Voxel Gearing

Abstract

This paper specifies the mechanics of voxel gearing:

A body-level rotational register for geared bodies carried by the cubic balanced-ternary substrate. The register is motivated by a dimensional gearing packet read as a 1:1:4 sequence of rotational closure,

where the three terms correspond to axial circular closure, precessional surface sweep and four coupled body gears.

A geared body is a local burdened body with an internal centred balanced-ternary register. One axial channel carries signed spin about the body axis, one precession channel carries motion of that axis, and four centred body channels carry residual internal orientation.

Each body channel can run negative, stall at zero or run positive. A separate projection parity may arrange those four centred channels into opposed field-expression pairs. The carrier count is hosted by the face-centred six-orbit of the 3×3×3 cube under the octahedral group \(O_h\).

Each channel stores a centred finite state over a chosen trit width. Negative and positive values encode opposite directed motion. Zero is a stall state, not absence. The register gives the body a discrete centred vocabulary for spin, precession and body-orientation motion without importing continuum angular-momentum machinery.

Geared interaction has three coupled stores. Internal register state can be exchanged between bodies at contact. Part of contact work can be written into the surrounding field. Part can become centre-of-burden motion when the contact geometry supplies a normal and tangent. Structured field pressure can also write back into the register through signed shell-drive projection. Voxel gearing is specified as a body-field-contact mechanism, not as a passive measurement label.

The four body channels are internal centred channels, not four ordinary spatial axes. This is required by dimension uniformity.
The same geared body must remain meaningful in a thin slab with \(n_z=1\) and also in shallow volumetric slabs such as \(n_z=3\), \(7\) and \(11\).

A direct spatial SO(3) or quaternion expression becomes underdetermined under thin-slab projection; an internal four-state register does not. The register is discrete and balanced-ternary. Continuous SO(4) language is used only as a counting scaffold for a four-component state with six pairwise modes.

Reading Guide

This paper is a mechanics paper.

Its object is the 1:1:4 geared-body register: what it is, why \(\pi+\pi^2+4\pi^3\) supplies its dimensional form, why a cubic substrate can host six carriers, and how the resulting register exchanges state with field and contact.

Bounded by Construction [1] supplies the field-transport substrate: the bounded-integer Laplacian, clamp accounting, register discipline and thin-slab interpretation. The present paper asks what rotational bookkeeping a geared body can carry on that substrate without importing continuum angular momentum or per-cell velocity populations.

Standard Model Structure from Stencil Geometry [3] supplies the cubic orbit setting. This paper uses only the face-centred six-orbit of the 3×3×3 stencil. The corner and edge orbits, wider 13-axis programme and any stabiliser-to-gauge mapping are outside the present scope.

The central mechanical idea is that a geared body carries a finite centred register whose shape is 1:1:4. Axial spin, precession and four internal body channels are encoded as centred balanced-ternary states. Contact can exchange those states. Field pressure can absorb energy from them or promote them. Depth changes field expression, not register count.

Positioning

Voxel gearing is a mechanics proposal for rotational state on a cubic balanced-ternary substrate.

It begins with that 1:1:4 packet and asks what kind of body-level register can carry it on a bounded cubic field.

Wilson loops provide a nearby lattice comparison [10,11]. They track gauge holonomy around closed lattice paths. Voxel gearing assigns its primary state to a geared body, not to a plaquette loop. The relevant loop is mechanical: body state, field expression and contact exchange.

Topological defect theory provides a nearby contrast [9]. A geared body is not a flaw in an ordered medium or a singularity in a continuous order parameter. It is a body carried by the substrate, burdened by its field coupling, and identified by register structure, field burden and contact behaviour.

Vortex mechanics is closer in spirit [4]. Vortex theories treat rotational structure as physically primary rather than secondary. Voxel gearing shares that instinct, but translates it into a discrete register language: axial closure, precessional sweep and four body-rotation gears as centred finite states rather than continuous circulation.

Lattice Boltzmann methods provide a contrast in local state design [5]. They attach population values to every cell along a chosen velocity set. Voxel gearing starts with a geared body carrying a six-channel internal register. A future voxel-level implementation may move register structure closer to local substrate state, but the mechanical object remains the geared register, not a per-cell fluid population.

Voxel gearing treats a geared body on the cubic substrate as carrying a six-channel rotational register. Its dimensional form is 1:1:4, its cubic host is the face-centred six-orbit, and its state is expressed as centred balanced-ternary motion.

Introduction

The starting point is the same 1:1:4 packet introduced above. It is read as a compact rotational sequence: circular closure, surface sweep and a fourfold body term.

The first term folds straight-line measure into an axial cycle.

The second carries that cycle around a second rotational degree.

The third treats body rotation as a volume-level term split into four coupled circular gears.

The packet is treated here as a mechanical register shape, not only as a scalar expression.

A geared body carried by the cubic balanced-ternary substrate can hold one axial spin channel, one precession channel and four centred body-orientation channels. Together these six channels form a body-level rotational register.

Bounded by Construction [1] supplies the field-transport substrate: bounded-integer Laplacian, clamp accounting, register discipline and thin-slab rule. The cubic host is supplied by the 3×3×3 stencil [3]. Under the full octahedral group \(O_h\), the face-centred neighbours form a six-element orbit [6,7]. The dimensional packet gives the register its 1:1:4 form; the face-centred orbit gives it a native cubic host.

