# One appearance, its parts and its bearer

Jeremy Rodgers · Independent Researcher

Version 1.4 · 16 October 2026

DOI: https://doi.org/10.5281/zenodo.23408737

Web edition: /consciousness/relational-awareness/module-b

## Occurrence, inclusion and ownership {#section-2}

Relational appearance requires an actual presentation of counterparts together with their relation. This chapter makes the distinctions exact: a shared description, a shared source and a shared occurrence are different achievements. Its mathematical results expose the extra commitments that turn evidence about descriptions into claims about jointness and ownership.

Compatible descriptions of two qualities do not establish relational appearance. The counterparts and their relation must be given in an actual presentation. This module distinguishes four questions: whether descriptions descend faithfully from occurrence tokens, whether several tokens have a common presentation, whether ownership is exclusive, and whether a physical relation supplies an occurrence law. These distinctions refine the source–aperture–readout framework of Rodgers (2026e) without assuming its constitutive assignment. A new presentation can meet the relational target; the original tokens need not fuse or continue.

Throughout, $P$ denotes a physical performance, $A$ an experience-inclusive act, and $Q$ its actual qualitative mode. A finite token set $E$, a partition $\Pi$, and a downward-closed family $K$ are mathematical objects. Their interpretation as experiential occurrences or domains requires independent grounds. In particular, a physical port, an equal content label, and a numerically identical qualitative occurrence are different things.

### 2.1 Faithful descent of occurrence identity {#section-2-1}

Let $E$ be a finite nonempty set, $\Pi$ a partition of $E$, and $q:E\to V$ a surjection. Write $\ker q$ for its fibre partition, and write $\Pi\preceq\Pi'$ when $\Pi$ refines $\Pi'$. A label partition is faithful if it preserves both sameness and difference of the nominated domains.

#### Proposition 2.1: Occurrence descent {#proposition-2-1}

There is a partition $\Lambda$ of $V$ satisfying

$$
e\sim_\Pi e'\quad\Longleftrightarrow\quad q(e)\sim_\Lambda q(e')
$$

if and only if $\ker q\preceq\Pi$. When it exists, it is unique and its blocks are $q(B)$ for $B\in\Pi$.

**Proof.** If $q(e)=q(e')$, reflexivity in $\Lambda$ gives $e\sim_\Pi e'$, proving necessity. Conversely, if $q(B)$ and $q(B')$ intersect, some members of $B,B'$ lie in one fibre. Fibre purity then gives $B=B'$. Thus the nonempty block images partition $V$, and they satisfy the asserted equivalence. Surjectivity proves uniqueness. $\square$

The finest coarsening that admits descent is $\Pi\vee\ker q$, the equivalence relation generated by both partitions. Every descending coarsening must contain both relations; their join satisfies the proposition. This is a repair of a description, not evidence that actual domains merged.

An alternative is to preserve the missing identity. For each $v\in V$, let $m_v$ count the $\Pi$-blocks meeting $q^{-1}(v)$. A refining description $q'=(q,\mathrm{tag})$ needs at least $m_v$ labels over $v$, because distinct blocks in that fibre cannot share a refined label. The assignment $q'(e)=(q(e),[e]_\Pi)$ attains the bound. The minimum total alphabet is therefore $\sum_vm_v$; a uniform reusable tag alphabet needs $\max_vm_v$ values, or $\lceil\log_2\max_vm_v\rceil$ binary bits. These are coding bounds conditional on the independently nominated partition.

For example, take

$$
E=\{a,b_1,b_2,c\},\qquad \Pi=\{\{a,b_1\},\{b_2,c\}\},
$$

and give $b_1,b_2$ the same label $b$. The projected domains $\{a,b\}$ and $\{b,c\}$ cannot form a faithful label partition. Their transitive completion merges all three labels only because the description identified two tokens. Two physically separate recurrent cores can instantiate the same error: with independent native bits,

$$
T(a,b_1,b_2,c)=(b_1,a\oplus b_1,c,b_2\oplus c)
$$

is the product of two bijective two-bit maps. Equal current $b$ values or copied descriptions do not introduce a cross-core operation. Assigning qualitative occurrences to these cores would be an additional interpretation, not a consequence of $T$.

