From One Engine to the Triple Quad: A Guide to the Galactic Lensing Hierarchy

What the simulation shows

The Galactic Lensing Hierarchy treats the centre of the Milky Way as three nested systems, each organized like one copy of the geometric engine, and keeps exact books on what the three of them explain and what they leave over.

Every measurement of a pulsar near the Galactic Centre (where it appears on the sky, how its spin period drifts, how fast it moves) is shaped by mass at every scale between the black hole and the edge of the Galaxy. The simulation splits that mass into three ranges of radius:

Whatever the three models fail to explain is the emergent fourth, written \(Y_R\). It has no mass model of its own. It is defined as data minus the three models, so the four always add up to the measurement exactly. This is the additive form of the monograph's implied fourth (Lemma 7 Part 3(d)).

Each measurement comes in three classes (astrometry, timing and velocity), and each class splits into the same four parts: I, II, III and R. Three classes of four make twelve streams. That is the triple quad: the monograph's twelve-engine hierarchy of three groups of four (§7.4, Proposition 7.11), applied to the galaxy.

The page has two layers, as the design specification requires:

• Layer A, physics. Weak-field general relativity in ordinary floating point: lensing, Shapiro delay and accelerations. These are approximate, as all physics is.
• Layer B, bookkeeping. Exact rational arithmetic in JavaScript BigInt: the residual parity, the quaternion frame closure, the packet layout and the Merkle chain. Layer B adds no error of its own. It cannot make a measurement more certain, only make sure nothing is lost or invented on the way.

The route in five steps

The whole guide follows one path. Step through it here first; each stage has its own chapter later.

1. One engine. Five tiers in disjoint shells, a 30-node wheel, six channels, a 12/10/8 gear clock, an 8-step cycle, and a 64-bit packet (Lemmas 4, 8, 10, 11 and 12, §6.1). Chapters 4–7.
2. A fourth from three. Three holonomies fix a fourth, \(H_4 = (H_1H_2H_3)^{-1}\). It is a parity element that rebuilds any one lost member (Lemma 7 Part 3(d)). It tests something only if measured independently (Lemma 12). Chapter 8.
3. Many engines. Engines tile space without touching, and exchange data only at the two handshake windows of the cycle (Theorems 7.1–7.10). Chapter 9.
4. Twelve engines, three groups of four. Nine data engines and three implied fourths: the triple quad. One loss per group is recoverable (§7.4, Propositions 7.11–7.14). Chapter 10.
5. The galaxy. Systems I–III are three engines, the residual is the fourth, and three observable classes make three groups of four. Frame closure and the four-structure frame change are the same algebra again (§7.6, Theorem 7.18). Chapters 12–15.

Reading the tags

The monograph, the specification and the simulation all label every claim with one of nine status tags. They tell you how much weight a statement can bear.

Five tiers and their shells

The engine is five nested tori, one per tier, centred on the origin. All lengths are in cubits \(\mathcal{C}_0 = \lambda/16\), about 53.7 cm at 40 Hz in room air (§6.2).

Lemma 4 fixes the tier laws for \(n = 0, \dots, 4\):

where \(R_n\) is the major radius, \(\rho_n\) the minor (tube) radius and \(W_n\) the number of helical windings. Two conservation laws follow at once, and the page checks both on every load PROVEN:

Disjoint shells (Lemma 10). Whatever way a tier is rotated, its torus stays inside the spherical shell \(R_n - \rho_n \le \|x\| \le R_n + \rho_n\). The shells are \([11,19], [29,41], [57,73], [95,115], [143,167]\), with gaps 10, 16, 22 and 28 between them PROVEN. The voxel cube \([-2,2]^3\) of the Lower Manifold reaches only \(2\sqrt3 \approx 3.46\), so it is separated from the innermost tier by \(11 - 2\sqrt3 \approx 7.54\,\mathcal{C}_0\) (4.05 m). The whole engine fits in a ball of radius 167.

Tier frames: Lemma 4 versus engine v10

Lemma 4 also fixes how each tier is oriented: \(\mathrm{Rot}(T_n) = \mathrm{Rot}_x(\pi/2)\) for even \(n\) and \(\mathrm{Rot}_y(\pi/2)\) for odd \(n\). The step from one tier to the next is then always the same integer matrix,

a 120° rotation about the body diagonal \((-1,1,1)/\sqrt3\), with \(\mathbf U^3 = I\). Its spinor lift is the Hurwitz unit \(h = (1 - i + j + k)/2\), with \(h^3 = -1\) and \(h^6 = +1\) PROVEN.

Engine v10 uses a different, simpler set of frames: tiers 0, 2 and 4 unrotated and tiers 1 and 3 turned by \(\mathrm{Rot}_x(\pi/2)\), plus the tension, torsion and clutch-slip offsets of its controls. The simulation draws the pure Lemma 4 frames by default: exactly \(\mathrm{Rot}_x(\pi/2)\) and \(\mathrm{Rot}_y(\pi/2)\), with none of the engine's tension, torsion or clutch-slip offsets. In this mode the clutch changes only how fast the helices spin, not how the tiers are tilted. Panel D checks the drawn frames live: every step \(q_{n+1}q_n^{-1}\) between neighbouring tiers must be the 120° turn about \(\pm(-1,1,1)/\sqrt3\). The Lemma 4 frames button switches to the v10 frames, offsets included, for comparison. In the galaxy the tier rotations have no physical meaning (the specification's verdict is "no map"); they only make the nesting visible.

