Bounded by Construction — CAELIX White Papers

Abstract

This paper specifies a bounded-integer field transport primitive developed as the constructive sequel to Lossy by Construction: A Substrate Audit of Computational Physics. The earlier audit asked what is actually computed when continuum equations are evaluated on finite binary hardware. The present paper gives one concrete answer: a discrete transport operator whose state is held in fixed-width signed integer registers, whose arithmetic budget is explicit, and whose failure modes are recorded rather than hidden.

The operator uses a von Neumann discrete Laplacian over balanced-ternary bounded registers. Its arithmetic is carried by three primitives: saturation clamp, ternary downshift and residue accumulation. The register has no mantissa, exponent, subnormal range or hidden continuum approximation. Its width is an experimental budget, not a physical constant.

The construction is presented as a CAELIX implementation pattern rather than as a completed theory of physical transport. It has been exercised in two current experimental regimes: a small-amplitude planar substrate-runtime regime and a large-amplitude thin-slab closure regime. In both cases, the important feature is not the interface code or visual presentation, but the register discipline: field state, velocity state, boundary treatment and clamp absorption are all explicit parts of the computation.

The central claim is that bounded integer transport exposes substrate behaviour that floating-point transport tends to absorb. Clamp events become measurable register-budget exceedances rather than silent numerical embarrassment. Boundary reflection survives as integer amplitude unless explicitly removed. Small activation terms require a residue ledger rather than being folded into the transport cadence. These features make the operator harder to use than its floating-point predecessor, but also more candid.

The paper specifies the register model, the integer Laplacian, the transport/activation separation, the per-tick update equation, the thin-slab interpretation of planar runs, the boundary construction required by large-amplitude tests, and two integer burden-coupling extensions. It does not claim that the present von Neumann stencil is the final CAELIX transport law. It claims that the register-bounded construction and its accounting rules are now stable enough to record, test and criticise.

Reading Guide

This paper follows directly from Lossy by Construction: A Substrate Audit of Computational Physics. The audit argued that numerical physics should report its computational substrate more candidly: what is continuous, what is discretised, what is borrowed from continuum mathematics, and what precision budget is spent by the machine.

The immediate upstream paper is Lossy by Construction: A Substrate Audit of Computational Physics.

The audit stops at methodology. The present paper gives one constructive response to that audit: a bounded-integer transport operator whose register width, clamp behaviour, boundary treatment and activation residue are explicit parts of the computation.

Two earlier CAELIX papers supply context for the construction, but are not required in full to follow the operator specification:

Balanced Ternary by Necessity: The Minimal Integer State Space for Directed Transitions.

Emergent Field Physics from Balanced-Ternary Microstates.

The first motivates the signed state alphabet \(\{-1, 0, +1\}\). The second records the earlier floating-point experiment programme from which the present integer operator developed. The present paper does not rely on the reader accepting the wider CAELIX interpretation. It specifies the operator as a computational object first.

Positioning And Prior Art

Balanced ternary, integer arithmetic, discrete Laplacians and cellular-automaton field methods are not new. Each has an established literature. The present claim is narrower: the paper specifies one bounded-integer transport operator assembled from those familiar pieces, with its register budget, saturation semantics, residue ledger and boundary behaviour treated as reportable parts of the computation.

Balanced-ternary floating-point formats such as Ternary27 and Tekum [4,5] address real-number representation. They define arithmetic formats. They do not define a field transport operator, nor do they treat clamp absorption or activation residue as observables.

Modular cellular-automaton Laplacians in the Dekking-family literature [7,8] iterate a neighbourhood difference through a finite cyclic ring. Their state semantics are modular. The operator specified here is signed and saturating: overflow clamps to a register limit rather than wrapping through a modulus.

Discrete Exterior Calculus [9] is a geometric calculus on meshes and differential forms. It is designed to preserve structure from smooth exterior calculus under discretisation. The present operator is a regular-lattice integer transport primitive. It does not operate on differential forms, and it does not treat the lattice as a mesh approximation to a smooth manifold.

Ternary and low-bit quantisation in machine learning [6] uses small discrete values to store or train neural-network weights. That is a representation and optimisation problem, not a live field-transport problem.

The contribution claimed here is not priority over the ingredients. It is the specification of a particular assembly: a signed bounded-register Laplacian with explicit trit-width budget, clamp accounting, ternary downshift, residue accumulation, burden-coupling extensions and a boundary treatment designed around integer-amplitude reflection.

Novelty Claim

The novelty claim is architectural.

The paper does not claim that saturation clamps are new, that residue accumulation is new, that von Neumann Laplacians are new, or that sponge boundaries are new. They are not. The claim is that this paper specifies a field transport primitive in which those pieces are made structurally visible rather than treated as incidental implementation details.

In the operator specified here, the register width is part of the experiment contract. Clamp events are recorded as register-budget exceedances. Sub-unit activation is carried by a residue ledger instead of being hidden in a floating-point mantissa. Boundary reflection is treated as integer amplitude that must be removed explicitly. Burden coupling is expressed through integer denominator modification and burden-driven negative damping rather than by introducing a continuous coefficient field.

The result is not a claim of final physical law. It is a substrate-honest transport primitive: narrow enough to specify, cheap enough to run in current prototypes, and explicit enough that its failure modes can be measured rather than absorbed.

Introduction

The Substrate Audit [1] argued that mainstream computational physics runs on a finite floating-point substrate whose precision cost is rarely reported as a first-class part of the result. That argument was diagnostic. It did not propose a replacement operator, and it did not need to; its claim was methodological.

The present paper is one concrete answer to what an alternative could look like. It describes a discrete field transport primitive whose state is held in bounded signed integer registers of configurable trit width, and whose update is expressed in primitives whose integer semantics are visible rather than hidden. The operator makes the substrate-honesty commitment of the audit paper concrete at the level of a running field solver. The width of the register is treated as an explicit experimental budget rather than as a constant of the framework.

