https://osf.io/tj2sx/files/osfstorage/68f3de3e3c15ecd6a0c3fec6
https://chatgpt.com/share/68f3e129-a2e8-8010-b19c-2127413c0d6b
Observer-Centric Neurocybernetics: Unifying Closed-Loop Control, Language-Game Semantics, and Hinge Hyperpriors for Brain Science
0. Executive Overview — Diagrams Pack
Figure 0-A. The Observer Loop (Measure → Write → Act)
Diagram (captioned flow):
World state x_t → Measure M → outcome label ℓ_t → Write W (append to Trace T_t) → Act Π (policy reads T_t) → World updates to x_{t+1}.
One-line mechanics (paste under the figure):
Trace update: T_t = T_{t−1} ⊕ e_t. (0.1)
Latching (fixedness): E[e_t ∣ T_t] = e_t and Pr(e_t ∣ T_t) = 1. (0.2)
Policy reads the record: u_t = Π(T_t); branch diverges if the write differs. x_{t+1} = F(x_t, u_t, T_t). (0.3)
Rigour anchor. Latching is “conditional-expectation fixedness” of past events in the observer’s filtration; policy-read causes branch-dependent futures.
Figure 0-B. Thermostat-with-a-Notebook Analogy
Diagram (captioned flow):
Read room → if cold, write “heat_on” in notebook → heater turns on because controller reads the notebook → tomorrow is warmer → the note can’t be “unwritten” for the controller’s next step.
Minimal math under the picture:
Notebook write: e_t = “heat_on”; T_t = T_{t−1} ⊕ e_t. (0.4)
Delta-certainty of your own note: Pr(e_t = “heat_on” ∣ T_t) = 1. (0.5)
Why tomorrow changes: u_t = Π(T_t), so F(x_t, Π(…⊕“heat_on”), T_t) ≠ F(x_t, Π(…⊕“off”), T′_t). (0.6)
Figure 0-C. “Agreement You Can Trust” Panels
Layout (four tiles):
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CSA@3 trend (top strip).
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ε heatmap (critic-pair order sensitivity).
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Redundancy index (fragments per claim in Trace).
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CWA lamp (green/red “Averaging is legal?”).
One-line definitions under the panel:
Commutation on item d: A∘B(d) = B∘A(d). (0.7)
Order-sensitivity: ε_AB := Pr[A∘B ≠ B∘A]. (0.8)
CSA majority (k=3 critics): CSA@3 = mean_d[ majority label unchanged by any order ]. (0.9)
CWA pass rule: CWA_OK ⇔ [ CSA@3 ≥ 0.67 ] ∧ [ p̂ ≥ α ] ∧ [ max ε_AB ≤ 0.05 ], α=0.05. (0.10)
Operational note. These panels are the “go/no-go” for pooling across participants or sessions; if CWA fails, report per-case (SRA) only.
Figure 0-D. The Trace Ledger (Immutability at a Glance)
Diagram (captioned flow):
Append-only Trace with hash chain inside each session; daily Merkle root; dataset root for exports.
One-line formulas under the figure:
Hash chain: h₀ := 0; h_t := H(h_{t−1} ∥ canonical_json(e_t)). (0.11)
Verify trace: VerifyTrace(T) = 1 iff recomputed h_T equals stored h_T. (0.12)
Why it matters. Hash-chained writes make latching operational (tamper-evident past), so conditioning on your own record is well-defined for the controller Π.
Figure 0-E. System Planes (Where Each Step Runs)
Diagram (captioned boxes):
Data Plane — measurement hot-path, Trace writes, CWA-gated pooling.
Control Plane — Ô policy, tick cadence, safety gates.
Audit Plane — immutable ledger, certificate logs, exports.