A geared body is a local burdened body with an internal centred balanced-ternary register. Each register value is centred and finite. Negative and positive values encode opposite directed motion; zero is a genuine stall state. The register is a discrete internal state for spin, precession and body-orientation motion, not an imported continuum angular-momentum vector.

The mechanical loop is body-field-contact. The body register shapes the field expression around the body. Field pressure loads the register and can promote or damp its centred state. Contact between geared bodies exchanges register state and can convert part of the internal rotational state into centre-of-burden motion when the contact geometry supplies a normal and tangent.

The four body channels are internal centred channels with separate projection parity.

They are not four ordinary spatial axes or four unsigned quadrants. A direct spatial SO(3) expression becomes underdetermined under slab reduction. An internal four-state reading preserves the six-channel count. SO(4) is retained only as a counting scaffold for a four-component state with six pairwise modes.

1. The Dimensional Gearing Packet

The dimensional packet is

The expression is read mechanically. It is not introduced as a free numerical fit. Each term marks a stage in rotational closure, and the coefficient of each term sets the number of gear channels of that kind. The packet is a compact way of saying that rotational state does not arrive as a single undifferentiated spin. It arrives as axial closure, precessional sweep and body-level closure.

The powers of \(\pi\) are treated here as dimensional closure markers:

The first closure folds a line into a cycle.

The second carries that cycle through a swept surface.

The third lifts the construction into body-scale rotational closure.

The coefficient 4 on the cubic term is the point at which the packet stops being a simple scalar ladder and becomes a gearing proposal: the body term is not one more axis, but four coupled internal gears.

The first closure: π

The first term represents circular closure. A line acquires a rotational return path and becomes a cycle. Mechanically, this is the axial gear: the body can rotate about a chosen axis.

This is the simplest rotational act available to the body. It does not yet describe how the axis itself moves, nor how the body carries orientation around that axis. It is spin closure: one circular gear.

The word axial is used because the closure has a distinguished centre line. A body may spin around that line in either direction, or stall at zero. The axial gear is the register channel that stores this centred spin state. It is the first part of the drivetrain, not the whole drivetrain.

The second closure: π²

The second term represents surface sweep. A circular closure is carried around a second rotational degree, so the axis of the first closure can itself move. Mechanically, this is the precession gear.

Precession is not another copy of axial spin. It is the motion of the spin axis. In a continuous picture it would be visualised as a cone or swept surface. In the register picture it is a second centred register channel coupled to axial spin but not identical to it.

This second closure matters because axial spin alone cannot describe how a geared body presents its spin state to the surrounding field. The precession channel carries the changing presentation of the axis. It is the bridge between a body simply spinning and a body whose spin state has orientation, sweep and field-facing structure.

The body closure: 4π³

The third term represents body-level closure. Once circular closure and precessional sweep exist, the remaining rotational state of the body cannot be reduced to a single extra axis. It is a four-channel internal body register.

The coefficient 4 carries the central mechanical move. The body term decomposes into four coupled circular gears, rather than one body channel scaled by four. Each body gear is centred in its own right. A separate projection parity can organise the four gears into opposed field-expression pairs when the internal state is written into the surrounding field. The gears carry residual internal orientation, phase opposition and body-rotation balance.

The body term is the point at which voxel gearing differs from a simple three-axis rotation model. It is not read as ordinary SO(3) rotation. It is read as an internal four-channel register whose expression may be projected into a thin slab or distributed through a deeper field.

The 1:1:4 packet

Combining the three stages gives the mechanical register shape

The packet specifies six gears in total:

Those six gears are not introduced as arbitrary control variables. They are the dimensional sequence of closure written as a proposed register: one circular spin closure, one precessional surface closure and four coupled body closures.

This interpretation also fixes the order of explanation. The 1:1:4 form is not derived first from the cube. The packet supplies the mechanical form; the cubic substrate must then provide a native six-carrier host. The cube is not asked to invent the gearing packet. It is asked whether it can carry it without importing an external continuum frame.

The remaining question is where such a six-channel packet can live on a cubic substrate.

The answer proposed here is the face-centred six-orbit of the 3×3×3 cube.

2. The Cubic Voxel Under \(O_h\)

The 3×3×3 cube of integer voxels admits a natural action of the full octahedral symmetry group \(O_h\), of order 48 [6,7]. The 27 voxel positions decompose into four orbits, fixing the carrier-count question before any dynamical mechanism is introduced.

The decomposition is elementary finite-group bookkeeping [8]. The centre is fixed by all of \(O_h\). A face-centred neighbour is fixed by the eight symmetries of a square face, a corner by six symmetries about a body diagonal, and an edge-centred neighbour by four symmetries about an edge axis. These stabiliser orders give orbit sizes 1, 6, 8 and 12, accounting for all 27 positions.

The present paper uses only the face-centred six-orbit. The corner and edge orbits are relevant to other CAELIX questions, including the wider 13-axis species programme and possible stabiliser-to-gauge mappings. They are outside the present gearing mechanism.

The cube supplies the six-carrier host; the 1:1:4 partition comes from the dimensional packet.

Why the face-centred six-orbit

The face-centred orbit consists of the six axial neighbours of the centre voxel. In coordinate form, these are

They are the only six-element orbit in the 3×3×3 neighbourhood. If a body-level register requires six carriers while remaining native to the cubic stencil, this is the unique orbit-count candidate.