The paper is narrowly scoped. It is about one operator and its register discipline. It is not a claim that continuous physical equations are wrong, that floating-point methods should be retired, or that the final CAELIX field engine will look like the construction presented here. The four-neighbour and six-neighbour von Neumann stencils used throughout are development choices. They are useful enough to specify, but they may be superseded by balanced-ternary-native stencils in later work.

The paper proceeds as follows:

Section 1 specifies the balanced-ternary bounded integer register and its three arithmetic primitives.

Section 2 specifies the integer Laplacian as a von Neumann stencil on that register and explains the thin-slab collapse from six neighbours to an effective four-neighbour planar operator.

Section 3 states the transport-cadence / activation-fraction separation rule.

Section 4 assembles the primitives into the per-tick update equation with clamp tracking on both state channels.

Section 5 treats register width as an experimental budget rather than a fixed physical choice.

Section 6 presents two integer burden-coupling extensions: refractive burden coupling and burden-driven negative damping.

Section 7 presents the three-layer hidden-band boundary construction required by the large-amplitude regime.

Section 8 works through a small integer example by hand.

Section 9 describes the current experimental implementations without treating their code names as part of the specification.

Section 10 states the validation checks and falsifiers for the operator.

Section 11 states the costs and open questions.

Section 12 closes the argument.

1. The Balanced-Ternary Bounded Integer Register

Let \(n\) be a positive integer: the trit width of the register. The register holds signed integer values drawn from the symmetric range

\mathcal{R}_n = \left\{-L_n, -L_n + 1, \ldots, -1, 0, +1, \ldots, L_n\right\} \] \[ L_n = \frac{3^n - 1}{2}

The trit width is not a fixed physical value. It is a chosen register budget. A narrow register gives cheap contained dynamics and makes overflow easy to provoke. A wide register gives more amplitude headroom and delays saturation. The chosen width must be reported as part of the experiment contract.

For example, a 9-trit budget gives \(L_9 = 9841\), while a 17-trit budget gives \(L_{17} = 64\,570\,081\).

Those values are examples of development budgets, not preferred constants. Other widths such as 12, 15, 27 or wider are equally valid if the experiment calls for them.

The register is not a balanced-ternary floating-point representation. It has no mantissa, no exponent, no sign flag and no subnormal range. The signed integer value is the state, and the trit width is the available amplitude budget. This distinguishes the representation from the Ternary27 and Tekum formats [4,5], which are floating-point formats modelled on IEEE 754, and from the BitNet weight-quantisation format [6], which is a static three-level weight for neural networks rather than a dynamic state register.

Current browser implementations use ordinary machine integer storage for development convenience, with balanced-ternary register discipline enforced by clamp, downshift and residue primitives at every update step. This is an implementation bridge. The paper specifies the register semantics, not a claim that the host representation is already trit-native.

Three Arithmetic Primitives

The register discipline is carried by three primitive operations. They are specified here as mathematical objects first and as pseudocode second, because the integer semantics are easy to misread in pure mathematical notation.

Saturation clamp. The clamp operator enforces the register bound:

\mathrm{clamp}_n(v) = \begin{cases} -L_n & \text{if } v < -L_n \\ \phantom{-}L_n & \text{if } v > +L_n \\ v & \text{otherwise} \end{cases}

The clamp produces no wraparound. Values outside the register range are saturated to the register limit, and the absolute amount of saturation can be recorded as a diagnostic.

In pseudocode:

function clamp_bt(v):
    if v >  L: return  L
    if v < -L: return -L
    return v

Ternary downshift. The downshift operator divides an integer by a power of three and truncates the result toward zero:

\mathrm{down}_t(v) = \left\lfloor \frac{v}{3^t} \right\rfloor_{\!\text{trunc}}

Truncation toward zero is load-bearing. Rounding toward negative infinity would introduce a sign-dependent bias that breaks the symmetry of the signed register. The operator is used to reduce the scale of intermediate quantities produced by the Laplacian before they enter the update equation.

In pseudocode:

function downshift_bt(v, t):
    denom = 3^t
    if denom <= 1: return v
    return truncate(v / denom)

Activation residue accumulation. Small kicks below the single-trit floor would be lost to truncation if applied directly each tick. The residue accumulator preserves them across ticks:

\mathrm{accum}(r, v, t, k) = \left(\left\lfloor \frac{k\cdot v + r}{3^t} \right\rfloor_{\!\text{trunc}}, \; (k\cdot v + r) \bmod 3^t \right)

returning a kick/residue pair. The kick is applied to the state on the current tick; the residue is stored per cell and used as the starting value of the next tick's accumulator. This preserves small activations over many ticks without inflating the transport cadence.

The three primitives are the arithmetic contract for the integer Laplacian update. No floating-point arithmetic is required in the transport step, and no operation other than these primitives modifies the register state.

The register is a signed bounded integer state of configurable trit width, updated only by clamp, truncating ternary division and integer residue accumulation. No calibration against a continuous distribution is required. The representation carries no hidden continuum content, and its precision is an explicit register budget rather than a rounding property of a floating-point substrate.

2. The Integer Laplacian

The Operator

Let \(\phi : \mathbb{Z}^3 \to \mathcal{R}_n\) be a scalar field on the cubic voxel lattice, held in the balanced-ternary bounded integer register of Section 1. The integer Laplacian at a site \((x, y, z)\) is the six-neighbour von Neumann stencil:

\begin{aligned} \nabla^2_\mathrm{int} \phi(x, y, z) &= \phi(x-1, y, z) + \phi(x+1, y, z) \\ &\quad + \phi(x, y-1, z) + \phi(x, y+1, z) \\ &\quad + \phi(x, y, z-1) + \phi(x, y, z+1) - 6\,\phi(x, y, z) \end{aligned}

The operator is purely integer-valued. Its absolute value is bounded by \(12\,L_n\), since each of the six neighbour terms contributes at most \(L_n\) and the centre term contributes at most \(6\,L_n\). For any chosen trit width, this bound is part of the arithmetic budget and must leave headroom for subsequent integer multiplications.

