A particle-supported slicing event on a fixed spinless cone

We construct an explicit one-parameter family of embedded spacelike polyhedral slices of a fixed spinless conical spacetime with deficit $\mu =\pi /2$. The family agrees with a prescribed resting exterior and has exactly one incidence event: the link of the particle changes from three incident faces to six. Six vertices, twelve edges, and six faces are born at the particle; their coordinates, lengths, areas, and one-sided laws are given exactly. A finite table of adapted affine charts yields identity products around regular vertices and the full spinless conical identification around a primitive meridian. The incoming germ, outgoing germ, and annular collar have the same framed affine peripheral holonomy. For an explicitly stated geometry-only interface, the event has spectator-blind control, a forward-domain decoder, and an exact NOT–CNOT–Toffoli implementation with separately reserved location and history storage. The result is a kinematic consistency certificate for one prescribed slicing event.

The object of this paper is one incidence event involving the actual singular worldline of a fixed conical spacetime. The event is not obtained by attaching a particle label to a vacuum exchange, and its support is not enlarged merely to include an otherwise unused particle. The particle's combinatorial link changes from a three-cycle to a six-cycle.

The conical description of a stationary spinless particle is classical DJH1984. Particle-associated polygon transitions also occur in the classical polygon literature Hooft1992,Hooft1993. We do not claim to introduce particle-supported transitions as such. The contribution here is an explicit construction together with a complete geometric and finite-interface certificate. Polygon and holonomy terminology is compatible with the language of Kadar2005; no constraint-algebra result from that work is used.

All mathematical assumptions needed below are stated explicitly. In particular, the construction does not depend on an unpublished length table or an unspecified reversible compiler.

A configuration contains:

1. a labelled polygonal slice or a labelled one-sided slice germ;

2. a geometric particle packet specifying an identifier, deficit, worldline, reference rest vector, and particle-cell label;

3. an exact clock and a finite permission register;

4. for a fixed finite list of original matter names, one declared cell location per name;

5. an internal matter state.

A geometric rule, called a $G$-rule, may inspect the declared geometric data and geometric control registers. Its guard, geometric output, certificates, and cell-label transport must not depend on the number of matter names or on their internal state. Applying a fixed cell-label transport to each original name is permitted.

At the event considered here:

1. no measurement or coupling rule is applied;

2. the internal matter factor is acted on by the identity;

3. the particle identifier is geometric data, not a test of matter occupancy;

4. rounded reports are auxiliary and do not determine the event guard;

5. implementation history is not a geometric observation;

6. decoder correctness is required on a declared forward event domain, not on intermediate compiler states.

The three-way distinction between geometric, measurement, and coupling rules is retained. The measurement primitive $\mathrm{Enter}$ is not used.

Faces are embedded flat polygonal disks in spacelike affine planes. They need not be convex. The fixed exterior may contain unbounded polygonal faces. Regular vertices need not be trivalent.

At a singular event parameter, collapsed cells are not positive dimensional cells of the slice. Their labels and one-sided formulas may be retained as germ metadata. A change between two such labelled germs is an admissible event when the two germs have the same underlying limiting slice and the prescribed common exterior.

The interface theorem below concerns Contract contract:G and Definition def:objects. It is not an admission theorem for a separate calculus that imposes additional restrictions, such as trivalent regular vertices, elementary four-face exchanges, convex faces only, or equal proper-time increments in every face frame. Compatibility with such an additional restriction would require a separate check.

Thus the geometric existence theorem is unconditional for the objects defined here; its use in any frozen external framework is conditional only on that framework admitting these explicitly described germs. No unprovided admission predicate is silently enlarged.

The construction is a three-to-six particle-link expansion. For negative parameters, three sloping faces meet at the particle. For positive parameters, six different sloping faces meet there. The old three faces remain in the cellulation, but cease to be incident to the particle.

This is one isolated event in the prescribed one-parameter family. It is a simultaneous local expansion, not a claim of genericity in the space of all slicings and not an identification with one particular elementary transition from a classical catalogue.

