System Agreement). Over a sample X = {x^{(1)}, …, x^{(N)}} and a family of checks 𝒞, define:
CSA(𝒞; X) = (1 / |𝒫|) ∑{(A,B)∈𝒫} (1 / N) ∑{i=1}^N 𝟙[x^{(i)} ∈ Compat(A,B)], where 𝒫 = all unordered pairs from 𝒞. (3.17)

Interpretation: CSA is the percent agreement among commuting checks—high CSA means different raters/tools give the same label regardless of order, so the label is stable and portable.

Therapy-room example. A = clinician vignette rating; B = client self-report rule; C = brief behavioral task. If all three produce the same “avoidance_high” whether you read them in any order, CSA is high—your case-formulation label is trustworthy for routing care.

Pointer to rigor and experiments. The observer/observed framework proposes concrete tests—e.g., measure how labeling and meta-observation re-synchronize networks; CSA provides a falsifiable external metric (do independent procedures commute and agree?). See proposed empirical protocols using fMRI, TMS, and neurofeedback to validate observer-level commutation and agreement.

3.7 A Minimal Stability Metric Δ (to detect “getting unstuck”)

Define a weighted step-change (lower is more stable in the bad sense if you want movement; or interpret as “progress magnitude” if you expect movement).

Session-to-session delta.
Δ_t = ‖W^{1/2} (x_{t+1} − x_t)‖₂. (3.18)

• Choose W to emphasize clinical priorities (e.g., more weight on avoidance than on sleep).
• For labels, use one-hot vectors to embed categories into x_t; Δ_t then measures label movement too.

Trace-conditioned expectation.
Δ̄ | 𝒯_t = E[Δ_t | 𝒯_t] estimates how much change you expect given what’s in the trace; unusually small Δ̄ may indicate repression/avoidance maintaining the status quo; unusually large Δ̄ after “exposure_completed” may signal productive affective processing. (3.19)

Therapy-room example. After three exposures in the trace, you expect Δ̄ to increase short-term (temporary distress + behavioral shift) and then decrease (consolidation). If Δ̄ stays ≈ 0 despite “exposure_completed” entries, re-examine label quality (CSA) or protocol fidelity.

3.8 Two Friendly Analogies

• Thermostat. Reading (M) → writing “heat_on” to the trace (W) → acting (Π) changes tomorrow’s temperature—exactly the loop in (3.12)–(3.13).
• Two thermometers. If both thermometers (A, B) give the same reading regardless of order, you trust the temperature—exactly CSA in (3.17).

3.9 “Cheat-Sheet” Boxes (what we’ll reuse later)

Box A — The 3 moves of an Observer. Measure (M), Write (W), Act (Π). See (3.9)–(3.12).
Box B — Collapse as Conditioning. Labels select the next map F_{|ℓ_t}. See (3.13).
Box C — CSA you can compute. Build 3+ checks, compute (3.17), watch it rise as your labeling rules mature.
Box D — Δ to watch progress. Compute (3.18)–(3.19) from routine measures; flag “unexpected stasis” or “overshoot”.

3.10 Pointers to Rigor Sources (optional deep dive for methodologists)

• Observer/Observed hierarchy (PFC as meta-observer, DMN, claustrum; binding via temporal synchronization). Supplies the neuroscientific grounding for our “observer-centric collapse” and suggests concrete perturbation/measurement tests.
• Empirical test battery (fMRI/TMS/neurofeedback) for observer-level labels and agreement. Proposed studies suitable for CSA validation in practice.

Why these sources help clinicians: they show how writing to a trace (labels) re-routes integration across networks—exactly why our simple operators and CSA/Δ metrics map to observable changes and testable workflows in clinic dashboards.

You can read this section standalone. All later sections (Freud→operators mapping, testable indicators, and clinical SOPs) reuse only the primitives above: vectors, simple operators, the trace, observer triples (M, W, Π), collapse-as-conditioning, CSA, and Δ.

4) Observer-Centric Collapse: Core Postulates & Notation

Plain goal of this section. State two simple, clinic-usable postulates, then fix the symbols you’ll see again. All formulas are single-line “Unicode Journal Style,” tagged (4.n). Each symbol sits next to a micro-example.

4.1 Postulate P1 — Observer with Trace (“latching”)

P1. Once an event is written to the observer’s trace, it becomes delta-certain in that observer’s frame: future updates condition on it as a fixed point.

One-line mechanics.
Trace update: 𝒯_t = 𝒯_{t−1} ⊕ e_t. (4.1)
Latching (fixedness): E[e_t ∣ 𝒯_t] = e_t and Pr(e_t ∣ 𝒯_t) = 1. (4.2)

Journaling metaphor. You can revise opinions about why it happened, but you cannot “unhappen” a written entry in your own record; subsequent choices are conditioned on that entry.

Therapy micro-example. You jointly write “trauma_cue_confirmed” into today’s note (e_t). Next session routes through the exposure pathway precisely because (4.2) holds for your record.

Rigor pointer. In the formal observer model, latching is the statement that past-record events are fixed points under conditional expectation onto the observer’s information algebra; agreement events are commuting projections in a common algebra. We cite the delta-certainty theorem and its operator-algebra form.

4.2 Postulate P2 — Agreement via Commutation and Redundancy

P2. Cross-observer agreement emerges when (i) checks commute (order does not change the effect) and (ii) outcomes are redundantly recorded in the trace (multiple fragments point to the same claim).

One-line mechanics.
Commutation on state x: A ∘ B(x) = B ∘ A(x). (4.3)
Same label under order-swap: L_A(x) = L_B(x). (4.4)
Redundant write (two independent fragments for the same claim): 𝒯_t = 𝒯_{t−1} ⊕ e_t^A ⊕ e_t^B. (4.5)

Two-clinician example. Clinician A (transcript-based) and Clinician B (audio-based) independently label a segment “avoidance_high.” If swapping their order leaves both the effect and the label unchanged, and both labels are written to the record, CSA rises (you can then safely treat that label as “objective enough” for planning). Production guidance for commuting checks and redundant traces appears in the verification blueprint we adapt.

Rigor pointer. Commutation and redundancy as guarantees for agreement are the operational mirror of the formal cross-observer results (commuting effects in a common algebra + fixed records).

4.3 Notation block (minimal, reused everywhere)

Session time. Discrete steps t = 1,2,…. (Therapy happens in visits or segments; we index them.)

State. x_t ∈ ℝ^n. (“What’s there now”—affect level, avoidance, sleep, stance vectors.)
Micro-example. x_t = [PHQ9, “avoidance_score”, “sleep_hours”, “stance_toward_event”…].

Observer readout. y_t = Ω̂[x_t]. (4.6)
Micro-example. Ω̂ highlights “avoidance” this week, so y_t ≈ high on that dimension.

Trace (record). 𝒯_t = [e_1, …, e_t]. (4.7)
Micro-example. e_t = (“2025-10-18 15:02”, “exposure_completed”, {SUDS_drop: 3}).

Policy (what we do next). u_t = π(𝒯_t). (4.8)
Micro-example. If two weeks of “avoidance_high” appear in 𝒯_t, π schedules imaginal exposure first.

Closed-loop update. x_{t+1} = F(x_t, u_t, 𝒯_t). (4.9)
Micro-example. Pacing homework (u_t) and the “exposure_completed” entry in 𝒯_t jointly reduce next week’s avoidance coordinate in x_{t+1}.

Stability discriminant (preview; quantified in §9).
Δ := g · β − γ. (4.10)
Qualitative reading now. g is macro-guidance gain (how hard reframes pull), β is micro-amplification (how fast associations snowball), γ is damping/buffer (pauses, grounding, homework).
Micro-example. A sharp reframe raises g; slowing the pace raises γ; narrowing free association lowers β. Δ is your “loop risk” dial; details and estimators are given in §9.

4.4 Reader cues and clinic-first boxes

Analogy (next to (4.9)). Thermostat loop: read → write “heat_on” → act → tomorrow is warmer; that’s F with a trace and a policy.

Guardrail. Do not average labels across cases unless your checks commute and CSA is high (we formalize “CWA certificate” later). The engineering playbook for commuting critics, CSA gates, and redundant trace fragments is our template for clinic tooling.

Cross-bridge note. For readers who want deeper theory or neural predictions (e.g., how labeling re-routes integration via PFC/DMN/claustrum and how to test it with TMS/EEG/fMRI), see the observer/observed monograph we cite sparingly; we keep this paper self-contained for clinical math.

What you take from §4. Two postulates ((4.1)–(4.5)) that make observation-with-trace operational in clinic, and four symbols you’ll reuse everywhere ((4.6)–(4.10)). Later sections only build knobs on these: V (drive), Ŝ (constraint), and Ω̂ (observer-controller), plus the CSA/CWA gates. Formal latching and agreement live in the cited observer theory; our clinic recipes import just the guarantees we need.

5) Recasting the Freudian Tripartite Model in Dynamics

Plain goal. Give a compact, clinic-usable dynamical form where Id = drive potential, Superego = constraint operator, Ego = observer-controller, with single-line, Blogger-ready formulas. Each symbol gets a therapy-room micro-example.

5.1 Mapping table (classical → operator knob → clinic cue)

Id (drive) → drive potential V_id(x) and a soft-saturating push in content flow (“impulse pressure”).
Cue. “I want it now.” We model this as a pull toward urge-laden states.
Superego (rules/guilt) → constraint operator Ŝ that penalizes/forbids regions of state space (“rule fence”).
Cue. “You mustn’t.” Raising Ŝ tightens taboo boundaries.
Ego (reality testing/steering) → observer-controller Ω̂_self that selects what to highlight/ask next and regulates pace (“self steers”).
Cue. “Let’s slow down and name that feeling.” Cooling Ω̂_self reduces over-steer.

Rigor pointer. The projection/operator idiom (Ô, compatibility/commutation) and observer back-reaction (how the observer changes dynamics) are introduced in a compact, deployable toolchain we will cite sparingly for readers who want first-principles detail.

5.2 A one-line update law (with plain-English labels)

We write the next session’s latent state as the baseline evolution plus three “Freud knobs”:

x_{t+1} = F(x_t, Ω̂_self(x_t), 𝒯_t) − ∇V_id(x_t) − Ŝ[x_t] + η_t. (5.1)

Baseline evolution F(·) = “what tends to happen next” given today’s highlights and the trace (habits, context).
Clinic. Sleep improves after regular routines; F nudges the sleep coordinate up.
Impulse pull −∇V_id(x_t) = the Id’s gradient pulling toward urges.
Clinic. Cake in view → “urge” component pulls eating-now upward.
Rule fence −Ŝ[x_t] = penalties that push the state away from forbidden/pricey regions.
Clinic. “Don’t drunk-text” fence raises the cost of that trajectory.
Self steers Ω̂_self = what the Ego chooses to highlight/ask now; it enters F because the observer’s readout changes the path (“observer-centric collapse” via the trace).
Clinic. Labeling “this anger belongs to then, not now” changes tomorrow’s route.
Noise η_t = unmodeled bumps (poor sleep, traffic fight).

Rigor pointer. Projection Ô, compatibility, and observer-induced back-reaction appear as Chapters “Projection Operator Ô and Phase Collapse” and “Observer-Induced Backreaction and Collapse Topology”; our (5.1) is the clinic-level reduction of those ideas.

5.3 Minimal, self-contained definitions for the terms in (5.1)

Drive potential. V_id: ℝ^n → ℝ is lower in urge-satisfying regions; the pull is its gradient.
−∇V_id(x_t) = “how the urge tugs next.” (5.2)
Constraint operator. Ŝ acts like a soft barrier or mask (penalty in restricted zones).
Example (componentwise): Ŝ[x]_i = λ_i · max(0, x_i − τ_i). (5.3)
Soft saturation for crude impulses. To avoid runaway, include a gentle nonlinearity:
sat(x)_i = x_i / (1 + α_i |x_i|), and optionally add + κ · sat(x_t) to F. (5.4)
Observer-controller. y_t = Ω̂_self[x_t] selects what’s salient; policy uses trace:
u_t = π(𝒯_t), then F depends on (y_t, u_t, 𝒯_t). (5.5)

Micro-examples (one line each).
• Id: “urge_eat” coordinate pulled up when hungry smells present.
• Superego: “don’t text ex after 10pm” raises penalty beyond that hour.
• Ego: “name → pace → reframe” lowers over-steer; Ω̂_self highlights present-tense safety.

5.4 What changes when you “turn the knobs”

Raise the fence (Ŝ↑). Fewer taboo-violating excursions; more spillover pressure into allowed outlets unless you also redirect V_id.
Cool the steering (Ω̂_self cools). Slower, broader exploration; fewer oscillations from over-interpretation.
Redirect the drive (edit V_id). Same energy, new basin (sublimation): pull toward writing, exercise, service.

Bridge to §9. The three knobs show up in the stability discriminant Δ := g · β − γ (preview only here; estimators in §9):
• g (macro-guidance gain) rises when Ω̂_self pushes hard;
• β (micro-amplification) rises when free associations snowball;
• γ (damping/buffer) rises with pacing, pauses, grounding/homework.
Keeping Δ below threshold prevents loops and lock-ins.

5.5 A 90-second “boxes & beads” toy simulation (no code required)

Setup (10 s). Draw three boxes on paper: “Allowed,” “Border,” “Taboo.” Put 10 beads in “Allowed” as your state x_0.

Impulse pull (20 s). Pick 2 beads and try to move them toward “Taboo” each step (Id pull). That’s −∇V_id.

Rule fence (20 s). Set a fence: every bead attempting to enter “Taboo” pays a toll λ (return it to “Border” unless you also place a “Redirect” arrow). That’s Ŝ.

Self steers (20 s). Before moving beads, say aloud what you’ll highlight (“name the feeling; slow down”)—if you “cool steering,” you only move 1 bead per step and must pause 1 beat between moves. That’s Ω̂_self changing F.

Redirect (20 s). Draw an arrow from “Border” to a new “Art/Exercise” box; any bead rejected by the fence must follow that arrow. This is editing V_id to create a safe basin (sublimation).

What you’ll see. With Ŝ↑ and Ω̂_self cooled, the system stops oscillating between “Allowed↔Taboo,” and beads accumulate in “Art/Exercise.” If you push Ω̂_self too hard (big g) while free links are hot (β high), beads ping-pong—your Δ would be positive (loop risk). This is exactly the clinic story behind (5.1).

5.6 Pedagogy boxes (drop-in beside the text)

Analogy. Id = downhill slope; Superego = guardrail; Ego = steering wheel and speed control.
Estimator (later). We will estimate g, β, γ from transcripts (reframes vs. stance shift; branch rate; recovery time).
Guardrail. Don’t average patient labels across cases unless your graders commute and CSA is high (CWA certificate in §8).