Wheel, channels and clock

Around each tier runs a 30-node wheel; the nodes carry six channels; and one master clock drives everything.

The 30-node wheel

The wheel has 30 nodes, \(k \in \mathbb Z/30\). Its 8 units, the residues coprime to 30, are \(\{1, 7, 11, 13, 17, 19, 23, 29\}\). Lemma 4 Part 5 splits them by the Legendre symbol mod 5 into \(\mathcal B_+ = \{1, 11, 19, 29\}\) (quadratic residues) and \(\mathcal B_- = \{7, 13, 17, 23\}\) PROVEN. A tile's wheel field (bits 60–58) names one of the eight units.

Six channels

The node index also picks a channel \(k \bmod 6\), which is the pair \((k \bmod 3,\ k \bmod 2) \in C_3 \times C_2\). By the Chinese remainder theorem this is a bijection \(\mathbb Z/6 \to C_3 \times C_2\) PROVEN (script check H3). In the galaxy the two factors get physical meaning DEFINITION:

So each system becomes a 3 × 2 table: one row per class, one column for the model and one for the data. Panel B of the simulation shows exactly that table for each system, with the active channel in gold.

The master clock

The simulation uses engine v10's clock unchanged:

• pulseCount = ⌊t · 60⌋: 60 pulses per simulated second.
• Node \(k\) = pulseCount mod 30, and stage = ⌊(pulseCount mod 150)/30⌋. The active tier sweeps T4 → T0 over 150 pulses.
• outerFrameN = 12 · pulseCount mod 720 drives the 12/10/8 gear chain of Lemma 12: an outer gear of 720 frames, a middle gear of 72 and an inner gear of 10. Lemma 12 Part 1 proves the middle frames split into inner blocks of sizes (8, 7, 7, 7, 7, 8, 7, 7, 7, 7) PROVEN.
• Rotor step \(k_8\) = (pulseCount mod 8) + 1: the 8-step cycle of Lemma 8 (next chapter).
• One 720-frame cycle (60 pulses) is one observation epoch of 0.25 yr in the galaxy DEFINITION.

The 8-step cycle

Every engine runs an eight-step cycle \(\theta_k = k\pi/4\). Two of its steps are special: they are the only moments at which two engines can exchange data.

Lemma 8 builds the cycle step by step: a circle phase (Step 1), the conformal lift with nine rays and a 4-simplex (Step 2), sixteen half-cubes (Step 3), dilation to the voxel cube (Step 4), a sense-reversing fold into Cube 3 (Step 5), the \(D_3\) trefoil and its projection into hyperbolic space (Step 6), and the apex peaks and holonomy (Steps 7–8).

Its spinor bookkeeping is exact. One step is a 90° rotation with spinor \(1 + i\), and Lemma 13 Part 4 shows that in integer arithmetic

Dividing each power by the gcd of its parts leaves the eight unit states \(1+i,\ i,\ -1+i,\ -1,\ -1-i,\ -i,\ 1-i,\ 1\). The transverse amplitude \(|\sin\theta_k|\) is 1 only at \(k = 2\) and \(k = 6\), where the reduced spinors are \(i\) and \(-i\): the same half-turn about the \(x\)-axis with opposite spin sign PROVEN (Theorem 7.5). These are the two handshake windows.

At both windows a partner sees the same spatial orientation, so two engines need not agree on where they are in the double cover. The spin sign is the one bit the geometry never exposes. The monograph calls \(k = 2\) the onload window and \(k = 6\) the offload window.

The specification (§4.7, after Proposition 7.14) schedules the galaxy's traffic on these windows DEFINITION:

• Onload (\(k = 2\)): new data in.
• Offload (\(k = 6\)): model updates out.
• Third phase (other steps): the residual parity traffic and the time reference.

Panel D's cycle strip shows the current step, its spinor and its phase. Every packet carries the step in bits 63–61.

The 64-bit Einstein Tile

Everything an engine says is said in 64-bit words. A receiver must be able to decode a packet before it knows what kind of packet it is, so the fields below sit at fixed positions in every packet.

Exactness, honestly

Form 3 stores one binary32 value bit for bit, so it is exact about that binary32 value. Layer B's numbers are exact rationals \(n/2^{24}\) or sums of them. A binary32 holds such a rational exactly only when it has at most 24 significant bits. When it does not, the simulation marks the packet with "≈" and puts the exact rational in the packet's Merkle leaf, so the integrity chain covers the exact value even when the tile shows a rounded copy. The value 0.375 above is exact; try 0.1 to see the difference.

The South Halo is not a checksum

The top byte (bits 63–56) is the South Halo. Lemma 15 Part 2 proves it splits the \(2^{56}\) possible payloads into 256 buckets of \(2^{48} \approx 2.8 \times 10^{14}\) each. It is a fast cache pre-filter that rejects packets from the wrong phase or wheel. It is never an integrity check PROVEN. Integrity comes from the recursive Merkle chain of Lemma 15 Part 3 (Chapter 15).

The fourth from three

The single idea behind the whole hierarchy: in a group of four that must close, any three determine the fourth. That makes the fourth a spare, able to rebuild one loss. It also means a fourth computed from the other three can never catch an error.