Use Minkowski coordinates $z=(T,X,Y)$ and metric

Fix

The spatial cone $C_{\mu }$ is obtained from

by identifying $(r,0)$ with $(r,L)$ and identifying all points with $r=0$. Its metric away from the apex is

The fixed spacetime is

Its singular worldline is

In the reference development, $\mathcal{W}$ is the line $\{(T,0,0)\}$.

For later use set

The cone consists of three developed quadrants

The ray in direction $v_3$ is identified with the ray in direction $v_0$. Indices on cone rays and cell labels are taken modulo three; the developed vectors $v_0,\dots ,v_3$ are kept distinct.

The upper seam is mapped to the lower seam by

Thus $J(T,0,-r)=(T,r,0)$, and $J$ fixes $\mathcal{W}$ pointwise.

The cone is defined by gluing the two boundary rays of the $3\pi /2$ wedge. It is not the quotient of the entire Minkowski plane by the finite group generated by $R(\pi /2)$. The latter construction would have a different opening angle.

A positive meridian increases the lifted angular coordinate by $L$. Its holonomy will mean the ordered product of affine coordinate transitions encountered along the positively oriented loop, with the last transition acting on the left.

With this convention, crossing the upper seam into the lower seam contributes $J$. The increasing-angle developing map on the universal cover satisfies the inverse equivariance relation

There is no sign contradiction: deck equivariance and the ordered coordinate-transition product use inverse conventions.

The affine reference coordinates retain their origin on $\mathcal{W}$. A basepoint for a loop does not mean that the affine coordinates are re-centred at that basepoint.

Consider

on the axis-aligned wedge with seam $(r,0)\sim (0,-r)$. If $p\ne q$, this function does not descend continuously to the cone at $s=0$.

In particular, the choice $p=3/5$, $q=5/13$ does not define a particle-supported slice on this seam. A positive upper cap does not repair the defect near the apex. If $p=q$, the displayed family has no rate-driven incidence event.

Proof. For every $r>0$,

These values must agree at identified points. They do not agree when $p\ne q$.

A fixed positive upper cap is inactive for sufficiently small $r$, so the same mismatch persists near the apex. If $p=q$, then $a=b$ and

The graph changes only by time translation; its incidence structure is independent of $s$.

This is a restriction on one named ansatz and one seam. It is not an obstruction to other slicing families.

In each sector $S_i$, with coordinates $(\xi ,\eta )$, define

These definitions agree on every common ray. In particular, on each ray,

They therefore define continuous single-valued functions on $C_{\mu }$.

Fix the parameter window $-1\le s\le 1$ and set

The slices are

The constants $2/3$ and $3/5$ are chosen for this construction. They are not inherited from the obstructed vacuum maximum.

For all $x\in C_{\mu }$,

Consequently:

1. if $s\le 0$, then

   $$F_s=s+\min \{2,h\};$$

2. if $s>0$, the branch $s+k$ can dominate $h$ only where

   $$\rho \le \frac{30s}{11}\le \frac{30}{11}<3;$$

3. for all $s\in [-1,1]$ and all $\rho \ge 3$,

   $$F_s=2+s.$$

Proof. Since $\min \{\xi ,\eta \}\le (\xi +\eta )/2$,

This proves the first two assertions. For $\rho \ge 3$, $h\ge 2$, so the outer minimum in (eq:family) equals $2$.

We use the following elementary graph criterion.

Let $F_s$ be continuous functions on a Euclidean cone, piecewise affine in finitely many developed sectors. Suppose that every affine branch has spatial gradient of norm at most $\lambda <1$. Then the graph of $F_s$ is embedded and spacelike on each face, and

If, in addition,

for $s_2>s_1$ and $c>0$, then the graphs are strictly time ordered. If $F_s=H+s$ near the boundary of a compact spatial patch, they match the prescribed translating resting exterior there.

