Research: Brownian scale, endpoint fibres, and heat-kernel lowering #158

Question

Can Brownian motion be recalculated as an independent calibration of the Process Geometry scale/chart/fibre programme, starting from raw increment histories rather than from a supplied Gaussian, heat equation, Fourier transform, or diffusive exponent?

This is the next independent representation family requested by #140. It depends methodologically on the native-method firewall merged in #157. Draft PR #141 remains the scale-compiler dependency; this task must not copy its implementation silently while that PR is unmerged.

Frozen task ladder

S0 — blind scale discovery

Primitive input: a finite increment law and chronological independent-increment composition.

For the first positive control use the symmetric law P(+1)=P(-1)=1/2. The compiler may read the raw law, its task observer, and a scale grammar, but it may not receive:

• the exponent 1/2;
• a Gaussian target;
• the heat equation or heat kernel;
• a Taylor/power-series expansion;
• Fourier/spectral diagonalization;
• a matrix/Koopman/Carleman lift.

The first certificate should derive the active centered response order and solve the population/scale balance. This is a scale certificate, not yet a central-limit theorem.

S1 — exact endpoint fibres and composition

For the nearest-neighbour walk on Z^d, define finite histories, endpoint map, fibre counts, and pushforward probabilities exactly. History concatenation must induce the exact Chapman–Kolmogorov convolution on endpoint fibres.

This is the first native executable carrier:

history -> endpoint task -> fibre count -> pushforward law -> concatenation certificate.

S2 — recurrence observer boundary

Compute exact finite return probabilities and then freeze the limit theorem needed for the dimension boundary. Keep separate:

• lattice return to the exact origin;
• Brownian return to a neighbourhood;
• Brownian hitting of a singleton.

The 2D singleton/neighbourhood distinction is a mandatory red team.

S3 — certified Brownian/heat-kernel lowering

Only after S0–S2, add a task-scoped lowering witness from the finite process carrier to the Brownian/heat-kernel presentation. Gaussian, PDE, Fourier, and Monte Carlo methods remain baseline/certificate lanes unless task adequacy and forgotten information are explicit.

S4 — physical observer

Later calibrate Stokes–Einstein and dynamic-light-scattering observables. Report hydrodynamic radius/diameter under the physical model; do not identify this automatically with a literal molecular diameter.

Initial positive, held-out, and negative cases

• Positive: symmetric finite-variance nearest-neighbour law.
• Held-out: biased finite law; blind uncentred scaling must return a typed centering-required result.
• Negative/outside grammar: heavy-tailed infinite-variance law; the first diffusive grammar must refuse or return inconclusive rather than force exponent 1/2.
• Semantic red team: same endpoint law is not automatically the same path task, first-passage task, or physical decoder.

Method lanes

Use the merged research tool from #157:

• native discovery: raw increment law, process composition, observer, scale obligations;
• native evaluation: endpoint fibres and exact process recurrence;
• certificate: exact exhaustive small cases and theorem-backed residuals;
• baseline: Gaussian/heat-kernel, Fourier, PDE, Monte Carlo, and ordinary matrix/polynomial methods.

No baseline event may be relabelled as native evidence.

Acceptance gate for the initial PR

• frozen S0 solver/method contract;
• executable blind scale certificate from the raw symmetric increment law;
• typed biased-law failure;
• exact finite endpoint-fibre counts and concatenation certificate;
• source firewall excluding series, Fourier, and matrix lowering from native source;
• machine-readable method trace and multi-axis cost;
• documentation of claim ceiling and dependency on Research: observer-driven scale and chart compiler for effective asymptotic analysis #140/research: add observer-driven scale compiler S0–S2 (NARROW) #141.

Claim ceiling

No new Brownian theorem, central-limit theorem, heat-kernel theorem, stochastic-calculus replacement, general stochastic solver, complexity speedup, physical particle-size inference, surreal-number necessity, Theory Map promotion, Experimental API, or Public API is claimed by S0/S1.

Programme effect

Potential pressure: U1 (task-sufficient presentation), U2 (native effective analysis), U3 (measured fibres), U4 (macroscopic response), and Effective Analysis.

Current status: T1 research calibration / stable Theory Map unchanged.