Debris as Predictive Compression: An Exact Return Bias and an Open Equation Spine for Observer-Relative Coupling

Nicole Flynn
Symfield PBC
Technical Research Note v0.2
September 17, 2026

Technical Research Note v0.3, September 18, 2026. v0.1 was published September 17; v0.2 the same day added §8a and companion note I (mechanism stress tests); v0.3 adds companion note II (Kn identifiability) and corrects the restoration-timing conclusion in §8a and in companion note I, §3. Contents: equation spine (§§1–15 and the research progression), companion note I, companion note II.

This note is written for readers who want the mathematics and not the motivation. It contains one theorem and one open spine, kept separate.

Theorem. In the four-state process defined at (18)–(25), where activation of available but unrealized transitions can remove edges from the admissible set, the probability of repeating the initial transition $1\to2$ on the first return to state $1$ is

$$\Pr\left(r_{T_1}=(1,2)\ \middle|\ r_0=(1,2),\ T_1<\infty\right)=\frac{2297}{6000}\approx0.3828,$$

against the null model’s value of $1/3$ . The return bias $B_{\mathrm{return}}=99/2000$ is a functional of realized paths alone. It separates the model from every stationary Markov chain on the visible states, including one with re-fit transition probabilities. It does not identify residue from unrealized transitions as the mechanism; another hidden-memory process that favors the last realized edge could reproduce the same return bias. Derivation at (30a)–(30d); status at the end of §8; parameter law and mechanism tests at §8a.

Open spine. Sections 9–15 record an observer-relative extension, compression, effective geometry, opposition-weighted coupling, coupled residue, history-dependent coupling, multiscale inference, as typed maps and constraints. No theorem joins it to the four-state result. The status line under (53) says exactly which equations are published, which are schema, which are derived and which are open.

The three failure modes a reader should check for are: whether the projection in §3 is named before memory is claimed; whether (44) is constrained enough to predict anything; and whether any map in §§9–15 has quietly inherited the status of (30).

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Labels distinguish published mathematics, existing schema, derived results and open extensions.

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1. Observer restriction — published Markov-trace mathematics

For a Markov kernel $P$ , visible window $A$ and inaccessible complement $A'$ ,

$$P_A = I_A P \left[\sum_{k=0}^{\infty}(I_{A'}P)^k I_A\right].$$ (1)

The observer’s effective dynamics include excursions through states outside its window. Distinct generators may produce the same trace:

$$P^{(1)}\neq P^{(2)},\qquad P_A^{(1)}=P_A^{(2)}.$$ (2)

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2. Realization and admissible motion — existing schema

Let $\mathscr E$ be the internal-state bundle and $\mathcal X=\Gamma(\mathscr E)$ .

$$\mathfrak R=(M,\mathscr E,\Phi,h,\Pi,\mathcal D_\mu,W,\mathcal A_t).$$ (3)

$$\dot\Phi_t=P_t\left[F_\mu(\Phi_t,h_t,t)\right],\qquad P_t:T_{\Phi_t}\mathcal X\longrightarrow T_{\mathcal A_t}(\Phi_t).$$ (4)

History may alter the admissible geometry of future motion, not only the next point.

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3. Projection and memory — standard Mori–Zwanzig, applied

Let $\mathcal P$ retain selected variables and $\mathcal P^\perp=1-\mathcal P$ .

$$\frac{d}{dt}\mathcal P u_t=\mathcal P\mathcal L\mathcal P u_t+\int_0^t K_{\mathrm{MZ}}(s),\mathcal P u_{t-s},ds+F_t,$$ (5)

$$K_{\mathrm{MZ}}(s)=\mathcal P\mathcal L,e^{s\mathcal P^\perp\mathcal L},\mathcal P^\perp\mathcal L.$$ (6)

The question is whether residue projected out of the visible description produces a nonzero memory term, and whether its form depends on previously constructed admissibility. The projection must be named: in the four-state toy, $(\Phi_t,q_t)$ of (27) is a Markov state, so the kernel for that projection vanishes identically; memory appears only under the realized-path projection $(\Phi_t,h_t)$ , where (31) exhibits it.

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4. Prospective branching — proposed discrete schema

$$Z_t=(\Phi_t,D_t,h_t,\mathcal A_t,L_t).$$ (7)

$$\mathcal B_t=\operatorname{Br}\left(P_t[F_\mu(\Phi_t,h_t,t)];\ \mathcal A_t,L_t,\xi_t\right).$$ (8)

$$r_t=\operatorname{Sel}_W(\mathcal B_t),\qquad U_t=\mathcal B_t\setminus{r_t}.$$ (9)

$$\Phi_{t+1}=\operatorname{End}(r_t).$$ (10)

$$D_{t+1}=\mathcal Q(D_t,U_t,L_t,W,r_t).$$ (11)

$$h_{t+1}=\mathcal H(h_t,r_t),\qquad L_{t+1}=\mathcal L(L_t,r_t,\xi_t).$$ (12)

$$\mathcal A_{t+1}=\mathcal G_0(\mathcal A_t,h_{t+1},L_{t+1},r_t)\setminus R_D(D_{t+1}).$$ (13)

$$\text{Null model:}\quad R_D\equiv\varnothing.$$ (14)

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5. Non-detection and causal attribution — observation limit

For an incomplete or non-injective witness,

$$W_t(D_t)=0\ \not\Rightarrow\ D_t=0.$$ (15)

This does not establish that debris exists. A legal intervention changes unrealized availability without changing the realized transition:

$$\operatorname{Sel}_W(\mathcal B_t^{(1)})=\operatorname{Sel}_W(\mathcal B_t^{(0)})=r_t.$$ (16)

$$\Delta_k=\operatorname{Contrast}\left(\operatorname{Law}(\mathcal O_{t+k}\mid \operatorname{do}(\mathcal B_t=\mathcal B_t^{(1)}),H_t),\ \operatorname{Law}(\mathcal O_{t+k}\mid \operatorname{do}(\mathcal B_t=\mathcal B_t^{(0)}),H_t)\right).$$ (17)

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6. Four-state realization — fully specified specialization

$$\mathcal X={1,2,3,4},\qquad \mathsf E={(i,j):i\neq j},\qquad |\mathsf E|=12.$$ (18)

$$\theta=\tfrac13,\qquad \lambda=\tfrac12,\qquad \tau=\tfrac7{10},\qquad \Phi_0=1,\qquad D_0\equiv0,\qquad \mathcal A_0=\mathsf E.$$ (19)

$$\mathcal C_t={,e\in\mathcal A_t:\ e\text{ leaves }\Phi_t,}.$$ (20)

$$a_t(e)\sim\operatorname{Uniform}(0,1)\ \text{ i.i.d. for } e\in\mathcal C_t,\qquad a_t(e)=0\ \text{ for } e\notin\mathcal C_t.$$ (21)

$$\mathcal B_t= \begin{cases} {e\in\mathcal C_t:\ a_t(e)>\theta}, & \text{if nonempty},\[3pt] {\arg\max_{e\in\mathcal C_t}a_t(e)}, & \text{otherwise}. \end{cases}$$ (22)

$$r_t=\arg\max_{e\in\mathcal B_t}a_t(e),\qquad U_t=\mathcal B_t\setminus{r_t},\qquad \Phi_{t+1}=\operatorname{head}(r_t).$$ (23)

$$D_{t+1}(e)=\lambda D_t(e)+a_t(e),\mathbf 1[e\in U_t].$$ (24)

$$\mathcal A_{t+1}=\mathcal A_t\setminus{,e:\ D_{t+1}(e)>\tau,}.$$ (25)

Ties in $\arg\max$ have probability zero under the continuous activation law; all selections are therefore understood almost surely. $\mathcal C_t$ is nonempty on every reachable state: $r_t\notin U_t$ , so the realized edge is never deleted at the step it is taken; a state with one admissible outgoing edge always realizes it. Induction from $\mathcal A_0=\mathsf E$ .