Proof. On a face with gradient $a$, the induced metric is

For every spatial tangent vector $v$,

Integrating the gradient bound along piecewise linear spatial paths and taking the infimum of path lengths gives the global Lipschitz bound. The projection of a graph to the cone is a homeomorphism, so the graph is embedded.

In the ultrastatic product, a causal curve between two points requires its time difference to be at least the spatial distance. The strict Lipschitz bound therefore also gives acausality between distinct graph points. At the apex, spacelikeness is understood in this uniformly Lipschitz, stratified sense.

The ordering and exterior statements follow directly from their respective hypotheses.

The functions (eq:family) descend through the seam and define embedded spacelike slices. Uniform constants are

More precisely, for $s_2>s_1$,

at every spatial point.

Proof. Single-valuedness was established before (eq:family). The nonconstant gradients are

up to sector rotations. Their norms are bounded by $2\sqrt{2}/3$.

For fixed $x$, the function

is nondecreasing and $1$-Lipschitz. Adding $s$ gives the displayed two-sided ordering bound. Lemma lem:graph applies.

Let

The moving core is $\{\rho <3\}$. Choose the collar

It is an annulus, disjoint from $\mathcal{W}$, and agrees with the fixed resting exterior translated in time by $s$.

Here and below, “fixed exterior” means fixed spatial cellulation, fixed transition data, and the prescribed equation $T=2+s$. It cannot mean an identical embedded subset of spacetime for every $s$, since the slices are strictly ordered.

The particle intersection is

Thus $P_s\in K_s$ for the entire window, including $s=0$. The worldline itself has not changed.

Write $A_i=3v_i$ and $D_i=4v_i$ for spatial vertices on the cone. Their lifted time coordinate is $2+s$.

For $s>0$, set

and introduce

All coordinates in this section are spatial unless a time coordinate is explicitly supplied.

The complete vertex data in one sector are

The last three rows are regular vertices only for $s>0$.

For $s<0$, the core face in sector $i$ is the triangle

For $s>0$, replace it by the following three faces:

The face $V_i$ is a simple, generally nonconvex pentagon.

For every $s$, the collar-side face is

Beyond $\rho =4$, retain the fixed unbounded face

The three exterior rays starting at the $D_i$ are also retained.

The faces just listed form the graph of $F_s$ without overlap of face interiors or omitted regions.

Proof. For $s\le 0$, Lemma lem:active-region gives the asserted three core triangles and resting exterior.

For $s>0$, in the half-sector $\xi \ge \eta$,

Hence the branch boundary is

It joins

In the other half-sector the boundary is

joining $C_i$ to $B_{i+1}$.

The region where $s+k$ dominates is precisely the union of the two triangles $[P,B_i,C_i]$ and $[P,C_i,B_{i+1}]$. The remainder of the sector triangle $\rho \le 3$ is the stated pentagon $V_i$. Since

the inner boundary is strictly separated from the outer crest. The cap is active exactly in the prescribed exterior.

Use the following persistent edge labels:

The new edge labels for $s>0$ are

The superscript on ${v_i}^{\mathrm{edge}}$ distinguishes an edge label from the developed direction vector $v_i$.

The induced lengths of all finite edges in $K_s$ are:

Every active edge has positive length on its open parameter interval. Every new edge vanishes linearly at $s=0$.

Proof. For an edge in a developed chart,

Along $e_i$, the spatial displacement is $3v_i$ before the event and $(3-\frac{3}{2}s)v_i$ after it, while $\Delta T=\frac{2}{3}|\Delta x|$. This gives the first row.

For $u_i$, both endpoints have time coordinate $2s$, so its length is $3s/2$.

For ${v_i}^{\mathrm{edge}}$, the spatial displacement is $(u_s,u_s)$ and the time displacement is $3u_s/5$. Hence

For $w_i^{-}$, the spacetime displacement from $B_i$ to $C_i$ is

Therefore

The other perimeter edge is symmetric. The remaining rows have $\Delta T=0$ and follow immediately.