Three facts matter for the present construction.

First, the orbit has the right count. The 1:1:4 register requires six carrier slots: one axial channel, one precession channel and four body channels. The face orbit supplies six slots without adding an external continuum frame.

Second, each face-centred element has a square stabiliser \(C_{4v}\). Choosing one face as an axial representative leaves a residual square symmetry around that chosen axis. This local symmetry fits the four-channel residual body register supplied by the dimensional packet.

Third, the six face elements are already the neighbour directions of the integer Laplacian.

The proposed register is not imported from an unrelated graph or a secondary display layer. It is attached to the same cubic adjacency that transports the field. This makes the register native to the body's field coupling rather than a decorative overlay.

3. The 1:1:4 Register

The dimensional packet supplies the form of the register:

The face-centred six-orbit supplies the carrier count. The register is a six-carrier object with a 1:1:4 internal partition. This partition should be read mechanically. It is not a convenient sorting of six numbers. It is a drivetrain split to be tested: one channel stores spin closure, one stores motion of the spin presentation, and four store residual body orientation.

Each channel is centred. Positive and negative values encode opposite directed motion; zero is a stall state. The register can store reversal, load and dead-centre behaviour without replacing the channel or changing its identity.

The axial channel

The axial channel is the first circular closure. It carries spin about the chosen body axis.

It is not the axis itself. It is the state of rotation around that axis.

Mechanically, the axial channel gives the geared body a directed spin register. Positive and negative values encode opposite spin directions. Zero is a stall state. In isolation, axial spin has no preferred direction of travel. It can load the body and the surrounding field, but it cannot choose a translational direction by itself.

Contact changes that. When two geared bodies touch, the contact normal supplies a tangent around the point of interaction. Axial slip can then be read against that tangent. Part of the slip may be redistributed between the two registers, part may be written into the field, and part may become centre-of-burden motion. This is the cleanest mechanical route from internal rotational state to translational response.

The axial channel has two roles in the proposed mechanism. It stores spin, and it supplies the most direct contact route by which internal rotational state can become motion.

An isolated spinning body has no preferred direction of travel; a contacting body pair does.

The precession channel

The precession channel is the second closure. It carries motion of the spin axis. It is not a faster or slower copy of axial spin. It is the register by which the axis itself can sweep, wobble or drift relative to the body's field expression.

In geometric language, axial spin closes a circle. Precession carries that closure around a second rotational degree. In mechanical language, precession lets the geared body change how its axial state is presented to the surrounding field.

This distinction matters because a body can spin without changing the presentation of that spin, and it can change the presentation of spin without simply increasing axial speed. The precession channel is the geared body's proposed register for that second behaviour. It turns axial closure into an oriented field-facing state.

Precession must remain distinct from axial spin because the body can rotate about an axis and also change the orientation or presentation of that axis. Treating those as one register would collapse the first two terms of the packet into a single gear, losing the 1:1 part of the structure.

The four body channels

The four body channels are the cubic body term. They carry the residual internal orientation state that remains after axial spin and precession have been separated. They are not four ordinary spatial axes, four quadrants on a screen or four independent continuum generators. They are four coupled body gears.

Each body channel is centred:

so every body gear can run negative, stall at zero or run positive. The channel sign is not fixed by the label \(B_i\).

A separate projection parity may be used when the four body gears are written into, or read back from, the surrounding field. The present parity convention is

This parity does not mean that \(B_0\) and \(B_3\) are positive-only gears, or that \(B_1\) and \(B_2\) are negative-only gears. It means that the four centred gears are projected into the field with opposed orientation signs. The exact labelling is conventional; the important distinction is between the centred body-register values \(B_i\) and the projection parity \(\sigma_B\).

An unsigned four-bin reading would count activity but would not encode directed body state. A fixed-sign reading would also be wrong, because it would prevent each body gear from reversing through zero. Voxel gearing needs centred body gears plus a field-expression parity.

The partition as a mechanical object

The 1:1:4 partition should be read as a drivetrain rather than as a list of observables. Axial spin, precession and body orientation are coupled but not interchangeable. Each channel has its own role in how the body stores rotational state and how that state can be expressed through the field.

This is the core mechanical proposal: \(\pi+\pi^2+4\pi^3\) is read as a six-channel register, not as an arbitrary grouping of carrier slots.

Status of the partition

The partition is a structural hypothesis with a specific mechanical origin. It is motivated by the dimensional packet, hosted by the cubic six-orbit and constrained by dimension uniformity.

The cube supplies six available carrier slots; the dimensional packet supplies the 1:1:4 drivetrain.

4. The Internal Four-State Register

The four body channels of Section 3 are not ordinary three-dimensional spatial rotations.

They are internal body gears carrying residual orientation after axial spin and precession have been separated, and they must survive changes in field expression. If the four body channels were visible rotation axes, the register would inherit the embedding limits. Voxel gearing treats them as internal components projected through the available field.

Why ordinary spatial rotation is insufficient

A three-dimensional rigid body is normally described using SO(3), or quaternions as its convenient double cover. That language is useful for objects embedded in continuous three-dimensional space, but not for a body carried by a bounded cubic substrate whose field expression may be reduced to a thin slab.

