examples: motion = bit shift × transform no-op × shaping — measured

The operator's factorization of M1: motion compensation in this substrate is
bit shift (integer motion = addressing, zero arithmetic — the deterministic
PLACE / VPGATHERDD) × transform no-op (the DCT a real codec puts after MC
collapses to identity when prediction is good) × shaping (the residue applied
directly as a shape, not transform-coded). M1 already IS this: recon =
shifted_ref + residual, no transform, and 62.5% of blocks had residual == 0
(transform fully absent).

This probe MEASURES the factorization as a tradeoff curve. All three codings
are orthonormal (equal PSNR at a fixed quant step Q); bytes = order-0 Shannon
entropy (ideal rANS). Residual lag-1 autocorrelation ρ is the motion-quality
proxy (good motion → white residual → ρ→0), because a transform cannot compress
white noise.

Measured (128x128, 8x8 blocks, Q=6):
   residual   ρ(meas)   direct    wht     dct   dct/dir  wht/dct
   ρ≈0.00     -0.001     9135    9579    9581   0.95x    1.00x
   ρ≈0.30      0.053     8386    8609    8608   0.97x    1.00x
   ρ≈0.60      0.419     7664    7147    7146   1.07x    1.00x
   ρ≈0.90      0.961     7641    3846    3754   2.04x    1.02x

Both falsifiable claims hold:
  1. transform no-op grows with motion quality — dct/dir → 1 (even below, 0.95x)
     as the residual whitens; the transform is a literal no-op in the good-motion
     regime and only earns its keep (2.04x) on structured residuals (poor motion),
     the regime good bit-shift avoids. The transform's value is inversely coupled
     to motion quality.
  2. a multiply-free sign transform suffices — wht/dct ≈ 1.00–1.02x across the
     whole sweep. The ±1 add/subtract Walsh–Hadamard transform (the shader's
     "zero FP, zero matmul" idiom) matches the full multiply DCT within 2%, even
     where the transform matters most. The substrate never needs a DCT.

So motion = bit shift (free) × transform no-op (WHT, vanishing as motion
improves) × shaping (direct residue) — one tradeoff curve, not three stages.
Ties to H-7 via the residue CellMode histogram. fmt + clippy clean, gated.

Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_01MLBnPuScZy6w9di2QEjsXM

Author: claude

Cargo.toml

@@ -87,6 +87,10 @@ required-features = ["codec"]
 name = "mc_via_shader"
 required-features = ["codec"]
 
+[[example]]
+name = "motion_transform_noop"
+required-features = ["codec"]
+
 [[example]]
 name = "entropy_ladder_probe"
 required-features = ["std"]

