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# Debris as Predictive Compression: An Exact Return Bias and an Open Equation Spine for Observer-Relative Coupling
- URL: https://www.symfield.ai/debris-as-predictive-compression/
- Published: 2026-09-17T05:34:48.000Z
- Updated: 2026-09-17T06:04:22.000Z
- Description: Can unrealized transitions leave detectable residue? A four-state stochastic model yields an exact 4.95% return bias, then maps an open path from predictive compression to observer-relative geometry, coupled debris fields and changing admissibility.
- Author: Nicole Flynn
- Tags: Mathematics & Geometry, Physical Sciences and Mathematics, Physics, Article, Blog

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→21\\to2 on the first return to state 11 is

Pr(rT1\=(1,2)|r0\=(1,2),T1<∞)\=22976000≈0.3828,\\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 1/31/3. The return bias Breturn\=99/2000B\_{\\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).

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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 PP, visible window AA and inaccessible complement A′A',

PA\=IAP\[∑k\=0∞(IA′P)kIA\].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)≠P(2),PA(1)\=PA(2).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 ℰ\\mathscr E be the internal-state bundle and 𝒳\=Γ(ℰ)\\mathcal X=\\Gamma(\\mathscr E).

ℜ\=(M,ℰ,Φ,h,Π,𝒟μ,W,𝒜t).\\mathfrak R=(M,\\mathscr E,\\Phi,h,\\Pi,\\mathcal D\_\\mu,W,\\mathcal A\_t). (3)

Φ̇t\=Pt\[Fμ(Φt,ht,t)\],Pt:TΦt𝒳→T𝒜t(Φt).\\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 𝒫\\mathcal P retain selected variables and 𝒫⊥\=1−𝒫\\mathcal P^\\perp=1-\\mathcal P.

ddt𝒫ut\=𝒫ℒ𝒫ut+∫0tKMZ(s)𝒫ut−sds+Ft,\\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)

KMZ(s)\=𝒫ℒes𝒫⊥ℒ𝒫⊥ℒ.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, (Φt,qt)(\\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 (Φt,ht)(\\Phi\_t,h\_t), where (31) exhibits it.

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### 4\. Prospective branching — *proposed discrete schema*

Zt\=(Φt,Dt,ht,𝒜t,Lt).Z\_t=(\\Phi\_t,D\_t,h\_t,\\mathcal A\_t,L\_t). (7)

ℬt\=Br(Pt\[Fμ(Φt,ht,t)\];𝒜t,Lt,ξt).\\mathcal B\_t=\\operatorname{Br}\\left(P\_t\[F\_\\mu(\\Phi\_t,h\_t,t)\];\\ \\mathcal A\_t,L\_t,\\xi\_t\\right). (8)

rt\=SelW(ℬt),Ut\=ℬt\\{rt}.r\_t=\\operatorname{Sel}\_W(\\mathcal B\_t),\\qquad U\_t=\\mathcal B\_t\\setminus\\{r\_t\\}. (9)

Φt+1\=End(rt).\\Phi\_{t+1}=\\operatorname{End}(r\_t). (10)

Dt+1\=𝒬(Dt,Ut,Lt,W,rt).D\_{t+1}=\\mathcal Q(D\_t,U\_t,L\_t,W,r\_t). (11)

ht+1\=ℋ(ht,rt),Lt+1\=ℒ(Lt,rt,ξt).h\_{t+1}=\\mathcal H(h\_t,r\_t),\\qquad L\_{t+1}=\\mathcal L(L\_t,r\_t,\\xi\_t). (12)

𝒜t+1\=𝒢0(𝒜t,ht+1,Lt+1,rt)\\RD(Dt+1).\\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)

Null model:RD≡⌀.\\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,

Wt(Dt)\=0⇏Dt\=0.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:

SelW(ℬt(1))\=SelW(ℬt(0))\=rt.\\operatorname{Sel}\_W(\\mathcal B\_t^{(1)})=\\operatorname{Sel}\_W(\\mathcal B\_t^{(0)})=r\_t. (16)

Δk\=Contrast(Law(𝒪t+k∣do(ℬt\=ℬt(1)),Ht),Law(𝒪t+k∣do(ℬt\=ℬt(0)),Ht)).\\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*

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

θ\=13,λ\=12,τ\=710,Φ0\=1,D0≡0,𝒜0\=𝖤.\\theta=\\tfrac13,\\qquad \\lambda=\\tfrac12,\\qquad \\tau=\\tfrac7{10},\\qquad \\Phi\_0=1,\\qquad D\_0\\equiv0,\\qquad \\mathcal A\_0=\\mathsf E. (19)