The problem is dependency, not the mathematics of SO(3). If the body register is identified with ordinary spatial rotation planes, its field expression depends on the available embedding depth. In a thin slab, the body does not lose abstract rotational degrees of freedom; the field loses the ability to express rotations whose consequences would lift state out of the slab. A register built from visible spatial expression would become indistinguishable when the field is measured with \(n_z=1\).

That would put the register vocabulary under the control of measurement geometry. The same geared body would appear to have one vocabulary in a full volume and a poorer one in a thin slab, which is unsuitable for a body-level register whose identity should precede its field expression.

Voxel gearing requires a register whose identity belongs to the geared body rather than to field-display thickness. The body may express its state through a slab or through a deeper volume, but the internal count must remain fixed.

Four internal components

The proposed replacement is a four-component internal state attached to the geared body. The four body channels are centred components of this internal state. They are not screen quadrants and not visible spatial axes.

A four-component internal state has six pairwise planes:

This six-plane count matches the six carriers hosted by the face-centred orbit. The 1:1:4 partition then assigns two of those modes to axial and precessional behaviour, leaving four modes as the body register.

This count is used as a structural correspondence. A four-component internal state supplies the right number of pairwise modes while remaining independent of visible slab thickness. It gives the body register enough structure to retain opposition, balance and residual orientation without tying those features to ordinary spatial axes. The stricter next step is a discrete substrate derivation of the same count.

This is the mechanical reason the body term is fourfold. The four channels are residual coupled gears of an internal four-state body. They preserve opposition and balance through centred state and projection parity rather than visible spatial orientation.

SO(4) as a counting scaffold

The continuous group associated with rotations of a four-component Euclidean state is SO(4), with six independent generators. It is a useful counting scaffold for the register: four components, six pairwise rotational modes.

The geared-body register is not identified with a continuous SO(4) representation.

The proposed register is discrete, finite and balanced-ternary. Its natural form is a centred integer structure, not a smooth rotation group. SO(4) is used only to make the count and internal-state logic legible.

The analogy separates carrier count from visible spatial axes. A six-mode register can be associated with pairwise relations inside a four-component internal state, rather than with three ordinary rotation planes in an embedding volume. This bridges the fourfold body term and the six-carrier cubic host.

A stricter future treatment may identify a discrete subgroup, a centred balanced-ternary analogue or a typed state machine that replaces the continuous scaffold entirely. The present paper needs only the weaker point: four internal components naturally supply six pairwise modes without tying the body register to ordinary spatial SO(3).

Dimension-uniform expression

The internal-register framing lets the same geared body express itself through different field depths. In a thin slab, the four body gears project into the available field surface. In a deeper field, their projection parity can distribute pressure through the additional depth. The register remains the same object in either case.

This matters because field depth is an expression channel, not the source of the register.

A deeper field can give the body gears more room to write burden, and a thin field can compress that expression, but neither case should add or remove gears. The body keeps the same vocabulary.

This is dimension uniformity in mechanical form. Depth changes expression, not identity. The body register is internal; the field supplies the space through which that state is written, loaded and exchanged.

5. Three Scales of Relevance

Voxel gearing has three relevant scales: the geared body, the local substrate and the surrounding field shell. These are mechanical roles, not competing explanations.

Body scale: the register bearer

The geared body bears the 1:1:4 register. It carries the axial, precession and four body channels as centred internal state, with projection parity handled separately when that state is expressed through the field. This is where the dimensional packet becomes a mechanical object.

At body scale, the register answers three questions: how the body spins, how the spin axis moves, and how residual body orientation is stored. The answer is a finite centred register,

not a continuum angular-momentum vector. The geared body is structured state whose rotation can load, stall, reverse, exchange and re-express.

Substrate scale: the future local law

The substrate scale is the bounded-integer update itself. The question is whether centred gearing state can be attached to voxels, macro-cells or coherent regions in a way that follows from the update rule rather than body-level bookkeeping.

That is the natural target for stricter implementation. A substrate-scale version would give each relevant local region a typed register slot that participates directly in the integer update. Geared bodies would then emerge from local register state and field burden, placing the 1:1:4 register closer to the law of the medium.

The present mechanics can be stated at body scale first. The stronger question is whether that register can later be pushed down into typed local substrate state without changing its 1:1:4 identity.

Field-shell scale: expression and exchange

The field shell is the expression layer: the contact surface between internal state and field expression. It is where the body's register becomes visible to the surrounding field and where field pressure writes back into the body.

At shell scale, axial state appears as circular closure around the body. Precession appears as motion of that closure. The four body channels appear through projection parity in the surrounding field pressure. In a thin slab this expression is compressed into the available surface; in a deeper slab the same register can distribute expression through depth.

The field shell also mediates interaction. Contact between two bodies is body-to-field-to-body exchange as well as body-to-body exchange. The shell carries burden, absorbs contact work and supplies signed drive back into the register.

Why the scales must remain separate

The body carries the register. The substrate supplies the update law. The field shell expresses and loads the register.

A shell measurement does not prove a substrate-scale law. A future voxel-level law is not needed to define the body-scale mechanism. A body-level register does not mean every voxel already carries six typed gears. Voxel gearing is coherent at body scale, naturally hosted by the cubic substrate, and ready for stricter local implementation where those slots must be made explicit.

