Tri-Engine Soliton: The Operating Manual

Quick start

Part 2 runs three copies of the geometric engine, one for each system around Sgr A* (I: the black hole's domain, II: the nuclear region, III: the Galaxy). On every pulse it compares the three copies exactly. Everything they fail to account for collects in one place, the emergent fourth, drawn as a crystal at the centre.

1. Open the page and press Play. The page opens paused at time zero, like every engine page on the site. Three.js loads, both 3D views appear, and the verification runs on its own. The log ends with 19 checks PASS, 0 FAIL; 66/66 lines identical, and the badge next to the log title turns green.
2. Read the badge in the crystal's top-left corner. At start it says LOCKED · Y_R = 0, H = 1. With no noise and no signal the model accounts for every measurement exactly, and the frame legs close exactly.
3. Look at Panel C's first table. All three rows say LOCK with \(\chi = 0.577350 = 1/\sqrt3\). Each class's three nodes, one per engine, make an exactly equilateral triangle.
4. Move Noise to 1σ. The badge changes to BREATHING, the triangles deform (\(\chi\) drops just below \(1/\sqrt3\)), the trefoils' streams pull together and the core drifts off centre. The residuals are inside the tolerance that noise allows.
5. Press Inject error. One of the four measured frame legs is perturbed. The badge changes to ACTIVE, the holonomy row says FAIL with the closure angle, and the three trefoils shear against each other. Press it again (it now reads Clear error) and set Noise back to 0 to return to the lock.

New to the series? Part 1 builds the three systems and the residual from one engine. Its ideas (residual parity, frame closure, the tiles) are reused here without change. The companion guide explains both parts lemma by lemma.

The screen

From top to bottom: the site bar, the title, the control bar, four panels, the telemetry log and the notes.

Tags, tooltips and theme

Every claim on the page carries one of the monograph's epistemic tags. The legend under the title lists them, and hovering a tag (or tapping it on a phone) shows its basis: which lemma it rests on, or why it is only a design choice.

Most numbers in Panel C also have tooltips: hover a quadrance, a residual or a tile word to see its exact rational value. The Light / Dark button switches the theme. Your choice is stored in the browser and shared with Part 1.

The control bar

The bar under the title holds everything that changes the simulation's state. The crystal's own display controls sit under the crystal (Chapter 7).

The clock

Part 2 runs on engine v10's master clock. One simulated time \(t\) drives every readout, so the engines, the crystal, the packets and the log always agree.

Simulated time advances by \(0.5 \times \text{Speed}\) per real second. Panel A's nodes glide smoothly from one pulse's position to the next and ride their spinning helices. Layer B always evaluates the whole pulse, in the helices' co-rotating frame DEFINITION.

Panel A: three active engines

A 3D view of the three engines nested around Sgr A*, each a five-tier engine at its own system's radii.

What is drawn

Moving the camera

• Orbit: drag with the mouse or one finger.
• Zoom: mouse wheel or pinch. The zoom runs from radius 2 to 4000. The default is 880 with log radii and 620 with linear radii.
• The camera always looks at Sgr A*. Toggling Log radii resets the zoom.

Panel B: the crystal

The emergent fourth as a Tier 3 crystal, ported from torsional_twistor_tier_3_crystal_2.html and set on the monograph's geometry. It has its own 3D view: drag to orbit, wheel or pinch to zoom (radius 3 to 24, default 6.4). The view turns the body diagonal (1, 1, 1) upright.

What is drawn

How it responds

• A class residual (\(Y_R/\sigma = r\)) shifts that pair along its own 120° direction by \(0.25\tanh(r/3)\) cubits and pulls its two streams together. They merge fully at \(|r| = 5\).
• The sum \(s = a + b + c\) of the three residuals lifts the poles along the axis by \(0.25\tanh(s/3)\).
• The core moves to \((z \text{ in the plane} + s \text{ on the axis})/3\), through a saturating scale that keeps it inside the cube.
• A frame error (open closure failing) twists the three trefoils against each other about the axis, by up to 0.7 rad.
• All of these ease toward their targets (12% per frame), so the crystal glides rather than jumps.