𝒞t\={e∈𝒜t:e leaves Φt}.\\mathcal C\_t=\\{\\,e\\in\\mathcal A\_t:\\ e\\text{ leaves }\\Phi\_t\\,\\}. (20)

at(e)∼Uniform(0,1) i.i.d. for e∈𝒞t,at(e)\=0 for e∉𝒞t.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)

ℬt\={{e∈𝒞t:at(e)\>θ},if nonempty,{argmaxe∈𝒞tat(e)},otherwise.\\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)

rt\=argmaxe∈ℬtat(e),Ut\=ℬt\\{rt},Φt+1\=head(rt).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)

Dt+1(e)\=λDt(e)+at(e)𝟏\[e∈Ut\].D\_{t+1}(e)=\\lambda D\_t(e)+a\_t(e)\\,\\mathbf 1\[e\\in U\_t\]. (24)

𝒜t+1\=𝒜t\\{e:Dt+1(e)\>τ}.\\mathcal A\_{t+1}=\\mathcal A\_t\\setminus\\{\\,e:\\ D\_{t+1}(e)>\\tau\\,\\}. (25)

Ties in argmax\\arg\\max have probability zero under the continuous activation law; all selections are therefore understood almost surely. 𝒞t\\mathcal C\_t is nonempty on every reachable state: rt∉Utr\_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 𝒜0\=𝖤\\mathcal A\_0=\\mathsf E.

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### 7\. Predictive compression,derived Markov representation

Because deletion is permanent, each edge is represented by

qt(e)∈\[0,τ\]∪{†},q\_t(e)\\in\[0,\\tau\]\\cup\\{\\dagger\\}, (26)

with †\\dagger denoting deletion, and

(Φt,qt)∈𝒳×(\[0,τ\]∪{†})𝖤(\\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

unrealized activation history→(Dt,𝒜t)→qt→future realized-path law.\\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*

T1\=inf{t≥1:Φt\=1}.T\_1=\\inf\\{t\\ge1:\\ \\Phi\_t=1\\}. (29)

Pr(rT1\=(1,2)|r0\=(1,2),T1<∞)\=22976000.\\Pr\\left(r\_{T\_1}=(1,2)\\ \\middle|\\ r\_0=(1,2),\\ T\_1<\\infty\\right)=\\frac{2297}{6000}. (30)

Breturn\=22976000−13\=992000\=0.0495.B\_{\\mathrm{return}}=\\frac{2297}{6000}-\\frac13=\\frac{99}{2000}=0.0495\. (31)

Condition on r0\=(1,2)r\_0=(1{,}2), i.e. x:=a0(1,2)\=max{a0(1,2),a0(1,3),a0(1,4)}x:=a\_0(1{,}2)=\\max\\{a\_0(1{,}2),a\_0(1{,}3),a\_0(1{,}4)\\}. A competitor e∈{(1,3),(1,4)}e\\in\\{(1{,}3),(1{,}4)\\} is deleted at t\=1t=1 iff a0(e)\>τa\_0(e)>\\tau, which implies a0(e)\>θa\_0(e)>\\theta and a0(e)<xa\_0(e)<x. Let N∈{0,1,2}N\\in\\{0,1,2\\} count deleted competitors. Edges out of 11 can be deleted only while Φt\=1\\Phi\_t=1, and the deletion indicators are decided at t\=0t=0 independently of the excursion on {2,3,4}\\{2,3,4\\}, hence independently of {T1<∞}\\{T\_1<\\infty\\}. Before the first return, residue on edges leaving state 11 can only decay and cannot newly cross τ\\tau; the first-return result is therefore independent of λ\\lambda. On return, activations are fresh and selection is uniform over the 3−N3-N surviving edges:

Pr(rT1\=(1,2)|r0\=(1,2),T1<∞)\=𝔼\[13−N|x is the maximum\].\\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,a0(1,3),a0(1,4))(x,a\_0(1{,}3),a\_0(1{,}4)) is 11 on the unit cube. Equation (30b) integrates its unnormalized restriction to the event that xx is maximal; division by the event probability 1/31/3 supplies the final factor of 33. Given xx, each competitor independently lies in (τ,x)(\\tau,x) with measure m:=max(0,x−τ)m:=\\max(0,x-\\tau) and in (0,τ\](0,\\tau\] with measure x−mx-m. Therefore

𝔼\[13−N;x is max\]\=∫0τx23dx+∫τ1\[(x−m)23+2m(x−m)2+m21\]dx.\\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 (τ,1\](\\tau,1\], x−m\=τx-m=\\tau. With τ\=710\\tau=\\tfrac7{10} the four pieces are

