An operator, not a neuron map
We construct a context-dependent response matrix H(c) = V G(c) Kᵀ. This resembles query–key–value factorization, but it is a signed linear response—there is no softmax, spike model, or learning here.
FIELD NOTE 001 12 OCTOBER 2026
Can you recover
the inside from
the answer?
One hidden circuit. Three observable signals. Four places to poke. The same measured response can hide radically different internal coordinates.
Open the experiment ↘FIG 01 · SIMULATED INPUT / OUTPUT
04Input ports
03Output channels
02Latent coordinates
∞Equivalent bases
01 / EXPERIMENT
The observer's desk
Switch hidden hypotheses. Apply controlled inputs. See whether the evidence can choose between them.
OBSERVER STATE 0 PULSES
BELIEF IN CIRCUIT A
50.0%
UNRESOLVED
A · 50.0%B · 50.0%
A passive observation at context 0 cannot tell these two hypotheses apart.
RESPONSE OPERATOR 3 OUTPUTS × 4 PORTS
Not a correlation matrix.
Each cell answers: If I perturb input port j, how much does output i change? Hypotheses are known to the observer; the hidden truth is not.
HYPOTHESIS A
HYPOTHESIS B
SIGNED RESPONSE
NEGATIVEPOSITIVE
MEASURED OUTPUTS
Only probed columns are filledOBSERVATION LEDGER RECENT PULSES
| # | Context | Port | Measured outputs | P(A) |
|---|---|---|---|---|
| No pulses recorded yet. The instrument is waiting. | ||||
02 / NEW EXPERIMENT · BLIND INVERSE
Learn the operator.
Never see the circuit.
Now there is no candidate catalog. The learner sees only the inputs it requested and the three noisy responses. Can it predict what a different intervention will do?
LEARNED FROM PULSES 0 MEASUREMENTS
POSTERIOR UNCERTAINTY2.00Estimated σ for unmeasured response, prior included
DESIGN IDENTIFIABILITY0/4 PORTSTwo distinct contexts per input port required
The observer estimates H(c) = H₀ + cH₁ by Bayesian linear regression. It learns 24 coefficients from the observations only; a rank-two projection is an optional compression, not a recovered anatomical graph.
INFERRED RESPONSES 3 OUTPUTS × 4 INPUT PORTS
BLINDLY LEARNED H(c)
LAST MEASUREMENT AT THIS CONTEXT
The measured grid contains only columns probed at the selected context. The predicted grid comes solely from fitted coefficients. Unmeasured columns are estimates—not observations.
SEALED GENERALIZATION AUDIT DOES NOT TRAIN THE LEARNER
Can it answer unseen questions?
Evaluate predictions at withheld contexts −0.5, +0.5 and +1.5, including extrapolation outside the trained context range. The reference is used only when you request an audit, never to choose a training probe.
No audit requested. A low training error is not evidence of future-answer accuracy.
TESTABLE CLAIMRecover a predictive response rule, then ask what new probes it gets wrong.
The Operaattori-equations fixture uses a seven-compartment illustrative morphology. All 72 stored scalar responses matched the original Operaattori runtime exactly in the pinned upstream validation run (maximum difference 0 mV). This validates the saved synthetic forward computations, not a real reconstructed neuron or biological QKV. Its three outputs are features of one soma voltage trace. The browser interpolates a saved response grid; it does not run the cable model live.
03 / TIME-DEPENDENT INVERSE GATE
The soma remembers what the instant hides.
Probe a synthetic cable neuron with sequences, not a static response table. Can a learned operator predict the complete voltage trace after histories it has never seen?
CONTINUOUS OUTPUT 32 ms TRACE
Unseen three-event sequence
Loading test trace…
This is not blind stimulus-source recovery: the drives are observed, the dendritic state is hidden. The held-out protocol changes pulse ordering, history count, and late-probe timing.
SOMA ALIASING KNOWN-ANSWER COUNTEREXAMPLE
Same measured soma voltage. Different future response.
Loading matched-history experiment…
Conditioner strength was deliberately tuned to equalize one pre-probe soma sample. The probe is identical for both histories, and each curve subtracts its own no-probe baseline. Matching one scalar does not equate full past soma traces or guarantee that the histories are observationally indistinguishable to stronger readers.
Loading the upstream temporal test receipt…
04 / INSTRUMENT
The heatmap can lie.
A known external source can manufacture correlations between sensors. Here we control that source.
PASSIVE OBSERVER NOT AN INTERVENTION
A common signal, three sensors.
These are Pearson correlations of simulated sensor recordings. A shared audio-like nuisance waveform is added to each sensor. Move its level and see what happens to the apparent network.
Right panel uses the exact known synthetic nuisance for subtraction (an oracle), not an estimated physical filter. Neither heatmap identifies neural wiring.
RECORDED CORRELATION
KNOWN AUDIO SUBTRACTED
05 / CLAIM BOUNDARY
What can be recovered?
The experiment distinguishes a measurable operator from the coordinates used to realize it.
Interventions beat passive viewing
Two circuits agree when c = 0. The active observer chooses the context and input port with the largest predicted response disagreement. Only pulses at informative contexts can identify the specimen.
Some insides are unidentifiable
Under any invertible latent basis change R, set V′ = VR, K′ = KR⁻ᵀ and G′ = R⁻¹GR. The observable H(c) stays exactly the same. No amount of repetition distinguishes those coordinates.
OUR EXPERIMENTAL CONTRACTChoose a probe that changes the answer—otherwise report not identifiable.
What this demonstrates
An experimentally controlled synthetic inverse problem, a Bayesian two-hypothesis observer, and a mathematical impossibility case. Deterministic seed, Gaussian noise, and known candidate models make the outcomes reproducible.
What it does not demonstrate
Not actual EEG, not a recovered biological QKV circuit, not a new theorem, and not blind neural source separation. Its aim is to calibrate an observer before applying inverse models to real recordings or detailed compartment simulations.
Built from questions raised by BrainLoops, HeadAsResonator, LensLuotainTarget, Operaattori and TransientWaveCompiler. The mechanisms here are self-contained toy equations, not imported biological results.