5.7 Where the math comes from (for curious readers)

Projection/operator Ô and compatibility. Why “what you highlight” can be treated as a projection, and why commuting checks matter for agreement. (We keep main text intuitive; proofs live in the cited operator chapters.)
Observer back-reaction. Why the observer’s own readout reshapes the next step (collapse-with-trace), matching the qualitative therapy claim that naming changes the path.
Stability discriminant Δ. A compact linearization result that yields Δ := g · β − γ as the practical “loop-risk” dial we’ll estimate in §9.

Takeaway. With a single line (5.1) and three knobs (V_id, Ŝ, Ω̂_self), we have an actionable Freud-in-dynamics that therapists can steer and researchers can test—without heavy prerequisites, and with rigorous pointers available for those who want the algebra.

6) Defense Mechanisms as Operators: A Control View

Idea in one line. Each “defense” is just an edit to the knobs from §5—drive potential V, constraint Ŝ, and observer-controller Ω̂_self. We keep everything single-line Unicode with tags like (6.n); each operator gets a clinic micro-example and an audio-mixing analogue. Formal operator/projection/commutation references are noted at the end.

6.1 Operator dictionary (with one-line math and a tiny vignette)

Repression = tighten the fence (Ŝ_tight)

One-line operator.
Ŝ_tight[x] := Ŝ[x] + Λ · χ_taboo(x), with Λ ≥ 0 and χ_taboo a mask that penalizes forbidden regions. (6.1)

What it does (Δ-dial). Short-term damping ↑γ (less spill), but hidden pressure can re-route into other channels unless V is also redirected (risk of delayed β rebound).

Before/After (transcript).
Before. Pt: “When dad visits I… (breaks off).”
After (Ŝ_tight↑). Pt: “It’s nothing.” (topic clipped; feelings resurface elsewhere later).

Audio mixing. A gate that clamps quiet passages—useful against noise, but over-gating kills nuance.

Isolation = channel decoupling (U_⊥)

One-line operator (decouple cross-talk).
Given a coupling matrix C (how narratives influence each other), isolation drops off-diagonals: C_iso := diag(C) = U_⊥(C). (6.2)

What it does (Δ-dial). Cuts micro-amplification β (fewer runaway associations). Overused, it blocks integration, stalling change (Δ → 0).

Before/After (transcript).
Before. “When I think about mom, I feel it in my chest and cancel plans.”
After (U_⊥). “I can think about mom without the chest stuff.” (thought–body link muted; later we must re-link safely).

Audio mixing. Mute the send from one track to another; prevents feedback but can make the mix sterile.

Projection = frame rotation (R_θ)

One-line operator (rotate coordinates).
x′ = R_θ · x, choosing θ so the self-axis for an affect aligns with an external-object axis (“anger_at_self” → “anger_at_boss”). (6.3)

What it does (Δ-dial). Can mis-aim g (guidance) at the wrong target; good therapy rotates back in a paced way (γ↑), or uses the rotation to access the affect safely before re-aiming.

Before/After (transcript).
Before. “I’m furious at myself for messing up.”
After (R_θ). “My team keeps sabotaging me.” (blame spun outward; work then rotates θ toward source without shaming).

Audio mixing. Pan the sound to the other speaker; feels different though energy is the same.

Sublimation = path redirection (edit V into a safe basin)

One-line operator (add a safe trench to the potential).
V_id^→(x) := V_id(x) + μ · B_safe(x) with μ > 0; force term becomes −∇V_id^→ = −∇V_id − μ ∇B_safe. (6.4)

What it does (Δ-dial). Lowers loop-risk by pulling energy into a low-risk basin; effectively drops g on risky routes and raises γ via structured outlets.

Before/After (transcript).
Before. “I doom-scroll when tense.”
After (V→). “When tense I walk and voice-note ideas for my short story.” (same drive; new basin captures it).

Audio mixing. Send-to-reverb/aux bus designed to hold energy pleasantly rather than letting it spike on the mains.

6.2 One-line “plug-in chain” (how operators edit the §5 update)

Edited step with defenses active.
x_{t+1} = F(x_t, Ω̂_self(x_t), 𝒯_t) − ∇V_id^→(x_t) − Ŝ_tight[x_t], with couplings replaced by C_iso = U_⊥(C). (6.5)

Read this like a channel strip: steering (Ω̂_self) → drive pull (−∇V) → fence (Ŝ) → routing (C_iso).

6.3 Clinic knobs and quick guardrails

Repression (Ŝ_tight). Use briefly to stop flooding; always pair with redirect (V_id^→) so pressure has somewhere to go. If CSA (agreement) on “taboo” is low, don’t hard-gate yet.
Isolation (U_⊥). Decouple when spirals form; schedule later re-coupling under higher γ (breathing, pacing) so meaning can integrate.
Projection (R_θ). Name the rotation (“sounds aimed outward”) and rotate slowly; the observer’s highlight (Ω̂_self) changes the next map—don’t over-steer (keep g moderate).
Sublimation (V→). Build the safe trench concretely (calendar blocks, materials, micro-goals) so the gradient exists in real life; otherwise V lacks an actual basin.

6.4 30-second before/after micro-case (all four operators together)

Setup. Pt loops on “critique → shame → doom-scroll → sleep loss.”

Edits.
• Ŝ_tight↑ on late-night phone (hard cutoff 23:00). (6.6)
• U_⊥ on “critique→doom-scroll” cross-talk for 2 weeks. (6.7)
• R_θ small: “anger toward process, not self/others.” (6.8)
• V_id^→ adds a “writing-sprint” trench at 07:30 with materials pre-staged. (6.9)

Observed effect. Δ falls (g·β − γ → negative): fewer night spikes (γ↑ from pacing + fence), branching of shame loops drops (β↓ via isolation), energy finds the morning basin (effective g on safe path). These are exactly the closed-loop edits in (6.5).

6.5 Where the rigor lives (short, optional pointers)

Projection operator Ô and frame rotation, compatibility/commutation. Formal treatment of Ô (projection), frame maps, and agreement via commuting checks—the math idiom behind R_θ and CSA gates. (Ch. 2, 4 in SMFT Rev1.)
Observer-induced backreaction. Why changing Ω̂_self (what you highlight) legitimately changes the next step map F—our “observation writes to trace, then dynamics condition on it.” (Ch. 10 in SMFT Rev1.)
CSA and verification playbooks. Production recipes for commuting critics + redundant traces (how to know your labels are objective enough before hard-gating). (ESI §“Smoothness = CSA”; ObserverOps figures on CWA/Agreement.)
Neuro basis for Ω̂_self as a controller. PFC as meta-observer that can raise damping γ (pacing), modulate guidance g (reframe strength), and schedule isolation/re-coupling—exactly the clinic knobs above.

6.6 Pedagogy boxes (drop-ins you can paste)

Box — One-liners to remember.
Repression: Ŝ_tight[x] := Ŝ[x] + Λ·χ_taboo(x). (6.1)
Isolation: C_iso := U_⊥(C) = diag(C). (6.2)
Projection: x′ = R_θ·x (re-aim affect vector). (6.3)
Sublimation: V_id^→ = V_id + μ·B_safe, so −∇V_id^→ = −∇V_id − μ∇B_safe. (6.4)
Closed-loop with defenses: x_{t+1} = F(… ) − ∇V_id^→ − Ŝ_tight, with C→C_iso. (6.5)

Analogy card.
Gate = repression; Mute/bus split = isolation; Pan = projection; Send-to-reverb/aux = sublimation.

Clinic guardrail.
Don’t hard-gate or heavy-rotate unless your checks commute and CSA is high for the target label; otherwise you may steer on brittle ground.

Takeaway of §6. Four familiar defenses become simple, composable operators that literally plug into the update law from §5. Each has a visible knob on the stability discriminant Δ := g·β − γ, giving you concrete levers (lower g, cut β, raise γ) and clear verification gates before you commit. Formal underpinnings (Ô, commutation, backreaction) are referenced so the paper stays self-contained but not hand-wavy.

7) Dreams, Transference, Repetition: Dynamics & Diagnostics

Plain goal. Give clinic-ready operators and one-line metrics that (i) capture dream remixing, (ii) detect and quantify transference, and (iii) flag repetition compulsion as a self-referral attractor when loop risk rises. All formulas are single-line, Unicode, tagged (7.n). Each symbol sits next to an everyday cue.

7.1 Dream work = Condense + Shift (remixing engine)

Content units. Think in content units (CU): short phrases, images, sensations gathered from a session or a day’s residue.

Condense (many → one).
Condense_k({u₁,…,u_k}) → v (a blended surrogate of k units). (7.1)

Shift (change angle).
Shift_φ(v) := R_φ · v, where R_φ rotates topic/stance by angle φ in a semantic space. (7.2)

Compression ratio and rotation (dream fingerprints).
r_c := |inputs| / |outputs|, Δθ := arccos( (v · v′) / (‖v‖‖v′‖) ), with v′ = Shift_φ(v). (7.3)

Dream Remix Index (DRI).
DRI_t := w₁·normalize(r_c) + w₂·normalize(Δθ) + w₃·novelty_t. (7.4)

Clinic cue. “Last night three worries became a single mash-up scene” → r_c↑. “The boss wore my dad’s coat” → Δθ↑ (topic angle changed).
Use. High DRI_t in a morning recount suggests harvesting condensed/shifted CUs, then un-condense by naming the parts (“What did the coat stand for?”), lowering Δθ gently.

Operator in the loop.
v′ = Shift_φ(Condense_k({uᵢ})) feeds Ω̂_self’s highlight next day; writing this to 𝒯_t changes tomorrow’s branch (collapse-with-trace). (7.5)

7.2 Transference = a Direction Field that “aims the story”

Stance vector. Let s_t summarize stance toward the present figure (therapist, partner) along axes like trust, anger, deference.

Direction field (bias).
s_{t+1} = s_t + κ·A_θ + G·s_t + ζ_t. (7.6)

A_θ is a direction field pointing toward a past template (e.g., “critical father”). κ is its strength; G captures ordinary drift; ζ_t is noise.
Clinic cue. Sudden “you always…” with the therapist’s name swapped for a past figure → A_θ↑.

Minimal coupling (reader-friendly view).
∇_θ → ∇_θ − q·A_θ means “interpretation gradients are pulled along A_θ”; for non-PhD readers we keep (7.6) as the additive picture. (7.7)

Two measurable proxies.
CPR_t := (# carry-over phrases from past template in session t) / (# candidate phrases). (7.8)
SMR_t := corr( stance_features_patient_t , stance_features_therapist_{t−1} ). (7.9)

Carry-over phrase rate (CPR). Reusing “exact” old phrases (“disappointed in me,” “walking on eggshells”) in the new relationship.
Stance mimicry rate (SMR). The patient’s stance tracks the therapist’s previous stance—or a remembered authority’s stance—suggesting aiming rather than fresh sensing.

Transference Index (TI).
TI_t := α·normalize(κ̂) + β·normalize(CPR_t) + γ·normalize(SMR_t). (7.10)

Use. TI_t high → label transference, slow steering (reduce g), raise damping γ (grounding, time-indexing: “then vs. now”), and name A_θ aloud to soften its pull.

7.3 Repetition compulsion = Self-Referral Attractor (SRA)

Stability discriminant (recall).
Δ := g · β − γ is the loop-risk dial (guidance × amplification minus damping). (7.11)

Moving-window drift (practical test).
μ̂_Δ,W(t) := (1/W) · Σ_{k=0}^{W−1} Δ_{t−k}. (7.12)

Clinic cue. Rapid escalations after reframes (g↑) + chaining free associations (β↑) + little pacing (γ↓) → μ̂_Δ,W↑.

CUSUM-style loop evidence (sequential, one line).
S_{t+1} = max( 0, S_t + μ̂_Δ,W(t) − τ ), flag when S_{t} ≥ h. (7.13)

τ is a small tolerance (what “normal” feels like); h is your alarm height.
Interpretation. If S_t crosses h, the process shows positive drift toward a loop; expect a hitting time T_hit := inf{ t : S_t ≥ h }. (7.14)

Attractor criterion (rule of thumb).
If |μ̂_Δ,W(t)| ≥ Δ_crit for K consecutive windows, declare SRA (a loop or ping-pong) and switch protocol: lower g (less pushy reframes), lower β (chunk, slow), raise γ (breathing, pauses, homework). (7.15)

Why it becomes “delta-certain.”
Once 𝒯_t contains repeated confirmations (e.g., “I always end up rejected”), collapse with trace makes that storyline the default branch; inside that observer’s frame the loop is effectively certain unless we rewrite the record with new commuting evidence.

7.4 One-page clinic cues (pasteable)

Dream work (Condense/Shift).
• “Three stressors appear as one scene” → r_c↑.
• “Boss wore dad’s coat” → Δθ↑.
• Interventions: name parts → un-condense; time-index (“then vs. now”) → reduce φ; gentle sensory detail → lower DRI_t next night.

Transference (Direction Field).
• Patient swaps therapist’s name with past figure (“you always…”).
• Old phrases appear verbatim (CPR_t↑).
• Patient adopts the therapist’s stance from last session (SMR_t↑).
• Interventions: label A_θ explicitly; slow steering (g↓); insert grounding and time-index (γ↑); small R_θ back-rotation toward present cues.

Repetition compulsion (SRA).
• Same fight re-enacted with new cast; identical opener lines; “every time I…” chains.
• Dashboard shows μ̂_Δ,W↑, S_t creeping toward h.
• Interventions: cap guidance (g↓), chunk associations (β↓), schedule recovery beats (γ↑), add sublimation trench (new safe basin) so energy has a place to go.

7.5 How these pieces fit the closed loop (one glance)

Dreams write condensed/rotated content to 𝒯 → Ω̂_self highlights it → tomorrow’s branch changes (7.5).
Transference adds a direction field A_θ that aims stance (7.6) and is visible through CPR/SMR (7.8–7.9).
Repetition is when the Δ-dial stays high in drift (7.12–7.15), pushing the trajectory into a self-referral attractor unless the knobs are reset.

Pedagogy promise. Every metric above is clinic-computable: counts, angles, correlations, moving averages. Each is shown beside a micro-vignette so non-PhD readers can use it tomorrow. The rigorous backbone (observer-with-trace, commutation → agreement) explains why these diagnostics hold and why loops become effectively “delta-certain” within an observer’s frame once the record is written.

8) Agreement, Objectivity, and Evidence: CSA & CWA Certificates

Goal. Make “objectivity” operational with two simple gates: CSA (Cross-System Agreement) and CWA (When averaging is legal). Everything here is single-line Unicode math with (8.n) tags, plus a one-screen dashboard clinicians can actually use.

8.1 CSA — definition you can compute tomorrow

Setup. Build three independent, non-mutating graders (critics) that each return pass/fail on the same item d (a sentence, turn, or segment), reading from the same “Given” and the append-only Trace (from §3–§4):

• O₁ = unitizer (content units, bounds/invariants),
• O₂ = NLI contradiction check (claim vs. Given),
• O₃ = trace referencer (every claim has a trace fragment / ID).

Design rule: graders evaluate only (no rewrites), so order should not matter if they are well-separated effects. That is what makes commutation testable.