The induced areas are

In particular, for $0<s\le 1$,

Proof. On a graph plane with gradient $a$, induced area is Euclidean projected area multiplied by $\sqrt{1-|a{|}^2}$.

For $V_i$, this factor is $1/3$. The original projected triangle has area $9/2$. Each new projected inner triangle has area

Subtracting the two inner projected areas and multiplying by $1/3$ gives the stated area of $V_i$.

For either inner face, the area factor is $\sqrt{1-9/25}=4/5$, giving $9s^2/11$. The collar quadrilateral has projected area $(16-9)/2=7/2$ and is horizontal.

The changing boundary in the half-sector $\xi \ge \eta$ obeys

Its Euclidean normal velocity with respect to the parameter $s$ is

In the other half-sector, the analogous equation is $\xi +10\eta =15s$. The rays, bisectors, and outer crest have fixed spatial directions.

The endpoint laws are

and the outgoing length derivatives follow directly from Proposition prop:lengths. Before the event, all finite edge lengths are constant.

These are slicing-parameter laws, not matter-velocity laws.

For $s<0$, the link at $P_s$ has the cyclic order

For $s>0$, its cyclic order is

The old edges $e_i$ and faces $V_i$ are no longer incident to the particle after the event.

The new bisector edges are genuine creases: their adjacent gradients are $(3/5,0)$ and $(0,3/5)$. The ray edges are also genuine creases; after unfolding across a ray, the transverse gradient changes sign. Thus the sixfold star is not produced by merely subdividing a single flat face.

Figure fig:split-VII-F records one view of the same separation. Figure fig:chain-VII-F records one view of the same separation. Figure fig:pair-VII-F records one view of the same separation. The drawings add no hypothesis.

The family (eq:family) has exactly one incidence-event parameter in $[-1,1]$, namely $s=0$. At that parameter:

1. the six vertices $B_i,C_i$ coalesce at $P_0$;

2. the twelve edges $u_i,{v_i}^{\mathrm{edge}},{w_i}^{\pm }$ vanish;

3. the six faces ${U_i}^{\pm }$ vanish;

4. the particle link changes from three to six incident faces.

There is no other collision, zero-length active edge, zero-area active face, or contact with the outer crest in the window.

Proof. The cellulation is independent of the sign-preserving choice of $s$. For $s<0$, all its finite lengths and areas are constant and positive.

For $s>0$, the vertex formulas show that the six new vertices are distinct and lie strictly inside $\rho =3$. Propositions prop:lengths and prop:areas give positivity of every active finite edge and face. Lemma lem:decomposition excludes self-intersection and omitted regions. The strict bound $\rho (C_i)\le 30/11<3$ excludes contact with the crest.

At $s=0$, the stated vertices, edges, and faces collapse according to their explicit formulas, while $e_i$ and $V_i$ have the incoming edge and face as their nondegenerate limits. The particle-link orders displayed above differ in valence, so the event cannot be a relabelling of an unchanged star.

For reference, the finite patch counts are

At $s=0$, the regular cellulation is the incoming one. The additional outgoing labels are germ metadata, not zero-length regular edges or zero-area regular faces.

Let

In sector coordinates define

and

Direct multiplication gives

They have determinant one and positive time orientation.

For a face in sector $i$, use the following affine coordinate changes from the cut-space reference coordinates:

Here $e_0=(1,0,0{)}^{\mathsf{T}}$. The same symbol $Q_i$ for a sector rotation and a collar face is distinguished by context: $Q_i{B}_h{Q}_i^{-1}$ is a matrix expression, whereas $Q_i=[A_i,D_i,D_{i+1},A_{i+1}]$ denotes a cell.

These charts are used on the regular part of the spacetime. Their formulas extend to face closures, but no claim is made that the cone apex has a regular affine neighbourhood.