∫07/10x23dx\=3439000,∫7/101τ23dx\=491000,∫7/101τ(x−τ)dx\=632000,∫7/101(x−τ)2dx\=91000.\\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 13\\tfrac13,

3(3439000+491000+632000+91000)\=22976000.3\\left(\\frac{343}{9000}+\\frac{49}{1000}+\\frac{63}{2000}+\\frac{9}{1000}\\right)=\\frac{2297}{6000}. (30d)

Equivalently, Pr(N\=0)\=98125\\Pr(N=0)=\\tfrac{98}{125}, Pr(N\=1)\=1891000\\Pr(N=1)=\\tfrac{189}{1000}, Pr(N\=2)\=271000\\Pr(N=2)=\\tfrac{27}{1000}, and 98125⋅13+1891000⋅12+271000⋅1\=22976000\\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 (2×1062\\times10^6 samples, 0.38270.3827).

The stationary Markov null gives 1/31/3. 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 𝒳\\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(next\=j∣at 1)\=13\\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 UtU\_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 UtU\_t requires the intervention (16)–(17). Also not established: that any physical system carries such residue; that τ\\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 (θ,τ)(\\theta,\\tau), including its independence from λ\\lambda; and later-return bias as a function of (θ,λ,τ)(\\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 θ<η<a0(r0)\\theta<\\eta<a\_0(r\_0), where η\\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.

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### 9\. Observer-relative compression — *open extension*

Let HtH\_t be a shared event record. Observer ii receives

Yρ,ti\=Ci,ρ(Ht).Y^i\_{\\rho,t}=C\_{i,\\rho}(H\_t). (32)

CF(Ht)\=Ht,CC(Ht)\=π(CF(Ht)).C\_F(H\_t)=H\_t,\\qquad C\_C(H\_t)=\\pi\\left(C\_F(H\_t)\\right). (33)

If π\\pi retains repeat-versus-change information it preserves the return bias; if it retains only state 11 versus “not 11,” the return bias is not observable. Detectability depends on which distinctions survive Ci,ρC\_{i,\\rho}.

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### 10\. Effective observer geometry — *open extension*

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

εi,t:𝖤→Vti,εi,t\=εi,t\[Yρ,ti\].\\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 VtiV^i\_t is the geometry of an observer-independent substrate. Let VtiV^i\_t be an inner-product space, v̂ee′i\\hat v^{\\,i}\_{ee'} the unit vector from εi,t(e)\\varepsilon\_{i,t}(e) to εi,t(e′)\\varepsilon\_{i,t}(e'), and Nk(e)N\_k(e) the kk nearest neighbors of ee in VtiV^i\_t with ⟨⋅⟩n\\langle\\cdot\\rangle\_n the uniform average over n∈Nk(e)n\\in N\_k(e). Then

⊗ti(e,e′)\=−⟨v̂ee′i⋅v̂eni⟩n.\\otimes^i\_t(e,e')=-\\left\\langle \\hat v^{\\,i}\_{ee'}\\cdot\\hat v^{\\,i}\_{en}\\right\\rangle\_n. (35)

εi,t≠εj,t⇒⊗ti≠⊗tj in general,\\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 Vee′,ti:=𝟏\[Cee′(t)≠⌀\]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′e' indexes the source of residue and ee its destination:

𝐊ee′,ti\=Vee′,tig(⊗ti(e,e′))∑e‾Ve‾e′,tig(⊗ti(e‾,e′)),g\>0increasing,𝐊ti∈ℝ𝖤×𝖤,\\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′e' having at least one viable destination. Then ∑e𝐊ee′,ti\=1\\sum\_e\\mathbf K^i\_{ee',t}=1 for each such column, so 𝟏𝖳𝐊ti\=𝟏𝖳\\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\>0g>0 makes every entry of 𝐊\\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.

𝐊ti\\mathbf K^i\_t couples transitions, not objects; this is what types (41) below.

A candidate rule, not a derived result. εi,t\\varepsilon\_{i,t} and gg must be declared independently of the coupling outcomes being predicted.

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### 12\. Relationship viability and composition — *existing relational proposal*

ρt(a,b)\=(Ra(t),Rb(t),Cab(t)),\\rho\_t(a,b)=\\left(R\_a(t),\\,R\_b(t),\\,C\_{ab}(t)\\right), (38)

where a,ba,b are objects and Cab(t)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 Cee′(t):=Chead(e)tail(e′)(t)C\_{ee'}(t):=C\_{\\operatorname{head}(e)\\,\\operatorname{tail}(e')}(t) when head(e)\=tail(e′)\\operatorname{head}(e)=\\operatorname{tail}(e') and ⌀\\varnothing otherwise, so that (39) restricts supp𝐊ti\\operatorname{supp}\\mathbf K^i\_t to the line-graph adjacency of 𝖤\\mathsf E: transitions couple only through a shared vertex. Any wider support requires a different declared lift.