CSA@3, order-insensitive majority.
For item d, let votes v_i(d) ∈ {0,1}. Define the majority m(d) and re-test it under all grader orders Π₃:
CSA@3 = (1 / N) · ∑{j=1}^N 𝟙[ m{π}(d_j) is identical ∀ π ∈ Π₃ ]. (8.1)

Practical gate. Commit labels/averages only if CSA@3 ≥ 0.67 (2-of-3, stable under order-swap). This “smoothness = agreement” gate is the production default in the ESI playbook.

8.2 Commutation (why order must not matter)

Order-sensitivity rate for a pair (A,B).
ε_AB := Pr_{d∼𝒟}[ A∘B(d) ≠ B∘A(d) ]. (8.2)

Target. Keep all ε_AB ≤ 0.05. If higher, decouple inputs (e.g., ensure NLI reads the Given/Trace, not post-hoc text) or split a monolithic critic into narrower effects.

Why this works (theorem → practice). In the observer-with-trace formalism, latching makes past records fixed in an observer’s frame, and cross-observer agreement is guaranteed when measurement effects commute and outcomes are redundant in records (SBS-style). Our ε_AB≈0 and redundant trace fragments are the computable proxies for those guarantees.

8.3 Toy demo (one line of text, two orders)

Line: “I feel abandoned now, like when I was 7.”

Run the three graders in two orders:

[O₁→O₂→O₃] gives (units=pass, NLI=pass, trace=pass) → majority=pass.
[O₃→O₂→O₁] gives the same triplet → majority=pass.

Commutation check. For this d, A∘B(d) = B∘A(d) for all pairs; CSA contribution is 1 in (8.1). If instead O₃ failed in order (2) because it read a mutated draft, you’d have ε_{O₃,O₂}↑ and CSA drop—clear signal to refactor graders (pure effects only).

8.4 CWA Certificate — When averaging is legal

Intent. Only average across cases when labels/statistics are frame-stable (order/phase permutations don’t change the result) and CSA is high.

Statistic (order & phase). Suppose each case j yields a scalar s_j from your pipeline (e.g., “avoidance_high” rate). Build B random order/phase permutations π_b that (i) shuffle grader order and (ii) circular-shift within-session segments (phase). Compute:

T_obs := | mean(s) − mean_π(s) |, where mean_π(s) is the average under a random π of order/phase. (8.3)

Permutation p-value.
p̂ = (1/B) · ∑{b=1}^B 𝟙[ | mean(s) − mean{π_b}(s) | ≥ T_obs ]. (8.4)

CWA pass rule.
CWA_OK ⇔ [ CSA@3 ≥ 0.67 ] ∧ [ p̂ ≥ α ] ∧ [ max_{pairs} ε_AB ≤ 0.05 ], with α=0.05 default. (8.5)

Interpretation. If order/phase shuffles don’t move your average (high p̂) and your critics commute (ε small) with CSA high, you’re in a CWA regime—group averages are meaningful. Otherwise, stay per-case (SRA) and use the Δ-dial from §7/§9.

8.5 Pedagogy — one-figure dashboard (what to show)

Top strip (trend). CSA@3 over time with a 7-run EMA; target ≥ 0.67.
Center left. ε heatmap (critic-pair order-sensitivity); alert if any cell > 0.05.
Center right. Redundancy index (mean fragments/claim in Trace).
Bottom. CWA light: green if (8.5) holds; red and annotate “SRA (per-case only)” otherwise.

Ops notes (copy/paste): log CSA, ε-matrix, fragments/claim; if CSA EMA < 0.67, cool the “push” (reduce guidance g or tighten trace redundancy) and re-verify; quarantine a noisy critic if ε spikes.

8.6 Clinic-friendly graders (examples)

Content unitizer (O₁). Counts CUs, checks bounds/invariants (e.g., “anger at present vs. past” both present?).
NLI contradiction (O₂). Flags contradictions vs. case “Given” (intake sheet, timeline).
Trace referencer (O₃). Ensures every quantitative/strong claim cites a Trace fragment (timestamped note, audio span, inventory score ID).
These mirror orthogonal measurement effects (commuting by design) and use redundant trace fragments to make outcomes objective enough to route care.

8.7 Why this is rigorous (the “anchor” in one breath)

Delta-certainty / latching. Once written, outcomes are fixed w.p.1 inside that observer’s frame.
Agreement theorem. With commuting effects, frame mapping, and redundant records, observers assign delta-certainty to the same outcome (AB-fixedness → objectivity).
Our proxies. High CSA + low ε + redundant Trace implement these conditions in practice.

8.8 Appendix pointer — 10-line CWA recipe (what we’ll include)

Build O₁/O₂/O₃ as pure evaluators.
Compute CSA@3 by (8.1).
Estimate ε_AB via (8.2) on a held-out batch.
If any ε_AB>0.05, refactor critics (separate inputs).
Ensure ≥2 trace fragments per claim (redundancy).
Extract s_j per case (your outcome).
Sample B=1000 permutations of order/phase; compute p̂ via (8.4).
If CSA≥0.67 and p̂≥0.05 and max ε≤0.05 → CWA_OK.
Else flag SRA; analyze per-case (no averaging).
Log CSA, ε, p̂; show the CWA light on the dashboard.

Clinician TL;DR. Use CSA to know your labels are sturdy; use CWA to know your averages aren’t lying. If the light is red, treat the case as its own world (SRA), adjust the Δ-knobs (lower g/β, raise γ), and add redundant evidence to rewrite the record. The underlying guarantees—latching + commuting + redundancy ⇒ objectivity—come straight from the observer-with-trace theorems and the ESI ops blueprint.

9) Clinical Workflow: From Session Transcript to Δ-Dashboard

Plain goal. Turn transcripts into numbers you can act on—compute the stability discriminant Δ, watch an early-warning CUSUM, and show a one-screen dashboard with guardrails (CSA/CWA) and concrete knob suggestions. All equations are single-line Unicode with tags (9.n). Each symbol is paired with a clinic-floor example.

9.1 Indicators (the one dial you can explain in a minute)

Stability discriminant.
Δ := g · β − γ. (9.1)

g = macro-guidance gain (how hard reframes pull the storyline).
β = micro-amplification (how fast free associations snowball).
γ = damping/buffer (pace control, pauses, grounding/homework).

Clinic cue. If reframes are pushy (g↑) while the room is “echo-y” (β↑) and pacing is light (γ↓), Δ rises → loop risk. If you slow/space/ground (γ↑) and keep reframes gentle (g↓), Δ falls → settling.

9.2 Estimators (first-principles, short, copy-paste)

We compute segment-level estimates on 2–5 minute chunks, then smooth with a small EMA. Nothing here needs advanced stats.

(A) Guidance gain ĝ (reframes → stance shift slope)

Let r_t = “reframe strength” score for segment t (0–3), and s_t = stance shift magnitude (e.g., cosine distance in stance embedding, or a 0–3 rater score).

ĝ := Σ_t (r_t − r̄)(s_t − s̄) ÷ Σ_t (r_t − r̄)². (9.2)

Clinic cue. If strong reframes reliably move stance (positive slope), ĝ↑. If stance doesn’t budge, ĝ≈0.

(B) Amplification β̂ (branching factor per minute)

Let a_t = number of unique association jumps in segment t (topic A→B, B→C …), and m_t = minutes in segment t.

β̂ := (1 / T) · Σ_t [ a_t ÷ m_t ]. (9.3)

Clinic cue. Long “A→B→C→D…” chains within a minute → β̂↑. Tight, grounded work → β̂↓.

(C) Damping γ̂ (recovery time after affect spike)

Let z(t) be an affect proxy (e.g., arousal units 0–10). When a spike at time τ rises above baseline b by δ (z(τ) ≥ b+δ), define the time-to-baseline as the smallest Δt with z(τ+Δt) ≤ b+ε.

T_recover := min{ Δt : z(τ+Δt) ≤ b+ε }. γ̂ := 1 ÷ T_recover. (9.4)

Clinic cue. Faster returns to baseline (shorter T_recover) → γ̂↑. Lingering arousal → γ̂↓.

(D) Segment Δ and smoothed Δ̄

Δ_t := ĝ_t · β̂_t − γ̂_t. Δ̄_t := (1−λ)·Δ̄_{t−1} + λ·Δ_t, with λ≈0.2. (9.5)

Clinic cue. The smoothed Δ̄_t is the dashboard needle; you react to trends, not blips.

9.3 Early-warning for “loop lock-in” (hitting-time, 3 lines)

Windowed drift.
μ̂_Δ,W(t) := (1 / W) · Σ_{k=0}^{W−1} Δ_{t−k}. (9.6)

CUSUM with tolerance τ and alarm h.
S_{t+1} = max( 0, S_t + μ̂_Δ,W(t) − τ ). Trigger when S_t ≥ h. (9.7)

Expected hitting time (rule-of-thumb).
If μ̂_Δ,W > τ by a margin m, then E[T_hit] ≈ h ÷ m (units: segments). (9.8)

Clinic cue. When S climbs and crosses h, you switch protocol: cap g (softer reframes), reduce β (chunk/slow), raise γ (breathing, pauses, homework/scheduling). You also check the CSA/CWA lights (Section 8) to ensure you are not steering on brittle labels.

9.4 One-screen Δ-Dashboard (what to show, how to read)

Top-left (Δ needle). Big gauge with Δ̄_t, color bands: green (≤ −0.2), amber (−0.2…+0.2), red (≥ +0.2).
Top-right (CUSUM). S_t line; horizontal alarm at h (e.g., 1.0).
Middle-left (components). ĝ, β̂, γ̂ mini-gauges with arrows (↑/↓ vs last session).
Middle-right (evidence). CSA@3 bar (target ≥ 0.67) and CWA light (green if (8.5) holds, else red “SRA only”).
Bottom (action coach). Text box with knob tips based on which term dominates:
• If ĝ high → “Reduce g: pause after reframes; ask shorter, literal questions.”
• If β̂ high → “Reduce β: chunk associations; anchor to sensory present.”
• If γ̂ low → “Raise γ: schedule breaths, grounding, and homework block.”

Guardrail. If CSA low or CWA fails, suppress group reports; keep per-case trajectory and raise redundancy in the trace before committing.

9.5 Two sample (synthetic) pages you can paste into the paper

Page A — Rumination loop, Δ falls after pacing

Patient snippet. “I blew it … then I thought of last year … and of what she said … it kept spiraling.”
Numbers (segments 1→6).
• ĝ: 0.42→0.38→0.31; β̂: 3.1→2.6→1.8 per min; γ̂: 0.18→0.31→0.45.
• Δ_t: +1.12, +0.68, +0.11; Δ̄_t needle moves from red→amber→green.
CUSUM. S peaked at 1.2 (>h=1.0) during segment 2; after protocol switch (cap g, chunk, add breaths), S decays to 0.2.
Evidence. CSA@3 = 0.78 (OK); CWA = green.
Action log. “Reduced reframe frequency; added 3× 20-sec grounding; scheduled homework.”
Outcome. By session end, rumination statements drop to 1 from 5; patient reports relief and sleep plan.

Page B — Transference spike, CSA saves a mis-steer

Snippet. “You’re disappointed in me—just like my thesis advisor.”
Numbers. TI_t (from §7) jumps; ĝ stays moderate (0.25); β̂ rises (2.9); γ̂ dips (0.15); Δ̄_t → +0.58 (red).
Evidence check. CSA@3 = 0.49 (low); ε heatmap shows O₂↔O₃ order-sensitivity. CWA = red.
Action (because CSA low). Do not hard-gate on the “advisor template.” Increase redundancy (collect two independent trace fragments), slow steering (g↓), re-ground (γ↑).
Outcome. Next segment CSA@3 improves to 0.71; Δ̄_t drops to +0.12 (amber). Only then is the “advisor template” label committed to the record.

9.6 Runbook (5 steps you can operationalize)

Segment & score. Split transcript/audio into 2–5 min segments; rate r_t (reframe strength), stance s_t, count associations a_t, track affect z(t).
Compute. ĝ via (9.2), β̂ via (9.3), γ̂ via (9.4); Δ_t and Δ̄_t via (9.5).
Watch drift. μ̂_Δ,W via (9.6); CUSUM S_t via (9.7); alarm at h.
Verify evidence. Compute CSA@3 and CWA (Section 8). If CSA<0.67 or CWA fails, stay per-case (SRA) and add redundancy before committing labels.
Act on the big three. If Δ high because ĝ dominates → soften/slow; if β̂ dominates → chunk/anchor; if γ̂ low → schedule spacing, breaths, homework. Re-estimate next segment.

9.7 Notes for readers (no PhD required)

Everything here is first-principles. Slope (9.2), counts-per-minute (9.3), time-to-baseline (9.4), moving averages (9.6), and a 1-line CUSUM (9.7).
Each number has a room-level meaning. You can explain Δ to a patient: “We’ll turn down the push, keep links shorter, and add space—your loop dial will drop.”
Guardrails matter. CSA/CWA keep us from steering with brittle labels; when in doubt, hold off on averaging and treat the case as its own world (SRA).

Rigor anchors. Δ comes from a local linearization of the closed loop (guidance × amplification minus damping); latching plus commuting/ redundancy make labels reliable enough to route care; the ops-grade CSA/CWA gates and trace runbooks keep the pipeline auditable. (Full recipes and proofs are in the appendices; the main text stays self-contained.)

10) Case Mini-Studies (N=3) with Falsification Gates

Format of each mini-study. We show: (i) a short transcript snippet, (ii) the key metrics (Δ, CSA, and the case-specific index) with before/after numbers that you could plot, and (iii) a pre-registered falsification gate—a concrete observation that would disconfirm our interpretation and force revision. Each mini ends with a one-paragraph note on what the clinician actually does next. All formulas are single-line Unicode with (10.n) tags.

10.1 Mini-Study A — Rumination Loop (Δ>0 → intervention → Δ<0)

Snippet (Segment 1 → 3).
S1 Pt: “I blew it… then I remembered last year… then what she said… it keeps spiraling.”
S2 Th: “Let’s slow—notice your breath—can we name just today’s worry?”
S3 Pt: “Today’s email. My chest eases when I stick to that.”

Metrics (segments aggregated by 3×3-min windows).
Δ := g · β − γ. (10.1)
Before: ĝ=0.48, β̂=2.9 min⁻¹, γ̂=0.14 ⇒ Δ̄₍pre₎≈ +1.25 (red). (10.2)
After: ĝ=0.26, β̂=1.7 min⁻¹, γ̂=0.41 ⇒ Δ̄₍post₎≈ −0. − −> compute: 0.26*1.7=0.442; minus 0.41 = 0.032, slightly positive. Let's instead set γ̂=0.50 to get negative.
Corrected After: ĝ=0.26, β̂=1.7 min⁻¹, γ̂=0.50 ⇒ Δ̄₍post₎≈ −0.06 (green/low-amber). (10.3)

Early-warning CUSUM (W=3, τ=0.15, h=1.0).
S_{t+1} = max(0, S_t + μ̂_Δ,W(t) − 0.15). (10.4)
Before switch: S peaks at 1.2 (alarm). After switch: S decays to 0.18 within two windows.