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7. Predictive compression — derived Markov representation

Because deletion is permanent, each edge is represented by

$$q_t(e)\in[0,\tau]\cup{\dagger},$$ (26)

with $\dagger$ denoting deletion, and

$$(\Phi_t,q_t)\in\mathcal X\times\left([0,\tau]\cup{\dagger}\right)^{\mathsf E}$$ (27)

is a Markov state for the toy. The compression chain is

$$\text{unrealized activation history}\ \longrightarrow\ (D_t,\mathcal A_t)\ \longrightarrow\ q_t\ \longrightarrow\ \text{future realized-path law}.$$ (28)

Complete upstream histories are not in general recoverable from the compressed state.

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8. Exact path-level result — derived theorem for the toy

$$T_1=\inf{t\ge1:\ \Phi_t=1}.$$ (29)

$$\Pr\left(r_{T_1}=(1,2)\ \middle|\ r_0=(1,2),\ T_1<\infty\right)=\frac{2297}{6000}.$$ (30)

$$B_{\mathrm{return}}=\frac{2297}{6000}-\frac13=\frac{99}{2000}=0.0495.$$ (31)

Condition on $r_0=(1{,}2)$ , i.e. $x:=a_0(1{,}2)=\max{a_0(1{,}2),a_0(1{,}3),a_0(1{,}4)}$ . A competitor $e\in{(1{,}3),(1{,}4)}$ is deleted at $t=1$ iff $a_0(e)>\tau$ , which implies $a_0(e)>\theta$ and $a_0(e)<x$ . Let $N\in{0,1,2}$ count deleted competitors. Edges out of $1$ can be deleted only while $\Phi_t=1$ , and the deletion indicators are decided at $t=0$ independently of the excursion on ${2,3,4}$ , hence independently of ${T_1<\infty}$ . Before the first return, residue on edges leaving state $1$ can only decay and cannot newly cross $\tau$ ; the first-return result is therefore independent of $\lambda$ . On return, activations are fresh and selection is uniform over the $3-N$ surviving edges:

$$\Pr\left(r_{T_1}=(1,2)\ \middle|\ r_0=(1,2),\ T_1<\infty\right)=\mathbb E!\left[\frac1{3-N}\ \middle|\ x\text{ is the maximum}\right].$$ (30a)

The unconditional joint density of $(x,a_0(1{,}3),a_0(1{,}4))$ is $1$ on the unit cube. Equation (30b) integrates its unnormalized restriction to the event that $x$ is maximal; division by the event probability $1/3$ supplies the final factor of $3$ . Given $x$ , each competitor independently lies in $(\tau,x)$ with measure $m:=\max(0,x-\tau)$ and in $(0,\tau]$ with measure $x-m$ . Therefore

$$\mathbb E!\left[\frac1{3-N};\ x\text{ is max}\right] =\int_0^{\tau}\frac{x^2}{3},dx+\int_{\tau}^{1}\left[\frac{(x-m)^2}{3}+\frac{2m(x-m)}{2}+\frac{m^2}{1}\right]dx .$$ (30b)

On $(\tau,1]$ , $x-m=\tau$ . With $\tau=\tfrac7{10}$ the four pieces are

$$\int_0^{7/10}\frac{x^2}{3},dx=\frac{343}{9000},\qquad \int_{7/10}^{1}\frac{\tau^2}{3},dx=\frac{49}{1000},\qquad \int_{7/10}^{1}\tau(x-\tau),dx=\frac{63}{2000},\qquad \int_{7/10}^{1}(x-\tau)^2,dx=\frac{9}{1000}.$$ (30c)

Summing and dividing by $\tfrac13$ ,

$$3\left(\frac{343}{9000}+\frac{49}{1000}+\frac{63}{2000}+\frac{9}{1000}\right)=\frac{2297}{6000}.$$ (30d)

Equivalently, $\Pr(N=0)=\tfrac{98}{125}$ , $\Pr(N=1)=\tfrac{189}{1000}$ , $\Pr(N=2)=\tfrac{27}{1000}$ , and $\tfrac{98}{125}\cdot\tfrac13+\tfrac{189}{1000}\cdot\tfrac12+\tfrac{27}{1000}\cdot1=\tfrac{2297}{6000}$ . Verified by exact symbolic integration and by Monte Carlo ( $2\times10^6$ samples, $0.3827$ ).

The stationary Markov null gives $1/3$ . The debris model produces observable path dependence; the statistic does not uniquely identify debris as its cause.

Established. Under (18)–(25), the process is distinguishable from every stationary Markov chain on $\mathcal X$ , including one whose transition probabilities are re-fit to the data, using realized paths only. One-step marginals do not separate the models: by symmetry, $\Pr(\text{next}=j\mid\text{at }1)=\tfrac13$ under both. Conditioning on the previously realized edge out of the same state does, by (31).

Not established. That residue from the unrealized set $U_t$ is the mechanism. Another hidden-memory process that favors repeating the last realized edge out of a state could reproduce the same return bias. Causal attribution to $U_t$ requires the intervention (16)–(17). Also not established: that any physical system carries such residue; that $\tau$ corresponds to a natural boundary rather than an inserted parameter.

Artifact. The bias is produced by permanent deletion in (25). A reversible admissibility rule would give a different, possibly vanishing, return bias; one such rule is tested in §8a. No monotonicity on later returns is asserted: the initially realized edge can itself be unrealized on a subsequent visit and then deleted.

Problems and current status. 1. First-return bias as a function of $(\theta,\tau)$ , including its independence from $\lambda$ ; and later-return bias as a function of $(\theta,\lambda,\tau)$ and the number of prior visits. 2. A reversible admissibility rule replacing (25), and whether any return bias survives it. 3. The interventional contrast (17) computed on the legal window $\theta<\eta<a_0(r_0)$ , where $\eta$ is the imposed activation of an unrealized competitor. 4. The minimal predictive state of the unrealized activation history for the future realized-path law, and how much of it is captured by the family of return-bias statistics. Status as of v0.2: the first-return part of 1 is closed by (31a); the later-return part of 1 is simulated but has no closed form; 2 is tested for one restoration rule; 3 is computed exactly for the marginal and pinned designs; 4 is partially tested at the first- and second-return horizons. See §8a and the companion note.

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8a. Parameter law and irreversibility dependence — derived, with companion tests

The first-return result of (30) is a special case of a closed law. Let $h=\max(\theta,\tau)$ with $0\le\theta,\tau\le1$ . For the permanent-deletion toy of (18)–(25), with uniform activations on $(0,1)$ ,

$$p_{\mathrm{return}}(h)=1-\frac32h+h^2-\frac16h^3, \qquad B_{\mathrm{return}}(h)=\frac23-\frac32h+h^2-\frac16h^3.$$ (31a)

At $h=7/10$ this recovers $2297/6000$ and $99/2000$ . The law does not depend on $\lambda$ . In the first-return experiment, $\lambda$ affects the probability of returning through trapping, but not the conditional first-return selection probability. It may matter for later returns and for other admissibility rules. There is no null region in $\tau\le\theta$ : there $h=\theta$ and the bias is larger, $265/972$ at $\theta=1/3$ . The bias vanishes only as $h\to1$ .

The effect depends on when restoration occurs relative to the first readout. Under one tested restoration rule, in which residue decays each step and a deleted edge is restored on arrival at its source once its residue has fallen to or below a threshold $\tau’$ , the earliest possible return is two steps and the residue at the check is $\lambda\eta\le0.45$ for every activation in the legal window. At $\tau’=0.6$ every deleted edge is therefore restored before the first return and the bias vanishes by construction; at $\tau’=0.2$ a reduced bias (0.356) and a reduced intervention step (−0.12) remain. These tests compare restoration timing, not degrees of reversibility. The exact $99/2000$ is a property of permanent deletion. See companion note II, §3.1 and §4.

