Methodology & Reproducibility

Program History. This documents the earlier single-spin prototype engines (planck.c / planck3.c): still the correct spec for reproducing the Hopfion Zoo and Hopf's Loom milestones. The active engine is src/moore26.c — see moore26 Findings; the meso3i era is archived in the Friction Substrate monograph and the Multi-Channel Bridge

This document specifies the formal mathematical architecture, state-space invariants, causal thermodynamic laws, and deterministic reproduction protocols for the Langevin's Wake (planck.c) and Hopf's Loom (planck3.c) discrete lattice engines.

All macroscopic behaviors observed in The Hopfion Zoo and Hopf's Loom, including 3D topological soliton stability, inertial rigidity under gradient stress, and multi-knot scattering cascades, emerge strictly from local, integer-only transition operators executing across a stationary 12-neighbor lattice without floating-point dynamics, pseudo-random thermal baths, or continuous differential solvers.

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1. Lattice Topology & Adjacency Invariants

The 3D simulation domain is a finite, three-dimensional True Face-Centered Cubic (FCC) Eulerian lattice of runtime dimension (planck3_init() defaults to nodes in browser; planck3_init_dims(256, 256, 256) scales to nodes for headless sweeps and collision basins).

• Coordination Number (): Every node communicates exclusively with its 12 nearest neighbors via fixed directional spin ports (0 = stationary/unaligned):
  • Ports 1–6 (Equatorial): 6 planar hex-grid neighbors handling toroidal circulation.
  • Ports 7–9 (Zenith): 3 upper-layer neighbors () handling ascending poloidal flow.
  • Ports 10–12 (Nadir): 3 lower-layer neighbors () handling descending poloidal return.
• Parity-Free A-B-C Stacking: Unlike hexagonal close-packed (HCP, A-B-A-B) lattices where vertical port azimuths invert on alternating layers, every layer in the True FCC substrate offers zenith ports and nadir ports . A single port index hops the identical offset layer-to-layer, tracing a straight 3D diagonal through the A→B→C→A cycle with 0 broken reciprocal edges (verified across 4,740,000 directed edges at ).
• Honest Spatial Embedding: Because the A-B-C stagger lives in the port offsets rather than shifted layer origins, nodes embed into Euclidean render space at where .
• No Global Coordinates: The transition operator is strictly translation-invariant. Nodes have no access to their global index or global tick ; state updates depend solely on the registers of the node and its 12 immediate neighbors.

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2. Discrete Node State Space (Node3)

To eliminate IEEE-754 floating-point truncation drift and chaotic rounding divergence over multi-thousand-tick integrations, each lattice site in planck3.c is represented as a compact 6-byte integer struct (Node3):

Register	Type	Range	Physical / Algorithmic Role
quanta ()	uint8_t		Resident discrete packet count (local mass density; ports hard-refuse once occupancy ).
spin ()	uint8_t		Active directional port index (1–6 equatorial, 7–9 zenith, 10–12 nadir, 0 stationary).
heat ()	uint16_t		Accumulated local routing friction and angular shear excitation.
buffer ()	uint8_t		Staged in-flight flux arriving during the current synchronous tick.
tension ()	uint8_t		Diffusing scalar congestion field emitted by near-deadlock nodes (occupancy T_EMIT_MIN ).

(Note: Load-time provenance masks painted are tracked externally by the analysis harnesses sweep.mjs, boost.mjs, and collide.mjs so the 6-byte core node carries zero particle-ID register overhead.)

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3. Synchronous Transition Operator ()

Time advances in discrete ticks . The engine supports three memory traversal modes sharing the identical planck3_node_tick physics kernel: flat (canonical deterministic global order), halo ( interior / ghost-ring chunks for cache acceleration in laminar regimes), and ordered-chunk.

Phase A: Local Flux, Congestion & Tension Shear

1. Transport Bandwidth Cap (FLOW_CAP): An active node () transfers a bounded integer count of quanta per tick along port , enforcing a strict causal speed limit of .
2. Near-Field Congestion & Far-Field Tension:
   • Congestion Routing (Near-Field): When a target neighbor approaches the deadlock occupancy threshold (DEADLOCK = 200), blocked flux deflects across adjacent reciprocal ports rather than clipping.
   • Propagating Tension Field (Far-Field): Nodes at or above T_EMIT_MIN = 200 emit scalar tension (), which diffuses across neighbors and decays per tick, creating a graded potential well of characteristic reach cells. Arriving flux inside a tension gradient shears up-field into flowing neighbor buffers (T_DIVERT = 4).
3. Phase Lock & Angular Phase Friction (): A node's painted phase steers flux cleanly when incoming momentum lies within the lock tolerance of the lattice port geometry. Misaligned handoffs and head-on conflicts incur a discrete integer penalty added to the heat register:
4. Conservative Unwinding Spray (+spray): When local heat breaches thermal_limit (default ), the node undergoes topological unwinding. Mass is never deleted; the failing knot's quanta sprays into neighboring buffers as un-actualized radiation, preserving strict global conservation.