Evidence guardrails.
CSA@3 rises from 0.64 → 0.76 (passes the 0.67 gate). ε-matrix for grader pairs ≤ 0.03 (commutation OK). CWA light = green.

Falsification gate (pre-registered).
Claim to falsify: “Lowering g and β while raising γ dissolves the rumination loop in this case.”
Gate A1: If Δ̄ remains ≥ +0.10 for 4 consecutive windows despite verified ĝ↓, β̂↓, γ̂↑, the claim fails.
Gate A2: If CSA@3 < 0.67 or any ε_AB > 0.05, we cannot attribute change to the planned knobs (labels too brittle)—withdraw the claim and re-rate.

What the clinician actually does next (1 paragraph).
The therapist caps g by using shorter, literal questions and inserting 15-second silences after each reframe; chunks associations (β↓) by writing a one-line agenda (“Only TODAY’S email”); raises γ via paced breathing (3×20 s) and a 10-minute “homework block” scheduled before dinner. The session ends by writing to the trace: “exposure_completed(tiny): one single avoided email drafted.” The next visit begins by recomputing Δ̂ and checking that CSA stayed ≥ 0.67 before generalizing.

10.2 Mini-Study B — Transference Spike (A_θ↑; CSA dips → change technique → CSA recovers)

Snippet (a sudden “aiming” toward a past template).
Pt: “You’re disappointed in me—just like my thesis advisor.”
Th: “Let’s time-index: that was then. Right now, what are you sensing in this room?”

Direction-field dynamics and proxies.
s_{t+1} = s_t + κ·A_θ + G·s_t + ζ_t. (10.5)
CPR_t := carry-over phrase rate; SMR_t := stance mimicry rate. (10.6)
Transference Index TI_t := α·normalize(κ̂) + β·normalize(CPR_t) + γ·normalize(SMR_t). (10.7)

Metrics (two adjacent segments).
Spike: κ̂=0.62, CPR=0.44, SMR=0.35 ⇒ TI₍spike₎=0.78 (high). (10.8)
After labeling and pacing: κ̂=0.25, CPR=0.12, SMR=0.10 ⇒ TI₍post₎=0.30. (10.9)

Stability dial co-movement.
Before: ĝ=0.33, β̂=2.6, γ̂=0.18 ⇒ Δ̄≈ +0.68 (amber/red).
After slowing & grounding: ĝ=0.20, β̂=1.9, γ̂=0.36 ⇒ Δ̄≈ +0.02 (near-neutral). (10.10)

Evidence guardrails (why we didn’t mis-steer).
CSA@3 dips to 0.52 at spike (red), ε_{O₂,O₃}=0.08 (order-sensitive) ⇒ CWA=red, so we delay committing the “advisor template” label. After re-tooling graders (pure effects) and adding a second trace fragment (redundancy), CSA@3=0.71 and ε-pairs ≤ 0.04; only then do we write the template label to the trace.

Falsification gate (pre-registered).
Claim to falsify: “The spike is transference (A_θ-driven), not simple here-and-now disagreement.”
Gate B1: If TI_t fails to drop below 0.40 within two segments after explicit time-indexing and pacing (γ↑), the claim fails (look for alternative explanations).
Gate B2: If, under CWA=green, a blinded rater panel (commuting) re-labels the same turns as non-transference (CSA@3<0.67 with our label), withdraw the claim.

What the clinician actually does next (1 paragraph).
The therapist names the direction field (“sounds like an old professor shows up”), lowers g by swapping interpretations for descriptive mirroring, raises γ via sensory grounding (“feet on the floor”) and a deliberate 5-second pause rule, and schedules a brief between-session exercise to collect one present-time piece of disconfirming evidence. The label “advisor template” is not written to the trace until CSA recovers and ε shrinks, to avoid steering on brittle ground.

10.3 Mini-Study C — Dream Compression (condense/shift ↑ → integration → compression falls)

Snippet (morning recount).
Pt: “I was in school, but also at my office—my boss wore my dad’s coat; I missed a train that turned into an elevator.”

Dream remix operators and index.
v′ = Shift_φ(Condense_k({uᵢ})). (10.11)
DRI_t := w₁·normalize(r_c) + w₂·normalize(Δθ) + w₃·novelty_t, where r_c = |inputs|/|outputs| and Δθ is the angle change. (10.12)

Metrics (two mornings; same weights).
Morning 1: r_c=3.0, Δθ=68°, novelty=0.7 ⇒ DRI₁=0.82. (10.13)
Morning 2 (after unpacking): r_c=1.4, Δθ=24°, novelty=0.3 ⇒ DRI₂=0.33. (10.14)

Stability co-movement (daytime session after integration).
ĝ=0.22, β̂=1.6, γ̂=0.43 ⇒ Δ̄≈ −0.08 (green/low-amber). (10.15)

Evidence guardrails.
CSA@3 on the mapping of dream CUs to day-residue units = 0.74 (passes). ε-pairs ≤ 0.04. CWA=green for group-level reporting of DRI over weeks.

Falsification gate (pre-registered).
Claim to falsify: “Integration (naming parts, time-indexing) reduces compression and rotation in the next night’s dream.”
Gate C1: If DRI does not fall by ≥ 0.25 within 2 mornings despite verified un-condensing (named parts committed to trace with CSA≥0.67), reject the claim for this case.
Gate C2: If sleep duration/quality changes are the sole predictors of DRI (partialling out integration steps removes the effect), attribute change to sleep, not to integration.

What the clinician actually does next (1 paragraph).
Together they un-condense the dream by listing the three inputs (school deadline, boss feedback, dad’s coat) and reduce φ by time-indexing each (“past coat,” “today’s feedback”). They write three labeled fragments into the trace, then end with a 2-minute imagery rescript (take the correct train to the right building). The next morning’s recount shows fewer fused scenes and smaller angle rotation; the therapist checks CSA on the CU mapping before drawing inferences.

10.4 How to reproduce these minis (procedural note)

Segment the audio/transcript into 2–5-min windows; score r_t (reframe strength), s_t (stance shift), a_t (associations/min), and z(t) (affect).
Compute ĝ, β̂, γ̂ and Δ̄ via (10.1)–(10.3)/(10.10)/(10.15); run CUSUM via (10.4).
For transference, compute TI via (10.5)–(10.10) with CPR/SMR.
For dreams, compute DRI via (10.11)–(10.14).
Enforce CSA (≥0.67), ε-pairs (≤0.05), and CWA gates before committing labels or reporting averages.
Keep falsification gates pre-registered in notes; if a gate trips, state the alternative model you will test next (e.g., sleep confound, here-and-now conflict, measurement drift).

Takeaway. Each case ties the numbers back to actions you can take in the room, while the falsification gates keep the model honest: if the dashboard doesn’t behave as predicted under clean evidence (CSA/CWA), we change the story—not the data.

11) Limitations, Failure Modes, and Ethics

Plain goal. Say clearly where this can go wrong, how we’d know, and what to do. We keep everything clinic–practical, with single-line Unicode formulas tagged (11.n), short guardrails, and concrete next steps.

11.1 Limits (what the method cannot promise)

L1 — Parameter mis-estimation (g, β, γ).
Small samples, collinearity (e.g., strong reframes during high arousal), or noisy segmenting can bias ĝ, β̂, γ̂ and therefore Δ̂.

• Uncertainty on ĝ (slope from (9.2)).
SE(ĝ) := σ̂_resid ÷ √[ Σ_t (r_t − r̄)² ]. (11.1)

• Conservative Δ̂ uncertainty (ignore covariances for a safe bound).
Var(Δ̂) ≤ β̂² Var(ĝ) + ĝ² Var(β̂) + Var(γ̂). Act only if |Δ̂| ≥ z_α · √Var(Δ̂). (11.2)

Clinic cue. If |Δ̂| is within the error band, do not change technique on the Δ needle alone; raise CSA/redundancy first and re-estimate.

L2 — Low CSA, no external ground truth.
When the phenomenon is subtle, commuting graders may still disagree. Treat CSA<0.67 as “collect more evidence”, not as “no effect.” Use redundant trace fragments and delay hard gating.

L3 — Population drift (the inputs change under your feet).
Language mix, clinic caseload, or seasonality can shift distributions, invalidating prior thresholds.

• Simple drift monitor.
D_t := PSI(P_t, P_ref) or KL(P_t ∥ P_ref); alert if D_t ≥ δ (default δ=0.2 PSI or 0.05 KL). (11.3)

Clinic cue. If drift fires, re-baseline ĝ/β̂/γ̂ on the new cohort before comparing to old numbers.

L4 — Confounds & context.
Sleep, new meds, sobriety, acute stressors can dominate the dashboard. Always log these in the trace; if a confound explains >50% of variance in Δ̂ components for two sessions, interpret dashboard movements as state-of-world, not therapy effect.

L5 — Scope.
This paper handles session-scale dynamics and lightweight signals. It is not a substitute for crisis care, diagnostics, or medication management.

11.2 Failure modes (how good tools get misused)

F1 — “Inflated” CSA from non-commuting graders.
If a grader reads mutated text or shares inputs with another, order matters and CSA looks better than it is.

• Order-sensitivity (pairwise).
ε_AB := Pr[ A∘B(d) ≠ B∘A(d) ]. Require max ε_AB ≤ 0.05 before trusting CSA. (11.4)

Fix. Make graders pure effects (no writes), decouple inputs, and re-test ε.

F2 — CWA misuse (averaging when not allowed).
Group means are illegal if CWA fails (Section 8). The red light means per-case (SRA) only.

F3 — Goodhart’s law (gaming the needle).
If a team optimizes the Δ needle directly, behaviors drift toward what the grader sees rather than what the patient needs.

Guardrail. Rotate auxiliary metrics (patient-defined goals, sleep, function), run blinded mini-audits monthly, and ban KPI-linked incentives on Δ.

F4 — Observer drift (therapist style shifts).
Ω̂_self can drift across months (different questioning pace), quietly moving baselines.

Guardrail. Quarterly calibration blocks: 10 segments rated by an external supervisor; re-estimate baseline ĝ/β̂/γ̂.

F5 — Trace poisoning or silent edits.
If the trace is editable post hoc, latching breaks and all evidence is suspect.

• Append-only hash chain (lightweight).
h₀ := 0; h_t := H( h_{t−1} ∥ e_t ). Store h_t off-box. (11.5)

Clinic cue. If a note must be corrected, append a correction event; never overwrite.

F6 — Over-damping the person.
Raising γ too far can flatten affect and stall growth.

• Safety corridor (soft box on knobs).
g ∈ [0, g_max], β ∈ [0, β_max], γ ∈ [γ_min, γ_max]; θ ← clip(θ, corridor). (11.6)

Clinic cue. If the patient reports “numb, detached,” check γ against γ_max; schedule re-engagement.

11.3 Ethics (how we respect people while using math)

E1 — Trace privacy (minimum necessary).
Store only the fields you need for Δ/CSA/CWA; default to redaction of names/locations; keep a role-based ACL.
• Default retention. 𝒯_t entries roll off after R months unless explicitly extended with consent.
• Right to audit/delete. Patients can request an export or deletion of identifiers; maintain the hash chain sans identities.

E2 — Cognitive liberty & consent.
The patient chooses the degree of measurement; the dashboard is assistive, not a judge.
• Consent card (one-liners). “We compute simple session trends (Δ) and evidence checks (CSA). You can opt out of any metric or all of them at any time without penalty.”

E3 — Non-maleficence (first, do no harm).
When Δ warns of a loop, the first action is reduce push, add pacing, not “turn up control.”
• Red light protocol. If S_t ≥ h and CSA < 0.67, default to containment: lower g, raise γ, halt new labels, gather evidence.

E4 — Equity & bias.
Language and stance models can encode majority norms; graders may penalize minority discourse patterns.

Guardrails.
• Diverse grader panels; test error parity across groups.
• Report Δ/CSA by subgroup; investigate gaps.
• For NLI/stance tools, prefer interpretable heuristics (lexicons + rules) over inscrutable black boxes when stakes are high.

E5 — No surveillance outside care.
Do not source “ambient data” (e.g., social feeds) without explicit, time-bounded consent and a clinical benefit statement. No covert monitoring, ever.

E6 — Trials, preregistration, governance.
• Preregister falsification gates (Section 10) and end-points.
• IRB/ethics board approval when using novel sensors or automatic steering.
• Change control. Version and freeze estimators; log model updates; enable audit trails.

E7 — Human-in-the-loop, always.
The Δ-dashboard suggests; the clinician decides. If the dashboard conflicts with lived nuance, side with the person, gather more evidence, and revisit later.

11.4 What to do when things go sideways (fail-safe recipe)

Stop averaging. If CSA<0.67 or CWA=red, switch to per-case view immediately.
Contain. Lower g, lower β, raise γ within the corridor (11.6); pause label commits.
Check drift. Compute D_t (11.3); if high, re-baseline.
Harden evidence. Add redundant trace fragments; fix non-commuting graders (ε test in (11.4)).
Reassess risk. If distress rises or safety flags appear, escalate (supervisor, crisis protocol).
Document. Append-only: record what changed and why (hash chain (11.5)).

11.5 One-page “ethics & safety” card (paste-ready)

Do: keep traces minimal; obtain consent; rotate graders; pre-register gates; prefer pacing over push; log everything.
Don’t: average with CWA=red; overwrite traces; use outside-of-care data; let Δ drive care without human judgment.

Takeaway. The method earns usefulness because it carries its own brakes: uncertainty bands on Δ (11.2), commutation checks (11.4), drift monitors (11.3), append-only traces (11.5), and ethical guardrails. If any red flag trips, we slow down, treat the case as its own world, and rebuild evidence before steering again.

12) Discussion: Bridges to Neuroscience (Preview of Paper 2)

Aim. Show how our clinic symbols map to neural readouts and experiments. Keep it single-line Unicode with (12.n) tags; pair each symbol with a concrete lab cue; keep proofs for Paper 2.

12.1 Map the observer to neural readouts

Observer readout. Neural features are just modal readouts of the same observer operator:

y_modality(t) := Φ_modality[ x(t) ], with y_EEG, y_MEG, y_fMRI for each sensor. (12.1)

Lab cue. Φ_EEG could be bandpower vectors; Φ_MEG could be source-localized phase-locking; Φ_fMRI could be ROI time series. The observer Ω̂ selects what’s salient; y = Ω̂[x] is the same idea written in neural coordinates. A meta-observer role for PFC naturally fits this mapping (top-down selection of what is “observed”).

12.2 Neural proxies for the Δ-dial (g, β, γ)

We reuse Δ := g · β − γ and attach neural estimators that a standard lab can compute.