Dimensional Degeneration

Dimensionless configurations are not admitted on a voxel substrate. Every spatial axis must have a positive integer extent, and the minimum is one. The nominally two-dimensional case nx×ny is a thin-slab configuration nx×ny×1, not a plane.

The thin-slab case degenerates the operator as follows. With \(nz = 1\), the \(\pm z\) neighbours of every site are evaluated by the boundary rule rather than by interior grid content. If the boundary rule is Neumann-reflective, contributing the centre value, the two z-neighbour terms produce \(2\phi(x, y, z)\) and cancel two of the six centre subtractions, leaving the familiar four-neighbour expression as an effective operator on the live plane. The operator itself is unchanged; what has changed is which neighbours are interior and which are boundary-evaluated.

A single operator specification serves both the thin-slab case and the full 3D case. The planar regime remains a depth-one 3D slab rather than a separate two-dimensional ontology. Increasing \(nz\) extends the substrate depth; it does not change the operator class.

Not A Continuum Laplacian

The operator should not be treated as a finite-difference approximation first. It is specified here as a discrete curvature measure on the register state. Its resemblance to the continuum Laplacian comes from the local linear form of the stencil. Continuum equations are downstream descriptions that may or may not be recovered from the discrete operator in particular limits; whether they are recovered is an empirical question for each regime.

A related distinction is useful here. Mod-\(k\) discrete Laplacians in the cellular-automaton literature [7,8] compute the same neighbourhood difference and reduce the result modulo \(k\), producing state values that cycle through a finite ring. The CAELIX integer Laplacian does not reduce its output modulo anything. It produces a signed integer curvature measure that is subsequently clamped against the register bound if the state update would otherwise overflow. The saturation semantics is different from the modular-ring semantics, so the two operators should not be expected to produce the same dynamics on identical initial conditions.

The integer Laplacian is a six-neighbour von Neumann stencil on the balanced-ternary bounded integer register. It produces a signed integer curvature whose interpretation is set by the update equation in which it appears. It collapses to a four-neighbour effective operator in the depth-one thin-slab case, while remaining a single 3D operator specification. It is distinct from both the continuum finite-difference Laplacian and the mod-\(k\) cellular-automaton Laplacian.

3. Transport Cadence And Activation Fractions

The register discipline imposes a structural constraint that is not present in a floating-point implementation. Every arithmetic operation must produce an integer result, and tiny inputs may truncate to zero if multiplied by a small denominator before being applied.

This is not a cosmetic implementation detail. It is a structural distinction between two kinds of kick applied to the register state.

Transport Cadence

The transport cadence is the rate at which field curvature propagates across the lattice. It is carried by the integer Laplacian, and its natural scale is the register range itself. A typical wave kick produced by the Laplacian is comparable to the centre value, and its truncation by a power-of-three divisor reduces but does not zero the kick.

In an early version of the implementation, the transport cadence was artificially reduced by placing a tiny fractional factor of \(1/3^9\) directly inside the wave kick. The consequence was that the integer Laplacian produced values that truncated to zero at almost every site, and the field appeared to freeze. The propagation cadence had been destroyed by a factor intended to regulate a different quantity entirely.

Activation Fractions

Activation fractions are small corrections applied to the register state that are natural below the single-trit floor. Burden and charge contributions, for example, are fractional quantities whose per-tick contribution is much smaller than the Laplacian kick. Applied directly by truncation, these would be lost to the integer floor.

The residue accumulator of Section 1 preserves them. Sub-unit activation is accumulated over many ticks, and a unit-scale kick is produced only when the cumulative residue exceeds the threshold. The activation is thus preserved in aggregate even when no single tick carries a visible integer update.

The Separation Rule

The rule that emerged from this distinction is:

Transport cadence should remain at lattice speed. Tiny fractional factors belong in the activation path, not in the transport operator.

This is the separation rule. In negative form, it records a failure mode: placing a tiny fractional factor directly inside the transport operator can erase propagation by truncating the wave kick to zero. In positive form, the Laplacian operates at the full register cadence of the lattice, while sub-unit effects are carried in a separate residue ledger whose contribution to the state is unit-scale when it fires and zero otherwise.

The formulation has no direct analogue in a floating-point Laplacian. In the floating-point regime the mantissa absorbs small fractional factors without an immediate visible cost, and transport and activation can be combined into one real-valued kick. The separation is specific to the integer-bounded-register regime: the register discipline changes how the update must be organised.

Transport and activation are held in separate channels. The integer Laplacian carries the transport cadence at lattice rate. The residue accumulator carries sub-unit activation. Mixing them was the failure mode that motivated the separation rule.

4. The Per-Tick Update Equation

The operator of Section 2 and the primitives of Section 1 assemble into the per-tick state update. Let \(\phi\) and \(v\) be the field and velocity registers respectively. Each tick proceeds in the following order.

1. Laplacian evaluation.

\ell = \nabla^2_\mathrm{int}\,\phi(x, y, z)

2. Wave kick. The Laplacian is scaled and downshifted:

w = \mathrm{down}_1\!\left(\left\lfloor \frac{\ell \cdot p}{q} \right\rfloor_{\!\text{trunc}}\right)

where \(p/q\) is a numerator-denominator pair that sets the effective transport gain, chosen to keep \(w\) well within the register range for the expected Laplacian amplitudes. A typical development choice is \(p/q = 1/8\).

3. Damping.

\delta = \left\lfloor \frac{v \cdot d_n}{d_d} \right\rfloor_{\!\text{trunc}}

where \(d_n/d_d\) is the damping fraction. A small integer ratio such as \(1/512\) gives a slow damping that does not overwhelm the transport cadence.

4. Source injection.

s = \mathrm{src}(x, y, z)

For driven experiments the source term is a per-cell integer produced by a driver function; for free-evolution experiments it is zero.