6. Registers, Loads and Signed Drives

Voxel gearing separates three quantities: register state, load and signed drive. Register state is what the geared body carries internally. Load is how much field pressure is present in a channel. Signed drive is the direction-sensitive part of that pressure, capable in principle of writing back into the register.

Centred balanced-ternary registers

A geared body carries six centred register values:

where \(A\) is axial spin, \(P\) is precession and \(B_0\ldots B_3\) are the four body gears. Each value is a centred balanced-ternary state over some chosen trit width:

The exact width is an implementation choice. The mechanical requirement is that the range is finite, centred and signed. Negative and positive values encode opposite directed motion.

The value \(r_i=0\) is a real stall state. It is not the deletion of the channel.

Normalising by the channel limit gives a dimensionless register activity,

with \(-1\leq\rho_i\leq 1\). This is the natural way to compare registers with different trit widths.

A one-trit microscopic body and a many-trit macroscopic body can share the same sign logic even though their precision differs.

Loads

A load is not a register value. It is the field pressure measured through a channel.

Axial load measures pressure associated with circular closure around the body.

Precession load measures pressure associated with motion of that closure.

Body loads measure pressure associated with the four centred body gears.

Loads are generally non-negative magnitudes. They say how much burden a channel is carrying, not necessarily which way the register wants to move. A stalled register may still sit under high load, just as a locked mechanical part can hold stress without turning. A moving register may temporarily carry low load if the surrounding field offers little resistance.

This distinction is central. Load does not equal motion.

Register state, load and field pressure exchange energy through a bounded mechanism.

Signed drives

Signed drive is the directional part of a load measurement. Where load is magnitude, drive is tendency: if field pressure writes back into the register, does it push the channel positive or negative?

In a discrete register, signed drive cannot be treated as an infinitesimal continuum force. It must accumulate until it is large enough to change the finite register state. A schematic update is

where \(R_i\) is the centred register value and \(\Delta_i\) is an integer step produced by sustained signed drive.

This gives the gearing loop its return path. The register writes to the field through its expression; the field writes back through signed drive. Uniform pressure mainly becomes load. Coherent asymmetric pressure can become register motion.

Contact transfer

Contact is where register state can be exchanged directly between geared bodies. A contact has a normal direction and a tangent. This matters because an internal spin state cannot choose a translation direction in isolation, but contact geometry can supply one.

At contact, the interacting registers of the two bodies define a slip: the mismatch or joint motion that cannot be satisfied by both bodies at once. In the present prototype, raw slip is an internal diagnostic rather than a primary evidence metric. The more useful scalar is contact work: accumulated register/contact pressure routed into field excitation and contact response. That work can be redistributed between registers, written into the field as local excitation, or become centre-of-burden motion when the contact geometry allows it.

Axial spin is the cleanest example. A spinning isolated body does not translate itself. When it touches another body, the contact normal supplies a tangent. Signed axial slip can then become tangential impulse. That is the mechanical bridge from internal rotational state to centre motion.

Precession and body-channel contact are more subtle and should not be forced into the same tangential rule without derivation. Their likely roles include normal response, torsion, field excitation and register redistribution. A typed implementation should distinguish damping from exchange and signed register transfer from unsigned contact work.

The body-field-contact loop

The proposed mechanism is a loop:

with contact providing a second route:

These two routes are the drivetrain of voxel gearing. The body carries centred finite state; the field loads it; contact exchanges it or dissipates part of it as work. The field can return coherent pressure back into it.

A complete law must derive the register update, load projection and contact transfer from the bounded-integer substrate update. The mechanics can still be specified before that derivation is complete: these are the moving parts the law must connect.

7. Instantiating the Register

The preceding sections define voxel gearing as a mechanical structure: a dimensional 1:1:4 packet hosted by a cubic six-orbit, encoded as a centred register and coupled through body, field and contact. To instantiate that structure, any concrete model must preserve the vocabulary of the mechanism rather than replacing it with unrelated bookkeeping variables.

The mechanical contract has five parts:

First, a geared body must carry six centred register channels. Positive, zero and negative states must all remain valid.

Second, the four body channels must remain centred in themselves, with projection parity handled separately when the body writes to or reads from the field.

Third, contact must have access to register slip, because slip is the route by which internal rotational state can be exchanged, damped or converted into contact work. Raw slip is not by itself the evidence metric; the implementation must also account for accumulated work routed through contact and field response.

Fourth, the surrounding field must carry both load and signed drive.

Fifth, depth must change expression rather than register identity.

This distinction between register and expression is the main guardrail. A model may display the register in two dimensions, distribute it through a shallow slab or embed it in a deeper volume. Those choices may alter load, projection and field storage. They must not alter the fact that the geared body carries one axial channel, one precession channel and four centred body channels.

The same rule applies to contact.

A valid instantiation does not need to solve the full contact law in one step, but it must preserve the exchange routes. Register state may be redistributed between bodies. Contact work may be written into the field. Field pressure may return as signed drive. A stricter implementation must distinguish damping from exchange, signed register transfer from unsigned contact work, and visual overlap from mechanical contact. If contact only moves body centres while leaving the register untouched, the model is not instantiating voxel gearing. If the field only colours the background while never loading or driving the register, it is also outside the proposed mechanism.

These requirements do not prescribe a particular programming language, renderer or timestep.