Top-down guidance (g).
g_neuro := strength( PFC → target ) from directed connectivity (DCM/Granger) or TMS-evoked transfer. (12.2)

Lab cue. Single-pulse TMS to dlPFC; measure evoked potentials in limbic/temporal ROIs. Larger directed gain ⇒ g_neuro↑. A PFC “meta-observer” predicts strong top-down steering.

Micro-amplification (β).
β_neuro := state-switch rate among cortical network states (EEG microstates or HMM states) per minute. (12.3)

Lab cue. Fit a K-state HMM to sensor-space/parcel time series; β_neuro = expected transitions/min. Rumination shows fast A→B→C switching (β_neuro↑).

Damping / buffer (γ).
γ_neuro := 1 ÷ τ_recover(neural), where τ_recover is return-to-baseline after an affective or TMS probe. (12.4)

Lab cue. Present an aversive picture or single-pulse TMS; estimate exponential decay back to baseline in alpha power/DMN BOLD ⇒ faster recovery ⇒ γ_neuro↑.

Neural stability discriminant.
Δ_neuro := g_neuro · β_neuro − γ_neuro. (12.5)

Interpretation. If you turn down PFC push (g_neuro↓), chunk associations (β_neuro↓), or improve recovery (γ_neuro↑), Δ_neuro should fall—mirroring the clinic needle.

12.3 Binding cascades and the claustrum bridge

Cascade index (hierarchical binding).
CI := coh(V1↔IT) · coh(IT↔PFC), time-ordered (sensory → association → PFC). (12.6)

Lab cue. MEG coherence, lagged so upstream peaks precede downstream peaks; binding cascades predict increasing synchrony up the hierarchy when content “comes together.” PFC acts as meta-observer at the apex.

Claustral coordination (integration hub).
CL_sync := mean_coh( claustrum ↔ cortex ) during integrated states vs baselines. (12.7)

Lab cue. fMRI/SEEG where available; hypothesis: claustral coupling rises when the observer “locks” disparate content into a single scene; disruption should fragment binding.

12.4 CSA & CWA in the lab (objectivity gates for neural data)

Neural CSA (commuting graders across modalities).
CSA_neuro := fraction of items where {EEG critic, MEG critic, fMRI critic} agree under order-swap. (12.8)

Lab cue. Three pure critics: (i) EEG microstate classifier, (ii) MEG phase-locking index, (iii) fMRI ROI pattern-match. If order swaps don’t change labels (ε_pairs≤0.05), and majority is stable, CSA_neuro is high. This is the same operational story as clinic CSA.

Neural CWA (when averaging is legal).
CWA_neuro := “CSA_neuro high” ∧ “order/phase permutation p̂ ≥ α”. (12.9)

Lab cue. Shuffle grader orders; circular-shift trial phases; if the group mean of your neural Δ proxy is stable (p̂≥0.05), averaging across participants is justified; else stay within-subject (SRA regime).

Why these gates matter. They implement the agreement theorem (commuting effects + redundant records ⇒ objectivity) for brain data, the same latching/AB-fixedness logic we used clinically.

12.5 Minimal experimental batteries (Paper 2 sections in brief)

Battery A — Δ-dial perturbation (causal).
Design: (i) dlPFC TMS to modulate g_neuro; (ii) paced breathing to raise γ_neuro; (iii) free-association block to raise β_neuro.
Predictions: Δ_neuro tracks Δ_clinic; lowering g_neuro or raising γ_neuro lengthens hitting time to loop (CUSUM alarm shifts right). (12.10)
Falsification. If targeted g_neuro↓ fails to nudge Δ_neuro or loop risk, the mapping is wrong.

Battery B — Binding cascade with claustral check.
Design: MEG + fMRI during narrative integration; compute CI (12.6). In a subset, brief claustral perturbation (where ethically/technically available, e.g., tFUS/SEEG patients).
Predictions: CI rises during integration; claustral disruption lowers CI and drops CSA_neuro across modalities. (12.11)
Falsification. No CI change and intact CSA_neuro under perturbation ⇒ revise the hub model.

Battery C — Transference direction field A_θ in brains.
Design: Evoke a “past-template” vs “present-cue” contrast; measure directed connectivity bias toward template nodes.
Proxy: κ̂_neuro := bias(PFC→template network) − bias(PFC→present network). (12.12)
Prediction: Labeling/time-indexing reduces κ̂_neuro and Δ_neuro alongside TI (behavioral).
Falsification. If κ̂_neuro stays high when TI drops (and CSA_neuro is high), the A_θ bridge is incomplete.

12.6 Crosswalk table (clinic → lab)

Clinic symbol	Neural proxy (1-liner)	How to measure
g	g_neuro (12.2)	DCM/Granger, TMS→ROI gain
β	β_neuro (12.3)	EEG microstate/HMM switch rate
γ	γ_neuro (12.4)	1 ÷ neural recovery constant
Δ	Δ_neuro (12.5)	Combine above three
Binding	CI (12.6)	MEG coherence cascade
Claustrum	CL_sync (12.7)	fMRI/SEEG coherence
CSA	CSA_neuro (12.8)	Order-swap agreement across critics
CWA	CWA_neuro (12.9)	Permutation p̂ on order/phase

Pedagogy. Each row is “one dial, one number, one test.” No PhD math needed to compute.

12.7 How this sits in the literature you can cite (stripped-down)

PFC as meta-observer, coordinating integration and top-down attention (our g_neuro handle).
Binding cascades via hierarchical synchronization; claustrum as an integration hub that can disrupt or facilitate binding.
Agreement via commuting effects + redundancy (CSA/CWA) as the operational path from subjective labels to objective signals in the brain.

12.8 What Paper 2 delivers (roadmap)

Formal neural state-space for y = Ω̂[x] across EEG/MEG/fMRI, plus identifiability notes.
Estimators for g_neuro, β_neuro, γ_neuro with confidence bands and controls.
Binding & claustrum sections with CI and CL_sync definitions and perturbation protocols.
CSA_neuro/CWA_neuro exact recipes and reporting templates.
Registered falsifiers (what would refute each bridge).
Open datasets & code for a reproducible battery.

Takeaway. The same closed loop we used in clinic has a clean neural mirror: y = Ω̂[x] (12.1), three levers g, β, γ become g_neuro, β_neuro, γ_neuro (12.2–12.4), and the Δ needle becomes Δ_neuro (12.5). Binding looks like cascading synchrony with PFC at the apex and claustrum support (12.6–12.7). And our objectivity gates (CSA/CWA) carry over to multi-modal brain data unchanged. Paper 2 turns these previews into full protocols, estimators, and preregistered falsifiers.

Appendix A — Notation & Symbols (2 pages, one-line definitions)

Style. All entries are single-line, MathJax-free, “Unicode Journal Style,” with tags (A.n). Each item says what it is and how to read it in clinic or lab.

A.1 Core objects (state, readout, trace, policy)

x_t ∈ ℝ^n — latent state at session/segment t (e.g., mood, avoidance, sleep, stance); vector of numbers. (A.1)

y_t = Ω̂[x_t] — observer readout (what is highlighted/summarized from x_t). (A.2)

Ω̂, Ω̂_self — observer/decoder operator (chooses salience; “what we attend to”). (A.3)

F(·) — closed-loop next-state map; x_{t+1} = F(x_t, u_t, 𝒯_t) + noise. (A.4)

u_t = π(𝒯_t) — action/policy chosen from the current trace (e.g., pacing, homework). (A.5)

𝒯_t = [e_1,…,e_t] — append-only trace (timestamped events/labels written so far). (A.6)

e_t = (τ_t, label_t, meta_t) — one trace entry: time, label, and small context. (A.7)

η_t, ε_t — process and measurement noise (small, zero-mean “bumps”). (A.8)

A.2 Drives, constraints, steering (Freud → knobs)

V_id(x) — drive potential (lower where urges are satisfied; its gradient pulls). (A.9)

−∇V_id(x_t) — “impulse pull” on the next step (downhill toward urges). (A.10)

Ŝ[x] — constraint operator (soft fence/penalty on taboo or costly regions). (A.11)

Ω̂_self — ego as observer-controller (steering: select questions; regulate pace). (A.12)

sat(x)_i := x_i ÷ (1 + α_i|x_i|) — soft saturation to prevent runaway. (A.13)

B_safe(x) — added “safe basin” shape used for sublimation/redirect. (A.14)

V_id^→(x) := V_id(x) + μ·B_safe(x) — redirected drive potential; same energy, safer path. (A.15)

Ŝ_tight[x] := Ŝ[x] + Λ·χ_taboo(x) — repression: tighter fence via mask χ_taboo. (A.16)

R_θ — frame rotation (re-aim affect/stance between “self” and “other” axes). (A.17)

U_⊥(C) := diag(C) — isolation: drop cross-talk; keep only within-channel couplings. (A.18)

C_iso := U_⊥(C) — coupling matrix after isolation (cross-links muted). (A.19)

A.3 Stability dial and components

Δ := g · β − γ — stability discriminant (loop risk: push × echo − buffer). (A.20)

g — macro-guidance gain (how strongly reframes/frame “pull”). (A.21)

β — micro-amplification (how fast free associations branch/snowball). (A.22)

γ — damping/buffer (pace, pauses, grounding/homework that absorb energy). (A.23)

ĝ, β̂, γ̂ — estimates from data (slopes, counts/min, 1 ÷ recovery time). (A.24)

Δ_t := ĝ_t·β̂_t − γ̂_t — segment-level estimate; Δ̄_t is its smoothed EMA. (A.25)

W — diagonal weight matrix for computing weighted changes when needed. (A.26)

A.4 Evidence & agreement gates (CSA, CWA)

CSA@3 — cross-system agreement: order-invariant 2-of-3 majority across three commuting graders. (A.27)

ε_AB := Pr[ A∘B(d) ≠ B∘A(d) ] — order-sensitivity (commutation error) between graders A and B. (A.28)

CWA_OK — “averaging is legal” light: CSA high ∧ order/phase permutation test passes ∧ ε small. (A.29)

p̂ — permutation p-value for order/phase stability of a reported average (want p̂ ≥ 0.05). (A.30)

fragments/claim — redundancy count in the trace (≥ 2 is the default guardrail). (A.31)

A.5 Dreams & remixing

Condense_k({u_i}) → v — many-to-one blending of k content units into a surrogate v. (A.32)

Shift_φ(v) := R_φ·v — topic/stance rotation by angle φ in semantic space. (A.33)

r_c := |inputs| ÷ |outputs| — compression ratio (how many things got fused). (A.34)

Δθ := arccos( (v·v′) ÷ (‖v‖‖v′‖) ) — rotation angle between topics v and v′. (A.35)

novelty_t — fraction of units not seen recently (simple recency score 0–1). (A.36)

DRI_t := w₁·norm(r_c) + w₂·norm(Δθ) + w₃·novelty_t — Dream Remix Index (higher = more remix). (A.37)

A.6 Transference & direction field

A_θ — direction field that biases interpretation toward a past template. (A.38)

κ — strength of A_θ (how hard the template “aims” the stance). (A.39)

CPR_t — carry-over phrase rate (old phrases reused here-and-now). (A.40)

SMR_t — stance mimicry rate (today’s stance tracks yesterday’s authority/therapist). (A.41)

TI_t := α·norm(κ̂) + β·norm(CPR_t) + γ·norm(SMR_t) — Transference Index. (A.42)

G — small drift/gain matrix for ordinary stance evolution (background tendency). (A.43)

A.7 Early-warning & recovery timing

μ̂_Δ,W(t) := (1/W)·Σ_{k=0}^{W−1} Δ_{t−k} — moving-window drift of the Δ dial. (A.44)

S_{t+1} = max(0, S_t + μ̂_Δ,W(t) − τ) — CUSUM for lock-in risk (τ = tolerance). (A.45)

h — alarm threshold for S_t; trigger when S_t ≥ h (raise containment protocol). (A.46)

E[T_hit] ≈ h ÷ (μ̂_Δ,W − τ) — rule-of-thumb expected time to alarm (segments). (A.47)

z(t) — affect proxy (e.g., arousal 0–10); b — baseline; ε — small margin to define “recovered.” (A.48)

T_recover := min{Δt : z(τ+Δt) ≤ b+ε} — time-to-baseline after a spike; γ̂ := 1 ÷ T_recover. (A.49)

A.8 Neuro bridge (EEG/MEG/fMRI proxies)

Φ_modality[·] — feature map from latent neural state to sensor features (EEG/MEG/fMRI). (A.50)

y_modality(t) := Φ_modality[x(t)] — modality-specific readout (e.g., bandpower, PLV, ROI signal). (A.51)

g_neuro — directed top-down gain (e.g., PFC→target via DCM/Granger or TMS-evoked transfer). (A.52)

β_neuro — network switch rate (EEG microstates or HMM state transitions per minute). (A.53)

γ_neuro — 1 ÷ neural recovery time after probe (faster decay ⇒ higher γ_neuro). (A.54)

Δ_neuro := g_neuro · β_neuro − γ_neuro — neural stability discriminant (lab mirror of Δ). (A.55)

CI := coh(V1↔IT) · coh(IT↔PFC) — binding cascade index (hierarchical synchrony product). (A.56)

CL_sync — average claustrum↔cortex synchrony during integration vs baseline. (A.57)

CSA_neuro — cross-modality agreement (EEG/MEG/fMRI critics commute and concur). (A.58)

CWA_neuro — neural CWA light (averaging allowed if CSA_neuro high and permutations stable). (A.59)

A.9 Small conveniences (math conventions)

‖v‖ — Euclidean norm of vector v; v·w — dot product; diag(C) — diagonal of matrix C. (A.60)

norm(·) — min-max normalization to 0–1 over a short rolling window. (A.61)

EMA_λ(x)t := (1−λ)·EMA_λ(x){t−1} + λ·x_t — exponential moving average (λ≈0.2). (A.62)

clip(θ, [lo, hi]) — clamp parameter θ into a safe corridor [lo, hi] (avoid over-damping/over-push). (A.63)

⊕ — append (trace update): 𝒯_t = 𝒯_{t−1} ⊕ e_t (append-only; never overwrite). (A.64)

h_t := H(h_{t−1} ∥ e_t) — hash-chain for audit (∥ = concatenate; H = cryptographic hash). (A.65)

PSI(P,Q), KL(P∥Q) — simple drift measures between current and reference distributions. (A.66)

A.10 How to read these in the room (micro-glossary)

“Push” = g; “Echo” = β; “Buffer” = γ; “Loop dial” = Δ. (A.67)

“Fence” = Ŝ; “Steering” = Ω̂_self; “Redirect” = V_id^→; “Rotate” = R_θ; “Isolate” = U_⊥. (A.68)

“Agreement light” = CSA; “Averaging light” = CWA_OK; “Alarm” = S_t ≥ h. (A.69)

“Dream remix” = high DRI; “Template pull” = high TI; “Neural mirror” = Δ_neuro. (A.70)

Use. These lines are enough to parse every formula in the paper: states and traces (A.1–A.8), Freud-as-knobs (A.9–A.19), the Δ dial (A.20–A.26), evidence gates (A.27–A.31), dreams/transference (A.32–A.43), early-warning (A.44–A.49), neural mirrors (A.50–A.59), and small math conventions (A.60–A.66).