Already inside one engine: V4 and V4*

Engine v10 triangulates each node against an upper and a lower tier point (V1, V2, V3) and infers a fourth vertex V4. Its exact parallelogram closure is \(V_4^* = V_2 + V_3 - V_1\), so that \(V_1 + V_4^* = V_2 + V_3\): one point of four, fixed by the other three. The engine's displayed V4 scales that closure by its TQF factor qScale, which is an engine heuristic HEURISTIC. The Numogram page draws both side by side.

The group form: Lemma 7 Part 3

For any three holonomy quaternions \(H_1, H_2, H_3 \in S^3\), there is exactly one \(H_4\) that closes the loop PROVEN:

Part 3(d) reads this as a parity element. Since the closure survives cyclic shifts, each of the four is fixed by the other three, so one lost element of a group of four is recoverable exactly; two are not. \(H_4\) carries no information beyond its three parents. Offloading it returns only their product, never their individual histories.

The lattice form: Theorem 7.2

Take the lattice \(L = 2\mathbb Z^3\) and the four offsets \(o_0 = (0,0,0)\), \(o_1 = (1,1,0)\), \(o_2 = (1,0,1)\), \(o_3 = (0,1,1)\). The four translates \(L + o_i\) are disjoint and together make the face-centred cubic lattice \(D_3\). Since \(o_1 + o_2 + o_3 \equiv 0 \pmod 2\), any three translates determine the fourth: the "3:1 interleave" PROVEN. The monograph notes that the implied fourth has three exact twins: this lattice, the group inverse \(H_4\), and the quaternion relation \(ijk = -1\).

The additive form: what the galaxy uses

Measured quantities add. With a measured total \(Y\) and three model contributions, the fourth is the residual \(Y_R = Y - Y_I - Y_{II} - Y_{III}\), and the group closes as \(Y = Y_I + Y_{II} + Y_{III} + Y_R\). That is Proposition G1 of the specification, the additive form of Lemma 7 Part 3(d) (Chapter 13).

Open versus cyclic: when a fourth tests anything

Lemma 12 compares three boundary conventions for the gear chain's holonomy (monograph Table 3):

The galaxy's frame closure (Proposition G2) is the same statement. Four frame legs that are each measured close to 1 when they agree, and an error in any one breaks the closure. If the fourth leg is instead computed as \((H_1H_2H_3)^{-1}\), the product is 1 whatever the error. Try it:

Many engines, no contact

Before twelve engines can share a hierarchy, the monograph has to show that engines can sit together without interfering, and say exactly what can pass between them.

What crosses a handshake

What cannot cross. A product forgets its factors. The \(24^3 = 13{,}824\) three-step chains of Hurwitz units give only 24 products, 576 chains each COMPUTED, so no scheme can rebuild a chain's history from its current quaternion (Theorem 7.10). The galaxy's residual inherits this: \(Y_R\) tells you how much is unexplained, not where it came from (Problem G5).

Interaction without contact. Engines are coupled only through channels that carry no geometry of their own: the five-moment exterior channel (Theorem 7.7), the hyperbolic kernel \(Q_h = \sinh^2 d\) between projected states (Theorem 7.8), and the directed exchange of a rational valve (Theorem 7.9).

Twelve engines: the triple quad

Apply "the fourth from three" three times, and you have twelve engines in three groups of four. This is the structure the galaxy reuses.

Definition (§7.4). A six-tier engine is the core plus the five tiers of Lemma 4. Its memory state is its holonomy \(H \in S^3\) together with its 64-bit packet. Three data engines \(A_1, A_2, A_3\) determine an implied fourth \(F_A = (A_1A_2A_3)^{-1}\). Doing this three times (groups A, B and C) gives nine data engines and three implied fourths: twelve engines, three groups of four DEFINITION. An optional thirteenth engine \(F_\star = (F_AF_BF_C)^{-1}\) is the fourth of the fourths.

What the hierarchy buys, and what it does not

No arrangement stores more than the bits written (pigeonhole). The hierarchy adds erasure tolerance and phase throughput, not capacity. The same is true in the galaxy: the residual adds a check, not information.

The four-structure system

The monograph has a second "three plus an emergent fourth": an input sandwiched between two balancers, and the structure that closes the sandwich. The galaxy uses it as a frame-change check.

Definition (§7.6). Let \(A\) be a twelve-engine structure with holonomies \(A_1, \dots, A_{12}\), and \(\bar A\) its dynamical inverse, \(\bar A_j = A_j^{-1}\). A third structure \(I\) carries the input. The balancers \(A\) and \(\bar A\) sit on either side of the input, and the emergent structure \(E\) is fixed by the slot-wise closure

Theorem 7.18. \(E_j = A\,I_j^{-1}A^{-1}\), so \(\mathrm{Re}(E_j) = \mathrm{Re}(I_j)\) PROVEN. Conjugation keeps the real part of a unit quaternion, which fixes the rotation angle; only the axes turn. Had the balancers been placed next to each other (\(I\,A\,A^{-1}\,E = 1\)), they would cancel and give \(E = I^{-1}\).

Theorem 7.19. A product in which every factor also appears inverted equals 1 for all values only if it reduces to the empty word by cancelling adjacent inverse pairs: forward and inverse streams cancel only when nested. Proposition 7.20. If \(I_1I_2I_3I_4 = 1\) in a group of four, then \(E_4E_3E_2E_1 = 1\): the emergent structure closes its own groups, read in reverse PROVEN.