They prescribe the mechanical contract. A valid instantiation may be a visual demonstration, a typed native model, a local voxel law or a macro-cell law, provided it preserves the 1:1:4 register and its body-field-contact exchange routes.

This section is deliberately placed before dimension uniformity. The question is not yet whether a particular run produces the right numbers. The question is what must remain invariant when the model changes representation. The answer is the register vocabulary: six centred channels, one projection-parity rule for body expression, contact slip, contact work, field load and signed drive.

Measured evidence is deferred to Section 9. At this stage, the register only has to be instantiated without changing what it is. That prepares the dimension-uniformity argument: the same six-channel object must survive thin and deeper field expressions.

8. Dimension Uniformity

Dimension uniformity means that the register remains the same object when the field is expressed through different available depths. A geared body may be embedded in a thin slab, a shallow volume or a deeper volume. Those cases change how the field carries the body's pressure, not how many internal gears the body owns.

The substrate condition

Bounded by Construction [1] established that the bounded-integer Laplacian is a six-neighbour operator. Thin-slab behaviour is handled by boundary evaluation and missing interior neighbours, not by replacing the operator with a different law.

Voxel gearing inherits this condition. The register should not change merely because an experiment presents the field as \(n_z=1\) rather than \(n_z=3\), \(7\) or \(11\). If the gearing law is native to the body-substrate relation, the body keeps the same six-channel identity across those expressions.

Why spatial rotation fails the test

A register identified directly with ordinary spatial rotation depends on the embedding frame. In full 3D the body may carry ordinary rotational state, but a thin slab cannot express the consequences of rotations that would lift field state out of the measured plane. A body register built directly from visible spatial expression would therefore lose distinguishability when depth is removed.

This dependency is unsuitable for a body-level register. The register would be controlled by measurement geometry rather than by the body. Voxel gearing treats the four body gears as internal centred components. Their field expression can flatten or thicken, but the internal count does not collapse.

Depth as expression, not identity

The correct distinction is between register identity and field expression. The register identity is

and remains six-channel. The field expression is the way that state is written into the surrounding substrate. In a thin slab, expression is compressed into the available plane. In a deeper volume, the body-channel projection parity can distribute expression through depth.

The phrase dimension uniformity does not mean that every depth should look the same. It means that the same internal geared body can be expressed at different depths without changing its register. In the current prototype, the practical browser ladder is \(n_z=1\), \(3\), \(7\) and \(11\). Depth changes projection, load distribution and field storage, not the 1:1:4 identity.

The mechanical consequence

The consequence is that body channels cannot be interpreted as ordinary visible axes. They must be internal body gears whose expression is mediated by the field. This is why the fourfold body term belongs after axial closure and precessional sweep: it is the internal body term that survives the loss or addition of embedding depth.

A later implementation may test this condition by comparing thin-slab and deeper runs. Such evidence would test the requirement, but the mechanical condition is already clear: a voxel-gearing register that collapses when depth changes is not a body-level register. It is a projection artefact.

9. Validation and Falsifiers

Validation asks whether voxel gearing behaves as a mechanical register when instantiated: whether driven controls separate, projection parity matters, the register survives depth changes, and contact or field pressure alters register state rather than only the visible field. The present browser implementation is body-level and provisional, so validation begins with controlled separation between live symmetric drive and geared drive across shallow slab depths.

Control structure

The measured model uses four live driven coupling controls:

The controls test whether the drivetrain has separable parts. The x:x:x control is the live body-field baseline: burden and continuous symmetric source drive remain, while geared modulation is suppressed. The x:x:x and Full rows share burden, source amplitude, contact rules, motion rules and timestep settings; they differ in whether the live source carries the gearing pattern. Full 1:1:4 must separate from x:x:x, 1:x:x and 1:1:x. Likewise, 1:1:x must separate from 1:x:x if precession is doing observable work.

A passive finite-probe mode was implemented as a sanity check. It uses burden plus a one-shot seed pulse, with no continuous source, register loading or contact-transfer work. It produces weak drift and little useful contact-work signal, so it is not a primary validation mode.

Metrics used as evidence

The measured model records field and register observables. The useful ones are scalar checks on the body-field-contact loop, not visual resemblance.

The falsifiers are failure of control separation, failure of projection-parity survival, failure under randomised registers, or collapse when depth changes. If adding depth changes the register itself rather than its field expression, the body-level mechanism fails.

The evidence sought here is separation of mechanism: axial closure, precessional sweep, body-channel expression, contact transfer and depth response must become distinguishable enough to be wrong under the revised run contract.

Fixed four-body depth test

A fixed four-body layout is used as the first depth test under the updated validation contract. The required depths are \(n_z=1\), \(3\), \(7\) and \(11\): small odd slab depths chosen to avoid even-plane centring artefacts while probing increasing expression thickness. The required live driven controls are x:x:x, 1:x:x, 1:1:x and Full 1:1:4.

The fixed four-body sample checks whether depth changes expression rather than register identity. The z=1 controls remain close together, while deeper Full runs shift the load pattern. The zero lower bound in all z=1 body-load ranges suggests that at least one body channel remains silent in the thin slab, while \(n_z\geq3\) gives all four body channels non-zero load. Peak Phi is not strictly monotonic, especially at \(n_z=11\), so the result is better read as depth-dependent redistribution than simple dilution. The main signal is that depth increases body-channel load capacity without replacing the six-channel register.