The readouts

• State badge (top left): LOCKED, BREATHING or ACTIVE (Chapter 12).
• Step readout (bottom right): step k = 3 · spinor 2i · sign +. The spinor is \((1+i)^k\) reduced to its direction, "window" marks steps 2 and 6, and the sign is the spinor's sign over the 720° turn (it flips at frames 180 and 540).
• STEP 000/720 bar: the outer-gear frame. Click anywhere on the bar to seek within the current epoch. The particle flow advances at most one step per display frame, so the streams never strobe at high speed, while the counter and the spinor turn follow the gear exactly.

Sliders

The sliders use the crystal file's units: 20 per cubit.

Checkboxes

The 4 × 4 layer and the gauge

The §7.7 layer

Below the crystal, a 4 × 4 grid lays out all 16 streams as §7.7 places a structure's engines. The 12 boundary cells hold the 12 streams in ring order, clockwise from the top-left corner. Ring cell \(j\) holds class \(j \bmod 3\) of structure \(\lfloor j/3 \rfloor\). The 4 interior cells hold \(w, x, y, z\) of the measured holonomy \(H_1H_2H_3H_4\).

• Cell borders use the class colours. Cells 9, 10 and 11 are the residuals: they turn amber within tolerance and red past it.
• The interior reads \(w = 1\), \(x = y = z = 0\) exactly when the legs close. Otherwise it is tinted and shows the approximate values. Hover any cell for its exact rational.
• Shifting the ring by 3 cells (the square's 90° turn) keeps every class in its own group of four. Shifting by 4 (the 120° step) moves each group to the next class. Panel D checks both.

The soliton gauge (Proposition 7.14)

The gauge readout turns the three residuals into the crystal's position, and back:

The guide has a gauge lab where you can type your own residuals.

Panel C: triple parity

Panel C holds the exact arithmetic, in four sections. Every value here is a BigInt rational. The decimals you see are rounded for display, and the tooltips give the exact value.

Cross-engine TQF per class

The note under the table gives the node and tier being sampled and the largest bit length involved.

The triple quad: residual parity

Each row is one class's group of four: the three systems' model values and the residual R. Their sum is the measured total \(Y\), which acts as the parity.

• Click a cell to erase it. With one loss in a row the cell shows ↺ value ✓: it was rebuilt exactly as \(Y\) minus the other three, and the group column reads rebuilt.
• Erase a second cell in the same row: both show lost and the group reads 2 lost. One total cannot fix two unknowns.
• You can lose up to three values at once, one per row, and still rebuild everything (Proposition 7.11). Restore all brings every cell back.

Holonomy: engine frames and measured legs

16 Einstein Tiles and the Merkle root

Each update packs 16 values into 64-bit tiles (monograph §6.1). The first three rows hold the twelve streams: astrometry, timing and velocity for System I, II, III and R. The fourth row holds the four holonomy projections \(w, x, y, z\). A ≈ mark means the 32-bit float in the tile is not exactly the rational it stands for, which is why each leaf also carries the exact value.

The line below gives the root \(\tau_1\) of the SHA-256 chain over all 16 leaves, the current step and its phase (onload at step 2, offload at step 6, third otherwise), and the South Halo bytes. The South Halo is a quick pre-filter, not an integrity check.

Tamper with one leaf adds \(2^{-24}\) to one random value and re-hashes. The root always changes. The tile word may not, because a float can be too coarse to show the change, and the South Halo never does. That is why the chain hashes the exact value.

Panel D: lemma rows

Each row names a monograph result, says what it governs on this page, and re-checks it live where a check applies. The summary line counts the results. At start it reads 19 live checks pass · 7 rows are definitions, heuristics, counts or not used.

The other seven rows are not tests. They record a definition (the engine-frame loop, the §7.5 bipyramid, the crystal-file port), a heuristic (the π/6 gate), a count-only correspondence (12 = 12), what was deliberately not adopted from the crystal file (its "ECC" bits), and the lemmas Part 2 does not use. A red FAIL anywhere means the page has a bug. The experiments in Chapter 14 never produce one.