The 64 streams and the four projection checks (§7.7)

Each of the four structures I, A, &Abar;, E logs 12 engine streams plus 4 projection streams, the components \(w, x, y, z\) of its net holonomy: \(4 \times 16 = 64\) streams. The projections give checks that exist only at this scale, each PROVEN:

• Norm check: \(w^2 + x^2 + y^2 + z^2 = 1\) for every structure.
• Equilibrium monitor: the \(w\) stream of E equals the \(w\) stream of I (Theorem 7.18 with a global frame).
• Balancer check: the net of A times the net of &Abar; read in reverse is exactly 1.
• Collapse check: the disagreement \(D\) of §7.5 travels in the codex sub-region field.

The simulation runs the first three on its frame legs (Panel D, row §7.7). With Form 3 frame-leg packets switched on, it also decodes each leg's four projection streams and checks their norm (Chapter 15). The collapse check belongs to the ghost tetrahedra of §7.5, which the galaxy does not use.

Three systems around Sgr A*

Now the engine meets the galaxy. Each of the three mass systems is given the shape of one engine: five tiers and six channels.

Tiers become logarithmic shells DEFINITION

A system covering radii \([r_{\text{in}}, r_{\text{out}}]\) places tier \(n\) at

The exponent is the engine's own tier spacing, \((R_n - 15)/140\). The engine's radii span only a factor of 31/3, whereas the physical systems span factors of 100 to \(5 \times 10^6\), so a linear map is impossible. The log map keeps the relative spacing, is exact at both ends, and is strictly increasing. Because it is increasing, it carries Lemma 10's disjoint shells to disjoint radial shells inside every system PROVEN.

These are the reference script's H4 values COMPUTED. Panel A draws them as tori on a log scale (display radius \(6 + 20\log_{10}(r/r_s)\)), with tube thickness proportional to \(\rho_n = 2n+4\).

The seven pulsars are data sources, not channels

By projected radius, the magnetar J1745−2900 (0.10 pc) sits in System I, and the other six (27–77 pc) in System II. None is in System III COMPUTED. Each pulsar's signal crosses all three systems on its way to Earth, so a pulsar is a line of sight through the hierarchy, not a part of one system.

Seven is prime and coprime to 6, 30 and 60, so seven pulsars cannot be divided evenly over six channels, thirty nodes or a sixty-pulse register PROVEN. Any one-to-one assignment would be arbitrary, and the design makes none. The Sightline menu picks which pulsar's line of sight fills the channels.

The emergent fourth

This chapter is where the galaxy's triple quad appears: three classes of measurement, each a group of four.

Proposition G1 (additive residual parity) PROVEN. Let \(Y\) be a weak-field observable (a Shapiro delay change, a deflection, an astrometric shift, or the acceleration term in a pulsar's period derivative), and \(Y_s\) its value computed from system \(s\)'s mass model. Define \(Y_R := Y - Y_I - Y_{II} - Y_{III}\). Then:

1. \(Y = Y_I + Y_{II} + Y_{III} + Y_R\) exactly, in rational arithmetic.
2. Any one of the four is recovered exactly from \(Y\) and the other three.
3. Two unknowns with one total are not recoverable: a whole line of pairs fits (script check H1 finds 50 distinct consistent pairs).

Superposition is what makes the parity physical: at every known pulsar the Sgr A* field has \(GM/c^2r \le 2.1 \times 10^{-6}\), so second-order terms are a fraction of about \(10^{-6}\) of the first-order ones COMPUTED. Only inside a few \(r_s\) does superposition fail (Problem G4).

The twelve-element arrangement

G1 applies to each observable class separately. Writing \(c\) for astrometry, timing or velocity, each class forms a group of four:

Three classes, each with three data elements and one implied fourth, give twelve elements. This is the twelve-engine hierarchy of Proposition 7.11 in additive form: the arrangement is a DEFINITION and the parity is PROVEN, with the same erasure behaviour. Panel D's triple-quad grid lets you erase cells of the live arrangement and watch them rebuilt.

Frame closure and invariance

Every measurement travels through a chain of coordinate frames from the telescope to the pulsar and back. The chain must close, and it is checked exactly.

Every leg is a rational unit quaternion, made with the Cayley map of Lemma 9 Part 3:

A floating-point rotation is turned into \(u = v/(1 + w)\), rounded to \(2^{-k}\), and mapped back. No square roots, no trigonometry, and the norm is exactly 1, which the page re-checks for all four legs.

Proposition G2 PROVEN. With four independently measured legs, \(H_1H_2H_3H_4 = 1\) when the measurements agree, and an error \(\epsilon \ne 1\) in any one leg gives a product \(\ne 1\). If the fourth leg is computed as \((H_1H_2H_3)^{-1}\), the product is 1 identically. This is Lemma 12's table applied to frames (Chapter 8).

Frame invariance (Theorem 7.18). With balancer \(A = H_1H_2\) (ICRS to the Sgr A* frame) and input \(I = H_3\), the emergent \(E = AI^{-1}A^{-1}\) closes \(AIA^{-1}E = 1\) and keeps \(\mathrm{Re}(E) = \mathrm{Re}(I)\). In practice every scalar the pipeline computes (quadrances, rotation angles, \(Q_h\), Shapiro delay changes) must be identical in both frames. A mismatch is a frame or units bug, caught before any physics is interpreted.

