Add Feynman path-integral shadow witness note

6-Documentation/famm/FEYNMAN_PATH_INTEGRAL_SHADOW_WITNESS_NOTE.md

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+# Feynman Path Integral Shadow Witness Note
+
+## Purpose
+
+Adapt the Feynman path-integral idea into the FAMM / BraidStorm / Shadow-Control math stack.
+
+The referenced notebook visualizes the path integral by generating many candidate histories from Point A to Point B with randomized harmonic deviations, then showing the stationary-action path emerge as the visually dominant classical trajectory. The project-useful interpretation is not the animation itself, but the structure:
+
+```text
+all possible histories
+→ action phase witness
+→ destructive shadow cancellation
+→ stationary survivor geodesic
+→ receipt-bearing classical path
+```
+
+## External reference
+
+Reference implementation:
+
+```text
+zombimann/Mathematical-video-animations-and-visualization
+Feynman_Path_Integral_Visualization.ipynb
+```
+
+The notebook states the key visual/theoretical frame: Feynman's formulation as a sum over all possible histories, non-classical paths destructively interfering, and the stationary-action/classical path emerging through constructive interference. It implements this with 400 randomized harmonic paths and a final stationary-action reveal.
+
+## Standard path-integral form
+
+```math
+K(B,A)
+=
+\int \mathcal D[x]\;e^{iS[x]/\hbar}
+```
+
+where:
+
+```math
+S[x]=\int L(x,\dot x,t)\,dt
+```
+
+The stationary-action condition is:
+
+```math
+\delta S[x_\star]=0
+```
+
+The classical path is the survivor path:
+
+```math
+x_\star
+=
+\operatorname*{arg\ stationary}_{x:A\to B} S[x]
+```
+
+## Project translation
+
+The path integral becomes a shadow-witness filter:
+
+```text
+candidate path          = hypothesis strand
+path action S[x]        = route cost / phase witness
+exp(iS/hbar)            = interference receipt
+non-stationary path     = shadow / coarsening contribution
+stationary path         = survivor geodesic
+endpoint condition A,B  = boundary invariant
+```
+
+## Universal Shortcut Center packet
+
+```math
+\Gamma_{\mathrm{path}}
+=
+(
+X_{\mathrm{paths}},
+\pi_{\mathrm{action}},
+W_{\mathrm{phase}},
+R_{\mathrm{stationary}},
+I_{\mathrm{endpoint}},
+G_{\mathrm{boundary}},
+K,
+\epsilon
+)
+```
+
+| Packet term | Meaning |
+|---|---|
+| `X_paths` | high-cost set of all histories from A to B |
+| `pi_action` | projection from path to action/phase |
+| `W_phase` | lower-cost interference/phase witness |
+| `R_stationary` | reconstruction/decision map selecting stationary path |
+| `I_endpoint` | endpoint invariant: path starts at A and ends at B |
+| `G_boundary` | boundary and admissibility guard |
+| `K` | cost of carrying path ensemble or phase witness |
+| `epsilon` | residual from non-stationary/shadow paths |
+
+## FAMM object
+
+```math
+\mathfrak C_{\mathrm{FeynmanShadow}}
+=
+A_{16}(u_{\mathrm{path}})
+\otimes
+[
+\Sigma_{\mathrm{paths}}
++
+\Sigma_S
++
+\Sigma_{e^{iS/\hbar}}
++
+\Sigma_{\mathrm{stationary}}
++
+\Sigma_{\mathrm{shadow}}
++
+\Sigma_{\mathrm{boundary}}
++
+\Sigma_{\mathrm{receipt}}
+]
+```
+
+## Shadow residual
+
+Let each candidate path carry phase:
+
+```math
+\Phi[x]=e^{iS[x]/\hbar}
+```
+
+Define stationary deviation:
+
+```math
+R_{\mathrm{stationary}}[x]
+=
+\|\delta S[x]\|
+```
+
+Define the shadow contribution:
+
+```math
+\Omega_{\mathrm{shadow}}
+=
+\left\|\sum_{x\in\mathcal P_{\mathrm{nonstat}}} e^{iS[x]/\hbar}\right\|
+```
+
+A good survivor geodesic has:
+
+```math
+R_{\mathrm{stationary}}[x_\star]\approx0
+```
+
+and the non-stationary family is either destructively cancelled or converted into a scar/coarsening field:
+
+```math
+\Omega_{\mathrm{shadow}}\to0
+\quad\text{or}\quad
+\Omega_{\mathrm{shadow}}\mapsto\Omega_{\mathrm{scar}}
+```
+
+## BraidStorm adaptation
+
+Each path is a braid strand:
+
+```math
+s_i
+=
+(x_i,S_i,\Phi_i,\epsilon_i,\Omega_i,\rho_i)
+```
+
+A crossing combines candidate histories:
+
+```math
+\beta_{ij}
+:
+(s_i,s_j)
+\to
+(s_i',s_j',\Delta S_{ij},\epsilon_{ij},\Omega_{ij},r_{ij})
+```
+
+Survivor rule:
+
+```text
+small action variation     → survivor candidate
+large phase mismatch       → destructive shadow / coarsening
+stable repeated phase      → center geodesic
+failed boundary condition  → Warden scar
+```
+
+## Navier-Stokes shadow-control adaptation
+
+For the NS16 witness route, the path-integral wrapper becomes a way to search over closure histories:
+
+```text
+candidate closure paths
+→ action / residual / witness phase
+→ unstable paths cancel or scar
+→ stationary witness route survives
+```
+
+This is useful because the project is already using shadows to locate where the witness packet fails to control the dangerous PDE term. The path-integral adaptation adds a principled language for treating failed/non-stationary routes as cancellation evidence rather than noise.
+
+## Builder-Judge-Warden mapping
+
+| Role | Path-integral use |
+|---|---|
+| Builder | proposes candidate path family / action functional / closure route |
+| Judge | checks endpoint boundary, stationary-action condition, invariant preservation, and receipt |
+| Warden | blocks false classical-path claims, unbounded path ensembles, hidden boundary failure, and empirical-only survivor selection |
+
+## Stack placement
+
+```text
+FEYNMAN_PATH_INTEGRAL_SHADOW_WITNESS_NOTE
+→ BraidStorm hypothesis strands
+→ Shadow Control Gap Map
+→ Golden Braid Centering Gate
+→ FAMM Scar Ledger
+→ NUVMAP Delta-DAG
+→ Builder-Judge-Warden
+→ survivor geodesic receipt
+```
+
+## Warden boundary
+
+This note does not claim the notebook is a rigorous numerical path-integral solver. It uses the path-integral structure as a project primitive:
+
+```text
+many candidate histories
+→ phase/action witness
+→ shadow cancellation or scar
+→ stationary survivor route
+```
+
+Allowed claim:
+
+```text
+The path-integral adaptation gives the project a way to treat non-surviving candidate routes as shadow/cancellation evidence, while stationary-action paths become survivor geodesics subject to Judge/Warden receipts.
+```
+
+Disallowed claim:
+
+```text
+A visualization of random harmonic paths proves quantum mechanics, Navier-Stokes regularity, or any project theorem by itself.
+```
+
+## Project sentence
+
+The Feynman path-integral wrapper turns all possible histories into a shadow-witness filter: every candidate path contributes an action phase, non-stationary paths cancel into the shadow/coarsening field, and the stationary-action path emerges as the receipt-bearing geodesic that survives interference.