5. Activation kicks. Burden and charge contributions are produced through the residue accumulator:

(b_{\mathrm{kick}}, b_{\mathrm{res}}') = \mathrm{accum}(b_{\mathrm{res}}, b(x,y,z), t_b, k_b)

(c_{\mathrm{kick}}, c_{\mathrm{res}}') = \mathrm{accum}(c_{\mathrm{res}}, c(x,y,z), t_c, k_c)

The residue states \(b_{\mathrm{res}}\) and \(c_{\mathrm{res}}\) are per-cell grids maintained between ticks.

6. Velocity update with clamp tracking.

v_\mathrm{raw} = v + w + s + b_{\mathrm{kick}} + c_{\mathrm{kick}} - \delta

v' = \mathrm{clamp}_n(v_\mathrm{raw}), \qquad A_v \mathrel{+}= \left| v_\mathrm{raw} - v' \right|

The accumulator \(A_v\) records the per-tick absolute clamp absorption on the velocity channel.

7. Field update with clamp tracking.

\phi_\mathrm{raw} = \phi + v', \qquad \phi' = \mathrm{clamp}_n(\phi_\mathrm{raw}), \qquad A_\phi \mathrel{+}= \left| \phi_\mathrm{raw} - \phi' \right|

The total clamp absorption per tick is \(A = A_v + A_\phi\), summed over all sites. A per-cell clamp map records whether either channel saturated at that site on the current tick; this map is used downstream as the closure observable.

Clamp Absorption As A Primary Observable

A floating-point field solver treats overflow as an error condition to be avoided. A fixed-point signal-processing implementation treats clamp events as a distortion source to be minimised. The present operator treats clamp events as a primary observable of the register-bounded computation.

The reasoning is structural. In the register-bounded regime, the clamp operator is the only mechanism by which the register state is prevented from encoding arbitrarily large quantities.

When a cell clamps, the register has failed to accommodate the local update at its current width. That failure is itself information: it locates where the dynamics have exceeded the register budget, and its accumulation in time measures how much register budget has been absorbed locally per tick.

Used this way, the clamp map is not noise. It is a field diagnostic whose structure shows where the bounded-register system is under the greatest local demand. The boundary treatment in Section 7 and the burden-driven negative-damping extension in Section 6 both rely on the clamp map as their primary diagnostic.

The per-tick update is a seven-step integer sequence from Laplacian evaluation through wave kick, damping, source, activation residues, and the two clamped state updates. Clamp absorption on both channels is recorded as a first-class diagnostic rather than suppressed, because the register-bounded regime makes saturation part of the experiment record rather than a hidden numerical artefact.

5. Register Width As An Experimental Design Parameter

The trit width \(n\) of the register is not a physical constant of the framework. It is an experimental budget whose value is set by the amplitude headroom, diagnostic purpose and runtime cost of the regime being tested.

Several widths have been used during development. The specific values below are examples of register budgets, not privileged constants.

Narrow registers. Appropriate for contained experiments in which field amplitudes are expected to remain well below the register limit. The clamp is used primarily as a drift backstop against runaway, and clamp events are rare. The register budget keeps small experiments cheap to run and easy to reason about. Subtle activation dynamics below the single-trit floor are preserved via the residue accumulator, which becomes the dominant mechanism for carrying fractional physics. A 9-trit budget is one current example of this regime.

Wide registers. Appropriate for experiments in which register saturation is the intended observable. The clamp is used as the closure diagnostic, and clamp events are both expected and measured. The register budget is deliberately made large enough that saturation occurs only where the dynamics actively drive the register to its limit. Saturation marks the location of a register-budget exceedance rather than a mere numerical mismatch. A 17-trit budget is one current example of this regime.

Moving from \(n\) trits to \(n+k\) trits multiplies the positive register limit by approximately \(3^k\). That scaling is native to the substrate and gives a simple headroom budget. The choice of \(n\) is part of the experiment contract and should be reported with the result.

The following table summarises the characteristic properties of representative register widths.

\(n\) (trits)	\(L_n = (3^n-1)/2\)	decimal digits	typical use
5	121	2.08	hand checks and toy traces
7	1,093	3.04	small contained tests
9	9,841	3.99	narrow-budget interactive runs
12	265,720	5.42	intermediate headroom
15	7,174,453	6.86	wider development headroom
17	64,570,081	7.81	saturation and closure tests
27	\(3.8 \times 10^{12}\)	12.58	high-headroom studies

Representative register widths. \(L_n\) is the positive-direction register limit. Decimal digits is \(\log_{10}(L_n)\). The use column is indicative only; the width is an experiment budget, not a fixed regime label.

The ability to configure register width is the integer-substrate analogue of precision allocation in mixed-precision floating-point computing. Where a mixed-precision FP code allocates FP32 or FP64 to different parts of the calculation [13], an integer-bounded code allocates trit widths. The two approaches are not identical, but they address the same underlying question: precision is a resource, not a background condition, and it should be budgeted where the physics requires it rather than uniformly at the widest available level.

Register width is an experimental design parameter. Narrower and wider budgets answer different diagnostic questions, and neither is privileged by the framework. The width is set by the intended observable, the amplitude headroom and the cost of the run rather than by a blanket policy.

6. Operator Extensions

The integer Laplacian of Section 2 is the minimal transport primitive. Two extensions are described here that are used in the large-amplitude closure regime. Both are expressible entirely in register-bounded integer arithmetic, and both illustrate that the register discipline is compatible with non-trivial coupling terms.

Refractive Burden Coupling

A static burden field \(b(x, y, z)\) is maintained alongside the state registers. It represents a local density-like scalar that affects propagation. The refractive coupling is introduced by modifying the transport gain denominator as a function of local burden:

q_\mathrm{eff}(x, y, z) = q + \left\lfloor \frac{b(x, y, z)}{2^{s}} \right\rfloor

where \(s\) is a fixed shift (a development value is \(s = 10\)), and the wave kick becomes

w = \mathrm{down}_1\!\left(\left\lfloor \frac{\ell \cdot p}{q_\mathrm{eff}} \right\rfloor_{\!\text{trunc}}\right)

The effect is that waves propagate more slowly through cells with high burden. A burden of \(5 \times 10^4\) with the development shift \(s = 10\) produces a \(q_\mathrm{eff}\) that is approximately fifty times the baseline, which gives roughly thirteen-fold slower propagation through the burden core.