Log and verification

The telemetry log at the bottom records what happens. Clear empties it.

Run Verification runs a line-by-line port of verify_galactic_hierarchy.py (the same code as Part 1) and compares each line with the script's own output under Python 3.13. The result goes into the badge next to the log title: 19/19 PASS · 66/66 lines match. A mismatch means the browser's arithmetic disagrees with the reference somewhere. The log then shows which line, and the badge turns red.

Locked, breathing, active

The soliton has three states. They are decided exactly, from Layer B's rationals.

The tolerances scale with the Noise slider DEFINITION:

So with Noise at 0 the tolerance is 0, and anything non-zero is ACTIVE. That is why an injected signal or error goes straight to ACTIVE, while noise alone stays BREATHING.

The lock, by hand

This lab rebuilds one class's triangle from the page's own formulas, in floating point, so you can see how Δφ deforms it. The page does the same in exact rationals.

Each engine \(s\) places its node on tier \(n\) at

then applies its tier's Lemma 4 frame and the rotation \(\sigma^s\). If the three \(\Delta\varphi_s\) agree, the three nodes are \(P, \sigma P, \sigma^2 P\) and the triangle is exactly equilateral.

Try equal residuals: all three engines turn by the same Δφ, so the triangle stays equilateral even though every residual is 2σ. Tension and torsion act the same way in all three engines, so they never break the lock either.

Experiments

Each recipe starts from a fresh page (reload, or set every control back to its default). The outcomes below were recorded on the J1745-2900 sightline at epoch 0 with the page paused. Noise values change from epoch to epoch, so the exact numbers will differ at other epochs, but the pattern will not.

1 · The baseline lock

Do nothing. The badge reads LOCKED. All three TQF rows read LOCK with χ = 0.577350 and Δφ = 0.0 · 0.0 · 0.0. The gauge is (0, 0, 0) with recovery exact, the measured legs read PASS exact, and Panel D shows 19 checks passing.

2 · Noise

1. Set Noise to 1σ. The badge changes to BREATHING. All three classes deform slightly (χ ≈ 0.57731, 0.57732, 0.57723). The Δφ column shows offsets of a few hundred mrad that differ between engines. The gauge reads about (−0.88, −1.62, 0.72) and the legs read within tolerance (about 161 µas).
2. Raise it to 3σ. The residuals triple (−2.65, −4.87, 2.17) but so does the tolerance, so the badge stays BREATHING. The streams of the timing pair nearly merge, since |b| approaches 5.

3 · Coupling off

With Noise at 3σ, set Coupling κ to 0. All three triangles snap back to LOCK (Δφ = 0) while the badge stays BREATHING and the gauge keeps its values. The triangles show the phase coupling; the soliton shows the residuals.

4 · Tension and torsion

Back at the defaults, set Tension to 40° and Torsion to 15°. Panel A's tiers tilt and the triangles change shape on screen, but Panel C still reads LOCK with χ = 0.577350: the turn is the same in all three engines. The π/6 gate reads closed (Δθ = 25°). Set Torsion to 10° and it opens (Δθ = 30°), and nothing else changes.

5 · A lensing event

1. Choose Lensing event. The log notes the injection and the closest approach about 3.57 years away. At first nothing changes: the event starts at its edge, where its signal is zero by construction.
2. Press Epoch ▶ fourteen times (to about 2030.3). The badge reads ACTIVE. Astrometry and timing are deformed, with Δφ = −523.6 · 0.0 · 0.0: System I's phase saturates at −π/6 while II and III stay at 0. Velocity stays LOCK. The gauge's timing residual is millions of σ, so the core sits at its saturated offset.

6 · A Shapiro event

Choose Shapiro event and step fourteen epochs. Now only timing deforms. Astrometry and velocity stay locked, because a Shapiro delay has no astrometric term.

7 · The scalar field and its range

1. Choose Scalar field on J1745-2900 (0.10 pc). Timing and velocity deform, with Δφ non-zero in Systems I and II and zero in III. Astrometry stays locked. The badge reads ACTIVE.
2. Switch the sightline to J1746-2850 (27.2 pc). Everything locks again. The Yukawa factor \(e^{-r/\lambda}\) with λ = 1 pc is about \(10^{-12}\) there, far below the 2⁻²⁴ grid Layer B uses, so the residual is exactly zero. The test can only see a short-range field on sightlines inside its range.