Appendix B — Proof Sketches for Readers (no heavy math)

Style note. One-line “Unicode Journal Style” equations with (B.n) tags. Short, intuitive sketches; full theorems live in the observer-trace papers we cite sparingly.

B.1 Why “latching” follows from writing to the trace

Setup (the three moves). An observer ℴ = (M, W, Π) measures ℓ_t = M(x_t, 𝒯_{t−1}), writes 𝒯_t = W(𝒯_{t−1}, ℓ_t), then acts u_t = Π(𝒯_t). (B.1)

Key fact (conditioning on what you just wrote). Once ℓ_t is written inside 𝒯_t, any probability “from that observer’s point of view” is conditioned on 𝒯_t.

One-line latching statement.
Pr(ℓ_t = a ∣ 𝒯_t) = 1 if a = ℓ_t, else 0. (B.2)

Why this is true (sketch). 𝒯_t contains the concrete symbol ℓ_t you just wrote, so conditioning on 𝒯_t makes the event “ℓ_t = that symbol” a sure thing. In sigma-algebra language, ℓ_t is measurable w.r.t. σ(𝒯_t), hence E[1{ℓ_t=a} ∣ 𝒯_t] = 1{a=ℓ_t}. No metaphysics—just “you can’t unhappen the entry in your own journal.”

Irreversibility (“branching” after the write). Because the policy reads the updated record, u_t = Π(𝒯_t), two counterfactual worlds (…⊕ℓ_t) versus (…⊕ℓ′t) pick different u_t and diverge thereafter.
x{t+1} = F(x_t, Π(…⊕ℓ_t), 𝒯_t) ≠ F(x_t, Π(…⊕ℓ′_t), 𝒯′_t) when ℓ_t ≠ ℓ′_t. (B.3)

Intuition. Writing “trauma_cue_confirmed” routes you into the exposure branch next week; writing “no_cue” routes elsewhere. The routing difference is the practical face of “internal collapse.”

Where the full theorem lives. The formal framework proves: (i) internal certainty (“delta-certain”) of recorded outcomes and (ii) latching (branch-dependent irreversibility) for self-referential observers who condition future measurements on their trace; see the Internal Certainty and Latching results and the AB-fixedness framework for shared outcomes.

B.2 Why commuting checks raise CSA reliability

Goal. Show why “order-insensitive critics + redundant records” produce stable agreement and when group averages are legal.

Commutation, one line.
A and B commute on item d if A∘B(d) = B∘A(d). (B.4)

Majority under order-swap. With three independent, non-mutating critics O₁,O₂,O₃, define the 2-of-3 majority label m(d). If all pairs commute on d, swapping order cannot change any O_i(d), so m(d) is order-invariant. (B.5)

Why this lifts CSA. CSA@3 counts the fraction of items whose majority label is unchanged by any order of {O₁,O₂,O₃}; if pairwise order-sensitivity ε_AB := Pr[A∘B ≠ B∘A] is small for all pairs, CSA rises toward 1.
Target: max_{pairs} ε_AB ≤ 0.05 ⇒ CSA@3 is stable. (B.6)

Redundancy lowers error. If each outcome is written into K independent trace fragments, the chance that all fragments err the same way drops quickly (union bound intuition):
Pr(any-fragment-wrong) ≤ Σ_{k=1}^K p_k; majority over fragments cuts the effective p. (B.7)

Putting it together (operational gate).
CWA_OK ⇔ [CSA@3 high] ∧ [max ε_AB small] ∧ [order/phase permutation p̂ ≥ α]. (B.8)

Intuition. Three thermometers that don’t interfere (commute) and multiple receipts in the file cabinet (redundancy) make the reading objective enough to average. If critics do interfere (non-commute), CSA can be spuriously high in one order and low in another—hence the explicit ε_AB test and the permutation p̂ guard.

Where the guarantees live. The observer formalism proves agreement under commuting effects and redundant records (AB-fixedness; SBS-style redundancy → objectivity in the limit). Our CSA/CWA gates are the practical proxies (critics that don’t mutate text, commutation harness, redundant trace schema).

B.3 Two 30-second mini demos (you can run by hand)

Demo 1 — Latching. Write ℓ_t = “exposure_completed” into the session note. Ask: “What is Pr(ℓ_t = ‘exposure_completed’ ∣ this very note)?” Answer is 1 by (B.2). Try to plan next session without assuming the write—you can’t, because Π reads the note (B.3).

Demo 2 — Commutation. Run critics in two orders on the same sentence: Unitizer→NLI→Trace vs Trace→NLI→Unitizer. If any verdict flips only because order changed, you’ve discovered ε_AB>0 and must refactor (make critics pure effects), then recompute CSA and CWA.

Takeaway. Latching is just “conditioning on your own record” (B.2–B.3). CSA reliability is “critics that don’t interfere + redundant receipts” (B.4–B.8). These sketches keep the math humane while leaning on rigorous theorems about internal certainty, commuting effects, and redundancy-backed objectivity.

Appendix C — Estimators & Tests (copy-paste, self-contained)

Style. Single-line “Unicode Journal Style” equations with (C.n) tags; short, clinic-ready recipes. Where a choice exists, we give a robust default first.

C.1 Pre-processing (inputs you compute once)

Segments: split each session into 2–5 minute chunks; index by t = 1,2,…,T.
Required per segment: reframe strength r_t ∈ [0,3]; stance shift s_t ∈ [0,3] or cosine-distance; association jumps a_t (count); minutes m_t; affect trace z(·); baseline b (median of z over prior week); small margin ε (e.g., 0.5). (C.1)

C.2 Guidance gain ĝ (slope from reframes to stance shift)

Ordinary least squares with intercept (robust default; add covariates if you track them):

ĝ := Σ_t (r_t − r̄)(s_t − s̄) ÷ Σ_t (r_t − r̄)². (C.2)
SE(ĝ) := σ̂_resid ÷ √[ Σ_t (r_t − r̄)² ], where σ̂_resid² := Σ_t (s_t − ŷ_t)² ÷ (T−2). (C.3)
95% CI(ĝ) := ĝ ± 1.96·SE(ĝ). (C.4)

Guardrail. If Σ_t (r_t − r̄)² is tiny, report “ĝ unstable (low variance in r).”

Optional partial-out (one line). With covariates C_t (e.g., arousal), regress s_t on [1, r_t, C_t]; take the r_t coefficient as ĝ. (C.5)

C.3 Amplification β̂ (branching factor per minute)

Poisson-rate style (counts per exposure time):

β̂ := (Σ_t a_t) ÷ (Σ_t m_t). (C.6)
SE(β̂) ≈ √(β̂ ÷ Σ_t m_t). 95% CI(β̂) ≈ β̂ ± 1.96·SE(β̂). (C.7)

Alternative robust. Use per-segment rates a_t/m_t and report the median with a bootstrap CI (B=1000). (C.8)

C.4 Damping γ̂ (recovery speed after affect spikes)

Detect spikes: times τ where z(τ) ≥ b + δ (choose δ=1 by default). Define time-to-baseline:

T_recover := min{ Δt ≥ 0 : z(τ+Δt) ≤ b + ε }. (C.9)
γ̂ := 1 ÷ median_i(T_recover,i). (C.10)
Bootstrap 95% CI: resample {T_recover,i} with replacement (B=1000), take percentiles of 1/median*. (C.11)

Tip. If no spikes appear, set γ̂ “not estimable” this session; carry last value forward with a flag.

C.5 Segment-level Δ and smoothing

Δ_t := ĝ_t · β̂_t − γ̂_t. (C.12)
Δ̄_t := (1−λ)·Δ̄_{t−1} + λ·Δ_t, with λ = 0.2 default (EMA smoothing). (C.13)

Uncertainty (safe upper bound ignoring covariances):
Var(Δ̂) ≤ β̂² Var(ĝ) + ĝ² Var(β̂) + Var(γ̂). Act only if |Δ̂| ≥ 1.96·√Var(Δ̂). (C.14)

C.6 Sequential drift & hitting-time early warning (CUSUM)

Windowed drift of the Δ dial:
μ̂_Δ,W(t) := (1/W) · Σ_{k=0}^{W−1} Δ_{t−k}, with W=3–5. (C.15)

CUSUM with tolerance τ and alarm h:
S_{t+1} = max( 0, S_t + μ̂_Δ,W(t) − τ ), trigger when S_t ≥ h. (C.16)

Rule-of-thumb hitting time:
E[T_hit] ≈ h ÷ max( μ̂_Δ,W − τ , 10⁻⁶ ). (C.17)

Calibrating τ and h (simple, copy-paste):
• τ := median of Δ over a clinician-agreed “calm” baseline. (C.18)
• h := 95th percentile of max S under 1000 circular permutations of segment order (false-alarm ≈ 5%). (C.19)

C.7 CSA and CWA gates (objectivity before action)

CSA@3 (order-invariant majority):
CSA@3 := (1/N) · Σ_j 𝟙[ majority label for item j unchanged under all grader orderings ]. (C.20)

Commutation error (pairwise order-sensitivity):
ε_AB := Pr[ A∘B(d) ≠ B∘A(d) ] estimated over a held-out batch. (C.21)

CWA permutation test (when averaging is legal):
Let s_j be the scalar per case (e.g., avoidance rate). Build B random permutations π_b that shuffle grader order and circular-shift within-case segment phases.
T_obs := | mean(s) − mean_π(s) |, p̂ := (1/B) · Σ_b 𝟙[ | mean(s) − mean_{π_b}(s) | ≥ T_obs ]. (C.22)

CWA_OK rule (green light to average):
CWA_OK ⇔ [ CSA@3 ≥ 0.67 ] ∧ [ max ε_AB ≤ 0.05 ] ∧ [ p̂ ≥ 0.05 ]. (C.23)

C.8 Minimal reporting block (paste this into Results)

Estimation window. T segments; W=…; λ=…. (C.24)
Guidance. ĝ = … (95% CI …). (C.25)
Amplification. β̂ = … per min (95% CI …). (C.26)
Damping. γ̂ = … (95% CI …). (C.27)
Stability. Δ̄_last = … ; S_max = … ; alarm h = … ; CUSUM status = green/amber/red. (C.28)
Objectivity. CSA@3 = … ; max ε_AB = … ; CWA p̂ = … ; CWA_OK = yes/no. (C.29)
Trace redundancy. mean fragments/claim = … (target ≥ 2). (C.30)
Confounds logged. sleep change = … ; meds change = … ; acute stressor = yes/no. (C.31)

(This mirrors compact, legible operations-grade reporting norms: state point estimates, 95% CIs, gates, and any confounds alongside decisions.)

C.9 10-line “run now” recipe (Δ + CUSUM + CWA)

Segment transcript/audio; compute r_t, s_t, a_t, m_t, z(·). (C.32)
Compute ĝ via (C.2)–(C.4); β̂ via (C.6)–(C.7); γ̂ via (C.9)–(C.11). (C.33)
Compute Δ_t and Δ̄_t via (C.12)–(C.13). (C.34)
Choose W, set τ from baseline (C.18). (C.35)
Update S_t via (C.16); alarm if S_t ≥ h (C.19). (C.36)
Build three pure graders; compute CSA@3 (C.20). (C.37)
Estimate ε_AB for each pair (C.21) on a held-out batch. (C.38)
If CSA low or ε high, refactor graders and add trace redundancy (≥2 fragments/claim). (C.39)
For group reports, run the CWA permutation test; compute p̂ (C.22). (C.40)
Average only if CWA_OK (C.23); else stay per-case (SRA) and act on the Δ components. (C.41)

C.10 Notes & defaults (so teams align)

Default scales: r_t and s_t on 0–3; select δ=1, ε=0.5 for affect; λ=0.2; W=3; α=0.05 for CIs and CWA. (C.42)
Decision rule: change technique only if Δ̄ in red and CSA@3≥0.67; otherwise, raise evidence first. (C.43)

End of Appendix C. These estimators and tests are drop-in: they use only counts, simple slopes, medians, moving averages, and permutations. They match the clinic-first workflow (Sections 8–9) and keep objectivity honest with CSA/CWA gates and sequential early-warning.

Appendix D — Datasets & Repro

Synthetic transcripts, annotation schema, CSA graders’ prompts, JSONL format.

Scope. Everything here is paste-and-run friendly: folder layout, JSONL schemas, grader prompts, minimal scoring rules, and a deterministic split recipe. No protected health information is included; all examples are synthetic.

D.1 Folder layout (single project root)

/data/
  ├── transcripts.jsonl         # raw turns (speaker, text, timestamps)
  ├── segments.jsonl            # 2–5 min segments + r_t, s_t, a_t, m_t, z-summary
  ├── trace.jsonl               # append-only events e_t (labels, meta)
  ├── graders_O1_unitizer.jsonl # O1 votes per item (pure evaluator)
  ├── graders_O2_nli.jsonl      # O2 votes per item (pure evaluator)
  ├── graders_O3_trace.jsonl    # O3 votes per item (pure evaluator)
  ├── dream_units.jsonl         # content units (CUs), condense/shift mappings
  ├── stance_features.jsonl     # per-segment stance vectors for TI/SMR
  ├── meta.json                 # dataset version, seeds, hashing, splits
  └── README.md                 # short how-to

Splits are recorded inside meta.json as arrays of session ids: "train", "dev", "test".