The coupling is purely integer, uses no floating-point division, and requires no calibration against a continuum sound speed.

This is a discrete refractive-index construction. Analogue-gravity methods in the continuum produce similar structure through a medium-loading framework [10,11]; here it is produced directly from an integer modification of the Laplacian denominator. The CAELIX position is that refractive coupling is a natural feature of the integer transport operator rather than an external overlay.

Burden-Driven Negative Damping

The second extension converts the clamp event from a numerical backstop into a physical observable. A burden-driven negative damping term is added to the damping of Section 4:

\delta_\mathrm{eff} = \delta - \left\lfloor \frac{v \cdot b(x, y, z)}{D} \right\rfloor_{\!\text{trunc}}

where \(D\) is a large integer divisor (a development value is \(D = 2^{20}\)) and \(\delta\) is the damping term defined in Section 4.

At high burden the second term exceeds the first, and the net effect is a negative damping on the velocity register. The physical reading is parametric amplification: the local oscillator is driven above its natural amplitude by coupling to the burden field. The amplitude grows exponentially until the register clamps.

The location where exponential growth meets register saturation is an observable of the bounded-register system. It is not an ordinary floating-point overflow artefact; it is a consequence of the balance between burden-driven amplification, damping and the chosen register budget.

A wider register delays saturation and may shift the measured saturation surface in finite-time runs. Convergence under register-width increase is a test requirement, not something assumed by the construction.

This gives a concrete integer-substrate mechanism for a closure-shell observable. Cells inside the shell are cells where the amplification has overwhelmed the register budget; cells outside are cells where standard damping still dominates. The shell location is read directly off the clamp map of Section 4.

Interpretive Status

The two extensions should not be read as completed physical laws. They are operator-level mechanisms that expose different roles for a static burden field. The first changes the transport denominator and acts like a discrete refractive load. The second changes the damping balance and controls whether local amplitude decays or grows toward saturation.

Keeping the two mechanisms separate matters. If a closure shell appears only when denominator loading and negative damping are both enabled, then the experiment has found a coupled behaviour but not yet isolated its cause. If the refractive term alone changes propagation delay without producing saturation, and the negative-damping term alone produces saturation without the same delay structure, then the two effects have distinct diagnostic signatures. That distinction is the basis for the ablation tests stated in Section 10.

Refractive burden coupling and burden-driven negative damping are expressible in the integer operator without any floating-point state in the transport step and without calibrating the live update against continuum quantities. The clamp event, which would be a nuisance in a floating-point solver, becomes an observable of the bounded-register experiment when amplification is driven by integer burden coupling.

7. Boundary Treatment

Boundary conditions are harder to tune on bounded integer registers than on floating-point fields. The reason is structural. In a floating-point solver, small reflected wave components are silently absorbed by mantissa rounding; the reflection is present but invisible at the precision the solver operates at. In the register-bounded solver there is no such absorption. Every integer quantum of reflected velocity survives exactly, and if the boundary does not explicitly remove it, it propagates back into the interior at full amplitude.

This section describes the boundary treatment used in the tested large-amplitude thin-slab regime. It is a three-layer construction: a deep hidden band beyond the visible experiment zone, a cubic-profile velocity drain across that band, and a phase-inverted ghost at the outermost ring. Each layer addresses a different failure mode observed during large-amplitude runs.

The Hidden Band

The simulation grid is larger than the visible experiment region. In one current test configuration, a 243×243 live region is embedded inside a 357×357 grid, leaving a 57-cell hidden band on every side. The hidden band is simulated by the full Laplacian and carries its own register state; it is outside the reported live region.

The purpose of the hidden band is physical space. Waves that leave the visible region enter the hidden band and continue to be integrated; they have somewhere to go before the outermost boundary is reached. Without this space, the drain profile of the next subsection has no room to act.

Cubic-Profile Velocity Drain

Within the hidden band, a per-cell velocity drain is applied. Let \(d\) be the depth into the hidden band measured from the live region boundary, so \(d = 0\) at the interface and \(d = 57\) at the outermost ring. The drain coefficient scales cubically with depth:

\mathrm{drain}(d) = \frac{D_{\max}\, d^3}{H^3}

where \(H = 57\) is the hidden-band width and \(D_{\max}\) is the peak drain fraction at the outer edge. A development value is \(D_{\max}/D_\mathrm{den} \approx 0.55\), giving a 55% velocity reduction per tick at \(d = H\).

The cubic profile is load-bearing. A linear profile produces a visible impedance lip at the interface, which itself acts as a reflection source. A quadratic profile is better but still produces observable reflection. The cubic profile starts nearly flat at the interface, roughly \(1/H^3\) at \(d = 1\), rising smoothly to the full drain at the outer edge. The interface is almost invisible to incoming waves, and the drain strength grows only after the wave has travelled far enough into the band that reflection at the interface is no longer a geometric concern.

The velocity drain is applied per cell per tick as a truncated integer fraction:

v_\mathrm{drained} = \mathrm{clamp}_n\!\left( v' - \left\lfloor \frac{v' \cdot \mathrm{drain}_\mathrm{num}(d)}{D_\mathrm{den}} \right\rfloor_{\!\text{trunc}} \right)

where \(\mathrm{drain}_\mathrm{num}(d)\) is the integer numerator produced by the cubic profile at depth \(d\), precomputed once per grid configuration.

Phase-Inverted Ghost

The outermost ring of cells is overwritten, after each step, with a sign-inverted snapshot of the cells one step inside:

\phi_\mathrm{edge}(i) = -\phi_\mathrm{inner}(i), \qquad v_\mathrm{edge}(i) = 0

The inner snapshot is taken after the current step and held for use as the ghost value at the next step. This gives a one-tick delayed opposite-phase reading at the outermost ring.