8 · A frame error

Press Inject error. The badge reads ACTIVE, and the measured legs read FAIL with an angle (the leg and the size are random: one run gave 12.15°). The four interior cells turn red, and the trefoils shear. All three triangles stay LOCK and the gauge stays at 0, because a frame error moves no node and adds no residual. Panel D's Lemma 12 row still passes: catching the error is the expected result. Press Clear error.

9 · Erasure

In the triple-quad grid, click one cell in each row: three cells show ↺ with ✓ and every group reads rebuilt. Click a second cell in one row: both cells in that row read lost. Press Restore all.

10 · Tampering

Press Tamper with one leaf a few times. The root changes every time. The South Halo never changes, and the tile word sometimes does not either, when \(2^{-24}\) is below the float's resolution at that value.

11 · Freeing the crystal

Untick Lemma 8.2 geometry and set Torus radius to 25 (the crystal file's own value). The lobes leave the rays. Panel D's Lemma 8 Step 6 row reads free with the offset (about 4.8 units), and the summary drops to 18 live checks. Tick it again to snap back.

12 · Windows and the spinor

Pause and click the STEP bar at about one-sixth of its length (frames 90–179). The readout says step k = 2 · spinor i · window. At about three-quarters (frames 450–539) it says step k = 6 · spinor −i · window, with the sign now −: the same orientation, the opposite spinor. Tick Spinor turn to watch the crystal make its 720° turn per epoch.

Troubleshooting

Numbers and constants

Tiers (Lemma 4, the same in all three engines)

\(R_n = 5n^2 + 15n + 15\), \(\rho_n = 2n + 4\), \(W_n = 16 - 2n\). The helix rate is in turns per 60 units of simulated time.

Systems

Sightlines

Measurement classes

Other constants

Glossary

Layer A / Layer B

Layer A is the floating-point physics. Layer B is the exact rational engine: values cross at a 2⁻²⁴ grid and from then on every operation is exact BigInt.

Quadrance Q

A squared distance, the rational-trigonometry replacement for length.

Quadrea A_R

Sixteen times the squared area of a triangle, computed from its three quadrances.

χ

\(\sqrt{A_R}/(Q_1+Q_2+Q_3)\). Never above \(1/\sqrt3\), and equal to it only for an equilateral triangle.

Lock (triangle)

\(3A_R = S^2\) exactly: the three engines agree for that class.

LOCKED (soliton)

Every residual is zero and the measured frame legs close exactly.

Residual Y_R

The measured total minus the three systems' models: the emergent fourth.

Δφ, κ

A node's phase offset, driven by its system's own residual. κ is the coupling strength.

P₄, P₄*

The inferred fourth vertex of a class's triangle (heuristic scale), and its exact closure \(P_{\text{II}} + P_{\text{III}} - P_{\text{I}}\).

Gauge (s, z)

Proposition 7.14's exact, invertible split of the three residuals into an axial part \(s\) and a planar part \(z\).

Open / cyclic closure

Open multiplies four independently measured legs and can fail. Cyclic computes the fourth from the other three and always closes, so it tests nothing (Lemma 12).

ω

The Hurwitz unit \((1+i+j+k)/2\): the 120° turn between engines as a quaternion. ω³ = −1.

Einstein Tile

A 64-bit word (§6.1) carrying a step, wheel sector, structure, gate, index, form, slot, scale and a 32-bit core.

South Halo

The tile's top byte, a quick pre-filter. It is not an integrity check; the Merkle root is.

Merkle root τ₁

The head of the SHA-256 chain over all 16 leaves, each the tile word plus its exact value (Lemma 15).

Pulse, stage, epoch, step

The clock's units (Chapter 5).

Cubit

The engine's length unit: tier radii are in cubits, and the crystal uses 20 display units per cubit.

The companion guide's glossary covers the theory terms.