examples/motion_transform_noop.rs

@@ -0,0 +1,311 @@
+//! Motion = bit shift × transform no-op × shaping — the measured factorization.
+//!
+//! The operator's reframe (2026-07-04): motion compensation in this substrate
+//! factors as **bit shift** (integer motion = addressing, zero arithmetic — the
+//! deterministic PLACE / `VPGATHERDD`) × **transform no-op** (the DCT stage a
+//! real codec puts after MC collapses to identity when prediction is good) ×
+//! **shaping** (the residue applied directly, coded as a shape — not
+//! transform-coded). M1 already IS this: `recon = shifted_ref + residual`, no
+//! transform; and M1 measured 62.5% of blocks with residual == 0 (transform
+//! fully absent — pure bit shift).
+//!
+//! The factorization is a **tradeoff curve, not three stages**: better bit-shift
+//! ⇒ whiter residual ⇒ the transform has less to compact ⇒ shaping is trivial.
+//! This probe MEASURES that curve, because a transform CANNOT compress white
+//! noise (its coding gain → 1). The falsifiable claims:
+//!
+//!   1. **transform no-op grows with motion quality** — the DCT's byte advantage
+//!      over *direct* shaping (no transform) → 1 as the residual whitens
+//!      (i.e. as motion improves). When it hits 1, the transform is a literal
+//!      no-op and can be dropped.
+//!   2. **a multiply-free sign transform suffices** — the Walsh–Hadamard
+//!      transform (±1 basis, add/subtract only — the shader's "zero FP, zero
+//!      matmul" idiom) tracks the DCT within a few %, so the substrate never
+//!      needs a multiply-based DCT. WHT is the honest "transform" for a codec
+//!      that IS the zero-FP cognitive shader.
+//!
+//! # Fairness
+//!
+//! All three transforms are **orthonormal**, so quantising coefficients with the
+//! same step `Q` yields the same L2 distortion (Parseval) — an equal-PSNR byte
+//! comparison. Bytes = order-0 Shannon entropy `H₀` of the quantised symbols
+//! (what an ideal rANS coder spends). "direct" = quantise the residual in the
+//! spatial domain (no transform = pure shaping); "wht"/"dct" = transform, then
+//! quantise the coefficients.
+//!
+//! Residual correlation `ρ` (lag-1 autocorrelation) is the stand-in for motion
+//! quality: good motion ⇒ white residual (`ρ → 0`); poor motion ⇒ structured
+//! residual (`ρ → 1`). This is the textbook "MC decorrelates, leaving a
+//! near-white residual" — here quantified on the substrate.
+//!
+//! Run: `cargo run --release --example motion_transform_noop --features codec`
+
+use ndarray::hpc::codec::CellMode;
+
+const W: usize = 128;
+const H: usize = 128;
+const B: usize = 8; // transform block edge (8×8), B = 2³ for the fast WHT
+const K: usize = B * B;
+const BX: usize = W / B;
+const BY: usize = H / B;
+const Q: f64 = 6.0; // quantisation step (same for all three → equal PSNR)
+
+fn mix(mut z: u64) -> u64 {
+    z = z.wrapping_add(0x9E37_79B9_7F4A_7C15);
+    z = (z ^ (z >> 30)).wrapping_mul(0xBF58_476D_1CE4_E5B9);
+    z = (z ^ (z >> 27)).wrapping_mul(0x94D0_49BB_1331_11EB);
+    z ^ (z >> 31)
+}
+
+/// A residual field at controlled lag-1 correlation `rho` — a blend of a smooth
+/// (correlated) component and white noise. `rho ≈ 0` = white (good motion);
+/// `rho ≈ 1` = structured (poor motion). Zero-mean.
+fn residual_field(rho: f64) -> Vec<f64> {
+    let mut f = vec![0.0f64; W * H];
+    for y in 0..H {
+        for x in 0..W {
+            let (xf, yf) = (x as f64, y as f64);
+            let smooth = 40.0 * (xf * 0.05).sin() * (yf * 0.045).cos() + 20.0 * (yf * 0.03).sin();
+            let white = ((mix((y as u64) << 20 | x as u64) & 0xFF) as f64) - 127.5;
+            f[y * W + x] = rho * smooth + (1.0 - rho) * (white * 0.5);
+        }
+    }
+    f
+}
+
+/// Measured lag-1 autocorrelation (horizontal + vertical averaged) — the honest
+/// whiteness readout, independent of the `rho` knob.
+fn autocorr_lag1(f: &[f64]) -> f64 {
+    let mean = f.iter().sum::<f64>() / f.len() as f64;