### 2.2 Pair witnesses, common witnesses and Fusion {#section-2-2}

Let $K\subseteq2^E$ contain every singleton and be closed under subsets. A face represents joint inclusion in at least one nominated domain; downward closure does not make every subset a separate subject. Define

$$
e\,R_K\,e'\quad\Longleftrightarrow\quad\{e,e'\}\in K.
$$

For $|E|\ge3$, the boundary family $2^E\setminus\{E\}$ and the full family $2^E$ agree on every proper-subset query. For three tokens, their pair relation is complete and transitive in both cases, but only one contains the common three-token face. The quantifiers differ:

$$
\forall e,e'\;\exists W\supseteq\{e,e'\}\qquad\text{versus}\qquad\exists W\supseteq E.
$$

Pairwise transitivity cannot exchange these quantifiers. For a fixed pair graph, every possible face must be a clique, but including every clique is an additional completion rule, and maximal cliques can overlap.

#### Proposition 2.2: Fusion and exclusive maximal domains {#proposition-2-2}

For finite $K$ as above, the condition

$$
F,G\in K,\quad F\cap G\ne\varnothing\quad\Longrightarrow\quad F\cup G\in K \tag{Fusion}
$$

holds if and only if the distinct maximal faces form a partition of $E$. Equivalently, $K$ consists of the subsets of partition blocks.

**Proof.** If two maximal faces intersect, Fusion places their union in the family, so maximality makes them equal. Finiteness and the singleton condition place each token in a maximal face. Downward closure supplies every subset of each block. Conversely, intersecting faces in a family of partition-block subsets lie in the same block, and their union is again a subset of it. $\square$

Fusion is a precise single-owner commitment about the same tokens. It does not follow from interaction or from a relation being computed. An overlapping-domain ontology can reject it without inconsistency. If a unique owner map is assumed at the outset, co-presence is equality of owners and a partition follows by definition; that assumption must not be mistaken for a physical discovery.

Uncertainty about token identity also yields a quantitative bound. On one common coupling, let $X$ indicate $a\sim b_1$, $Y$ indicate $b_2\sim c$, and $Z$ indicate $a\sim c$. Assume equivalence-domain ownership and $\Pr(b_1\ne b_2)\le\delta$. On the event that the middle token is identical, $XY\le Z$, hence $X+Y-Z\le1$; elsewhere its maximum is two. Therefore

$$
p_{ab}+p_{bc}-p_{ac}\le1+\delta. \tag{4}
$$

If each estimated marginal has absolute error at most $\epsilon_i$, add $\sum_i\epsilon_i$ to the right-hand side. Different trials do not automatically supply this coupling or stable middle-token identity. Passing the inequality also does not select a common higher face: the boundary-family example passes it.

### 2.3 Exact partial identification under exclusive ownership {#section-2-3}

For three nominated occurrences, permit absence but assume each present occurrence has at most one owner in the target occasion. The allowed pair-indicator patterns, ordered $(12,23,13)$, are

$$
000,\quad100,\quad010,\quad001,\quad111.
$$

Let their pair probabilities be $p_1,p_2,p_3$, and let $t$ be the mass of $111$.

#### Proposition 2.3: Three-pair identification {#proposition-2-3}

The exact identified interval is

$$
\max\left\{0,\frac{p_1+p_2+p_3-1}{2}\right\}\le t\le\min_i p_i.
$$

It is nonempty exactly when $0\le p_i\le1$ and $p_i+p_j-p_k\le1$ for all distinct $i,j,k$. If only jointly valid boxes $p_i\in[\ell_i,u_i]\subseteq[0,1]$ are available, the exact interval becomes $[L,H]$, where

$$
\begin{aligned}
L&=\max\left\{0,\max_{i<j}(\ell_i+\ell_j-1),\frac{\ell_1+\ell_2+\ell_3-1}{2}\right\},\\
H&=\min_i u_i.
\end{aligned} \tag{5}
$$

Compatibility is $L\le H$, equivalently $\ell_i+\ell_j-u_k\le1$ for all distinct $i,j,k$.