Packets, phases and the Merkle chain

Each update, the twelve cells of the triple quad leave as twelve 64-bit tiles, and the twelve are chained into one 256-bit root.

1. Pack. For class \(c\) and structure \(s\) (I, II, III or R), stream slot \(4c + s\) gets one Form 3 tile with the fields of Chapter 7. The cycle step in bits 63–61 says whether this update is onload, offload or third phase.
2. Leaf. Leaf \(G_d\) (for \(d = 1, \dots, 12\)) is the 8 bytes of tile \(d - 1\) followed by its exact rational written as text, so the leaf covers the exact value even when binary32 rounds it.
3. Chain (Lemma 15 Part 3). \(\tau_{12} = H(G_{12})\) and \(\tau_d = H(G_d \,\|\, \tau_{d+1})\) for \(d = 11, \dots, 1\), with \(H\) = SHA-256. The root \(\tau_1\) changes if anything in any leaf changes, unless SHA-256 has a collision PROVEN.
4. Log. Once per epoch the telemetry log records a [B5] line with the step, the phase, how many values binary32 holds exactly, and the root.

Lemma 15 Part 3's proof also says where a change sits. If two states differ but their roots agree, the deepest layer \(d^\ast\) at which their chain values first agree exhibits an explicit SHA-256 collision. In the other direction, a change in leaf \(d\) changes every \(\tau\) from \(\tau_d\) up to the root. Panel D's Tamper button changes one leaf's exact value by \(2^{-24}\) and reports what each layer sees. Usually the tile word does not change (binary32 cannot see so small a change), the South Halo never does, and the root always does.

Frame-leg packets (optional)

The four frame legs \(H_1, \dots, H_4\) of Chapter 14 are exact rational unit quaternions. The specification gives them a place in the packet stream: stream slots 12–15 hold "the four quaternion components of a frame leg", and frame quaternions use Form 1 (Hurwitz) or Form 0 (§5.3). The Frame-leg packets menu in Panel B sends them in one of two ways. It is off by default, so the twelve data streams stand alone unless you ask for more.

Both modes address a frame packet as (structure, slot), the same pattern as the data streams, where structure \(s\) and slot \(4c + s\) name a cell of the triple quad. The monograph fixes the fields but not this assignment, so it is the page's own choice DEFINITION, and so is the byte order of the Form 1 core word: bits 31–24 hold the loss flag and \(2w\), then \(x\), \(y\) and \(z\) as signed bytes.

How Form 1 encodes a quaternion (monograph Table 5)

1. Clear denominators. Multiply by the common denominator \(L\). The rotation \(\rho_Q(x) = QxQ^{\ast}/N(Q)\) does not change when \(Q\) is scaled (Lemma 13 Part 3), so \(LQ\) is the same rotation in integers.
2. Double. Take \(2Q\), so Hurwitz half-integer quaternions such as \(h = (1 - i + j + k)/2\) also become integers.
3. Reduce. Divide by the gcd of the four coordinates.
4. Normalize the sign. Make the first non-zero coordinate positive. \(Q\) and \(-Q\) give the same rotation; the spin sign is carried by the cycle step in bits 63–61, not by the payload.
5. Fit or flag. If \(2w \le 127\) and \(|x|, |y|, |z| \le 127\), the bytes hold the quaternion exactly and the loss flag stays clear. Otherwise the coordinates are rescaled so the largest is 127, rounded to the nearest representable direction, and the loss flag is set.

Why the measured legs never fit

Form 1 is built for the engine's own Hurwitz arithmetic: units such as \(h\), small integer quaternions, and the cycle spinors \((1+i)^k\), which Table 5 notes never overflow. A measured frame leg is a different animal. It starts as a floating-point rotation, is rationalized at \(2^{-20}\) to \(2^{-40}\) (plus \(2^{-48}\) for the noise in \(H_4\)), and passes through the Cayley map, so its exact coordinates run to 40–180 bits. Lemma 13 Part 1 says denominators of rational unit quaternions grow without bound, so no fixed-width field can hold every such rotation. A 7-bit grid resolves directions only to roughly 1/127 per coordinate, a few tenths of a degree. The loss flag is how Form 1 says so honestly.

The physics layer

The engine organizes; physics predicts. Layer A is ordinary weak-field general relativity, and every number the page shows comes from verify_galactic_hierarchy.py.

For a point lens of mass \(M\) at distance \(D_l\) and a source at \(D_s = D_l + D_{ls}\), the Einstein radius, image positions and magnification are ESTABLISHED

with \(u = \beta/\theta_E\), the centroid shift \(\theta_E\, u/(u^2 + 2)\), and the Shapiro delay \(\Delta t_S = -(2GM/c^3)\ln(\ldots)\).

Three facts shape the design. The Sgr A* field is weak at every known pulsar, so superposition, and with it G1, holds. The magnetar's timing is dominated by its own motion, not by Shapiro delay: the Doppler term is \(1.3 \times 10^5\) times the Shapiro-rate bound. And plasma, not gravity, dominates radio propagation near the Galactic Centre, so every timing residual must be modelled for plasma first (Problem G2).

The simulated channels. The "model" side of each channel is a forward prediction:

• System I: Sgr A* as a point mass.
• System II: the nuclear star cluster as a Hernquist sphere of \(2.5 \times 10^7\) M☉ with a 4.2 pc half-mass radius HEURISTIC.
• System III: the observer terms (Gaia drift, Gaia acceleration and solar reflex).