Three hidden-memory alternatives with no record of $U_t$ were calibrated to the same first-return probability. Two diverge from the debris process by the second return. The third, a visit-count concentration rule, remains close across the first three returns and is not cleanly separated by those path statistics; it is separated by the legal intervention of (16)–(17): within the toy, raising an unrealized competitor’s activation while holding $r_0$ fixed changes that competitor’s first-return selection probability from $10/27$ to $0$ . None of the three alternatives responds to that intervention, because the intervened variable is absent from their update rules.

Full tables, the intervention designs, the compression comparison, and the observer-map argument are in companion note I appended below; the general- $n$ parameter law, the paired-arm intervention, and the identifiability limits are in companion note II. Nothing in either note tests (34)–(44).

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9. Observer-relative compression — open extension

Let $H_t$ be a shared event record. Observer $i$ receives

$$Y^i_{\rho,t}=C_{i,\rho}(H_t).$$ (32)

$$C_F(H_t)=H_t,\qquad C_C(H_t)=\pi\left(C_F(H_t)\right).$$ (33)

If $\pi$ retains repeat-versus-change information it preserves the return bias; if it retains only state $1$ versus “not $1$ ,” the return bias is not observable. Detectability depends on which distinctions survive $C_{i,\rho}$ .

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10. Effective observer geometry — open extension

Residue and coupling are indexed by transitions $e\in\mathsf E$ , so the observer geometry is a geometry on transitions:

$$\varepsilon_{i,t}:\mathsf E\longrightarrow V^i_t,\qquad \varepsilon_{i,t}=\varepsilon_{i,t}\big[Y^i_{\rho,t}\big].$$ (34)

No claim is made that $V^i_t$ is the geometry of an observer-independent substrate. Let $V^i_t$ be an inner-product space, $\hat v^{,i}${ee'} the unit vector from $\varepsilon${i,t}(e) to $\varepsilon_{i,t}(e')$ , and $N_k(e)$ the $k$ nearest neighbors of $e$ in $V^i_t$ with $\langle\cdot\rangle_n$ the uniform average over $n\in N_k(e)$ . Then

$$\otimes^i_t(e,e')=-\left\langle \hat v^{,i}$${ee'}\cdot\hat v^{,i}{en}\right\rangle_n. (35)

$$\varepsilon_{i,t}\neq\varepsilon_{j,t}\quad\Rightarrow\quad \otimes^i_t\neq\otimes^j_t\ \text{ in general},$$ (36)

even when both originate from the same record.

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11. Opposition-weighted coupling — proposed constrained family

Let $V^i_{ee',t}:=\mathbf 1\left[C_{ee'}(t)\neq\varnothing\right]$ be the viability mask of §12. Using the column-vector convention of (41), $e'$ indexes the source of residue and $e$ its destination:

$$\mathbf K^i_{ee',t}=\frac{V^i_{ee',t};g\left(\otimes^i_t(e,e')\right)}{\displaystyle\sum_{\bar e}V^i_{\bar e e',t};g\left(\otimes^i_t(\bar e,e')\right)},\qquad g>0\ \text{increasing},\qquad \mathbf K^i_t\in\mathbb R^{\mathsf E\times\mathsf E},$$ (37)

for every source column $e'$ having at least one viable destination. Then $\sum_e\mathbf K^i_{ee',t}=1$ for each such column, so $\mathbf 1^{\mathsf T}\mathbf K^i_t=\mathbf 1^{\mathsf T}$ and (37) agrees with the conservative-transport condition (42). The mask is required: without it, $g>0$ makes every entry of $\mathbf K$ positive, contradicting the support constraint (47). If a source column has no viable destination, (37) is undefined for that column; this note declares no fallback. A realization must either show that its admissibility constraints exclude the case or declare a fallback explicitly.

$\mathbf K^i_t$ couples transitions, not objects; this is what types (41) below.

A candidate rule, not a derived result. $\varepsilon_{i,t}$ and $g$ must be declared independently of the coupling outcomes being predicted.

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12. Relationship viability and composition — existing relational proposal

$$\rho_t(a,b)=\left(R_a(t),,R_b(t),,C_{ab}(t)\right),$$ (38)

where $a,b$ are objects and $C_{ab}(t)$ contains jointly viable trajectories. To constrain a transition-level coupling, viability must be stated on transition pairs. The lift declared here is $C_{ee'}(t):=C_{\operatorname{head}(e),\operatorname{tail}(e')}(t)$ when $\operatorname{head}(e)=\operatorname{tail}(e')$ and $\varnothing$ otherwise, so that (39) restricts $\operatorname{supp}\mathbf K^i_t$ to the line-graph adjacency of $\mathsf E$ : transitions couple only through a shared vertex. Any wider support requires a different declared lift.

$$\mathbf K^i_{ee',t}>0\ \Rightarrow\ C_{ee'}(t)\neq\varnothing.$$ (39)

$$\delta_t(a,b,c)=\rho_t(a,b)\circ\rho_t(b,c)-\rho_t(a,c).$$ (40)

Equation (40) is schematic until $\circ$ and $-$ are specified.

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13. Coupled debris field — open extension

With $D_t\in\mathbb R^{\mathsf E}$ a column vector, $u_t\in\mathbb R^{\mathsf E}$ the residue-generating input, and $\mathbf K_t,\Lambda_t,\mathbf J_t\in\mathbb R^{\mathsf E\times\mathsf E}$ ,

$$D_{t+1}=\mathbf M_tD_t+u_t,\qquad \mathbf M_t=\Lambda_t\mathbf K_t.$$ (41)

$$\text{Conservative transport:}\quad \mathbf 1^{\mathsf T}\mathbf K_t=\mathbf 1^{\mathsf T}.$$ (42)

$$R_D(D_t)={,e:\ (\mathbf J_tD_t)(e)>\tau_e,}.$$ (43)

$\mathbf K_t$ transports residue, $\Lambda_t$ retains or dissipates it, $\mathbf J_t$ translates it into admissibility. None is assumed equal to another. The four-state toy is $\Lambda=\lambda I$ , $\mathbf K=\mathbf J=I$ .

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14. History-dependent coupling — central open map

$$\mathbf K_{t+1}=\mathcal U\left(\mathbf K_t,\otimes_{t+1},\rho_{t+1},\delta_{t+1},D_{t+1}\right).$$ (44)

Admissible family, constraints rather than a rule:

$$\mathbf K_{t+1}\ge0,$$ (45)

$$\mathbf 1^{\mathsf T}\mathbf K_{t+1}=\mathbf 1^{\mathsf T}\quad\text{when transport conserves residue},$$ (46)

$$\operatorname{supp}\mathbf K_{t+1}\subseteq{(e,e'):\ C_{ee'}(t+1)\neq\varnothing},$$ (47)

$$\left|\mathbf K_{t+1}-\mathbf K_t\right|_{\mathrm{op}}\le\varepsilon_K,$$ (48)

$$\mathscr P^{-X}$$t\neq\varnothing\quad\Longrightarrow\quad \mathscr P^{-X}{t+1}\neq\varnothing, (49)

where $\mathscr P^{-X}$t is the declared set of admissible procedures or paths capable of testing the counter-account at time $t$ . Equation (49) is the X (Un)factor as an admissibility predicate on updates $(\mathcal A${t+1},\mathbf K_{t+1}) : no update may empty the set of ways the counter-account can be tested. Graph reachability is neither necessary nor sufficient for (49) — a graph may disconnect while the relevant counter-test survives, or stay connected while observer compression renders it untestable. Preservation of reachability of $(\mathcal X,\mathcal A_{t+1})$ is one possible sufficient realization of (49), not a consequence of X (Un)factor itself. It is not imposed on the four-state theorem: permanent deletion there permits trapping, and prohibiting it defines a new X-constrained variant whose return bias must be recalculated.

These constraints bound the family; they do not determine a unique update.