Phase B: Degree-Normalized Diffusion & Quadrature Drain

1. Degree-Normalized Diffusion: To maintain identical diffusivity across Hex6 (), Oct8 (), and True FCC () substrates without bit-shift overflow, per-tick neighbor exports are normalized by coordination degree:
2. Quadrature Bandwidth Drain (BWF_LUT): Total node processing capacity is bounded by the vector budget . The available cooling fraction at routing load is resolved via a 16-bit integer lookup table (uint16_t BWF_LUT[256], widened from uint8_t to prevent wraparound at low load ):

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4. Causal Thermodynamics & Cold Matter Inversion

A foundational distinction of this framework is that the thermal register (heat, ) does not represent a statistical-mechanical ensemble average, stochastic Brownian velocity dispersion, or coupling to an external probabilistic heat bath. In classical continuum physics, standard parabolic heat and fluid equations (Fourier and Navier-Stokes) unphysically permit infinite propagation speeds, forcing relativists and aerospace engineers to bolt on second-order hyperbolic delay patches (such as Maxwell-Cattaneo or Israel-Stewart "extended causal thermodynamics") to prevent thermal flux from outrunning the speed of light. On our discrete 12-neighbor lattice, causal transport () is native to the adjacency graph by construction, requiring zero differential relaxation patches, and implements Discrete Causal Thermodynamics: an exact integer accounting of directional routing impedance and finite node bandwidth. The core transition operator is PRNG-free; the optional stochastic-foam scenarios (e.g., the RFC-042 floor) inject rand()-driven ambient flux on top of it.

Property	Classical / Continuum Thermodynamics	Discrete Causal Thermodynamics (planck3.c)
Microscopic Basis	Probabilistic ensemble over continuous phase space ().	Deterministic integer state transitions on a 12-port FCC lattice.
Causal Speed Limit	Infinite in classical Fourier/Navier-Stokes; patched via hyperbolic delay terms in Israel-Stewart.	Native (FLOW_CAP) enforced by neighbor adjacency.
Origin of Heat ()	Random microscopic kinetic jitter ().	Exact integer residual of angular port misalignment () and hop friction ().
Cooling Mechanism	Passive statistical diffusion into an infinite external reservoir.	Local per-tick quadrature bandwidth allocation (BWF_LUT).
Equilibrium Fate of Solitons	Entropic dissipation into uniform thermal noise unless externally pumped.	Cold Matter Inversion: Zero-friction () phase-locked cores quench to the thermal floor.

4.1 The Quadrature Causal Bandwidth Budget

Every lattice site possesses a fixed, finite causal processing capacity per discrete Planck tick (). A node cannot simultaneously route maximum mass flux () and execute maximum thermal relaxation (). When local routing load is below the saturation cliff (), unallocated causal capacity drains accumulated heat toward the register floor at the configured dissipation rate (default ). When routing congestion saturates the node (), , choking thermal drain and trapping heat until the node unwinds via radiation spray.

4.2 Cold Matter Inversion

In continuum statistical mechanics, localized matter is treated as a high-energy excitation that naturally decays into a lower-energy, high-entropy background. In Causal Thermodynamics, finite node bandwidth inverts this relationship:

1. Unstructured Vacuum Foam Is Warm: In an active zero-point or turbulent background (such as the RFC-042 reciprocating floor hexzpe), opposing streams continuously collide at , generating per cell and settling into a warm steady-state equilibrium floor ().
2. Phase-Locked Topological Solitons Are Cold: Within a coherent 3D Hopfion traced by pathfinder.mjs, every loop forms an exact in-degree-1 cycle where each hop stays within the Phase-Lock window at sub-deadlock density (). Because destructive head-on clashes vanish while BWF_LUT cooling capacity remains open, the soliton continuously drains its internal heat toward the baseline floor.

Consequently, a stable particle does not fight entropy through external energy pumping; it persists as a cold, phase-locked causal anchor carving a frictionless slipstream through a warm substrate.

4.3 Hopfions as Causal Thermodynamic Attractors

Because crossing trajectories incur heavy Phase Friction and thermal unwinding while closed, linked trajectories circulate without collision penalties, 3D Hopfions act as natural causal thermodynamic attractors of the True FCC substrate. Under perturbation or turbulent relaxation, misaligned packets heat up, lose Phase Lock, and shed radially as evaporative spray: leaving the surviving linked core to quench into a stable eigenstate.

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5. Conservation Laws & Quantitative Telemetry

Every headless run continuously audits four primary macroscopic observables:

1. Exact Mass Conservation: Across closed runs, total system occupancy (resident quanta + staged buffer) is strictly invariant:
2. Skeleton Centroid & Velocity (v10): Computed exclusively over surviving painted cells (painted != 0) inside the primary connected BFS component, making trajectory tracking immune to boundary wall-flood and ambient accretion bias.
3. Knot Retention (knotRetain ): The fraction of initial knot mass remaining bound within the live connected component.
4. Painted Coherence (coherence ): Structural retention of the original closed-streamline skeleton across the duration of the run.