All linear entries are rational. All translations are rational multiples of $s$. Consequently, at dyadic $s$, every chart entry is rational and exactly representable through the displayed symbolic table. No closure assertion for an unrestricted collection of Lorentz matrices is required.

For an oriented crossing from a face chart $f$ to a face chart $g$, define the seam exponent

If a chart overlap has more than one connected component, the component is part of the crossing data.

The affine transition is

This specifies every adjacency transition: ordinary adjacencies have $n=0$, and the marked-seam adjacencies have $n=\pm 1$.

For a closed chart word $f_0,f_1,\dots ,f_m=f_0$,

For a loop on a slice avoiding the particle, $N$ is its oriented winding number.

Proof. Substitution of (eq:cocycle) cancels every adjacent pair ${A_f}^{-1}{A}_f$ and gives the displayed product.

To identify $N$, lift the spatial projection of the loop to the angular cover with $\theta \in \mathbb{R}$. Each positive marked-seam crossing increases the sheet index by one, and each negative crossing decreases it by one. The terminal angular coordinate differs from the initial one by $NL$. Since the graph projection is a homeomorphism, this is also the winding number of the loop in the punctured slice.

Every sufficiently small loop around a regular vertex has affine product equal to the identity, including regular vertices lying on the marked seam.

Proof. Such a loop is contractible in the punctured cone and has winding number zero. Apply Lemma lem:product.

Choose the collar basepoint

in sector zero. Use the reference affine chart there, without re-centring its origin.

A collar meridian is the polygonal loop $\rho =15/4$, based at $b_s$, traversed in increasing angular order. In the collar face charts its ordered word is

and therefore its affine product is exactly $J$.

For a loop strictly inside the moving core, choose

Because

this loop lies in the $V_i$ faces for every $s\in [-1,1]$. Join it to $b_s$ along the bisector of sector zero. The resulting based meridian is denoted ${\gamma }_s$.

For every $s\in [-1,1]$,

The collar meridian and the core meridian are primitive and positively oriented. The same identity holds for both event germs at $s=0$.

Proof. The angular lift of either meridian increases by exactly $L$, so its winding number is one.

For the core meridian, write $A_i=A_{V_i}$. Starting in the $V_0$ chart, the three-face product is

The path from the collar reference chart into $V_0$ contributes $A_0$, and the return path contributes ${A_0}^{-1}$. The based product is therefore

This is an equality in the full affine group; the translations cancel as well as the Lorentz factors.

At $s=0$, the core meridian avoids every collapsed cell. Its charts and transitions are common to the two germs, giving the same identity.

The preceding common meridian provides a certificate for the whole window. The new star itself has the same nontrivial peripheral word.

For $s>0$, the loop $\rho =s$ lies strictly inside the six-face star, because

In angular order its charts are

Let $A_0,\dots ,A_5$ denote these six adapted coordinate changes. The ordered product is

Transporting the framing to the common collar again gives $J$.

For $s<0$, a sufficiently small meridian meets the three $V_i$ faces and has the three-factor product already computed. The local outgoing loop may shrink as $s\downarrow 0$; it is not used as the nonvanishing loop at the event parameter. Its correspondence with ${\gamma }_s$ is the explicit annular homotopy in the punctured slice.

For an affine change of reference coordinates $A=(C,a)$,

Moreover,

Thus the affine holonomy fixes the developed particle worldline pointwise in every presentation.

The literal equality $\mathrm{Hol}({\gamma }_s)=J$ uses the declared reference framing. Under an unrestricted common affine change of framing, the invariant statement is the corresponding conjugacy class together with its marked fixed axis.

Because $J^4=I$, a matrix product alone does not establish primitivity. A fourfold meridian is noncontractible but has identity holonomy. Primitivity in Proposition prop:holonomy was proved by the angular lift, independently of the matrix calculation.

The geometric particle packet is

It is the same packet on both germs.

The future unit normal of a $V_i$ face, in sector coordinates, is

The future unit normals of $U_i^{+}$ and $U_i^{-}$ are respectively

They are rotated into the developed sector by $Q_i$.