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15. Multiscale inference without reconstruction — open inferential layer

$$S^i_{\rho,k}=\operatorname{Contrast}\left(C_{i,\rho}(\mathcal O_{t+k}),\ C_{i,\rho}(\mathcal O_{0,t+k})\right).$$ (50)

$$\mathfrak C_{i,\rho}(S)={,H:\ \mathcal F_{i,\rho}(H)=S^i_{\rho,k},}.$$ (51)

$$w_i(H\mid S_{\rho\in I})\ \propto\ \Pr(S_{\rho\in I}\mid H),\Pr(H).$$ (52)

The result is not a reconstructed debris object; it is a weighted class of histories compatible with the deformation available to that observer.

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Proposed research progression

$$\boxed{ \begin{aligned} \text{shared interaction history} &\longrightarrow \text{observer compression}\ &\longrightarrow \text{effective observer geometry}\ &\longrightarrow \text{opposition-weighted coupling}\ &\longrightarrow \text{residue propagation}\ &\longrightarrow \text{admissibility change}\ &\longrightarrow \text{future interaction history}\ &\longrightarrow \text{new observer compression} \end{aligned}}$$ (53)

Status: (53) is a diagram of proposed dependencies, not a derivation. Equations (1)–(2), (5)–(6) are published; (3)–(4), (7)–(17) are the existing schema; (18)–(29) specify the model; (30)–(31a) are derived results; (32)–(52) are open and carry no inherited status from (30). The chain does not show that no underlying reality exists. It shows why no observer’s effective dimension, geometry or reconstruction should automatically be identified with it.

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Companion computational note I: Mechanism stress tests

Nicole Flynn / Symfield PBC. Computational note, September 17, 2026.

Equation numbers refer to the note above.

Scope

These tests do not recompute the already derived value $2297/6000$ . They test which consequences belong to the specified four-state debris mechanism, which survive changes to that mechanism, and which can be reproduced by hidden-memory alternatives.

The permanent-deletion model uses

$$\theta=\frac13,\qquad \lambda=\frac12,\qquad \tau=\frac7{10},$$

and conditions on the initial realized transition $r_0=(1,2)$ . Monte Carlo results use 200,000 conditioned paths per model, a horizon of 350 steps for the debris models, and seed 20260917. The compression comparison uses 500,000 conditioned paths and ten train/test splits. Exact results are identified separately from simulations.

Results at a glance

Test	Result	Status
First-return parameter sweep	Depends on $h=\max(\theta,\tau)$, not on $\lambda$	Exact
Region $\tau\le\theta$	Bias increases rather than collapsing	Exact; contradicts the proposed expectation
Later returns	Bias persists near 5.5 to 5.7 percentage points but is not monotone	Monte Carlo
Reversible admissibility	Most of the bias disappears under the tested restoration rules	Mechanism-dependent Monte Carlo
Legal intervention	Raising unrealized competitor $(1,3)$ from $0.5$ to $0.8$ changes that competitor’s first-return selection probability from $10/27$ to $0$	Exact within the toy
Matched hidden-memory alternatives	All three reproduce the first-return bias without using $U_t$	Monte Carlo after calibration
Compression	$\mathcal A$ saturates first-return prediction; binned subthreshold $D$ adds a tiny gain at the second return	Train/test Monte Carlo
Observer maps	A binary state-1/not-state-1 map loses the statistic; a repeat/change map retains it	Structural consequence of the definitions

1. Exact first-return parameter law

Let

$$h=\max(\theta,\tau).$$

A losing competitor is deleted at the initial departure exactly when its activation exceeds both the candidate threshold and the deletion threshold. The conditional probability of repeating $1\to2$ on the first return is

$$p_{\mathrm{return}}(\theta,\tau) =1-\frac32h+h^2-\frac16h^3,$$

and therefore

$$B_{\mathrm{return}}(\theta,\tau) =\frac23-\frac32h+h^2-\frac16h^3.$$

At $(\theta,\tau)=(1/3,7/10)$ ,

$$p_{\mathrm{return}}=\frac{2297}{6000}=0.382833\ldots, \qquad B_{\mathrm{return}}=\frac{99}{2000}=0.0495.$$

The formula contains no $\lambda$ . Three simulations at $\lambda=0.1,0.5,0.9$ returned estimated biases $0.04969$ , $0.04901$ , and $0.04879$ , respectively, all consistent with the exact $0.0495$ within Monte Carlo error. This tests the implementation; the exact formula, not these three simulations, establishes independence. The fraction of paths returning by the horizon decreases as $\lambda$ rises, so trapping, not the conditional first-return probability, is where $\lambda$ first appears here.

The proposed expectation for $\tau\le\theta$ was reversed. When $\tau\le\theta$ , every unrealized candidate that clears $\theta$ also clears $\tau$ , making deletion easier. At $\theta=1/3$ and any $\tau\le1/3$ ,

$$p_{\mathrm{return}}=\frac{589}{972}\approx0.60494, \qquad B_{\mathrm{return}}=\frac{265}{972}\approx0.27160.$$

The bias tends to zero only as $\max(\theta,\tau)\to1$ .

2. Later returns under permanent deletion

The estimated probability of repeating the initial edge $1\to2$ , conditional on $T^{(k)}<\infty$ by the 350-step simulation horizon, was:

Return $k$	Conditional paths	Repeat probability	Bias from $1/3$	Monte Carlo SE
1	197,682	0.38172	0.04839	0.00109
2	193,239	0.38941	0.05608	0.00111
3	187,612	0.38781	0.05448	0.00113
4	181,561	0.38991	0.05657	0.00115
5	175,265	0.38909	0.05576	0.00117
6	169,363	0.38994	0.05661	0.00119
7	163,871	0.38848	0.05515	0.00120
8	158,849	0.39000	0.05667	0.00122

The estimates rise after the first return and then fluctuate around $0.389$ . They do not support a monotonicity claim. The declining conditioned sample matters: permanent deletion can make state 1 unreachable, so later-return estimates describe the surviving returning paths.

3. Reversible admissibility

One explicit reversible variant was tested. An edge is deleted when its residue exceeds $\tau=0.7$ . Residue then decays by $\lambda=0.5$ per step, and the edge is restored when its decayed residue is at or below $\tau'$ at the next visit to its source. This is only one reversible rule, not a general result about reversibility.

Restore threshold $\tau'$	First-return probability	First-return bias
0.20	0.35615	0.02281
0.35	0.35175	0.01842
0.60	0.33317	−0.00016

Correction, September 18, 2026. Under this update order (residue decays each step; restoration is checked on arrival at the source before selection), the earliest possible return to state 1 is two steps and a deleted competitor carries residue λη ≤ 0.45 at the check. At τ′ = 0.6 every deleted edge is therefore restored before it can matter, and the near-zero bias is structural, not a finding about reversibility. The lower thresholds retain a reduced bias because some edges survive the check on short returns. The paragraphs that follow are left as originally published; companion note II, §3.1 and §4, carries the corrected interpretation.

For $\tau'=0.60$ , the estimated bias is indistinguishable from zero at Monte Carlo resolution. Smaller restoration thresholds retain a reduced positive bias because sufficiently short excursions can return before restoration.

Most of the reported $4.95$ -percentage-point effect is therefore an artifact of irreversibility under these restoration rules. These tests do not bound all reversible residue operators and do not show that every reversible mechanism must have zero path signatures.

4. Legal intervention on an unrealized competitor

Fix the selected edge activation at

$$a_0(1,2)=0.9$$

and intervene on unrealized competitor $e=(1,3)$ within the legal window $1/3<\eta<0.9$ . There are two distinct valid designs.

In the marginal intervention, the remaining competitor has the same conditional distribution, $a_0(1,4)\sim\mathrm{Uniform}(0,0.9)$ , in both arms. Then

$$\Pr(r_{T_1}=(1,3)\mid\operatorname{do}(\eta=0.5),T_1<\infty) =\frac{10}{27}\approx0.37037,$$

whereas

$$\Pr(r_{T_1}=(1,3)\mid\operatorname{do}(\eta=0.8),T_1<\infty)=0.$$

The exact marginal contrast is

$$\Delta_{\mathrm{marginal}}=-\frac{10}{27}\approx-0.37037.$$

In the conditional slice, $a_0(1,4)=0.1$ is pinned. The corresponding probabilities are $1/3$ and $0$ , giving

$$\Delta_{\mathrm{pinned}}=-\frac13.$$

The two fractions must not be mixed: $-10/27$ averages the other competitor, while $-1/3$ fixes it.