The one-tick delay is part of the tested boundary contract. Zero delay cancels the wave before it reaches the ring and destabilises the boundary; longer delays introduce phase slip and can turn cancellation into reflection. One tick is the minimum latency that cancels a lattice-speed outgoing wave in the present large-amplitude tests.

Four corner cells remain at \(\phi = v = 0\) throughout, because the corners have no well-defined single-axis inward direction and no natural ghost partner.

Why The Tested Regime Uses All Three Layers

The three layers address three separate failure modes. Without the hidden band, the drain has no room to act: a short drain is either too gentle to attenuate the wave or too harsh to avoid creating its own reflective impedance lip.

Without the cubic-profile drain, the transition from live region to hidden band is itself a reflection source. Linear and quadratic profiles produce observable reflection at the interface; cubic is the lowest power that makes the interface transparent within the tested amplitude range.

Without the phase-inverted ghost, the outermost ring acts as a weak reflective wall. The cubic drain brings the outgoing wave to a small fraction of its interior amplitude, but small is not zero. The ghost cancels the residual by destructive interference at the ring.

In the tested large-amplitude regime, each layer alone was insufficient. Two layers in combination reduced reflection but did not remove enough of it for the interior closure observable to remain clean. All three together produced a boundary quiet enough for the interior to run to register saturation without exterior reflection dominating the measurement.

The boundary treatment is not a refinement of floating-point practice. It is a response to integer substrate behaviour: residual boundary energy survives unless the construction explicitly removes it.

8. A Worked Example

A small example illustrates the operator at hand-computable scale. The point is not to derive a continuum speed or a closed-form wave law. The point is to show that the register state after one tick is an exact consequence of the initial condition and the integer update rule.

Setup

Consider a 5×5×1 thin-slab grid with reflective boundaries on all four in-plane edges. Use a 9-trit register,

\[ L_9 = 9841 \]

and seed a single excitation at the centre cell:

\phi(2,2,0) = 3000, \qquad v(x,y,z) = 0\quad\text{everywhere}

Use the transport parameters

p = 1, \qquad q = 8, \qquad d_n = 1, \qquad d_d = 512

with no source term and no activation kicks. The only active mechanism is the integer Laplacian with ternary downshift and clamp.

Step 1: Laplacian At The Centre

At the centre cell, all four in-plane neighbours are initially zero. The two z-neighbour contributions come from the reflective \(nz=1\) thin-slab rule, so both contribute the centre value:

\begin{aligned} &\nabla^2_\mathrm{int}\phi(2,2,0) \\ &= 0 + 0 + 0 + 0 + \phi(2,2,0) + \phi(2,2,0) - 6\cdot3000 \\ &= 2\cdot3000 - 18000 \\ &= -12000 \end{aligned}

The effective in-plane Laplacian is therefore \(-12000\), not \(-18000\). This is the thin-slab collapse of Section 2 written out explicitly.

Step 2: Laplacian At A Face Neighbour

At a face neighbour of the centre, such as \((1,2,0)\), the only non-zero contribution comes from the centre cell:

\begin{aligned} &\nabla^2_\mathrm{int}\phi(1,2,0) \\ &= 0 + 3000 + 0 + 0 + 0 + 0 - 0 \\ &= 3000 \end{aligned}

The four face neighbours of the centre therefore receive the same positive Laplacian contribution by symmetry. The remaining cells have zero Laplacian at tick 0 because their sampled neighbours are all zero.

Step 3: Wave Kicks

The centre wave kick is

\begin{aligned} w(2,2,0) &= \mathrm{down}_1\!\left(\left\lfloor \frac{-12000}{8} \right\rfloor\right) \\ &= \mathrm{down}_1(-1500) \\ &= -500 \end{aligned}

At a face neighbour,

\begin{aligned} w(1,2,0) &= \mathrm{down}_1\!\left(\left\lfloor \frac{3000}{8} \right\rfloor\right) \\ &= \mathrm{down}_1(375) \\ &= 125 \end{aligned}

All velocities are zero at tick 0, so the damping term is also zero everywhere:

\delta = 0

Step 4: Velocity Update

With no source and no activation kicks, the raw velocity update is simply the wave kick:

\begin{aligned} v_\mathrm{raw}(2,2,0) &= -500 \\ v_\mathrm{raw}(1,2,0) &= 125 \end{aligned}

Both values lie inside the 9-trit bound \([-9841,+9841]\), so no velocity clamp occurs:

\[ A_v = 0 \]

Step 5: Field Update

The field register is updated by adding the clamped velocity:

\begin{aligned} \phi'(2,2,0) &= \mathrm{clamp}_9(3000 - 500) \\ &= 2500 \end{aligned}

At a face neighbour,

\begin{aligned} \phi'(1,2,0) &= \mathrm{clamp}_9(0 + 125) \\ &= 125 \end{aligned}

No field clamp occurs, so the total clamp absorption for the first tick is

A = A_v + A_\phi = 0

After One Tick

The state after one tick is therefore:

site class	\(\phi'\)	\(v'\)
centre \((2,2,0)\)	2500	-500
four face neighbours	125	125
all other cells	0	0

The initial centre amplitude has decreased from 3000 to 2500, and the four face-neighbour cells have each picked up amplitude 125. The operator has moved support from the seeded cell into its face-neighbour stencil by integer arithmetic alone.

Second-Tick Check

At tick 2, the centre sees its four face neighbours at 125 and its two reflective z contributions at 2500:

\begin{aligned} \nabla^2_\mathrm{int}\phi(2,2,0) &= 4\cdot125 + 2\cdot2500 - 6\cdot2500 \\ &= 500 + 5000 - 15000 \\ &= -9500 \end{aligned}

The next centre wave kick is therefore

\begin{aligned} w(2,2,0) &= \mathrm{down}_1\!\left(\left\lfloor \frac{-9500}{8} \right\rfloor\right) \\ &= \mathrm{down}_1(-1187) \\ &= -395 \end{aligned}

Before damping, the centre velocity would update from \(-500\) to \(-895\). Again, this remains inside the 9-trit register bound.