+    let var = f.iter().map(|&v| (v - mean) * (v - mean)).sum::<f64>() / f.len() as f64;
+    if var < 1e-12 {
+        return 0.0;
+    }
+    let mut cov = 0.0;
+    let mut n = 0.0;
+    for y in 0..H {
+        for x in 0..W {
+            if x + 1 < W {
+                cov += (f[y * W + x] - mean) * (f[y * W + x + 1] - mean);
+                n += 1.0;
+            }
+            if y + 1 < H {
+                cov += (f[y * W + x] - mean) * (f[(y + 1) * W + x] - mean);
+                n += 1.0;
+            }
+        }
+    }
+    (cov / n) / var
+}
+
+/// Orthonormal 8-point DCT-II basis row `k`.
+fn dct_row(k: usize) -> [f64; B] {
+    let mut r = [0.0f64; B];
+    let alpha = if k == 0 {
+        (1.0 / B as f64).sqrt()
+    } else {
+        (2.0 / B as f64).sqrt()
+    };
+    for (n, rn) in r.iter_mut().enumerate() {
+        *rn = alpha * (std::f64::consts::PI * (2.0 * n as f64 + 1.0) * k as f64 / (2.0 * B as f64)).cos();
+    }
+    r
+}
+
+/// 2-D separable orthonormal DCT-II of an 8×8 block (full multiplies).
+fn dct2(block: &[f64; K]) -> [f64; K] {
+    let basis: Vec<[f64; B]> = (0..B).map(dct_row).collect();
+    // rows
+    let mut tmp = [0.0f64; K];
+    for r in 0..B {
+        for k in 0..B {
+            let mut s = 0.0;
+            for n in 0..B {
+                s += basis[k][n] * block[r * B + n];
+            }
+            tmp[r * B + k] = s;
+        }
+    }
+    // cols
+    let mut out = [0.0f64; K];
+    for c in 0..B {
+        for k in 0..B {
+            let mut s = 0.0;
+            for n in 0..B {
+                s += basis[k][n] * tmp[n * B + c];
+            }
+            out[k * B + c] = s;
+        }
+    }
+    out
+}
+
+/// 2-D separable orthonormal Walsh–Hadamard transform of an 8×8 block. The basis
+/// is ±1/√8 — the transform is pure add/subtract (a sign transform), ZERO
+/// multiplies in the butterfly. This is the codec transform that matches the
+/// "zero FP, zero matmul" cognitive shader. (Multiplies here are only the final
+/// 1/√8 normalisation, foldable into the quant step.)
+fn wht2(block: &[f64; K]) -> [f64; K] {
+    let mut m = *block;
+    let scale = 1.0 / (B as f64).sqrt();
+    // rows: in-place fast WHT (add/subtract butterfly), then normalise
+    for r in 0..B {
+        fwht(&mut m[r * B..r * B + B]);
+    }
+    // cols
+    let mut col = [0.0f64; B];
+    for c in 0..B {
+        for r in 0..B {
+            col[r] = m[r * B + c];
+        }
+        fwht(&mut col);
+        for r in 0..B {
+            m[r * B + c] = col[r] * scale * scale;
+        }
+    }
+    m
+}
+
+/// In-place fast Walsh–Hadamard transform (natural order), length 8 = 2³.
+/// Add/subtract butterflies only — no multiplies.
+fn fwht(a: &mut [f64]) {
+    let n = a.len();
+    let mut h = 1;
+    while h < n {
+        let mut i = 0;
+        while i < n {
+            for j in i..i + h {
+                let (x, y) = (a[j], a[j + h]);
+                a[j] = x + y;
+                a[j + h] = x - y;
+            }
+            i += 2 * h;
+        }
+        h *= 2;
+    }
+}
+
+/// Order-0 Shannon entropy of an integer symbol stream, in bytes.
+fn entropy_bytes(sym: &[i64]) -> f64 {
+    use std::collections::HashMap;
+    let mut hist: HashMap<i64, u64> = HashMap::new();
+    for &s in sym {
+        *hist.entry(s).or_insert(0) += 1;
+    }
+    let n = sym.len() as f64;
+    let bits: f64 = hist
+        .values()
+        .map(|&c| {
+            let p = c as f64 / n;
+            -(c as f64) * p.log2()
+        })
+        .sum();
+    bits / 8.0
+}
+
+/// Quantise a coefficient/sample to the nearest multiple of `Q` → integer symbol.
+fn quant(v: f64) -> i64 {
+    (v / Q).round() as i64
+}
+
+fn main() {
+    let block_at = |f: &[f64], bx: usize, by: usize| -> [f64; K] {
+        let mut blk = [0.0f64; K];
+        for j in 0..B {
+            for i in 0..B {
+                blk[j * B + i] = f[(by * B + j) * W + (bx * B + i)];
+            }
+        }
+        blk
+    };
+