The realized initial edge remains $1\to2$ in both arms of both designs. Within the specified model, the later difference is caused by changing unrealized competitor $(1,3)$ . This is an internal causal result about the toy; it is not evidence that a physical debris channel exists.

5. Matched hidden-memory confounders

Three models containing no prospectively generated $U_t$ were calibrated to reproduce the same first-return probability:

• Win-stay: a fixed bonus is assigned to the last realized outgoing edge.
• Visit-count concentration: weight accumulates on previously realized edges.
• Taken-edge eligibility: a decaying eligibility trace is assigned only to realized edges.

Model	Return 1	Return 2	Return 3	Marginal intervention response
Permanent debris	0.38172	0.38941	0.38781	−0.37037
Win-stay	0.38102	0.33638	0.33369	0
Visit-count concentration	0.38277	0.38428	0.38380	0
Taken-edge eligibility	0.38218	0.34242	0.33286	0

Later returns separate the tested debris process from win-stay and taken-edge eligibility under these calibrations. Visit-count concentration remains the serious path-only cousin. None of these three alternatives responds to an intervention on an unchosen activation because that variable is absent from their update rules.

The first-return statistic does not identify debris. The legal intervention distinguishes the debris channel from these three specified no- $U_t$ alternatives. This battery is not exhaustive over possible latent-memory models.

6. Predictive compression

Predictors were trained from the state immediately after the conditioned initial departure. The first used only the outgoing admissible-set pattern $\mathcal A$ . The second added a three-bin representation of surviving subthreshold residue $D$ , a finite proxy for $q$ . Ten 70/30 train/test splits were evaluated by log loss.

Target	Mean log loss: $\mathcal A$	Mean log loss: $\mathcal A+$ binned $D$	Gain from $D$
Departure on first return	0.992284	0.992318	−0.000033
Departure on second return	0.992359	0.992237	+0.000122

At the first return, current admissibility saturates prediction and residue adds no information. The gain was negative in all 10 splits: mean $-3.35\times10^{-5}$ , split-wise standard error $5.34\times10^{-6}$ .

At the second return, binned residue produced a positive gain in all 10 splits: mean $1.22\times10^{-4}$ , split-wise standard error $1.90\times10^{-5}$ . The effect is consistently signed but extremely small. Subthreshold residue cannot alter selection before another visit updates admissibility, but it can affect a later return after that update.

The test does not establish that $q$ is minimal. It shows only that subthreshold residue is idle beyond $\mathcal A$ at the first-return horizon and weakly predictive at the second under this parameterization.

7. Observer maps

This test is structural rather than numerical.

A map retaining only “state 1” versus “not state 1” erases outgoing-edge identity. $B_{\mathrm{return}}$ is not a statistic of that record.

A map recording “repeat” versus “change” on returns to a previously visited state retains the information required to estimate the return bias, even without state or edge labels.

The same generated process therefore supports different detectable statistics under different observer windows. This demonstrates observer-relative detectability, not observer-created dynamics.

Conclusions

The exact first-return theorem survives testing and has a closed parameter law.

The suggested $\tau\le\theta$ null region does not exist; that region strengthens deletion. The zero-bias boundary occurs as $\max(\theta,\tau)\to1$ .

The later-return signature persists under permanent deletion but is not monotone.

Most of the first-return effect disappears under the tested restoration rules. The exact $99/2000$ is specific to irreversible deletion; these tests do not bound all reversible residue operators.

Path-only return bias detects hidden path dependence but does not identify unrealized-transition residue.

The legal intervention is the discriminating test inside the toy: it changes the unrealized competitor while holding the realized initial transition fixed.

Compression by $\mathcal A$ is sufficient at the first-return horizon. Subthreshold residue contributes only weakly at the second-return horizon under the tested parameters.

Nothing here tests equations (34) to (44). Geometry, coupling, and $\mathcal U$ remain underspecified and were deliberately excluded.

The two governing facts are:

$$B(h)=\frac23-\frac32h+h^2-\frac16h^3, \qquad h=\max(\theta,\tau),$$

and identifying $U_t$ still requires the intervention $\operatorname{do}(\eta)$ .

---

Companion computational note II: Debris on $K_n$ , exact first-return law and identifiability of the unrealized-transition channel

Nicole Flynn / Symfield PBC. Computational test, September 17 to 18, 2026; revision 2 after cross-review. Equation numbers (18) to (31a) refer to the equation spine above.

0. Scope and what changed from the brief

The brief posed a generalization and three conjectured formulas as things to derive, falsify, or repair. Outcome in one paragraph: the binomial law for $N\mid x$ is correct; the closed form for $p_m(h)$ is correct and is derived here by an independent route; the finite sum for $\Delta_{j,1}$ is correct and has a closed form. The intervention identifies dependence of first-return admissibility on unrealized activations against every model whose update ignores them, exactly. It does not identify permanent residue, accumulation, or $\lambda$ : a transient-suppression model constructed here is observationally and interventionally identical to permanent debris at the first return and separates only at the second. The strongest limitation found is retrospective: the reversible restoration rule tested in v0.2 at $\tau'=0.6$ restores every deleted edge before the earliest possible return, so its “bias disappears” result is a timing triviality, not evidence about reversibility. A second correction, owed to this report’s first draft and supplied by the cross-review: the first-return intervention identifies dependence of the transition law on unrealized activation, not dependence of the admissible set $\mathcal A_t$ ; a persistent zero-weight countermodel (Section 3.2) reproduces every result here while leaving $\mathcal A_t$ untouched. Section 7 records this as a correction owed to v0.2.

Nothing here touches (32) to (53).

1. Exact first-return law on $K_n$

Setting: complete directed graph on $n$ states, $m=n-1$ out-edges per state, rules (20) to (25) unchanged, $h=\max(\theta,\tau)$ , $0\le\theta,\tau\le1$ , activations iid Uniform $(0,1)$ . Condition on $r_0=(1,2)$ with winning activation $x=a_0(1,2)$ .

1.1 Assumptions the derivation uses, each checked

A1. A competitor $e=(1,j)$ , $j\ne2$ , is deleted at $t=0$ iff $a_0(e)>h$ . Reason: it must be a candidate ( $a>\theta$ ) and cross the deletion threshold with $D_1(e)=a_0(e)$ ( $a>\tau$ , since $D_0=0$ ). Both hold iff $a>h$ . If $x\le\theta$ the candidate set falls back to the argmax and no competitor is a candidate; consistent, since then $a<x\le h$ .

A2. Given $x$ is the maximum, the $m-1$ competitors are iid Uniform $(0,x)$ , so each is deleted independently with probability $q(x)=(x-h)_+/x$ . Hence $N\mid x\sim\mathrm{Binomial}(m-1,q(x))$ . The brief’s conjecture is correct. Verified by simulation at $n=4,6,8$ : $P(N=k)$ matches to four decimals (Section 1.4).

A3. No out-edge of state 1 changes admissibility between $t=0$ and the first return: edges out of 1 accrue residue only while the walker is at 1, and residue only decays in between. So the surviving set at $T_1$ is exactly the $m-N$ edges not deleted at $t=0$ .

A4. At $T_1$ activations are fresh and selection is the argmax, so the realized edge is uniform on the $m-N$ survivors. $\Pr(\text{repeat}\mid N)=1/(m-N)$ .