Invariants under the figure:
Irreversible writes at tick τ_k; pooling only if CWA.score ≥ θ; slot conservation on buffers/tools. (0.13)
Figure 0-F. One-Minute CWA Certificate Checklist
Diagram (checkbox list beside the CWA lamp):
□ Three independent, non-mutating critics (units/invariants, contradiction vs Given, trace-referencer). (0.14)
□ Order test: all ε_AB ≤ 0.05 (swap critic order on a held-out batch). (0.15)
□ Permutation p-value: p̂ ≥ 0.05 under order+phase shuffles. (0.16)
□ CSA threshold: CSA@3 ≥ 0.67 (majority label stable). (0.17)
□ Redundant traces: ≥2 fragments/claim (e.g., tool log + hash). (0.18)
→ If all boxes ticked, CWA_OK = green; else SRA only (no group averages). (0.19)
Sources behind the figures (for readers without our library)
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Latching & filtration (Figures 0-A, 0-B): fixedness via conditional expectation; policy-read branching.
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Commutation, CSA, CWA (Figures 0-C, 0-F): order tests, thresholds, and the certificate recipe.
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Ledger & planes (Figures 0-D, 0-E): hash-chain trace, auditability, and three-plane ops.
1. Minimal Math, Human Words: The Observer Triplet
1.1 First principles (plain words → one-liners)
What is an observer?
Think “a lab team with three moves”: Measure the world, Write the result into a record, then Act using that record. We’ll call the trio ℴ = (M, W, Π). The running record is a Trace of timestamped events. This is enough to build experiments, dashboards, and controllers.
Single-line definitions (Blogger-ready).
Observer triplet: ℴ = (M, W, Π). (1.1)
Trace as an append-only list: T_t = [e₁,…,e_t] = T_{t−1} ⊕ e_t. (1.2)
Event schema (concept): e_t = (τ, channel, ℓ_t, meta). (1.3)
Analogy (you’ll reuse it): your lab log. You measure, you write, and next steps are planned from what’s written. You can debate why later, but you can’t unhappen an entry once it’s in your own log.
1.2 Latching = “conditioning on your own record”
Once you write today’s outcome ℓ_t into T_t, any probability “from your point of view” is now conditioned on T_t. That makes the just-written event a fixed point: inside your frame, it’s certain; and because the policy reads the trace, tomorrow branches based on what you wrote. This is the operational face of “collapse-as-conditioning.”
One-liners.
Delta-certainty of the written label: Pr(ℓ_t = a ∣ T_t) = 1 if a = ℓ_t, else 0. (1.4)
Fixed-point form: E[1{ℓ_t = a} ∣ T_t] = 1{a = ℓ_t}. (1.5)
Policy reads the record: u_t = Π(T_t). (1.6)
Next step is label-selected: x_{t+1} = F_{∣ℓ_t}(x_t, u_t, ξ_t). (1.7)
Everyday picture. “Write ‘exposure_ready’ in the note → the scheduler runs the exposure branch next session. If you had written ‘not_ready’, the future would route differently.”
1.3 Commutation = “checks that don’t interfere”
Two checks A and B commute on an item when order doesn’t matter; then majority labels are stable under order-swap, and cross-observer agreement (CSA) rises. In practice, you test order-sensitivity ε_AB and require it to be small before you average anything (the CWA rule).
One-liners.
Commutation on x: A ∘ B(x) = B ∘ A(x). (1.8)
Order-sensitivity (held-out set D): ε_AB := Pr_{d∼D}[ A∘B(d) ≠ B∘A(d) ]. (1.9)
CSA (3 critics, order-invariant majority): CSA@3 = mean_d[ majority label unchanged by any order ]. (1.10)
CWA pass (pooling is legal): CWA_OK ⇔ [CSA@3 ≥ 0.67] ∧ [p̂ ≥ 0.05] ∧ [max ε_{AB} ≤ 0.05]. (1.11)
Everyday picture. Three “thermometers” that don’t affect each other (commute) + multiple receipts (redundant traces) → the reading is stable enough to average; if they do affect each other, don’t average.