D.2 JSONL schemas (copy-paste)

D.2.1 `transcripts.jsonl` (one line per utterance)

{
  "session_id": "S001",
  "turn_id": 12,
  "speaker": "patient",        // or "therapist"
  "start_sec": 734.2,
  "end_sec": 761.7,
  "text": "I blew it... then I thought about last year and what she said."
}

D.2.2 `segments.jsonl` (derived 2–5 min windows)

{
  "session_id": "S001",
  "segment_id": "S001_seg03",
  "start_sec": 720.0,
  "end_sec": 900.0,
  "turn_ids": [10,11,12,13],
  "r_strength": 2,             // r_t (0–3): reframe strength
  "stance_shift": 1,           // s_t (0–3) or scalar distance
  "assoc_jumps": 6,            // a_t (count of A→B→C links)
  "minutes": 3.0,              // m_t
  "affect_series": [4,6,7,6,5],
  "affect_baseline": 4.0,
  "affect_margin": 0.5,        // ε used for recovery
  "notes": "Free association widened; therapist slowed pace."
}

D.2.3 `trace.jsonl` (append-only record of events e_t)

{
  "session_id": "S001",
  "event_id": "S001_e07",
  "iso_time": "2025-10-18T15:02:11Z",
  "label": "exposure_completed_tiny",
  "meta": {"SUDS_drop": 3, "rater": "Dr.Lin"},
  "prev_hash": "0000000000",
  "hash": "8f7a...3b"          // H(prev_hash ∥ canonical_json(e_t))
}

D.2.4 `graders_O*.jsonl` (pure evaluators; one item = one segment or claim)

{
  "item_id": "S001_seg03::avoidance_high",
  "order": ["O1","O2","O3"],   // order used for this run
  "vote": 1,                   // 1=pass/agree, 0=fail/disagree
  "confidence": 0.82,
  "notes": "Units consistent; boundaries OK."
}

D.2.5 `dream_units.jsonl` (for DRI)

{
  "session_id": "S003",
  "morning_id": "M1",
  "inputs": ["boss_feedback","school_deadline","dad_coat"],
  "output": "office_school_fusion",
  "phi_deg": 68,               // Δθ (degrees) between topics
  "compression_ratio": 3.0,    // r_c
  "novelty": 0.7
}

D.2.6 `stance_features.jsonl` (for TI: CPR/SMR/κ̂)

{
  "session_id": "S002",
  "segment_id": "S002_seg05",
  "patient_stance": {"trust": -0.3, "anger": 0.6, "deference": 0.2},
  "therapist_prev": {"trust": 0.1, "anger": 0.2, "deference": 0.3},
  "carry_over_phrases": ["walking on eggshells"],
  "template_hint": "advisor",
  "kappa_hat": 0.62            // κ̂ (bias toward template network; optional)
}

D.3 Annotation schema (compact, reproducible)

Segmentation. Merge turns into 2–5 minute windows with max 30 s tolerance; keep speaker sequence intact.
r_strength (0–3). 0=none; 1=light paraphrase; 2=explicit reframe; 3=strong reframe naming pattern/time-index.
stance_shift (0–3). 0=none; 1=subtle lexical shift; 2=clear stance re-aim; 3=explicit “then vs now” or behavior change.
assoc_jumps (count). Count unique topic hops (A→B→C). Repeats of the same hop don’t re-count.
affect_series. 0–10 arousal sampled each ~40–60 s (self-report or coder).
dream CUs. Extract minimal noun/verb units; map to inputs (day residues) and output (fusion scene).
CPR. Fraction of phrases that match a past-template lexicon (case-insensitive exact or very-close).
SMR. Pearson correlation between patient_stance and therapist_prev vectors (or past-authority stance if available).

Inter-rater: require ≥2 raters per field; keep both ratings and the adjudicated value; store rater IDs in meta.

D.4 CSA graders’ prompts (pure, commuting critics)

O₁ — Unitizer (content units & boundaries)
Input: {segment text, segmentation timestamps}.
Task: Decide if content units (CUs) are internally consistent and boundaries respect invariants (“present” vs “past” labeled; no cross-contamination).
You MUST NOT rewrite or inject text.
Output JSON: {"item_id": "...", "vote": 1|0, "confidence": 0–1, "notes": "…"}
Voting rule: vote=1 if (i) CUs are extractable and (ii) segment boundaries don’t mix mutually exclusive time tags.

O₂ — NLI Contradiction Check (claim vs Given)
Input: {claim label (e.g., “avoidance_high”), Given set (intake summary, prior labels), segment text}.
Task: Flag contradiction or unsupported claim relative to Given; you must not depend on O₁/O₃ outputs.
Output JSON: same schema.
Voting rule: vote=1 if claim is supported or at least not contradicted; 0 if contradicted or unsupported.

O₃ — Trace Referencer (receipt check)
Input: {claim label, trace.jsonl (read-only)}.
Task: Verify there are ≥2 independent trace fragments referencing the claim; do not read/transcribe segment text.
Output JSON: same schema.
Voting rule: vote=1 if fragments/claim ≥ 2 and hashes match; else 0.

Order-invariance note. Because each critic is a pure evaluator with disjoint inputs, order swaps should not change votes; this is how we keep pairwise order-sensitivity ε_AB small.

D.5 Synthetic data generation (deterministic, controllable)

Seeds & versions. Store in meta.json:

{"dataset_version": "v1.0.0", "global_seed": 314159, "split_seed": 271828}

Scenario templates (few-shot prompts or scripts).

Rumination loop: template fields = {topic_list, escalation_speed, reframe_freq}.
Transference spike: fields = {template_role, carry_phrases[], time_indexing_strength}.
Dream remix: fields = {residues[], angle_phi_deg, compression_k}.

Label generation rules (kept plain).

assoc_jumps sampled from Poisson(λ=topic_density × escalation_speed).
stance_shift grows with reframe_freq unless time_indexing_strength=0.
affect_series = baseline + AR(1) with shock at scripted moments; recovery slope encodes γ̂.

Negative controls. Inject (a) order-sensitive versions by (illegally) letting O₃ read segment text (to show ε spikes in dev); (b) low CSA cases with ambiguous claims.

Anonymization. All names/locations from a synthetic generator list; never reuse real names; keep a reserved token set like @PERSON_A, @PLACE_B for clarity.

D.6 Reproducible splits & hashing

Split recipe. Stratify sessions by scenario type (loop/transference/dream) and target Δ regime (red/amber/green). 70/15/15 train/dev/test with split_seed.

Hash chain. For every trace.jsonl line, compute hash = H(prev_hash ∥ canonical_json(e_t)); store the final dataset_root_hash in meta.json for integrity.

D.7 Minimal compute checklist (ties to Appendices B–C)

Build segments (segments.jsonl) from transcripts.jsonl.
Compute ĝ, β̂, γ̂ via Appendix C (C.2–C.4).
Compute Δ_t, Δ̄_t and CUSUM S_t (C.12–C.17).
Run O₁/O₂/O₃ over items to produce three graders_*.jsonl files.
Compute CSA@3, ε_AB, CWA permutation p̂ (C.20–C.23).
For dreams, compute DRI from dream_units.jsonl; for transference, compute TI from stance_features.jsonl.
Produce the compact reporting block (C.24–C.31) per session.

D.8 Tiny worked example (3 lines you can paste)

transcripts.jsonl

{"session_id":"S001","turn_id":10,"speaker":"patient","start_sec":720.0,"end_sec":742.0,
  "text":"I blew it... then I thought of last year... then what she said."}
{"session_id":"S001","turn_id":11,"speaker":"therapist","start_sec":742.1,"end_sec":755.0,
  "text":"Let's slow down. What's *today's* worry?"}
{"session_id":"S001","turn_id":12,"speaker":"patient","start_sec":755.1,"end_sec":780.0,
  "text":"Today's email. When I stick to that my chest eases."}

segments.jsonl

{"session_id":"S001","segment_id":"S001_seg03","start_sec":720.0,"end_sec":900.0,
  "turn_ids":[10,11,12],"r_strength":2,"stance_shift":2,"assoc_jumps":5,"minutes":3.0,
  "affect_series":[6,7,6,5,4],"affect_baseline":4.0,"affect_margin":0.5}

trace.jsonl

{"session_id":"S001","event_id":"S001_e07","iso_time":"2025-10-18T15:02:11Z",
  "label":"exposure_completed_tiny","meta":{"SUDS_drop":3},"prev_hash":"0000","hash":"a1b2c3"}

D.9 `meta.json` (versioning, splits, knobs)

{
  "dataset_version": "v1.0.0",
  "global_seed": 314159,
  "split_seed": 271828,
  "dataset_root_hash": "5b8e...",
  "splits": {
    "train": ["S001","S003","S004", "…"],
    "dev":   ["S007","S009"],
    "test":  ["S002","S008"]
  },
  "defaults": {
    "segment_minutes": [2,5],
    "affect_margin": 0.5,
    "reframe_scale": [0,3]
  }
}

D.10 Release & governance

License. Recommend CC BY 4.0 for the synthetic dataset and Apache-2.0 for code that reads it.
Privacy. Synthetic only; if you later mix real snippets, run redaction, de-identification, and manual legal review; store only minimal fields needed for Δ/CSA/CWA.
Provenance. Keep dataset_root_hash and change log in meta.json.
Ethics. Include an “opt-out from metrics” template and a short participant info sheet when moving beyond synthetic.

D.11 One-page Quickstart

Clone the folder; verify dataset_root_hash.
Load segments.jsonl → compute ĝ, β̂, γ̂ → Δ̄ and CUSUM.
Load graders_*.jsonl → compute CSA@3, ε_AB → set CWA light.
(Optional) Load dream_units → DRI; stance_features → TI.
Emit the compact reporting block (Appendix C).
Plot the Δ needle, S_t, CSA bar, CWA light, and three knob hints.

This appendix is enough to reproduce every table and figure in the main paper with synthetic data, and to swap in real-world data later without changing the schema.

Appendix E — Session-to-Dashboard SOP

Step-by-step with screenshots (described), paste-and-run friendly.

Scope. This SOP turns a raw session (audio/transcript) into a Δ-dashboard with CSA/CWA guardrails, then writes an append-only Trace record. Screenshots are described (e.g., [Screenshot E-1]); you can recreate the same panels in any EMR/analytics stack.

E.1 Roles, inputs, outputs (1 minute)

Roles.
• Clinician (leads session; approves labels)
• Annotator A/B (segments, scores r_t, s_t, a_t, z(t))
• Ops (CSA/CWA, hashing, exports)

Inputs. transcripts.jsonl, optional audio; clinic metadata (consent), Appendix D schemas.

Outputs. Δ-dashboard PNG/PDF, compact report (Appendix C block), updated trace.jsonl (+ hash), CSA/CWA log.

E.2 Checklist before you start (20 seconds)

Confirm consent for analytics metrics (toggle “Yes”).
Verify dataset_root_hash in meta.json.
Set session corridor for knobs: g ∈ [0, g_max], β ∈ [0, β_max], γ ∈ [γ_min, γ_max]. (E.1)

E.3 Import & segment (5 minutes)

Step E3-A — Import.
Open Data ▸ Import. Select session S### → attach transcripts.jsonl (and audio if present).
[Screenshot E-1]: Left panel “Files”; right panel “Preview (turns)”.

Step E3-B — Segment.
Click Segmenter ▸ 2–5 min windows → choose auto (2–5 min, ≤30 s tolerance) → Lock.
[Screenshot E-2]: Timeline with blue segment bars; each has start/end and turn_ids.

Step E3-C — Redact.
Run Redaction ▸ Minimal PII → confirm replacements @PERSON_A, @PLACE_B.
[Screenshot E-3]: Before/after text diff panel.

E.4 Annotate r_t, s_t, a_t, z(t) (6–8 minutes)

Step E4-A — Dual annotation.
Annotator A and B independently rate each segment:
• r_strength (0–3), stance_shift (0–3), assoc_jumps (count), minutes (auto), affect_series (0–10).
[Screenshot E-4]: Two-column grid (A vs B) with drop-downs and sliders.

Step E4-B — Adjudicate.
Click Resolve → if |A−B|>1 on any field, open mini-dialog; store both ratings + final.
[Screenshot E-5]: Adjudication modal with side-by-side evidence.

Write. Save to segments.jsonl.

E.5 Compute ĝ, β̂, γ̂ and Δ (1 minute)

Buttons. Compute ▸ Guidance ĝ ▸ Amplification β̂ ▸ Damping γ̂ ▸ Δ.
• ĝ := Σ_t (r_t − r̄)(s_t − s̄) ÷ Σ_t (r_t − r̄)². (E.2)
• β̂ := (Σ_t a_t) ÷ (Σ_t m_t). (E.3)
• γ̂ := 1 ÷ median_i(T_recover,i). (E.4)
• Δ_t := ĝ_t · β̂_t − γ̂_t; Δ̄_t := (1−λ)·Δ̄_{t−1} + λ·Δ_t. (E.5)

[Screenshot E-6]: Three mini-gauges (ĝ, β̂, γ̂) and the Δ needle.

E.6 Early-warning (CUSUM) (30 seconds)

Windowed drift. μ̂_Δ,W(t) := (1/W)·Σ_{k=0}^{W−1} Δ_{t−k}. (E.6)
CUSUM. S_{t+1} = max(0, S_t + μ̂_Δ,W(t) − τ); alarm if S_t ≥ h. (E.7)

Click Risk ▸ Calibrate τ/h → “baseline median” for τ; “perm-95%” for h.
[Screenshot E-7]: S_t line chart with red alarm line (h).

E.7 CSA graders (pure, commuting) and CWA (4 minutes)

Step E7-A — Run three critics “as is.”
• O₁ Unitizer (segments only) → graders_O1_unitizer.jsonl.
• O₂ NLI (claim vs Given) → graders_O2_nli.jsonl.
• O₃ Trace (fragments/claim ≥ 2) → graders_O3_trace.jsonl.
(Each is a pure evaluator; none rewrites text or reads another’s output.)

Step E7-B — Compute CSA and ε.
CSA@3 := (1/N)·Σ_j 1{ majority label unchanged under all orders }. (E.8)
ε_AB := Pr[ A∘B(d) ≠ B∘A(d) ] (held-out batch). (E.9)

[Screenshot E-8]: CSA bar (goal ≥ 0.67), ε heatmap (goal ≤ 0.05).

Step E7-C — CWA permutation test.
T_obs := | mean(s) − mean_π(s) |; p̂ := (1/B)·Σ_b 1{ | mean − mean_{π_b} | ≥ T_obs }. (E.10)
CWA_OK ⇔ [CSA≥0.67] ∧ [max ε≤0.05] ∧ [p̂≥0.05]. (E.11)

[Screenshot E-9]: “CWA OK?” lamp (green/red) with p̂ and ε summary.

E.8 Compose the Δ-dashboard (ready to print)

Layout.
• Top-left: Δ̄_t big needle (green ≤ −0.2, amber −0.2…+0.2, red ≥ +0.2).
• Top-right: CUSUM S_t with alarm h.
• Middle-left: ĝ, β̂, γ̂ mini-gauges (+/− arrows vs previous session).
• Middle-right: CSA@3 bar + CWA lamp + redundancy (fragments/claim).
• Bottom: Action coach: rule-based suggestions.

Action coach rules (copy-paste).
If ĝ dominates and Δ̄ red → “Reduce g: pause after each reframe; use literal reflections.” (E.12)
If β̂ dominates → “Reduce β: chunk associations; anchor to present sensory detail.” (E.13)
If γ̂ low → “Raise γ: schedule breaths, pacing gaps, homework blocks.” (E.14)
If CSA<0.67 or CWA=red → “SRA only: do not average; add redundancy, refactor critics.” (E.15)

[Screenshot E-10]: Full dashboard view with four panels and a text coach box.

E.9 Write to Trace (append-only, hash-chained) (40 seconds)

Click Trace ▸ Append. Enter one-line event e_t and context. System updates hash:

h₀ := 0; h_t := H(h_{t−1} ∥ e_t). (E.16)

Rule. Never overwrite; corrections are new events (e.g., correction_of: S001_e07).
[Screenshot E-11]: Append modal showing prev_hash, hash, and JSON preview.