+    println!("Motion = bit shift × transform no-op × shaping — measured");
+    println!("  {W}×{H}, {B}×{B} blocks, Q={Q} (orthonormal ⇒ equal PSNR); bytes = H₀ (ideal rANS)");
+    println!("  ρ = residual lag-1 autocorrelation (motion quality proxy: ρ→0 = good motion)\n");
+    println!(
+        "  {:<10} {:>7} {:>10} {:>10} {:>10}   {:>8} {:>8}",
+        "residual", "ρ", "direct", "wht", "dct", "dct/dir", "wht/dct"
+    );
+
+    for &rho in &[0.0, 0.3, 0.6, 0.9] {
+        let f = residual_field(rho);
+        let rho_meas = autocorr_lag1(&f);
+
+        let (mut direct, mut wht, mut dct) = (Vec::new(), Vec::new(), Vec::new());
+        for by in 0..BY {
+            for bx in 0..BX {
+                let blk = block_at(&f, bx, by);
+                // direct = pure shaping (quantise spatial residual, no transform)
+                for &v in &blk {
+                    direct.push(quant(v));
+                }
+                // wht = multiply-free sign transform
+                for &v in &wht2(&blk) {
+                    wht.push(quant(v));
+                }
+                // dct = full multiply transform
+                for &v in &dct2(&blk) {
+                    dct.push(quant(v));
+                }
+            }
+        }
+        let (db, wb, cb) = (entropy_bytes(&direct), entropy_bytes(&wht), entropy_bytes(&dct));
+        println!(
+            "  ρ≈{:<7.2} {:>7.3} {:>10.0} {:>10.0} {:>10.0}   {:>7.2}× {:>7.2}×",
+            rho,
+            rho_meas,
+            db,
+            wb,
+            cb,
+            db / cb, // transform's advantage over direct shaping (>1 ⇒ transform earns keep)
+            wb / cb, // multiply-free vs full transform (≈1 ⇒ WHT suffices)
+        );
+    }
+
+    // H-7 tie-in: for the white residual (good motion), how many transform coeffs
+    // survive quantisation? Near-white ⇒ energy spread ⇒ few zeros; the point is
+    // the transform buys nothing there (dct/dir ≈ 1 above), so the CellMode of the
+    // *direct* residue is what the codec actually stores.
+    let fw = residual_field(0.0);
+    let mut modes = [0usize; 3];
+    for by in 0..BY {
+        for bx in 0..BX {
+            for &v in &block_at(&fw, bx, by) {
+                let q = quant(v);
+                let mode = if q == 0 {
+                    CellMode::Skip
+                } else if q.abs() <= 127 {
+                    CellMode::Delta
+                } else {
+                    CellMode::Escape
+                };
+                modes[match mode {
+                    CellMode::Skip => 0,
+                    CellMode::Delta => 1,
+                    _ => 2,
+                }] += 1;
+            }
+        }
+    }
+    println!(
+        "\n  [H-7] white-residual (good motion) direct CellMode: skip={} delta={} escape={}",
+        modes[0], modes[1], modes[2]
+    );
+
+    println!(
+        "\n  MEASURED CONCLUSION:\n\
+         \x20 • dct/dir is the transform's byte advantage over direct shaping. As the\n\
+         \x20   residual whitens (ρ→0, i.e. motion improves) it approaches 1 — the\n\
+         \x20   transform becomes a literal NO-OP and can be dropped. It only earns its\n\
+         \x20   keep on structured residuals (ρ high = poor motion), the regime good\n\
+         \x20   bit-shift motion avoids. This IS 'transform no-op': the transform's\n\
+         \x20   value is inversely coupled to the motion's quality.\n\
+         \x20 • wht/dct ≈ 1 across the sweep — the multiply-free ±1 sign transform\n\
+         \x20   (add/subtract only, the shader's zero-FP idiom) matches the full DCT.\n\
+         \x20   The substrate never needs a multiply-based DCT; WHT is the codec\n\
+         \x20   transform that IS the cognitive shader.\n\
+         \x20 • So: motion = bit shift (free addressing) × transform no-op (WHT, and\n\
+         \x20   vanishing as motion improves) × shaping (the direct residue). The three\n\
+         \x20   factors are one tradeoff curve, not three independent stages."
+    );
+}