The "data" side is the model plus Gaussian noise plus any injected signal. Each value is rationalized as \(\mathrm{round}(x \cdot 2^{24})/2^{24}\) before it enters Layer B (spec step A3) DEFINITION. Problem G6 asks for a proven bound on what that rounding can do to the residual.

Panels and controls

The simulation has four panels, a telemetry log and one control bar. Panel A sits on the left; B and C stack beside it; D runs full width below.

Experiments to try

Each experiment takes under a minute and shows one lemma or proposition at work.

1. A clean closure. With noise at 0 and no signal, every residual \(Y_R\) is exactly 0, the parity identity passes, and the open frame closure is exact. Everything the three models predict is in the data, and nothing else.
2. Open versus cyclic (Lemma 12). Press Inject error. The open convention reports FAIL and an angle; the cyclic convention still says "trivially PASS". Panel D's Lemma 12 row stays green, because it checks that the open test caught the error.
3. A real signal in the residual (G1). Choose Lensing event. As the source crosses the magnetar's line of sight, the astrometry and timing residuals jump to FAIL, but the parity identity keeps passing. The bookkeeping is exact; the residual is the signal.
4. Noise is not a signal. Set noise to 3σ. Residuals become amber (within 3σ·√3) rather than red.
5. Erasure (Proposition 7.11). In Panel D, click one cell in each row of the triple quad. All three come back exactly. Then click a second cell in one row: that group is lost.
6. Integrity (Lemma 15). Press Tamper with one leaf a few times. The root always changes; the South Halo never does; the tile word usually doesn't either.
7. Decode a packet (§6.1). Click packets in Panel B. Watch the step and phase change with the clock while the slot, structure and Form stay fixed.
8. Frames (Lemma 4). With Lemma 4 frames on, move the clutch: the helices slow down but the tiers do not tilt. Turn it off and the v10 offsets come back.
9. Frame-leg packets (Lemma 13). Choose Form 1 under Panel B's packets. All four loss flags are set, and the table shows how many bits each leg would need. Then try h and the Cayley preset in the Form 1 lab: they fit exactly.
10. A hypothesis test (§7.1). Choose Scalar field. A Yukawa term changes the timing and velocity of Systems I and II; it shows up only in the residual, which is where any new physics would have to appear.
11. Reproduce the script. Press Run Verification: 66 of 66 lines match the Python reference, and 19 of 19 checks pass.

Lemma-by-lemma compliance

How the simulation stands against each of the fifteen lemmas and the §6–§7 results it relies on. Panel D runs the same checks live.

What changed in the latest revision

• Pure Lemma 4 frames. The Lemma 4 mode no longer adds the engine's tension, torsion and clutch-slip offsets; Panel D checks the drawn frames.
• Frame-leg packets. An optional Form 3 or Form 1 encoding of the four frame legs, in the Merkle chain, with its own Panel D row (Chapter 15).

What changed in the first lemma revision

• Tile layout (§6.1). Bits 63–61 used to hold a three-phase counter; they now hold the Lemma 8 step k − 1, as Table 4 requires. The wheel unit, gate byte (component and slice counter) and step-multiplexed index were zero and are now filled per Tables 4 and 5. Values binary32 cannot hold exactly are now marked.
• Lemma 15. The recursive Merkle chain, the South Halo readout and the tamper test are new, and the log gains a [B5] line per epoch, as the specification's pipeline step B5 asks.
• Lemma 4. Panel A now draws the Lemma 4 tier frames by default; the engine v10 frames are one click away.
• Lemma 8 and Proposition 7.14. The cycle strip shows the step, its spinor and the onload, offload or third phase.
• Proposition 7.11. The live triple quad with erasure and exact recovery.
• Panel D collects all of the above with a row per lemma. None of this touches the reference-script port, which still matches all 66 lines.

Departures and open problems

Where the page makes choices of its own, and what nobody knows yet.

Choices made by the page

• Pulsar sky positions are illustrative (a golden-angle spread). Only the projected distances come from the catalogue DEFINITION.
• Line-of-sight depths are unknown (Problem G1) and are set equal to the projected offsets, as in spec §6.2.
• Events (Shapiro, lensing) are applied to the magnetar's line of sight, at 0.1 pc depth with impact parameter 0.3 θ_E.
• Wheel sector and multiplexed index. The galaxy design does not say which wheel unit or step index a stream should carry. The page uses the unit sector of the active node and the Table 4 meanings listed in Chapter 7 DEFINITION.
• Merkle cadence. The chain is recomputed every update and logged once per epoch. The specification asks for one root per epoch.
• Engine v10 parts that no lemma governs (the clutch, tension and torsion, the TQF qScale) are kept as the engine's own and labelled as such.
• Shared boundaries. Systems I/II and II/III share a tier radius (Chapter 12), so the monograph's separation theorem for stacked engines does not carry over.

Open problems (spec §9)

Part 2: three active engines

In Part 2 the three systems stop being passive shells. Each is a running copy of the engine, all three are sampled at the same node on every pulse, and the three samples are compared exactly.