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6. Falsification Summary (Negative Results)

Before isolating the 37 stable 3D Hopfion eigenstates on the True FCC lattice, five alternative hypotheses were falsified and logged in The Open Frontier:

• Active ZPE Shielding [RFC-042, hexzpe]: A reciprocating floor maintained exact macro-isotropy () at , but caused 2D rings to absorb floor mass and sever earlier than sparse foam.
• Unclosed Vortex Envelopes [RFC-043, hexvort]: Continuum-quantized tangent rings leaked radially and clumped into "vortex necklaces" (ccirc ), proving that streamline closure by construction is required for zero-loss survival.
• Planar Flank Shielding [RFC-044, +flank]: Local angle refusal could not distinguish corner turns (dd=1) of closed hex orbits from lateral infiltration.
• Planar Z-Relief Channels [RFC-045 / RFC-046, +zshadow / +zhopfion]: Out-of-plane overflow drains suppressed deadlock knots (fdead ) but bled of band mass because displaced re-entry destroyed toroidal-poloidal winding topology.
• HCP Parity Frustration [RFC-047]: On an A-B-A-B stacked-hex 12-port grid, layer-alternating vertical azimuths deleted required descent ports on even layers (0/29 lock-tolerant closures), necessitating the parity-free A-B-C True FCC lattice [RFC-048].

Continuation note: this summary ends with the Hopfion-dictionary era. The subsequent grounded-measurement campaign characterized a single-spin prototype node whose constraints are implementation artifacts, not target physics; its record remains in git history. The multichannel findings that carry forward are consolidated in The Multi-Channel Bridge.

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7. Deterministic Reproduction Suite

All empirical claims on this site can be reproduced locally from source.

1. Build the Engines (Docker / Emscripten & GCC)

Always compile via src/build.sh so modularized exports (createPlanck, createPlanckHex, createPlanck3) and -DGIT_REV provenance stamps are preserved:

# Build 2D and 3D WebAssembly modules
cd src
docker run --rm -v "$PWD/..:/src" -w /src/src emscripten/emsdk:3.1.61 bash build.sh

# Run native C verification against reference state dumps (0 mass-channel diffs over 400 ticks)
docker run --rm -v "$PWD:/src" -w /src/src gcc:13 \
  sh -c "gcc -O2 -Wall planck3.c planck3_test.c -o /tmp/p3t -lm && \
         /tmp/p3t /src/test/loom/t0.bin 60 /src/test/loom/t400.bin"

2. Trace & Sweep the 37-Specimen Hopfion Dictionary

pathfinder.mjs runs a memoized Phase-Lock DFS (/hop, strict in-degree-1 vetting) on the 12-port FCC graph and writes seeds/hopfion_<R0>_<R1>.json. sweep.mjs loads each seed into planck3.wasm (dissipation = 15, thermal_limit = 50000) and writes knot_dictionary.json:

# Trace canonical 100x100x40 Hopfion (19 loops, 27,971 cells, 1,678,260 quanta)
node script/pathfinder.mjs --w=100 --h=100 --d=40 --r0=25 --r1=9

# Trace 256^3 dictionary candidates and run 400-tick stability sweep
node script/pathfinder.mjs --r0=32,64,72,80 --r1=8,12,16,20
node script/sweep.mjs --config=config/full-sweep.json

# Monitor live sweep/basin progress
node script/status.mjs --config=config/full-sweep.json

# Export settled ZOO3 binary specimens for the WebGL Hopfion Zoo
node script/export-zoo.mjs

3. Reproduce Kinematic Boost & S-Matrix Collision Basins

boost.mjs tests inertial stability under injected boundary flux envelopes; collide.mjs dual-spawns dictionary knots along FCC axes and classifies scattering outcomes via dual painted-mask BFS while exporting sparse ParaView .vtk frames (logs/vtk/<run>/frame_NNNNN.vtk):

# Kinematic interference basin (Stage 1 gentle / Stage 2 tension)
node script/boost.mjs --config=config/boost-pilot.json

# 256^3 S-Matrix collision runs (head-on b=0 and asymmetric glancing b=15)
node script/collide.mjs --config=config/collide-matrix.json

4. Binary Telemetry Formats (ZOO3 and P3ST)

• ZOO3 Sparse Specimen Format (/public/hopfion/*.bin):
  • Header (20 bytes): 0x5a4f4f33 ("ZOO3" big-endian magic, 4B) + w (u32 LE, 4B) + h (u32 LE, 4B) + d (u32 LE, 4B) + n (u32 LE active cells, 4B).
  • Records ( bytes): [x: u8, y: u8, z: u8, quanta: u8, spin: u8, tension: u8, painted: u8].
• P3ST Full Lattice Snapshot (/public/loom/hopfion.p3st):
  • Header (16 bytes) followed by of raw Node3 records.