Relaxation on a graph
Color similarity defines a local weighted graph. The lazy random walk moves a spatial signal through that fixed geometry.
xₜ₊₁ = P xₜThis is Sihti's image-style diffusion, not an object detector.
EXPERIMENT 01 / IMAGE ↔ NEURON
The residue
isn't the answer.
An image diffuses. A dendritic voltage relaxes. Both leave behind a sequence of differences. Can those Sihti residues help recover a hidden state—or do they merely rename the same information?
x0 = ∑ residues + terminal core EXACT ACCOUNTING · NOT SOURCE RECOVERY
THE QUESTION / 001
The same operation can be applied to image pixels and dendritic compartments. Does it recover anything that equally compact ordinary observations cannot?
01 / EXPLORE
Inside the sieve
One fixed initial state. Seven measured time checkpoints. Six octave differences and a surviving core.
IMAGE · DIFFUSION STATE● SYNTHETIC
NEGATIVE / LOWPOSITIVE / HIGH
An image-written graph with appearance-sensitive local conductances. Its displayed field is a signed perturbation, not the RGB image used to define the graph.
THE TIME AXIS PULSES OFF
SELECT CHECKPOINTt = 0
012481632
WHAT THE OBSERVER GETS
Eight fixed image-pixel sensors, sampled at all seven checkpoints.
THE RECONSTRUCTION LEDGER
Max |sum of bands − initial|—
Lossless bookkeeping requires all bands plus the terminal core. It is not evidence that hidden compartments can be recovered from the sensors.
The image uses an 8×8 color-derived diffusion graph. The cable is a seven-compartment passive model with one observed soma; its step matrix is checked against the original Operaattori compiler. The browser replays saved synthetic data, not a live simulation.
02 / FALSIFY
The three-band race
Same seven original measurement times. Same stored number of scalar values. Independent held-out states, five seeded runs.
GATE G2 · TRANSFER ADVANTAGE
LOADING
Comparing the Sihti code against matched compression baselines.
To pass: stored residues must beat both the best three raw time slices and train-fitted PCA in both systems.
HIDDEN STATE RMSE · HELD OUTLOWER IS BETTER ↓
G0Reversible
The complete difference stack reconstructs its original state.
LoadingG1Same information rank
An invertible difference transform cannot create observability.
LoadingG2No free recovery
Does a three-band residue memory outperform equally sized controls?
Loading04 / STATE-DEPENDENT DENDRITES
The missing history.
A nonlinear AMPA/NMDA cable. Hidden earlier input ports and amplitudes. Eight noisy soma readings. One future standardized probe.
G3-2 · RESIDUE ADVANTAGE ≥ 5%
LOADING
Loading the frozen, held-out result.
What is hidden?
Past input locations, stimulus amplitudes, and every dendritic voltage. The learner sees only eight pre-probe soma values, then learns to predict four effects of a later fixed pulse.
The test contains unseen ordered pairs of stimulation ports, not only new noise.
MEAN HELD-OUT PROBE-RESPONSE ERROR · mVLOWER IS BETTER ↓
SINGLE HELD-OUT HISTORY
Actual response Three raw soma readings Three Sihti residues Three train-fitted PCA coords
These four samples are differences between a future probe and the no-probe counterfactual in one soma trace. The plot is scored after training and is never used to select features. This is synthetic supervised prediction, not recovery of actual synapses.
G3-0Upstream physics
Checking original Operaattori source against the independent reference.
PENDINGG3-1Exact algebra
Eight differences plus the final core reconstruct the eight measured voltage readings.
LoadingG3-3Past information
Do both three-feature history models beat a single instantaneous soma snapshot by 5%?
LoadingReproduce: frozen protocol · machine-readable results · source and tests. The numerical evaluation uses a seven-compartment illustrative neuronal cable, not experimental data.
03 / MECHANISM
What really transfers?
One common mathematical instrument. Two different physical meanings.
Passive dendritic dynamics
Capacitances and conductances define the cable's linear operator. One channel observes the soma while six hidden voltages evolve.
vₜ₊₁ = P vₜBased on Operaattori's backward-Euler passive compiler. This gate has no NMDA nonlinearity.
What left between times
The same difference transform captures what changed from each dyadic checkpoint to the next.
rⱼ = x(tⱼ) − x(tⱼ₊₁)Full rank is preserved because the original seven observations can be exactly reconstructed.
The observer's most important limitation
The image has 64 hidden cells but only 56 scalar time measurements, so full-state identification is impossible even with noiseless data. The cable has seven measurements for seven compartments, but the inversion is ill-conditioned. With noise, theoretical rank is not practical recoverability. Compressing those measurements cannot manufacture additional information.
THE PROTOCOL
Nothing up the sleeve.
Five deterministic seeds. Each seed: 400 training states, 140 validation states, 180 independently held-out test states. A linear ridge decoder is trained on synthetic hidden-state labels; temporal band selection and regularization are tuned on validation only. An image uses eight spatial sensors; the cable exposes one soma sensor. Sensor noise is independent before applying the residue transform.
Fairness boundary: all seven checkpoints were sampled first; the three-band restriction is a storage budget, not a limit of three physical measurements. The PCA encoder learns from training examples; raw and residue approaches search the same 35 three-time combinations using validation. Full-history ridge variants are shown for reference, and may differ because regularization is not invariant to coordinate changes.
This is an exploratory synthetic experiment with a fixed linear decoder family. It does not prove a universal impossibility of residue-based gains, address nonlinear dendritic memories, or recover QKV or real neural sources.