The weak separation among the four \(n_z=1\) controls is also informative. The fixed layout is highly symmetric, so it is a poor place to expect dramatic geared modulation if the projection parity is partly self-cancelling on the surface. In that reading, the thin fixed-square case is a symmetry baseline rather than a failed positive run: it shows that the live driven field remains active, while the body-channel expression is constrained by slab depth and layout symmetry. The useful comparison is therefore not only Full versus x:x:x at \(n_z=1\), but the change from silent or partly silent body channels at \(n_z=1\) to non-zero body-channel loading once depth is available.

Random-register test

The random-register test asks whether the separation survives a less symmetric body layout. The model uses eleven seeded bodies with randomised register phases. The comparison keeps the same live field conditions while changing whether the source carries the gearing pattern.

The random-register sample is the stronger layout test. Full 1:1:4 produces higher Work and T Lock than the x:x:x live baseline at both 5k and 15k in \(n_z=1\). At 15k ticks, Work rises from 78.99k to 100.18k, about 27%, while T Lock rises from 0.837 to 0.900. Deeper Full runs then carry much larger body-channel loads while keeping lower peak Phi than the z=1 random runs. The same thin-to-depth pattern seen in the fixed layout also appears here: the \(n_z=1\) body-load ranges begin at zero, while the deeper random runs give all four body channels non-zero load. The effect survives a non-hand-arranged layout, which is the relevant test here.

Projection-parity checks

The current body-channel projection parity is \((+,-,-,+)\). It is a convention until it survives comparison with alternatives. The natural next test is to run matched seeds under alternative parity maps, such as \((+,+,-,-)\), \((+,-,+,-)\) and their sign-reversed equivalents.

If projection parity matters mechanically, those alternatives should change Work, T Lock, body-load distribution or depth response under otherwise matched conditions. If they do not, then the parity convention is notation rather than mechanism. That would not erase the 1:1:4 register, but it would weaken the present reading of how the four body channels write into the field.

Contact and register activity

Work and T Lock are the most direct checks that the register is participating. Work records accumulated contact pressure routed through register/contact response. T Lock records similarity of register state across body pairs. Raw slip remains an internal contact quantity, but it is not the headline evidence metric. The mechanism would weaken if Work stayed near zero during sustained geared contact, if T Lock never changed, or if centred channel speeds ignored field pressure and contact history.

Sub-trit residuals matter here. In a finite register, weak contact should not vanish merely because a single tick is too small to cross an integer boundary. Residual accumulation is part of the mechanical contract, not just an implementation trick. It is also why Work is the safer summary metric: it can reveal persistent small contact effects that raw per-tick slip may hide.

This is especially important near stalled states. A centred channel can sit at zero while still holding load, and repeated weak contacts may be the only route by which that state begins to move. Without residual accumulation, the model would falsely treat weak but persistent contact as no contact at all.

The present tables should therefore be read as contact-participation tests rather than as finished collision laws. The random-register sample is useful because it produces many unscripted contact geometries. If the Full run raises Work and T Lock under the same live field conditions as x:x:x, then the gearing pattern is altering register/contact behaviour rather than merely changing the colour or amplitude of the field. A stricter model should preserve that distinction while making the transfer route explicit: which part of contact work is damping, which part is exchange, and which part becomes field excitation or centre-of-burden motion.

Parameter and implementation tests

Display parameters should not change the existence of the effect. Transfer constants, contact skin and field-drive scale will change timescale and amplitude, but the separability of the register should deform continuously rather than switch on only at a narrow tuning.

A stricter implementation should reproduce the qualitative pattern with typed centred registers: live-control separation, non-zero contact work under geared contact, random-register stability, and depth expression that changes load distribution without changing register identity. Exact numerical agreement is not required. Mechanical agreement is. Clean separation of every register channel remains a target, not a result already established by the browser model.

The intended next test is not a prettier visual model. It is a typed model in which register state, projection parity, load, drive and contact residuals are separate objects with explicit update rules. That would make the current validation criteria harder to satisfy, which is the point.

A useful typed implementation would also make failure sharper. If alternative projection parities leave Work, T Lock and body-load depth response unchanged, then parity is notation rather than mechanism. If removing residual accumulation erases small-contact Work while leaving the visible field similar, then the residual path is mechanically active. If random layouts separate from x:x:x but symmetric layouts do not, then layout symmetry becomes a controlled variable rather than an annoyance. These are stronger outcomes than a better-looking render, because each one can be wrong in a specific way.

10. Costs and Open Questions

Voxel gearing is not a completed substrate law. It specifies a mechanical register and the exchange routes that a stricter law must support. Several costs remain.

Discrete derivation of the four-state register

The four-state internal register is presently justified by dimensional closure, carrier count and dimension uniformity. A stricter derivation should show why the bounded cubic substrate produces this internal four-state body register directly, rather than accepting it as the correct mechanical reading of the packet.

The SO(4) comparison is useful as counting scaffolding, but it is not the final algebra. The final object should be discrete, centred and native to the balanced-ternary substrate.

Typed local state

The current browser model is body-level. The natural next implementation is typed local state: register values, projection parity, load, signed drive, contact residuals and field excitation should be separate objects with explicit update rules.