𝐊ee′,ti\>0⇒Cee′(t)≠⌀.\\mathbf K^i\_{ee',t}>0\\ \\Rightarrow\\ C\_{ee'}(t)\\neq\\varnothing. (39)

δt(a,b,c)\=ρt(a,b)∘ρt(b,c)−ρt(a,c).\\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 Dt∈ℝ𝖤D\_t\\in\\mathbb R^{\\mathsf E} a column vector, ut∈ℝ𝖤u\_t\\in\\mathbb R^{\\mathsf E} the residue-generating input, and 𝐊t,Λt,𝐉t∈ℝ𝖤×𝖤\\mathbf K\_t,\\Lambda\_t,\\mathbf J\_t\\in\\mathbb R^{\\mathsf E\\times\\mathsf E},

Dt+1\=𝐌tDt+ut,𝐌t\=Λt𝐊t.D\_{t+1}=\\mathbf M\_tD\_t+u\_t,\\qquad \\mathbf M\_t=\\Lambda\_t\\mathbf K\_t. (41)

Conservative transport:𝟏𝖳𝐊t\=𝟏𝖳.\\text{Conservative transport:}\\quad \\mathbf 1^{\\mathsf T}\\mathbf K\_t=\\mathbf 1^{\\mathsf T}. (42)

RD(Dt)\={e:(𝐉tDt)(e)\>τe}.R\_D(D\_t)=\\{\\,e:\\ (\\mathbf J\_tD\_t)(e)>\\tau\_e\\,\\}. (43)

𝐊t\\mathbf K\_t transports residue, Λt\\Lambda\_t retains or dissipates it, 𝐉t\\mathbf J\_t translates it into admissibility. None is assumed equal to another. The four-state toy is Λ\=λI\\Lambda=\\lambda I, 𝐊\=𝐉\=I\\mathbf K=\\mathbf J=I.

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### 14\. History-dependent coupling — *central open map*

𝐊t+1\=𝒰(𝐊t,⊗t+1,ρt+1,δt+1,Dt+1).\\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:

𝐊t+1≥0,\\mathbf K\_{t+1}\\ge0, (45)

𝟏𝖳𝐊t+1\=𝟏𝖳when transport conserves residue,\\mathbf 1^{\\mathsf T}\\mathbf K\_{t+1}=\\mathbf 1^{\\mathsf T}\\quad\\text{when transport conserves residue}, (46)

supp𝐊t+1⊆{(e,e′):Cee′(t+1)≠⌀},\\operatorname{supp}\\mathbf K\_{t+1}\\subseteq\\{(e,e'):\\ C\_{ee'}(t+1)\\neq\\varnothing\\}, (47)

∥𝐊t+1−𝐊t∥op≤εK,\\left\\|\\mathbf K\_{t+1}-\\mathbf K\_t\\right\\|\_{\\mathrm{op}}\\le\\varepsilon\_K, (48)

𝒫t−X≠⌀⇒𝒫t+1−X≠⌀,\\mathscr P^{-X}\_t\\neq\\varnothing\\quad\\Longrightarrow\\quad \\mathscr P^{-X}\_{t+1}\\neq\\varnothing, (49)

where 𝒫t−X\\mathscr P^{-X}\_t is the declared set of admissible procedures or paths capable of testing the counter-account at time tt. Equation (49) is the X (Un)factor as an admissibility predicate on updates (𝒜t+1,𝐊t+1)(\\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 (𝒳,𝒜t+1)(\\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*

Sρ,ki\=Contrast(Ci,ρ(𝒪t+k),Ci,ρ(𝒪0,t+k)).S^i\_{\\rho,k}=\\operatorname{Contrast}\\left(C\_{i,\\rho}(\\mathcal O\_{t+k}),\\ C\_{i,\\rho}(\\mathcal O\_{0,t+k})\\right). (50)

ℭi,ρ(S)\={H:ℱi,ρ(H)\=Sρ,ki}.\\mathfrak C\_{i,\\rho}(S)=\\{\\,H:\\ \\mathcal F\_{i,\\rho}(H)=S^i\_{\\rho,k}\\,\\}. (51)

wi(H∣Sρ∈I)∝Pr(Sρ∈I∣H)Pr(H).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

shared interaction history→observer compression→effective observer geometry→opposition-weighted coupling→residue propagation→admissibility change→future interaction history→new observer compression\\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)–(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.