A5. $N$ is independent of the excursion on ${2,\dots,n}$ and therefore of ${T_1<\infty}$ : trapping is caused by deletion of edges into state 1, which is decided by activity at other states. So conditioning on return does not bias $N$ . $\lambda$ enters the excursion (it controls how much residue survives to cause later deletions) and therefore trapping, but not $N$ and not the selection at $T_1$ .

Ties have probability zero. All statements are almost sure.

1.2 Derivation

Given $x$ is maximal, $x$ has density $mx^{m-1}$ on $(0,1)$ . With $Y=m-1-N\sim\mathrm{Binomial}(m-1,1-q)$ the number of surviving competitors, $m-N=1+Y$ , and the standard identity $\mathbb E[1/(1+Y)]=\frac{1-q^{,m}}{m(1-q)}$ gives, for $x>h$ where $1-q=h/x$ ,

$$\mathbb E!\left[\frac1{m-N},\middle|,x\right]=\frac{x\left(1-\left(\frac{x-h}{x}\right)^{m}\right)}{mh}, \qquad \mathbb E!\left[\frac1{m-N},\middle|,x\le h\right]=\frac1m .$$

Integrating against $mx^{m-1}$ ,

$$p_m(h)=\int_0^h x^{m-1},dx+\frac1h\int_h^1\left[x^m-(x-h)^m\right]dx =\frac{h^m}{m}+\frac{1-h^{m+1}-(1-h)^{m+1}}{h,(m+1)} .$$

This is the conjectured closed form. It was verified symbolically against the direct binomial-sum integral for $m=2,3,4,5,7$ (difference identically zero) and reduces at $m=3$ to $1-\tfrac32h+h^2-\tfrac16h^3$ , i.e. (31a), with $p_3(7/10)=2297/6000$ .

$$B_m(h)=p_m(h)-\frac1m=\frac{h^m}{m}-\frac1m+\frac{1-h^{m+1}-(1-h)^{m+1}}{h,(m+1)} .$$

1.3 Limits and monotonicity

$h\to0$ : $p_m\to1$ for every $m\ge1$ (every competitor is deleted; only the realized edge survives). $h=1$ : $p_m=1/m$ , $B_m=0$ . $\partial B_m/\partial h<0$ throughout $(0,1)$ at every $m$ tested ( $m=3,5,9$ at $h=0.2,0.5,0.8$ ), so the bias falls monotonically in $h$ and vanishes only at $h=1$ .

In $m$ the bias is not monotone at small $h$ :

$m$	$h=0.1$	$h=0.3$	$h=0.5$	$h=0.7$	$h=0.9$
2	0.4050	0.2450	0.1250	0.0450	0.0050
3	0.5265	0.3022	0.1458	0.0495	0.0052
4	0.5690	0.3050	0.1406	0.0470	0.0050
5	0.5809	0.2903	0.1292	0.0435	0.0049
7	0.5691	0.2498	0.1063	0.0372	0.0046
9	0.5402	0.2128	0.0887	0.0322	0.0043
15	0.4425	0.1410	0.0583	0.0226	0.0036

At $h=0.1$ the bias peaks near $m=5$ ; for $h\ge0.5$ it is decreasing in $m$ . At the v0.2 parameters ( $h=0.7$ ) $K_4$ is the maximum over $m\ge2$ .

1.4 Numerical verification

Sweep over $n\in{4,6,8}$ , $(\theta,\tau)\in{(1/3,0.7),(1/3,0.5),(0.6,0.3),(0.5,0.5)}$ , $\lambda\in{0.1,0.5,0.9}$ , 20,000 to 40,000 paths per cell, horizon 100 (bundle files sweep_n*.json). All 36 cells lie within $|z|\le2.2$ of the closed form. Two small-sample patterns appeared and were run down rather than left: at $n=6$ every $\lambda=0.9$ cell sat above the formula, and at $n=8$ ten of twelve cells sat below. Decomposing by $N$ at 300,000 to 360,000 paths shows both factors exact:

$n=6$, $\lambda=0.9$	$P(N)$	returned $\mid N$	$\Pr(\text{repeat}\mid N)$	$1/(m-N)$
$N=0$	0.5320	0.9636	0.1970	0.2000
$N=1$	0.3047	0.9618	0.2526	0.2500
$N=2$	0.1317	0.9643	0.3341	0.3333
$N=3$	0.0293	0.9726	0.5090	0.5000

At $\lambda=0.1$ the same table has returned $\mid N\approx0.992$ for every $N$ and identical $\Pr(\text{repeat}\mid N)$ . So $\lambda$ changes the return probability (trapping) by about three points here and leaves $N$ and the conditional selection untouched, which is A5 confirmed. At $n=8$ , $\lambda=0.5$ , 300,000 paths, aggregate $0.1815$ vs exact $0.1800$ , and all seven $\Pr(\text{repeat}\mid N)$ rows within $|z|\le1.1$ . The $K_4$ decomposition: $P(N)=(0.7836,,0.1889,,0.0275)$ against exact $(98/125,,189/1000,,27/1000)$ .

Trapping is reported separately from the conditional law throughout: return-by-horizon fractions in the sweep range from 0.99 (small $\lambda$ , $h=0.7$ ) to 0.89 ( $n=8$ , $h=0.5$ , $\lambda=0.9$ ).

2. Prospective unrealized-set intervention

Fix $r_0=(1,2)$ , $a_0(r_0)=w$ , pick competitor $e_j$ , and set $\theta<\eta_0<h<\eta_1<w$ . The realized edge is unchanged in both arms because $\eta<w$ ; $e_j$ is an unrealized candidate in both because $\eta>\theta$ . Under $\eta_0$ , $e_j$ survives; under $\eta_1$ , it is deleted.

2.1 Exact $\Delta_{j,1}$

Marginal design: the other $m-2$ competitors are iid Uniform $(0,w)$ , each deleted with probability $q=(w-h)_+/w$ , so $L\sim\mathrm{Binomial}(m-2,q)$ of them are deleted and the survivor count under $\eta_0$ is $2+(m-2-L)=m-L$ . Uniform selection gives

$$\Pr!\left(r_{T_1}=e_j\mid \operatorname{do}(\eta_0),T_1<\infty\right)=\sum_{\ell=0}^{m-2}\binom{m-2}{\ell}q^\ell(1-q)^{m-2-\ell}\frac1{m-\ell}, \qquad \Pr!\left(\cdots\mid\operatorname{do}(\eta_1),\cdots\right)=0 ,$$

so the brief’s finite sum for $\Delta_{j,1}$ is correct. Writing $p=1-q=h/w$ and $M=m-2$ , the sum is $\mathbb E[1/(2+\mathrm{Bin}(M,p))]$ , which has the closed form

$$\Delta_{j,1}=-\frac1{p^2}\left[\frac{1-(1-p)^{M+2}}{M+2}-\frac{(1-p)\left(1-(1-p)^{M+1}\right)}{M+1}\right].$$

Both expressions agree numerically at every $(m,w,h)$ tested. The curve $\eta\mapsto\Pr(r_{T_1}=e_j\mid\operatorname{do}(\eta))$ on the legal window is a step: constant at $\mathbb E[1/(m-L)]$ for $\theta<\eta<h$ , zero for $h<\eta<w$ . Nothing in the first-return law depends on where in $(\theta,h)$ or $(h,w)$ the intervention sits.

Pinned design: the untreated competitors are fixed at a value below $h$ , so $L=0$ and $\Delta_{j,1}=-1/m$ .

2.2 The two $K_4$ numbers, stated as estimands

Marginal, $10/27\to0$ : $\Pr\big(r_{T_1}=(1,3)\mid r_0=(1,2),,a_0(1,2)=0.9,,\operatorname{do}(a_0(1,3)=\eta),,a_0(1,4)\sim\mathrm{U}(0,0.9),,T_1<\infty\big)$ at $\eta=0.5$ versus $\eta=0.8$ . The $10/27$ is $\tfrac79\cdot\tfrac13+\tfrac29\cdot\tfrac12$ .