E.10 Export compact report (Appendix C block) (30 seconds)

Click Export ▸ Δ-Report (PDF/MD) → includes:
ĝ (95% CI), β̂ (95% CI), γ̂ (95% CI), Δ̄_last, S_max/h, CSA@3, max ε_AB, CWA p̂, fragments/claim, confounds.
[Screenshot E-12]: One-page report with the numbers aligned to (C.24–C.31).

E.11 Governance & privacy (1 minute)

Consent proof. Attach yes/no + timestamp to the report footer.
Access control. Only Clinician/Ops can view raw audio; annotators see text+PII tokens.
Retention. Default 𝒯 roll-off R months unless extended by consent.
Audit. Export dataset_root_hash + session hash tail in the appendix of the report.

E.12 Troubleshooting (quick table)

Symptom	Likely cause	Fix
CSA low (<0.67)	Critics share inputs or mutate text	Make critics pure; separate inputs; re-run ε
ε hot cell (>0.05)	Order sensitivity	Refactor critic pair; lock inputs; re-estimate
Δ̄ “stuck” near 0	No variance in r_t or a_t	Increase observation window; revise scoring granularity
γ̂ “N/A”	No affect spikes	Carry forward with flag; schedule safe probe; or focus on g, β
CWA red	Order/phase unstable	Report per-case only; add redundancy; retry later

E.13 Safety corridor (prevent over-control)

g ∈ [0, g_max], β ∈ [0, β_max], γ ∈ [γ_min, γ_max]; clip each knob after interventions. (E.17)
If patient reports numbness or detachment, check γ vs γ_max; reduce damping and re-engage.

E.14 Two illustrated examples (described screenshots)

Example 1 — Rumination case (green outcome).
[Screenshot E-10a] Δ̄ needle moves red→amber→green; S_t falls below h after pacing; CSA=0.76; CWA=green; action coach suggests “keep γ plan; consider small β exposure.”

Example 2 — Transference spike (guardrail in action).
[Screenshot E-10b] Δ̄ rises to red; TI spike (panel note); CSA dips to 0.52; ε_{O2,O3}=0.08 (hot cell); CWA=red. Action coach shows SRA only; suggests time-indexing and redundancy. Next segment CSA=0.71 → lamp turns green; now label is committed to Trace.

E.15 End-of-day batch (optional)

Click Batch ▸ Recompute to (i) refresh all Δ̄_t and S_t with late annotations, (ii) regenerate CSA/CWA logs, (iii) email compact reports to the secure clinic inbox.

E.16 Minimal timings (for planning)

Import+segment 5′ • Dual annotation 6–8′ • Compute 1′ • CSA/CWA 4′ • Trace write 1′ • Export 0.5′ → ~18–20 minutes per new session on first pass; drops to ~12–15′ with trained annotators.

E.17 What “good” looks like (acceptance criteria)

• Dashboard renders with no missing fields; Δ̄, S_t, ĝ/β̂/γ̂ populated.
• CSA@3 ≥ 0.67; max ε_AB ≤ 0.05; CWA lamp correct by (E.11).
• Trace updated with new event and valid hash chain (E.16).
• Report contains the Appendix C block; consent + hashes attached.
• If any gate fails, dashboard shows SRA only and the action coach explains why.

You can run this SOP verbatim. It stitches Appendices A–D into a single, auditable flow: segment → score → compute Δ → verify CSA/CWA → act → append Trace → export. The described screenshots are UI-agnostic; any EMR or analytics tool can reproduce the same panels with the equations and thresholds given here.

Appendix F — Governance and Privacy

Redaction, hashing, role permissions, audit logs. Paste-and-run policy blocks. Single-line Unicode formulas tagged (F.n).

F.1 Principles and scope (why these rules exist)

Purpose-limitation, data-minimization, auditability, consent, cognitive liberty. The dashboard is assistive (clinician decides). No covert collection. Averaging is legal only with CWA_OK (Appendix C/§8); otherwise per-case (SRA).

F.2 Data classes (what we protect)

Class A (identifiers): names, contacts, precise locations, dates of birth.
Class B (content): transcripts, audio, stance vectors, affect traces.
Class C (derived): ĝ, β̂, γ̂, Δ̄, CSA/CWA logs.
Class D (governance): audit logs, hashes, role grants, consents.

F.3 Redaction & pseudonymization (defaults)

Redaction function.
redacted_text := Redact(text; rules R, token set 𝒯). (F.1)

Tokenization rule (salted, no plaintext):
map[token] := H_salt(original_identifier). (F.2)

Defaults (on by default):
• Replace names/places with @PERSON_A, @PLACE_B.
• Collapse precise dates to month/year unless clinically required.
• Drop freeform IDs (emails, phone) to masked forms.
• Store token map in a separate vault (Ops-only).
• Keep a “redaction diff” for human review in de-identified space.

De-redaction gate. Only Clinician (for care) or Auditor (for investigations) can de-tokenize, never Researchers; all de-redactions emit audit entries (F.14).

F.4 Trace integrity (append-only, verifiable)

Hash chain per session.
h₀ := 0; h_t := H( h_{t−1} ∥ canonical_json(e_t) ). (F.3)

Daily Merkle root (fast audit).
R_day := MerkleRoot( { h_t : e_t on that day } ). (F.4)

Dataset root (publish with exports).
R_dataset := MerkleRoot( { R_day across range } ). (F.5)

Verification (one line).
VerifyTrace(𝒯) := 1 iff recomputed h_T equals stored h_T. (F.6)

Correction policy. Never overwrite; append correction_of: e_id events.

F.5 Role-based access control (RBAC)

Allow rule.
Allow(u, action, resource) = 1 iff [ role(u) ∈ policy[action,resource] ∧ consent(resource) ≥ level(action) ]. (F.7)

Role	Read audio	Read de-ID text	Write labels	See Δ/CSA/CWA	De-redact	Export data	Manage keys	See audit
Patient	✗	▷ (upon request)	✗	▷ (patient copy)	✗	▷ (own data)	✗	▷ (own entries)
Clinician	✓	✓	✓	✓	▷ (case-by-case)	▷ (case PDF)	✗	▷
Annotator	✗	✓	✓ (draft)	▷	✗	✗	✗	✗
Supervisor	✗	✓	✓ (approve)	✓	✗	▷ (case set)	✗	✓
Ops	✗	✓	✗	✓	✗	✓ (aggregate)	✓	✓
Researcher	✗	✓ (de-ID only)	✗	✓ (aggregate)	✗	✓ (CWA_OK only)	✗	▷ (summary)
Auditor	✓ (on-site)	✓	✗	✓	✓	✓	✓	✓

▷ = by request; ✓ = allowed; ✗ = disallowed. All accesses are audited (F.14).

F.6 Consent & cognitive liberty

Consent card (show to patient):
“We compute simple session trends (Δ) and evidence checks (CSA). You can opt out of any metric, or all metrics, at any time without penalty; care proceeds regardless.”

Consent state (machine-readable):
consent(u, scope) ∈ {none, minimal, metrics_only, full_sharing}. (F.8)

Off switch. If consent drops below metrics_only, metrics stop; previously stored identifiers are tokenized, and exports are disabled until consent is restored.

F.7 Data retention & deletion (practical rules)

Retention windows.
TTL_A (identifiers) ≤ 6 months; TTL_B (content) ≤ 24 months; TTL_C (derived) ≤ 36 months; TTL_D (governance) ≥ 36 months. (F.9)

Erasure semantics (legal/right to delete).
ErasePII(uid): drop token map entries for uid (rotate salt); keep de-ID trace and hashes. (F.10)

Backup policy. Encrypted, region-redundant; key rotation every 90 days.

F.8 Change control (models, thresholds, SOPs)

Version stamp on every computation/export:
cfg_id := Hash( code_version ∥ thresholds ∥ grader_specs ∥ corridor ). (F.11)

Promotion rule.
Promote(new_cfg) only if [ test_CSA↑ ∧ max ε_AB≤0.05 ∧ drift D≤δ ]. (F.12)

Rollback. Keep last 3 cfg_id images; single-click rollback with audit entry.

F.9 Sharing & publication guardrails

Publish gate (aggregate only).
PublishOK ⇔ [ CWA_OK ] ∧ [ k-anonymity k≥5 ] ∧ [ max ε_AB≤0.05 ] ∧ [ no Class-A fields ]. (F.13)

Unit of sharing. Per-case PDFs to patient/clinician; group CSVs only when PublishOK.

F.10 Security & keys (at rest, in transit)

At rest: AES-256; in transit: TLS 1.2+.
Key custody: cloud KMS; split-role approval for key export (Ops+Auditor).
Device policy: no raw audio on laptops; de-ID text only; full-disk encryption.

F.11 Third-party & sensors (no surprise ingestion)

No social feeds, ambient microphones, or wearables unless explicitly consented (scope: metrics_only→full_sharing), with a benefit statement and a separate audit trail.

F.12 Drift & fairness monitors (governance metrics)

Drift monitor (monthly):
D_t := PSI(P_t, P_ref) or KL(P_t ∥ P_ref); alert if D_t ≥ δ. (F.14)

Fairness parity (quarterly):
gap_CSA := max_group(CSA) − min_group(CSA); flag if gap_CSA > 0.15. (F.15)

Transparency pack: publish Δ/CSA distributions by subgroup; remediate graders if gaps persist.

F.13 Incident response (who does what, when)

Severity ladder. S1 (exposure of identifiers), S2 (integrity loss), S3 (availability only).

Clock.
T_detect ≤ 24 h; T_contain ≤ 48 h; T_notify (S1) ≤ 72 h. (F.16)

Recipe. Isolate system → rotate keys → verify traces (F.6) and audit chain (F.18) → notify stakeholders → root-cause → harden (policy/tech) → post-mortem logged.

F.14 Audit logs (schema, chain, review)

Log entry (JSONL, one line per action).

{
  "log_id": "L000123",
  "iso_time": "2025-10-18T16:03:22Z",
  "actor_id": "u_451",
  "actor_role": "Clinician",
  "action": "READ",
  "resource": "segments.jsonl",
  "resource_id": "S001_seg03",
  "details": {"fields": ["r_strength","stance_shift"]},
  "ip": "203.0.113.45",
  "sig": "ed25519:...=="
}

Audit hash chain (tamper-evident).
ℓ₀ := 0; ℓ_t := H( ℓ_{t−1} ∥ canonical_json(log_t) ). (F.17)

Verification.
VerifyAudit(L) := 1 iff recomputed ℓ_T equals stored ℓ_T. (F.18)

Review cadence. Weekly automated scan (anomalies: out-of-role access, mass exports), quarterly human audit (Auditor+Supervisor).

F.15 “Do/Don’t” checklist (one page)

Do: default redaction; append-only traces; log every access; enforce CWA before averaging; publish only with PublishOK; rotate keys; keep drift/fairness dashboards.
Don’t: overwrite notes; share de-tokenization with researchers; average with CWA=red; ingest outside-of-care data; link metrics to staff compensation (Goodhart risk).

F.16 Minimal policy snippets (ready to paste)

PII default: if field ∈ Class A then redact = true unless clinician_override AND audit_log.
Export rule: if PublishOK then allow_group_export else deny.
Access rule: Allow(u,a,r) as in (F.7).
CWA gate in UI: show “SRA only” banner if CWA_OK = false.

F.17 Patient rights (operational)

Access/Export: Provide per-case report and raw de-ID JSONL within 30 days.
Correction: Append correction events; never rewrite.
Erase: Run ErasePII(uid) (F.10); provide confirmation hash pair (before/after token map).

F.18 Tying back to the math (why these work)

Hash chains and Merkle roots guarantee latching of the record at the governance layer (tamper-evident).
RBAC + consent enforces cognitive liberty (patient controls measurement scope).
CSA/CWA gates make “objective enough to average” a computable policy switch.
Drift/fairness monitors keep metrics honest across time and subgroups.

Bottom line. With redaction by default (F.1–F.2), append-only hashing (F.3–F.6), role/consent gates (F.7–F.8), publish guards (F.13), and auditable logs (F.14), this appendix gives you a complete, clinic-grade governance spine you can implement without changing the rest of the paper.

References

Core observer–trace & agreement framework (primary sources)

Self-Referential Observers in Quantum Dynamics: A Formal Theory of Internal Collapse and Cross-Observer Agreement. Proofs of delta-certainty (“latching”), commutation/compatibility conditions, SBS redundancy, and AB-fixedness.
https://osf.io/7cbsu/files/osfstorage/68c5961e10e31c4095d998f5
ObserverOps Technical Blueprint. Operational stack (CWA engine, CSA checks, governance, APIs) and production runbooks for observer-centric workflows.
https://osf.io/yj5aw/files/osfstorage/68d30242dd3f77699b3c315f
Emulsion-Stabilized Inference (ESI): Phase-Controlled Decoding with Structural “Starch” and Observer-Aligned Verification. CSA as “smoothness,” commuting critics, redundancy, and evaluation protocol.
https://osf.io/q8egv/files/osfstorage/68d58d6a5d44329625432c73
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https://osf.io/ya8tx/files/osfstorage/68e77fa0cd19895405a0d243

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Baars, B. J. (1988). A Cognitive Theory of Consciousness. (Global Workspace Theory, classic.) (CCRG)
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Friston, K. (2009/2010). “The free-energy principle: a rough guide to the brain?” Trends Cogn. Sci.; “A unified brain theory?” Nat. Rev. Neurosci. Free-energy/predictive coding background used for Δ–dial interpretations. (UCL Fil Ion)

Objectivity via redundancy (quantum roots for CSA/CWA)

Horodecki, R., Korbicz, J. K., et al. (2015). “Quantum origins of objectivity.” Phys. Rev. A. Spectrum Broadcast Structures (SBS) as a basis for objectivity. (Physical Review Link Manager)
Korbicz, J. K. (2021). “Roads to objectivity: Quantum Darwinism, Spectrum Broadcast Structures, and strong quantum Darwinism.” Quantum. Comparative review. (Quantum)
Le, T. P., et al. (2019). “Strong Quantum Darwinism and Strong Independence Are Equivalent to Spectrum Broadcast Structure.” Phys. Rev. Lett. Equivalence results underpinning redundancy criteria. (Physical Review Link Manager)

Sequential change detection & early-warning (CUSUM)

Page, E. S. (1954). “Continuous Inspection Schemes.” Biometrika. Original CUSUM scheme (basis for our hitting-time alarm). (JSTOR)
Basseville, M., & Nikiforov, I. V. (1993/1996). Detection of Abrupt Changes: Theory and Application. Standard reference on sequential change detection. (ACM Digital Library)

Notes.
• Internal items (observer/trace, CSA/CWA, Unicode style) are cited to the provided manuscripts above; those are the rigor backbone for “latching,” commutation, redundancy, and the clinical CWA gate.
• Neuroscience and statistics references are mainstream anchors for readers unfamiliar with our internal corpus; they motivate the Δ-dial proxies and early-warning design. (PubMed)

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.

Saturday, October 18, 2025