Projection Friction Calibrator

Raw = tone at reduced amplitude (friction applied). Compensated = tone restored to full amplitude (×6, ×2, or ×1).

The terminalColor regime hides 5/6 of the signal — barely audible. The ×6 compensator recovers it perfectly. This is NOT amplification — it's inverting a known geometric projection.

Toggle between raw and compensated on each regime to hear the difference. With dither ON in the Live tab, the compensation tracks the raw signal more closely.

Algorithm: With dither on, the input signal varies continuously. The Z6 projection maps this variation to 6 discrete levels. A regime with low friction (6→6) passes all the variation through → all 6 levels appear. A regime with high friction (6→1) averages the variation to 1 output → only 1-2 levels appear.

Formula: visible_fraction ≈ K / 6 where K = distinct projection values observed over N dither samples. The ratio K/6 directly estimates the information that survives the projection.

This works because the projection is linear in expectation for small signals with dither. The number of Z6 levels crossed per unit input variation is proportional to the projection's rank — which equals the output dimension.

Each Z6 projection is a lattice — the input space ℝ⁶ is partitioned into 6^N basins (traps), one per output point in Z6^N. The "depth" of each trap is the input volume that maps to that output. Shallow traps (BM: 6⁶ tiny basins) are easy to escape; deep traps (TC: 6 large basins) hold the signal constant.

Frequency response reveals the trap escape rate: at frequency f, the 6-sample buffer spans a fraction of a cycle. If that fraction < trap width, the signal stays trapped (high friction). If > trap width, it escapes (low friction). The cutoff frequency is where the buffer span matches the trap width — the regime's resonant escape frequency.

Friction kernel = the curve |Y(f)|/|X(f)|. A steep kernel (TC) means the regime resists information flow at all but the highest frequencies. A flat kernel (BM) means the regime passes all frequencies equally. The kernel shape is determined by the weight matrix's singular values in Z6.

The trapdoor writes, the calibrator reads. Both use the same Z6 projection:

Three regimes from three architectures:

• terminalColor (6→1): 6 inputs → 1 output. Friction = 5/6. Only 1/6 of signal visible. Compensator = ×6.
• couplingBoundary (6→3): 6 inputs → 3 outputs. Friction = 3/6. Half visible. Compensator = ×2.
• boundMatter (6→6): 6 inputs → 6 outputs (identity). Friction = 0. All visible. Compensator = ×1.

The compensation is exact for the deterministic component. With dither averaging, it approaches the true value within ADC quantization error.

Bit-exact on any hardware — same as Z6 engine. Integer multiply-add only.

• Sensor calibration — recover true value from partial projection, no reference standard needed
• Write-once weights — commit to a weight set via projection, verify integrity with forward pass
• Secure swarm voting — hash votes through lattice, collision detection prevents Sybil attacks
• Model watermarking — embed watermark in Z6 geometry, survives INT8 quantization
• Barometric weather — pressure trends are irreversible; trapdoor commits to "rain alert" decision

State definition: Each projected value y ∈ Z6^N is a state. The sequence y₀, y₁, ... is a walk on the lattice. Transitions from state a to state b are counted in the transition matrix T[a→b].

Prediction: Given current state y_t, predict y_t+1 = argmax_z T[y_t → z]. This is the MAP (maximum a posteriori) estimate from observed transitions.

Predictability score: S = 1 − H(p) / log₂(K) where H(p) is the entropy of the transition distribution and K is the number of reachable states. S=1 means every transition is deterministic (max predictability). S=0 means transitions are uniform random.

Regime change: When predictability suddenly drops, the input dynamics have changed. This is a model-free changepoint detector — works on any time series.