Consequently the particle's face-relative rapidities satisfy

so

The reference rest vector remains $e_0$. A stationary worldline therefore does not imply zero rapidity relative to every incident face.

Let $g_0^{-}$ be the limiting slice at $s=0$ equipped with the incoming one-sided formulas and labels, and let $g_0^{+}$ be the same limiting slice equipped with the outgoing one-sided formulas and labels. The additional outgoing cells are inactive germ slots at $s=0$.

The geometric particle realization transport is

with both points equal to $(0,0,0)$ on the fixed worldline.

The matter-name transport is a function on cell labels. Retain the labels

and the six exterior labels. Define

for every incoming label $\ell$. The geometric realizations of some persistent labels change: for example, $e_i$ is $[P,A_i]$ before the event and $[B_i,A_i]$ afterward. Cell-label transport is abstract location transport, not a trajectory for a material point.

Thus particle realization transport and matter-name transport are different typed maps, even though both preserve the particle-cell label $P$.

Let $s=j{2}^{-d}$ be the decoded clock, with

Let $m=0$ denote the incoming mode and $m=1$ the outgoing mode. Let $\nu$ be a permission counter.

The forward event domain is

with the declared canonical geometric data.

The rule is

No coupling or measurement occurs.

The guard, geometric output, particle packet, certificates, and transport $\tau$ of (eq:Grule) satisfy Contract contract:G. They are independent of the number and internal state of matter names.

Proof. The guard reads only $m,j,\nu$ and fixed geometric validity data. The output geometry is the explicitly prescribed outgoing germ. The particle packet is a fixed geometric packet. The holonomy and geometric certificates depend only on the displayed family and chart table. The transport is the fixed identity map on persistent labels.

None of these definitions reads a location occupancy count or an internal matter state. The same geometric event is defined when there are no matter names at all. Applying the same fixed map to each supplied name changes no geometric decision.

The condition $j=0$ is an exact clock test. It is not inferred from a rounded edge length. At the event, newly born edge slots have limiting length zero, but for sufficiently small positive dyadic $s$ their rounded reports can also be zero. The particle deficit is likewise a fixed symbolic identifier, not an estimate extracted from a length report.

There are $25$ cells in the incoming compact patch and $49$ in the outgoing compact patch. The fixed exterior adds three unbounded rays and three faces. Thus the global dictionary contains $55$ labels, and six bits suffice for a cell location.

One explicit dictionary is:

Codes $55,\dots ,63$ are invalid.

At $s=0$, codes $25,\dots ,48$ are germ metadata and are not valid regular locations. All incoming location labels remain valid outgoing locations at the event. In particular, $\tau$ is injective.

Fix integers

They specify report precision, clock precision, and permission width.

The continuous family (eq:family) is used to prove geometry and one-sided laws. The finite decoder represents only its dyadic subfamily. Rounded reports are additional finite data and do not define that subfamily.

For each of the $24$ finite edge slots in Proposition prop:lengths, store

An inactive slot has report zero. All finite lengths are less than $6$, so $b+3$ bits per report suffice.

The canonical report residual satisfies

All reports can be determined exactly from the symbolic length table: the lengths are rational multiples of square roots of fixed integers, and floor comparisons reduce to exact rational or integer comparisons.

The geometric control and report block has

bits, allocated as follows:

The clock uses signed two's-complement representation, restricted to $-2^d\le j\le 2^d$. Thus $j=0$ is an exact integer-zero test.

The particle block has nine bits: six for its cell label and three fixed identifiers for the particle, deficit, and reference rest/worldline data. Their semantic values are the fixed packet ${\Pi }_{\star }$; no numerical approximation to $\pi /2$ is stored.

Set

For each original matter name reserve six location bits and six history bits. These are not charged to the geometric budget.

A canonical geometric word determines:

1. $s=j{2}^{-d}$;

2. the exact function (eq:family);

3. the appropriate cellulation from Section sec:cells;

4. the affine chart table from Section sec:holonomy;

5. the fixed particle packet;

6. the auxiliary canonical reports.