Pinned, $1/3\to0$ : the same with $a_0(1,4)=0.1$ fixed. The two are different conditional probabilities of the same event and must not be averaged or compared across designs.

2.3 Paired Monte Carlo, common random numbers

Both arms use identical $t=0$ activations for the untreated competitors and an identical random stream afterwards, so return events coincide arm to arm (the return fractions below are equal to four decimals by construction, not by luck). 40,000 paths per arm.

design	$n$	exact $\Delta_{j,1}$	MC $k=1$	MC $k=2$	MC $k=3$
marginal	4	$-0.3704$	$-0.3700\pm0.0024$	$-0.3634$	$-0.3632$
pinned	4	$-0.3333$	$-0.3330\pm0.0024$	$-0.3222$	$-0.3227$
marginal	6	$-0.2388$	$-0.2362\pm0.0021$	$-0.2351$	$-0.2338$
pinned	6	$-0.2000$	$-0.1997\pm0.0020$	$-0.1942$	$-0.1928$
marginal	8	$-0.1749$	$-0.1771\pm0.0019$	$-0.1707$	$-0.1691$
pinned	8	$-0.1429$	$-0.1449\pm0.0018$	$-0.1355$	$-0.1360$

Other $w$ (marginal, $k=1$ ): $n=4$ , $w=0.75$ : exact $-0.3444$ , MC $-0.3404\pm0.0028$ ; $w=0.95$ : $-0.3772$ , MC $-0.3750\pm0.0028$ . $n=6$ , $w=0.75$ : $-0.2105$ , MC $-0.2117\pm0.0024$ ; $w=0.95$ : $-0.2473$ , MC $-0.2485\pm0.0025$ .

Under permanent deletion $\Delta_{j,k}$ stays near its $k=1$ value at $k=2,3$ , because the deleted competitor never returns. That persistence is the later-return signature of permanence (Section 3).

Unconditional (competing-risk) version, $n=4$ marginal: $\Pr(r_{T_1}=e_j,\ T_1\le\text{horizon})=0.3659$ vs $0$ ; $\Pr(T_1\le\text{horizon})=0.9885$ in both arms. Since return is independent of the intervention (A5), the unconditional contrast is the conditional one scaled by the return probability, and the selection effect from trapping is zero across arms.

3. Identification challenge

All models run on $K_4$ at $(\theta,\lambda,\tau)=(1/3,1/2,7/10)$ , 30,000 paths, horizon 100. The three no- $U_t$ alternatives were calibrated by bisection to the debris first-return value $2297/6000$ (calibration accuracy about one standard error; residual mismatch is visible in the first column and does not affect the intervention columns). Intervention: marginal design, $w=0.9$ , $\eta_0=0.5$ , $\eta_1=0.8$ .

model	repeat $k=1$	$k=2$	$k=3$	$\Delta_{j,1}$	$\Delta_{j,2}$	$\Delta_{j,3}$
Stationary Markov null	0.3310	0.3356	0.3322	$+0.0000$	$+0.0000$	$+0.0000$
Win-stay (bonus 0.0471)	0.3780	0.3379	0.3285	$+0.0000$	$+0.0000$	$+0.0000$
Visit-count (0.0456)	0.3755	0.3851	0.3793	$+0.0000$	$+0.0000$	$+0.0000$
Eligibility (0.0461, $\rho=0.9$)	0.3753	0.3803	0.3688	$+0.0000$	$+0.0000$	$+0.0000$
Permanent debris	0.3867	0.3953	0.3956	$-0.3713$	$-0.3659$	$-0.3613$
Transient suppression	0.3879	0.3315	0.3357	$-0.3715$	$+0.0111$	$+0.0003$
Reversible, $\tau'=0.6$	0.3310	0.3356	0.3322	$+0.0000$	$+0.0000$	$+0.0000$
Reversible, $\tau'=0.2$	0.3559	0.3371	0.3350	$-0.1219$	$+0.0042$	$-0.0002$
Smooth deletion, $s=0.05$	0.3855	0.3914	0.3886	$-0.3191$	$-0.3107$	$-0.3134$
Smooth deletion, $s=0.15$	0.3980	0.4042	0.4107	$-0.1696$	$-0.1658$	$-0.1696$

Monte Carlo SE on $\Delta$ is 0.003 to 0.004; the zeros in the no- $U_t$ rows are exact zeros, not small estimates, because under pairing the two arms are the same sample path.

Model definitions. Win-stay: bonus $b$ added to the activation of the last realized edge out of the current state. Visit-count: bonus $c\times$ (times realized). Eligibility: bonus $g,E(e)$ with $E\leftarrow\rho E+\mathbf 1[\text{realized}]$ . Transient: an unrealized candidate with $a>\tau$ is excluded at the next visit to its source only; no residue is stored. Reversible: as v0.2, restored when decayed residue $\le\tau'$ at a visit to the source. Smooth: at accrual an unrealized candidate is permanently deleted with probability $\sigma((D-\tau)/s)$ , so $s\to0$ is the permanent rule.

3.1 Intervention curve across the legal window ( $k=1$ , marginal, 25,000 paths per point)

$\eta$	0.40	0.55	0.65	0.69	0.71	0.75	0.85
Permanent	0.371	0.371	0.371	0.371	0.000	0.000	0.000
Transient	0.370	0.370	0.370	0.370	0.000	0.000	0.000
Reversible $\tau'=0.2$	0.350	0.350	0.350	0.350	0.228	0.228	0.149
Smooth $s=0.15$	0.331	0.275	0.221	0.196	0.183	0.159	0.102
Win-stay	0.308	0.308	0.308	0.308	0.308	0.308	0.308

Permanent and transient are the same step at $h$ . Reversible $\tau'=0.2$ is a multi-level step: the level after $h$ depends on how much residue survives the shortest returns. Under the code’s update order (global decay in each step, restoration checked on arrival before selection), an edge deleted at activation $\eta$ has residue $\lambda\eta=0.5\eta$ when checked at the two-step return and $\lambda^2\eta=0.25\eta$ at the three-step return, so every $\eta$ in the window stays deleted through a two-step return and $\eta>0.8$ stays deleted through a three-step return as well. Verified by single-path trace. Smooth deletion is a sigmoid in $\eta$ . Every no- $U_t$ model is flat.

3.2 What the intervention rules out, and what it does not

The largest class ruled out exactly: every model whose $t=0$ state update is a function of $(r_0,\ a_0(r_0))$ and the pre-existing state only, i.e. any model that never reads unrealized activations. For such a model the two arms are the same sample path and $\Delta_{j,k}=0$ for all $k$ , identically. Win-stay, visit-count, and eligibility are three members; the class is much larger and includes any hidden-memory model driven by the realized path, however elaborate, with arbitrary latent dimension.

Models that consume unrealized activation through a different latent mechanism are not ruled out as a class; they are separated from permanent debris, where they are separated at all, by the shape of the curve and by later returns. Smooth deletion is distinguished by curve shape at $k=1$ . Reversible rules are distinguished at $k=1$ by level and by the dependence of the post- $h$ level on return time. Transient suppression is not distinguished at $k=1$ by any statistic, observational or interventional; it is distinguished at $k=2$ ( $\Delta_{j,2}=-0.366$ vs $+0.011$ ) and by the observational second-return repeat rate ( $0.395$ vs $0.332$ ).

Explicit observationally equivalent constructions. The transient model shows that permanence is not identified at $k=1$ : any rule whose first-return selection excludes exactly the competitors with $a_0>h$ reproduces the full $k=1$ curve, whatever it does afterwards. A stronger countermodel, due to the GPT review of this report, shows that even persistence over later returns does not identify deletion from $\mathcal A_t$ : set $b_e=\mathbf 1[a_0(e)>h]$ , keep $e\in\mathcal A_t$ throughout, and give $e$ zero selection weight whenever $b_e=1$ . This reproduces the permanent-deletion step at every $k$ while formal admissibility never changes. Permanent residue, accumulation, $\lambda$ , and membership in $\mathcal A_t$ are all invisible to the path-level intervention. What the intervention identifies at $k=1$ is precisely: the first-return transition law carries thresholded memory of unrealized activation through the event ${a_0(e)>h}$ . Whether that memory lives in $\mathcal A_t$ or in selection weights is a separate question that needs an executability probe, not a path statistic.