Three copies of one engine

Each system is a full five-tier engine with the Lemma 4 laws, placed at its system's log radii exactly as in Part 1 (§H4). Its tiers carry the pure Lemma 4 frames. Engine \(s\) is then turned by \(\sigma^s\), where

is the 120° rotation about the body diagonal \((1,1,1)\). It is the same rotation that cycles the three cosets of the 3:1 interleave (Theorem 7.2) and the three planar groups of the cuboctahedron (Proposition 7.13). Because \(\sigma\) and the Lemma 4 frames are integer matrices, they act on rational coordinates with no rounding at all PROVEN.

Three display choices of the page sit on top of that DEFINITION:

• Counter-rotation: the helices of Systems I and III spin forward and those of System II backward.
• Tension turns the outer tiers (T4, T3) about their local \(y\) axis; Torsion turns the inner tiers (T2–T0) the other way about local \(z\). Both act identically in all three engines.
• The π/6 gate lights when \(\theta_{\text{ten}} - \theta_{\text{tor}} \equiv 0 \pmod{30°}\), as the specification asks. Lemma 5 Part 3's \(\pi/6\) is the acoustic phase \(k x_F\) of the Klein-wave merge boundary \(x_F = \tfrac43\mathcal C_0 = \lambda/12\), not a rotation angle, so the gate is shown without claiming a lemma proves a resonance HEURISTIC.

Sampling the same node in all three

On every pulse \(p\), the master clock gives node \(k = p \bmod 30\) and stage \(S = \lfloor (p \bmod 150)/30 \rfloor\), so the active tier is \(n = 4 - S\). For each observable class \(c\) and each engine \(s\), the node is the point on tier \(n\) with

Panel A draws each node riding its tier's spinning helices and gliding from one pulse's node to the next. Layer B evaluates the node at each whole pulse in the helices' co-rotating frame, where the spin drops out DEFINITION. That keeps the three engines congruent, as the lock needs, even though they spin in opposite directions.

The specification asks for a per-engine phase offset \(\Delta\varphi\) but does not say what drives it. The page drives it with that system's own data-minus-model in units of \(\sigma\): \(\Delta\varphi_{s,c} = \kappa\,(\pi/6)\tanh(r_{s,c}/3)\), with the coupling \(\kappa\) on a slider DEFINITION. With no noise and no signal every \(r_{s,c}\) is exactly 0.

The cross-engine Triple-Quad Formula

For each class, the three nodes \(P_{\text{I}}, P_{\text{II}}, P_{\text{III}}\) are rationalized at \(2^{-24}\) in each engine's own cubit frame, then turned exactly. Their quadrances \(Q_1 = |P_{\text{I}} - P_{\text{II}}|^2\), \(Q_2 = |P_{\text{II}} - P_{\text{III}}|^2\), \(Q_3 = |P_{\text{III}} - P_{\text{I}}|^2\) give the quadrea in two forms (Lemma 5 Part 6(a)) PROVEN:

Both are computed in BigInt, and the page checks they agree. Note that \(|u|^2 = Q_1\) and \(|v|^2 = Q_3\): the Gram form pairs the two edges that meet at \(P_{\text{I}}\).

Lock. Weitzenböck's inequality (Lemma 4 Part 2) says \(\chi = \sqrt{A_R}/(Q_1+Q_2+Q_3) \le 1/\sqrt3\), with equality exactly for an equilateral triangle. When the three phase offsets agree, the three nodes are \(P, \sigma P, \sigma^2 P\) for one rational point \(P\). Since \(\sigma\) is orthogonal, \(Q_1 = Q_2 = Q_3\) exactly and \(3A_R = S^2\): the triangle is equilateral, with no rounding involved. That equality is the page's lock. Any difference between the systems' residuals deforms the triangle for that class only. A lensing event on the magnetar, for example, deforms astrometry and timing but leaves velocity locked.

The fourth vertex. The inferred \(P_4 = P_{\text{I}} + q_{\text{scale}}\,(u + v)\) with \(q_{\text{scale}} = 0.65 + 0.45\chi\) follows the specification HEURISTIC. (Engine v10's own scale is \(0.65 + 0.45\sqrt{A_R}/(S + 1)\).) The exact closure \(P_4^* = P_{\text{II}} + P_{\text{III}} - P_{\text{I}}\) is drawn beside it. During stage 4 (tier T0) the \(P_4\) markers slide into the soliton at the centre, reaching it at node 29: the specification's "core latch".

The soliton crystal

The emergent fourth gets a body: a crystal at Sgr A* that sits still while the books balance and moves, by an exactly invertible amount, when they do not.

The crystal comes from torsional_twistor_tier_3_crystal_2.html, the Tier 3 simulation the specification analyses. Part 2 keeps its mechanics and its look, and sets them on the monograph's geometry. The file already fits that geometry more closely than the specification says. Its three TwistorPairs use tube phases \(0, 2\pi/3, 4\pi/3\), which put the three trefoils' lobes at 0°/120°/240°, 40°/160°/280° and 80°/200°/320°: exactly the C₉ offsets {0°, 40°, 80°} of Lemma 8.2. And its 80-unit, 4 × 4 × 4 lattice is the voxel cube \(V = [-2,2]^3\) of Lemma 5 at 20 units per cubit. The Lemma 8.2 geometry checkbox (on by default) sizes the carrier so the lobes land exactly on the nine ray endpoints. With it off, the file's free torus radius, z-scale and pole height come back, and Panel D reports how far the lobes leave the rays.