**Proof.** The atom masses must be $t$, $p_i-t$ for each single edge, and $1-\sum_ip_i+2t$ for $000$. Nonnegativity proves necessity and constructs every admissible exact solution. For boxes and fixed $t\le\min_i u_i$, choose $p_i=\max(\ell_i,t)$, which minimizes their sum. Feasibility is then $\sum_i\max(\ell_i,t)\le1+2t$. Expand the left side as the maximum, over $S\subseteq\{1,2,3\}$, of $\sum_{i\in S}\ell_i+(3-|S|)t$. Empty and singleton sets impose only $t\le1$ and $\ell_i\le1$; pairs and the full set give exactly the lower bounds in (5). The atom construction attains every resulting $t$. Taking an index minimizing $u_k$ proves the equivalent three box inequalities. $\square$

Triple tests do not characterize every global random partition. Split five tokens into $S$ of size two and $T$ of size three. Every partition satisfies

$$
\sum_{i\in S,j\in T}p_{ij}-\sum_{\{i,j\}\subset S}p_{ij}-\sum_{\{i,j\}\subset T}p_{ij}\le2. \tag{6}
$$

Indeed, a block containing $a$ members of $S$ and $b$ of $T$ contributes $ab-\binom a2-\binom b2\le\min(a,b)$. For $b\ge a$, subtracting $a$ gives $-(b-a)(b-a-1)/2\le0$, and the other case is symmetric. Summing blocks gives at most $\min(|S|,|T|)=2$; expectation proves the inequality. Set every cross marginal to $1/2$ and every within-side marginal to zero. All triple inequalities hold, but (6) has left side three. General finite compatibility is a linear programme over admissible partitions, or over a different domain family if exclusive ownership is rejected.

The probability polytope constrains one common target ensemble under the ownership premise. It neither infers that the nominated tokens exist nor imposes that premise on every possible presentation theory.

### 2.4 Reader validity, probe creation and context {#section-2-4}

Let $R_i$ be a positive assertion about a specified joint appearance, and $Z_i$ the occurrence at its readout time. Correct comparison of stimulus values does not by itself calibrate this assertion. Suppose, in the relevant assay context,

$$
\Pr(R_i=1\mid Z_i=0)\le\alpha_i<1.
$$

Writing $q_i=\Pr(R_i=1)$ and $z_i=\Pr(Z_i=1)$ gives $q_i\le\alpha_i+(1-\alpha_i)z_i$, hence

$$
z_i\ge\max\{0,(q_i-\alpha_i)/(1-\alpha_i)\}.
$$

No sensitivity premise is needed for this lower bound. Silence supplies no upper bound. A further guarantee $\Pr(R_i=0\mid Z_i=1)\le\beta_i<1$ gives $z_i\le\min\{1,q_i/(1-\beta_i)\}$.

Let $Y_i$ denote the same-episode earlier target, with independently defended creation/destruction allowances

$$
\Pr(Y_i=0,Z_i=1)\le d_i^+,\qquad \Pr(Y_i=1,Z_i=0)\le d_i^-.
$$

Then $\Pr(Y_i=1)\in[z_i-d_i^+,z_i+d_i^-]$, clipped to $[0,1]$. If the assay’s target distribution differs from a common reference ensemble by total variation at most $\rho_i$, subtract or add $\rho_i$ as well. Thus jointly valid response intervals $[q_i^-,q_i^+]$ give

$$
\ell_i=\max\left\{0,\frac{q_i^--\alpha_i}{1-\alpha_i}-d_i^+-\rho_i\right\}, \tag{7}
$$

$$
u_i=\min\left\{1,\frac{q_i^+}{1-\beta_i}+d_i^-+\rho_i\right\}. \tag{8}
$$

With specificity alone use $u_i=1$. These boxes can enter Proposition 2.3, but an accurate later report does not establish $d_i^+=0$: the probe may create the new third. Random selection after a completed episode is also insufficient unless its target and exogeneity contract are warranted.

For $m$ assays, with $n_i$ independent Bernoulli repetitions of common mean $q_i$ within assay $i$, the simultaneous Hoeffding radius

$$
s_i=\sqrt{\frac{\log(2m/\eta)}{2n_i}}
$$

gives $q_i^\pm$ by clipping $\widehat q_i\pm s_i$, with total failure probability at most $\eta$. Independence between assays is unnecessary for the union bound. Calibration uncertainty needs a separate joint budget. Repeated observations cannot calibrate the phenomenal specificity they presuppose. For illustration only, $q=.80$, $\alpha=.05$, $d^+=.02$, $\rho=0$ give a lower target probability $.7694736842$. These are stipulated assay parameters, not measurements of consciousness.