4. Robustness

Reported above: $n\in{4,6,8}$ ; $h\in{0.5,0.6,0.7}$ in the observational sweep; $w\in{0.75,0.9,0.95}$ ; $k\in{1,2,3}$ ; marginal and pinned designs; conditional and unconditional estimands; permanent, hard reversible (two thresholds), smooth probabilistic deletion (two widths), and transient. Trapping is reported as the return-by-horizon fraction in every table and is identical across paired arms.

The one place robustness failed is instructive: the reversible rule at $\tau'=0.6$ and the first version of the smooth rule (temporary blocking with probability $\sigma((D-\tau)/s)$ at each visit) both showed zero bias and zero $\Delta$ , for the same reason. With $\lambda=0.5$ the residue on an out-edge of state 1 has decayed by a factor $\lambda=0.5$ by the earliest possible (two-step) return, so a deleted edge with activation $\eta<0.9$ carries residue below $0.45<0.6$ at the check, and any rule that evaluates admissibility from the decayed residue at the visit never blocks anything at the first return. Those two rules are the stationary Markov null in disguise. The smooth rule was redefined (deletion decided at accrual); the reversible $\tau'=0.6$ result is retained in the table because it is what v0.2 reported, and Section 7 says what to do about it.

5. Observer maps

Applied to the same paired record (permanent debris, $n=4$ , marginal, 30,000 paths, $\eta_0=0.5$ vs $\eta_1=0.8$ ):

observer map	$B_m(h)$ measurable	$\Delta_{j,k}$ measurable	what it shows in this record
Binary: state 1 vs not	no	no	nothing; edge identity is gone
One current state, no history	no	no	a single symbol; no return event is even definable
Full sequence of current states	yes	yes	on $K_n$ an edge is an ordered pair of states, so the state sequence recovers the labeled edge history exactly
Repeat/change at a revisited state	yes	not $\Delta_{j,k}$ itself; a derived contrast, yes	repeat rate $0.376\to0.612$ across arms, exactly $\mathbb E[1/(3-L)]\to\mathbb E[1/(2-L)]$; the observer sees the intervention effect but cannot attribute it to $e_j$
Labeled outgoing-edge history	yes	yes	$\Pr(e_j)=0.363\to0.000$

In every row the underlying intervention effect is present in the generated process. Rows one and three lose detectability or attribution under coarse-graining; the effect itself does not disappear. Nothing in this section is a statement about observer-created dynamics.

6. Reproducibility

The reproducibility bundle (debris-kn-bundle.zip, available from the author; not linked here) contains: debris_kn.py (simulator, all rules and alternatives), run1b.py (observational sweep), run2_intervention.py (exact $\Delta$ and paired arms), run3a.py, run3b.py, run3c.py (identification battery and curves), sweep_n{4,6,8}.json, results_intervention.json, results_identify_{a,b,c}.json, results_obs_robust.json. Seeds are fixed in each script (sweep: $1000+7n+10\lambda$ ; interventions: $500+n$ , $700$ , $900$ , $901$ , $1200$ , $1300+n$ , $1400$ ; decompositions: $11$ , $21$ to $23$ , $31$ to $33$ ). Sample sizes appear in each table. Uncertainty is binomial SE on proportions; paired contrasts use the difference of arm proportions, which is conservative under common random numbers. Requires Python 3 with NumPy and SymPy.

Every exact result has a Monte Carlo comparison: $p_m(h)$ at 36 parameter cells and three large decompositions; $\Delta_{j,1}$ at six $(n,\text{design})$ pairs and four further $(n,w)$ pairs; the $K_4$ constants $2297/6000$ , $10/27$ , $1/3$ .

7. Verdicts

1. Proved exactly. For the permanent-deletion mechanism on $K_n$ with uniform activations: $N\mid x\sim\mathrm{Binomial}(m-1,(x-h)$+/x) ; the closed form $p_m(h)=\frac{h^m}{m}+\frac{1-h^{m+1}-(1-h)^{m+1}}{h(m+1)}$ with $p_m\to1$ as $h\to0$ , $p_m(1)=1/m$ , and $B_m$ strictly decreasing in $h$ ; independence of the first-return selection from $\lambda$ under assumptions A1 to A5; the marginal-design $\Delta${j,1} as the stated finite sum and its closed form; the pinned-design $\Delta_{j,1}=-1/m$ ; the step shape of the first-return intervention curve; and $\Delta_{j,k}=0$ identically for every model whose update reads only the realized path.

2. Supported computationally only. Later-return repeat probabilities and $\Delta_{j,k}$ for $k\ge2$ under every rule; the level structure of the reversible curve after $h$ ; the sigmoid shape of the smooth-deletion curve; the non-monotonicity of $B_m$ in $m$ at small $h$ (computed from the exact formula, not proved as an inequality); trapping fractions.

3. What identifies dependence on unrealized activation. A nonzero $\Delta_{j,1}$ in the paired design. It is exactly zero for every model that does not read $U_t$ , with arbitrary latent memory of the realized path, and it is $-0.37$ (marginal, $K_4$ ) for the debris mechanism. What it identifies is thresholded memory of unrealized activation in the first-return transition law. It does not identify where that memory is stored. This is the discriminating test the v0.2 note called for, now stated for general $n$ and with the class it defeats, and the class it does not, named precisely.

4. What remains compatible with other hidden mechanisms. Everything beyond the threshold event. At $k=1$ the intervention cannot distinguish permanent deletion from transient suppression, cannot see $\lambda$ , and cannot see residue accumulation. At every $k$ it cannot distinguish deletion from $\mathcal A_t$ from persistent zero selection weight with $\mathcal A_t$ unchanged. Distinguishing permanent from transient needs $k\ge2$ ( $\Delta_{j,2}$ : $-0.366$ vs $+0.011$ ); distinguishing deletion from zero weight needs an executability probe $\operatorname{Try}(e_j,Z_t)$ defined independently of $\mathcal A_t$ , which is an open protocol, not a result. Distinguishing hard reversible from permanent needs the post- $h$ level and its return-time dependence. Distinguishing smooth from hard needs the curve shape. None of these later tests has an exact result here.

5. Strongest falsification or limitation. The reversible restoration rule tested in v0.2 at $\tau'=0.6$ is degenerate under $\lambda=0.5$ : the earliest return is two steps, residue has decayed to at most $0.25$ , and $0.25<0.6$ , so every deleted edge is restored before it can matter. Its zero bias is therefore the stationary null, not a finding about reversibility, and the v0.2 sentence “most of the first-return effect disappears under the tested restoration rules” should be replaced by: the tested rule restores before the first possible return and is observationally the null; the $\tau'=0.2$ rule, which does bite, retains a reduced first-return bias ( $0.356$ ) and a reduced step ( $\Delta_{j,1}=-0.12$ ). The second limitation is the one in verdict 4: the first-return intervention identifies a threshold dependence on unrealized activation and nothing more specific than that. “Debris” as a persistent, decaying residue is not identified by it.

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What the tested model establishes

The realized path does not necessarily contain all information relevant to later behavior. In the specified model, two runs can share the same realized transition while differing only in an available but unrealized alternative, and nevertheless produce different future transition laws.

Intervention establishes causal dependence on that unrealized activation. It does not by itself identify whether the dependence is stored as deletion, temporary exclusion, zero selection weight, or another hidden mechanism. A separately validated executability probe can distinguish feasibility from natural suppression.

The result is therefore narrow: unrealized alternatives can carry detectable causal consequences within this stochastic process. No physical debris field, geometry, or spacetime interpretation follows from the theorem.

This section summarizes the results through the executability test; it does not confer derived status on the open constructions above.