For $j<0$, only incoming mode is valid; for $j>0$, only outgoing mode is valid. At $j=0$, the two mode values select the two declared germs.

The decoder reconstructs coordinates and exact lengths from the symbolic formulas, not from report words. It ignores retained implementation history. Intermediate compiler states are not assigned geometric interpretations.

Let $f$ be the map on canonical $B$-bit geometric words at the forward event that changes

and leaves the clock and particle packet fixed. Its outgoing reports are the canonical limiting reports at $s=0$. Then, on the forward event domain, decoding $f$ gives precisely the geometric part of (eq:Grule).

Proof. At $s=0$, all persistent finite edges have identical incoming and outgoing limiting lengths. Every new slot has limiting length zero. Thus the two canonical limiting report arrays agree. The mode change selects the outgoing germ rather than the incoming germ, and the permission field has the required decrement. All other decoded data agree with (eq:Grule).

Although the forward map in this example is already injective, we retain complete input history. This gives a simple, explicit implementation certificate and makes no minimal-memory claim.

Let $D_{\mathrm{geo}}$ be the finite set of canonical forward geometric words. For the fixed family, its only free geometric field is $\nu \in \{1,\dots ,2^r-1\}$, so

On two $B$-bit blocks prescribe

Inputs and outputs are disjoint because their first-block mode bits are respectively zero and one. Distinct inputs have distinct outputs because the second output block retains $x$.

Define a permutation $P_{\mathrm{geo}}$ on all $2B$-bit words by exchanging each pair

and fixing every other word.

A permutation of $N$ logical bits consisting of $q$ specified transpositions can be implemented using NOT, CNOT, and Toffoli gates, with $N-2$ clean work bits returned to zero. For the transpositions above, a sufficient gate bound is

Proof. Connect the two words of a transposition by a shortest Hamming path of length $k\le N$. The endpoint transposition is a product of $2k-1$ adjacent-word transpositions along that path.

An adjacent-word transposition is a NOT on the differing bit controlled on the specified pattern of the other $N-1$ bits. Negative controls are obtained by conjugating the relevant controls with NOT gates. The positive multiple-control NOT is implemented by computing the control conjunction into clean work bits with Toffoli gates, performing the target flip, and reversing the conjunction computation. At most $N-2$ work bits suffice. For $N\ge 3$, fewer than $4N$ elementary gates suffice for each adjacent-word transposition, including the NOT conjugations.

Applying this construction to all $q$ transpositions gives the bound. Every conjunction computation is uncomputed, so the work bits return to zero.

Only this elementary finite-permutation argument is needed; no efficiency claim is made.

Apply Lemma lem:synthesis to $P_{\mathrm{geo}}$ with

This geometric circuit is independent of the number of names.

For a supplied list of $n$ original names, append six location and six history bits per name. On the forward domain the location transport is the identity. After the geometric circuit, copy each location bit to its initially zero history bit, controlled by the outgoing mode bit. This uses six Toffoli gates per name and never feeds a name bit into a geometric output.

A sufficient total bit count, including clean work, is

A sufficient gate count is

The geometric budget is independent of $n$. The statement concerns a family of finite circuits, one for each fixed finite name list, not a single bounded circuit accepting unboundedly many names.

For every $b,d\ge 0$, $r\ge 1$, and fixed finite original-name list, there is an exact NOT–CNOT–Toffoli circuit $U$ with the bit bound (eq:totalbits) such that

for every canonical forward event code $c$ with clean history and work registers.

The circuit retains input history, returns all work bits to zero, and acts on the internal matter space as

Its geometric output and geometric control are independent of spectator number and spectator state.

Proof. The geometric permutation sends $(x,0_B)$ to $(f(x),x)$ on the forward domain. Proposition prop:decoder identifies its decoded geometric output with the geometric part of $G$.