Reading The Example

The trace can be extended indefinitely. The support expands through repeated face-neighbour coupling, while the amplitude and velocity history are determined by the integer gain, downshift and damping terms. It should be read as a local update audit, not as a derived speed law.

No arithmetic operation in the update has been evaluated in floating point. The computation is reproducible by hand, and its register state is an exact function of the initial condition and the update rule. The local propagation pattern is visible without appeal to any continuum-limit interpretation.

9. Implementations And Scope

The operator is currently exercised in two experimental CAELIX implementations. Both are browser-based prototypes, and both target different amplitude regimes. Their code names are not part of the specification. The mathematical construction in the present paper is sufficient to reimplement the operator without reference to the interface code.

Small-Amplitude Planar Regime

The smaller-amplitude implementation is a planar substrate-runtime test regime used for substrate settling, motif inference and propagation over power-of-three grids. It uses a narrow register budget, reflective boundary sampling of the Laplacian neighbours, and the full residue-accumulation machinery for burden and charge activation. Clamp events are tracked but are primarily a backstop against drift; they are rare in normal operation.

The reflective-sampling boundary is adequate for this regime because the substrate amplitude remains well below the register limit by design. Reflected waves at the boundary are small, and their propagation back into the interior is not a structural observable.

Large-Amplitude Thin-Slab Regime

The larger-amplitude implementation is a thin-slab closure test under driven external forcing. It uses a wider register budget, the full six-neighbour Laplacian (which collapses to effective four-neighbour behaviour at \(nz = 1\)), both operator extensions of Section 6, and the three-layer boundary construction of Section 7. Clamp events are the primary closure observable and are expected to occur in structured spatial patterns.

The large-amplitude regime required the full boundary machinery. The test drives register saturation inside the core by design, and the outward-radiating energy from the driving ring would contaminate the interior observable if it were not absorbed cleanly at the boundary. The three-layer construction is the boundary contract that made the large-amplitude test usable.

What The Two Implementations Do Not Yet Share

The two test regimes have architectural differences beyond register width and boundary treatment. The smaller-amplitude planar regime carries activation-residue machinery for sub-unit burden and charge. The larger-amplitude thin-slab regime operates with driver amplitudes well above the single-trit floor and does not currently use residues. The smaller-amplitude regime has an explicit motif-interaction and annihilation layer built around the Laplacian. The larger-amplitude regime is currently pure Laplacian plus burden-coupling extensions. Neither difference is a structural problem; both regimes point toward a common core operator, with activation residue and motif layers belonging to higher tiers above the transport operator.

At the time of writing, the two implementations described above are the current operator test regimes. The remainder of the CAELIX simulation suite (wave-propagation demonstrations, soliton experiments, interference studies) still uses floating-point telegraph and diffusion solvers and has not yet been migrated to the integer operator. That wider migration is not a prerequisite for the present paper.

Two current test regimes are enough to specify the operator contract: one small-amplitude, one large-amplitude. They use different register budgets and different boundary treatments, but they instantiate the same bounded-integer transport specification. The wider CAELIX simulation suite can migrate case by case where the integer operator adds diagnostic value.

10. Validation And Falsifiers

The operator is useful only if its diagnostics survive controlled changes to the register budget, boundary construction and coupling terms. The purpose of this section is to state what would weaken or falsify the present construction as an operator, rather than to defend every current parameter choice.

Register-Width Convergence

Clamp maps produced in saturation studies must be tested across increasing register widths. A wider register delays saturation by construction. That is expected. What must converge is the qualitative structure of the saturation region after adjusting for the changed amplitude budget and run time.

If the measured closure surface, clamp-shell geometry or exterior response changes without convergence as \(n\) increases, then the observed structure is dominated by register width rather than by the intended dynamics. In that case the clamp map remains a valid register diagnostic, but it cannot be interpreted as a stable closure observable.

Boundary Contamination

Large-amplitude runs must be repeated with altered hidden-band width, drain profile and ghost timing. If the interior clamp map changes materially when the boundary is moved farther away or when the drain is retuned, then the claimed interior observable is contaminated by exterior reflection.

The three-layer boundary is not validated by looking quiet. It is validated only if the interior diagnostic remains stable under boundary-parameter variation.

Thin-Slab Collapse

The six-neighbour operator should collapse to the effective four-neighbour planar operator at \(nz=1\) under the stated reflective boundary rule. That collapse is a direct algebraic check. If a depth-one slab produces behaviour that cannot be accounted for by the corresponding four-neighbour effective stencil, then either the boundary rule is being misapplied or the implementation is not following the operator specification.

The next depth test is also simple: increasing \(nz\) should extend substrate depth rather than change the class of the operator. If early \(nz>1\) runs require a different transport law rather than a depth extension of the same law, the thin-slab interpretation has failed.

Transport And Activation Separation

A direct ablation should compare the separated transport/activation update against the known failure mode in which tiny factors are placed directly into the transport kick. If both versions produce equivalent propagation, then the separation rule is unnecessary. If the mixed version freezes or suppresses propagation while the separated version preserves it, the rule is validated as an integer-register constraint.

The residue ledger must also be checked for bias. Over long runs, positive and negative sub-unit activations should not drift asymmetrically under otherwise symmetric conditions. A persistent sign bias would indicate an error in truncation, residue handling or modulo convention.

Burden-Coupling Ablation

The two burden extensions should be tested separately. Refractive burden coupling changes the effective transport denominator. Burden-driven negative damping changes local amplification. If a saturation shell appears only when both are active, but neither component can be isolated, then the mechanism has not been separated cleanly.

The minimum useful ablation set is: baseline integer transport, refractive coupling only, negative damping only, and both together. The clamp map, exterior reflection, interior saturation history and field recovery should be compared across all four.

Isotropy And Lattice Artefacts

The current von Neumann stencil is axis-preferential. That is not a defect by itself, but it must be measured. Circularity, D4 residue, radial error and orientation-dependent propagation speed should be logged as first-class diagnostics. If large-scale observables remain locked to grid axes beyond the expected stencil anisotropy, the operator is not yet suitable as an isotropic transport primitive.