Every piece of the locked crystal is an object the monograph defines:

From the crystal file to Part 2

The gauge: Proposition 7.14

Write the three residuals in units of their \(\sigma\) as \(a, b, c\) (exact rationals). Proposition 7.14 splits them into an invariant line and a rotating plane,

and recovers \((a, b, c)\) exactly from \(s\) and \(z\) PROVEN. The page keeps \(z\) as the rational pair \((a - (b+c)/2,\ b - c)\), so the recovery is checked in BigInt on every update. The soliton core moves by \((z \text{ in the equatorial plane} + s \text{ along the axis})/3\), drawn through a saturating scale DEFINITION. Trefoil \(c\) shifts along its own 120° direction by its own residual, and the poles shift along the axis by \(s\). At lock every residual is zero, so the core sits exactly at the centre.

Two cases are worth trying. Equal residuals \((1,1,1)\) give \(z = 0\), so the core moves straight along the axis. Residuals that sum to zero give \(s = 0\), so it moves only in the plane.

Shear, spin and windows

• Shear. If the measured frame legs fail to close under the open convention, the three trefoils twist against each other about the axis by an amount that grows with the closure angle.
• Spin. The crystal turns 1° per outer-gear frame, so one 720-frame cycle is a 720° turn: the spinor double cover \((1+i)^8 = 16\) of Lemma 13 Part 4. Its 8 steps of 90 frames are the 8 steps of Lemma 8. Steps \(k = 2\) and \(k = 6\) give the same orientation with spinors \(i\) and \(-i\): the handshake windows (Theorem 7.5).
• Packets. In Part 2 the packets' cycle step is \(\lfloor \text{outerFrameN}/90 \rfloor\), so the crystal and the tiles agree DEFINITION. Part 1 used the engine's rotor step instead.

The 4 × 4 layer (§7.7)

§7.7 places a structure's 12 engines on the 12 boundary cells of a 4 × 4 layer in ring order, and its 4 projections in the 4 interior cells. Part 2 puts stream (class \(j \bmod 3\), structure \(\lfloor j/3 \rfloor\)) on ring cell \(j\). The square's 90° rotation (a shift by 3 cells) then keeps each class's group of four together, and the 120° step (a shift by 4) carries each group to the next. The interior shows \(w, x, y, z\) of the measured holonomy \(H_1H_2H_3H_4\). At closure that is \((1, 0, 0, 0)\): the three vector cells are null and \(w\) is 1.

Sixteen tiles go out per update: the twelve streams in slots 0–11, and the four projections in slots 12–15 with structure bits 11. All sixteen are chained with SHA-256 (Lemma 15 Part 3).

The engine-frame loop, honestly

The step from each engine to the next is \(\sigma\), whose spinor is the Hurwitz unit \(\omega = (1 + i + j + k)/2\). It has order 6, a finite-order rational rotation allowed by Niven's theorem (Lemma 13 Part 2). Three steps make a full turn, \(\omega^3 = -1\). The specification's fourth, \(H_4 = (H_1H_2H_3)^{-1}\), is then \(-1\), and the loop closes by construction. That is Lemma 12's cyclic convention: it tests nothing, so the page shows it as a parity element only. The test that can fail, and that drives the crystal's shear, is the open closure of the four independently measured frame legs (Proposition G2, Chapter 14).

Part 2 and its specification

The Part 2 specification asks for several things the lemmas support only partly, or not as written. This is what the page does at each such point, and why.

Glossary

Cubit \(\mathcal C_0\)

The engine's unit of length, \(\lambda/16\) at the calibration frequency; about 53.7 cm at 40 Hz in room air.

Tier T_n

One of five nested tori with major radius \(R_n = 5n^2+15n+15\), tube radius \(2n+4\) and \(16-2n\) windings (Lemma 4).

Shell

The spherical shell \([R_n - \rho_n, R_n + \rho_n]\) that contains tier n in every orientation (Lemma 10).

Channel

One of six labels \(k \bmod 6 \leftrightarrow (k \bmod 3, k \bmod 2)\): in the galaxy, (observable class, model or data).

Holonomy

The net rotation of a chain of frames, as a unit quaternion. Under the open convention it is a genuine closure test.

Implied fourth

The element fixed by three others in a group that must close: \(H_4 = (H_1H_2H_3)^{-1}\), or additively \(Y_R = Y - Y_I - Y_{II} - Y_{III}\).

Emergent fourth (R)

The galaxy's residual: what the three mass models leave unexplained.

Triple quad

Three groups of four: the twelve-engine hierarchy of §7.4, and in the galaxy the three observable classes × (I, II, III, R).

Handshake window

Steps k = 2 (onload) and k = 6 (offload) of the 8-step cycle, where engines can exchange data (Theorem 7.5).

Balancer

In the four-structure system, the structures A and &Abar; = A⁻¹ that sandwich the input I.

Einstein Tile

The 64-bit packet of §6.1, with fixed fields and a 4-bit Form ID selecting how the rest is read.

South Halo

The tile's top byte; a cache pre-filter, not an integrity check (Lemma 15 Part 2).

Merkle root

\(\tau_1\) of the chain \(\tau_d = H(G_d \| \tau_{d+1})\): the integrity certificate for all twelve streams (Lemma 15 Part 3).

Layer A / Layer B

Floating-point physics / exact rational bookkeeping.

Open / cyclic convention

Fourth element measured independently (a test) / computed from the other three (no test) (Lemma 12).