This would remove a common ambiguity in visual prototypes. A change in colour, field intensity or rendered motion would no longer be allowed to masquerade as register exchange unless the typed state actually changed.

Contact law

The contact law is the hardest open part of the mechanism. Contact must distinguish register exchange, damping, field excitation and centre-of-burden motion. Those are not the same operation. Axial slip has the cleanest tangent route, but precession and body-channel contact need a stricter derivation before they can be treated as anything more than proposed transfer channels.

The Work metric is useful because it records participation of the contact route, but it does not yet separate every destination of that work.

Projection parity

The current body-channel projection parity is a working convention. It may be mechanically real, or it may only be a labelling device. The needed test is simple: rerun matched seeds under alternative parity maps and ask whether Work, T Lock, body-load distribution or depth response changes.

If parity changes nothing, the register may still be 1:1:4, but the proposed field-expression map is too strong. If parity changes the result sharply, it becomes part of the mechanism.

Depth ladder

The present depth ladder uses small odd slab depths, \(n_z=1\), \(3\), \(7\) and \(11\), as practical browser-scale probes. These depths are enough to show the thin-to-depth expression change, but they are not a final sampling theory.

A native implementation should test a wider ladder, separate boundary artefacts from true depth expression, and check whether the non-monotonic Phi behaviour at higher depth is refocusing, storage redistribution or a numerical boundary effect.

Scope

The paper reads \(\pi+\pi^2+4\pi^3\) as a 1:1:4 rotational gearing packet. The cubic substrate supplies a natural six-carrier host [3,6,7], and a geared body is treated as carrying an internal centred register rather than as an ordinary SO(3) rotor.

Contact and field pressure are treated as the places where register state is exchanged, loaded and re-expressed. The bounded-integer derivation is not complete, and the browser prototype has not yet separated every register channel cleanly. The present task is to specify the mechanism and the conditions under which it should survive or fail.

11. Conclusion

Voxel gearing reads the dimensional packet as a 1:1:4 sequence of rotational closure: axial circular closure, precessional sweep and four coupled body gears. The result is a six-channel body-level register: one axial channel, one precession channel and four centred body channels.

The cubic substrate supplies a native six-carrier host through the face-centred orbit of the 3×3×3 voxel neighbourhood under \(O_h\) [3,6,7]. The internal four-state reading keeps the body register dimension-uniform: depth changes field expression, not register identity.

The central mechanism is that a geared body carries centred internal state: a balanced-ternary register whose values can stall, reverse, load, exchange and be re-expressed through the field. The four body channels are internal body gears, not spatial axes. Projection parity is a field-expression rule, not the sign of the register values themselves.

The measured evidence is used here as an instantiation test, not as the definition of the mechanism. The updated tables in Section 9 report live driven controls, contact work rather than raw slip, and slab depths \(n_z=1\), \(3\), \(7\) and \(11\). The results support a narrow reading: Full 1:1:4 separates from the live x:x:x baseline in the random-register test, and deeper slabs carry larger body-channel loads without changing register identity. This justifies a stricter typed implementation.

Voxel gearing defines how a geared body stores, exchanges and re-expresses rotational state through a 1:1:4 register on the cubic balanced-ternary substrate. The derivation is not complete. The useful result is narrower and more immediate: the mechanism is now specified well enough to test directly.

References

1. Alan Ball, Bounded by Construction: The Balanced-Ternary Integer Laplacian as a Field Transport Primitive, Zenodo, 2026.
2. Alan Ball, Lossy by Construction: A Substrate Audit of Computational Physics, Zenodo, 2026.
3. Alan Ball, Standard Model Structure from Stencil Geometry, Zenodo, 2026.
4. Alan Ball, Emergent Field Physics from Balanced-Ternary Microstates, Zenodo, 2026.
5. Alan Ball, Balanced Ternary by Necessity: The Minimal Integer State Space for Directed Transitions, Zenodo, 2026.
6. C. J. Bradley and A. P. Cracknell, The Mathematical Theory of Symmetry in Solids: Representation Theory for Point Groups and Space Groups, Oxford University Press, 1972.
7. T. Hahn (editor), International Tables for Crystallography, Volume A: Space-Group Symmetry, Wiley, 2005.
8. W. Burnside, Theory of Groups of Finite Order, Cambridge University Press, 1897.
9. N. D. Mermin, The topological theory of defects in ordered media, Reviews of Modern Physics, 51(3):591--648, 1979.
10. K. G. Wilson, Confinement of quarks, Physical Review D, 10(8):2445--2459, 1974.
11. J. Kogut and L. Susskind, Hamiltonian formulation of Wilson's lattice gauge theories, Physical Review D, 11(2):395--408, 1975.

Further Reading

The following texts are not needed for the gearing mechanism, but provide useful background for readers approaching the construction from adjacent literatures.

1. K. G. Wilson, Confinement of quarks, Physical Review D, 1974.
2. A. M. Polyakov, Gauge Fields and Strings, Harwood Academic Publishers, 1987.
3. N. D. Mermin, The topological theory of defects in ordered media, Reviews of Modern Physics, 1979.
4. P. G. Saffman, Vortex Dynamics, Cambridge University Press, 1992.
5. S. Succi, The Lattice Boltzmann Equation for Fluid Dynamics and Beyond, Oxford University Press, 2001.