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

Nicole Flynn
Symfield PBC
Technical Research Note v0.1
September 17, 2026

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.

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).

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

1. Observer restriction — published Markov-trace mathematics

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

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

2. Realization and admissible motion — existing schema

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

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

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

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

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.

4. Prospective branching — proposed discrete schema

5. Non-detection and causal attribution — observation limit

For an incomplete or non-injective witness,

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

6. Four-state realization — fully specified specialization

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 $\mathcal A_0=\mathsf E$.

7. Predictive compression,derived Markov representation

Because deletion is permanent, each edge is represented by

with $\dagger$ denoting deletion, and

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

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

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

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:

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

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

Summing and dividing by $\tfrac13$,

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. No monotonicity on later returns is asserted: the initially realized edge can itself be unrealized on a subsequent visit and then deleted.

Open problems for this section. 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.

9. Observer-relative compression — open extension

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

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 $C_{i,\rho}$.

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:

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

even when both originate from the same record.

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:

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.

12. Relationship viability and composition — existing relational proposal

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.

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

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 $\mathbf K_t,\Lambda_t,\mathbf J_t\in\mathbb R^{\mathsf E\times\mathsf E}$,

$\mathbf K_t$ 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 $\Lambda=\lambda I$, $\mathbf K=\mathbf J=I$.

14. History-dependent coupling — central open map

Admissible family, constraints rather than a rule:

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.

15. Multiscale inference without reconstruction — open inferential layer

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

Proposed research progression

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)–(31) are specified and (30)–(31) derived; (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.