This falsifier is especially important for the thin-slab regime, where depth-one geometry can hide or exaggerate anisotropy depending on the boundary contract.

The operator is not validated by running. It is validated by surviving changes that should not alter the intended observable: wider registers, deeper boundaries, altered drain parameters, separated burden couplings and depth extension beyond \(nz=1\). Failure under those checks would not make the arithmetic invalid, but it would narrow what the current operator can honestly be claimed to measure.

11. Costs And Open Questions

A balanced account requires stating what the operator does not yet do and what it costs.

Costs

The integer operator on commodity hardware is not free. Integer arithmetic on 64-bit host integers is fast, but the per-cell accounting associated with the register discipline (clamp tracking, residue grids, drain coefficient evaluation, ghost-buffer snapshots) is real work that a naive floating-point Laplacian does not carry. In current browser prototypes, the integer Laplacian plus clamp tracking is broadly comparable in cost to a floating-point Laplacian of the same stencil, but that comparison is implementation-dependent and should not be treated as a general performance claim. The main argument here is diagnostic clarity, not speed.

The operator does not inherit native continuum special functions. Where a floating-point solver can call sin, exp or sqrt at library precision, the integer operator must either avoid those functions or supply bounded-precision approximations. Current burden and source profiles are constructed by ordinary floating-point arithmetic at initialisation, then rounded into integer register form; the running solver does not invoke transcendental functions in its inner loop.

The operator does not currently have an established isotropy measurement protocol comparable to those used for the hexagonal and cubic lattice gas automata [12]. Rotational isotropy is a tractable open question, but it has not yet been measured with the methodology the floating-point programme used in the Emergent Field Physics paper [3].

Open Questions

Several pieces are explicitly provisional.

The four-and-six-neighbour von Neumann stencil is the current transport primitive but may not be the final CAELIX stencil. A balanced-ternary-native stencil using all 26 neighbours of the 3×3×3 cube is a candidate, with weights derived from the cube's octahedral orbit decomposition rather than chosen by calibration. Whether such a stencil is worth the additional computational cost is open.

The integer bridge is not trit-native arithmetic. Host integer storage with clamp and downshift enforcement is a development convenience. A stricter implementation could store and operate on explicit trit-register representations, but that is a separate implementation question. The present paper specifies the operator semantics rather than a host-language storage strategy.

The burden-driven negative-damping extension of Section 6 has not been analytically characterised. Its amplification rate and saturation shell location are determined empirically. A closed-form expression for the clamp-shell radius as a function of the burden profile and amplification divisor would make the extension quantitatively predictive rather than demonstratively observable.

The three-layer boundary construction has not been characterised across the full range of amplitudes for which it is used. It is quiet enough for the current experiments, but the margins are not mapped. A quantitative boundary-reflection measurement as a function of outgoing wave amplitude, hidden-band width, cubic-profile parameters and ghost delay would turn the construction into an engineering specification rather than a working recipe.

The activation residue machinery is used in the smaller-amplitude regime but not currently in the larger-amplitude closure regime. It should eventually become part of a common operator core. That unification can wait until the larger-amplitude regime needs sub-unit activation physics, which it does not yet exercise.

Scope Of The Claim

The paper does not claim that the integer operator is superior to floating-point equivalents in all regimes. It claims that the integer operator makes a specific class of substrate behaviour (clamp absorption, boundary reflection, saturation-shell physics) visible in a way that floating-point equivalents do not. The floating-point programmes are not being retired by the present work; they are being complemented by it.

The paper does not claim that any current register width is final. The 9-trit and 17-trit values discussed above are examples of development budgets chosen for specific diagnostic targets. Wider registers may be required for experiments that need greater amplitude headroom, and narrower registers may be adequate for smaller-scale studies.

The paper does not claim that the three-layer boundary is the only viable boundary construction. It claims that the three-layer construction was sufficient for the tested large-amplitude thin-slab regime, and that each of its three layers addresses a distinct failure mode visible in the integer-bounded substrate. Simpler boundary constructions work in the smaller-amplitude regime and should be used where they are sufficient.

12. Conclusion

The Substrate Audit argued that numerical physics should report the substrate on which its results are computed. This paper has taken the next step: it specifies one transport primitive whose substrate budget is not hidden. The state is held in bounded signed integer registers; the trit width is an explicit experimental budget; overflow is recorded as clamp absorption; sub-unit activation is carried by a residue ledger; and boundary treatment is exposed as part of the result rather than treated as background plumbing.

The main claim is methodological before it is physical. The operator does not prove that a balanced-ternary Laplacian is the final transport law for CAELIX, nor that floating-point solvers are obsolete. It shows that a field update can be run in a regime where the finite substrate is visible enough to audit. The cost of that honesty is also visible: register width must be chosen, saturation must be interpreted carefully, boundary reflection must be measured, and integer coupling terms must be ablated rather than assumed.

The two current test regimes are sufficient to state the operator contract. The smaller-amplitude regime exercises residue accumulation, motif interaction and ordinary propagation without making saturation the main observable. The larger-amplitude thin-slab regime stresses the same register discipline under closure-style loading, where clamp maps and boundary behaviour become central diagnostics. Their shared value is not that either is final, but that both make the same arithmetic commitments visible under different stress conditions.

The strongest future work is not to add more visual demonstrations. It is to falsify the operator cleanly: widen the register and test convergence, move the boundary and measure contamination, separate refractive burden from negative damping, extend the slab depth beyond \(nz=1\), and quantify anisotropy rather than hand-waving it away. If those tests fail, the operator remains a useful integer experiment but its physical interpretation narrows. If they survive, the bounded-register construction earns a stronger role in the CAELIX field programme.

The operator is not finished work. It is a specified primitive with costs, open questions and falsifiers. Its value is that it makes register-level substrate behaviour observable rather than absorbed. That is the methodological commitment of the Substrate Audit expressed as a concrete running operator.

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