### 2.5 Fixed content maps and support intersection {#section-2-5}

Let a finite physical domain $X$ have independently warranted allowed-content sets $\Gamma(x)$, and let $\Delta:X\to Z$ claim sufficient target description. A function $h$ with $h(\Delta(x))\in\Gamma(x)$ exists exactly when every nonempty $\Delta$-fibre has nonempty intersection of its $\Gamma$ sets. Necessity follows from the common value $h(z)$; choosing one value per fibre proves sufficiency. Pairwise intersections are insufficient: $\{a,b\}$, $\{b,c\}$, $\{a,c\}$ have no common member.

If independent counterpart choices have $m,n$ values and the target preserves both projections, all $mn$ complete target values differ. A sufficient aperture needs at least $mn$ mathematical values. If each pair also has $r$ independently different presented relations, the count is $mnr$. This is not a Hilbert-dimension, orthogonality or accessible-bit theorem.

#### Proposition 2.4: Same-function support intersection {#proposition-2-4}

On a full Cartesian domain $X=\prod_{i=1}^nX_i$, suppose the same function $f$ factors through coordinate supports $S$ and $T$. Then it factors through $S\cap T$.

**Proof.** For $x,y$ agreeing on $S\cap T$, the Cartesian domain admits $w$ with $w_S=x_S$ and $w_T=y_T$. The prescriptions agree on the intersection. $S$-sufficiency gives $f(x)=f(w)$ and $T$-sufficiency gives $f(w)=f(y)$. Thus $f$ is constant on every intersection fibre. $\square$

Two separately feasible maps cannot be substituted for this premise. On binary $(x,y)$, take $\Gamma(0,0)=\{a\}$, $\Gamma(1,1)=\{b\}$ and both off-diagonal sets $\{a,b\}$. A map through $x$ and another through $y$ are feasible, while no constant map is. On a correlated diagonal $x=y$, even one nonconstant function has either coordinate as a sufficient description; the mixed point needed by the proof is absent. Nor is a quantum tensor factorization automatically a Cartesian domain of independently variable complete states.

### 2.6 Regularity and correlated indicators {#section-2-6}

On a fixed metric space with probability anchors $p(x_i)\in[\ell_i,u_i]$ and an independently warranted upper Lipschitz constant $L\ge0$, define

$$
\begin{aligned}
\underline p(x)&=\max\{0,\max_i[\ell_i-Ld(x,x_i)]\},\\
\overline p(x)&=\min\{1,\min_i[u_i+Ld(x,x_i)]\}.
\end{aligned}
$$

An admissible function exists exactly when $\ell_i\le u_j+Ld(x_i,x_j)$ for every pair, with valid anchor intervals. Necessity is the Lipschitz inequality. For sufficiency, maxima and minima of these cones are $L$-Lipschitz; the pair conditions place both envelopes within every anchor interval. The triangle inequality makes $\underline p\le\overline p$. Each envelope is itself admissible, and every admissible function lies between them, proving sharpness. The consistency lower bound

$$
L\ge\max_{i,j:d(x_i,x_j)>0}\frac{(\ell_i-u_j)_+}{d(x_i,x_j)}
$$

does not warrant the upper regularity used to transport an anchor.

For two indicators $Y_1,Y_2$ of one binary target $J$, independently justified error budgets imply

$$
\Pr(Y_1\ne Y_2)\le\Pr(Y_1\ne J)+\Pr(Y_2\ne J)\le\eta_1+\eta_2.
$$

A disagreement rate $.12$ contradicts two $.05$ budgets even when the indicators’ marginals agree. The exact unrestricted latent test uses nonnegative cells $\pi_{j,y_1,y_2}$, fixes their observed $(y_1,y_2)$ marginal, constrains $\sum\mathbf1_{y_i\ne j}\pi\le\eta_i$, and minimizes/maximizes $\sum\pi_{1,y_1,y_2}$. Every feasible cell table defines a joint law, so the linear-programme bounds are sharp for that model class. Without target-error restrictions, any target prevalence is compatible: draw $J$ independently of the complete observed pair. Hardware independence is not independence conditional on actual appearance, and an experienced word such as “different” need not present the original sensory counterparts.

### 2.7 Native articulation, manifestation and categorical grounding {#section-2-7}

The earlier proposals motivate distinct constitutive laws. The following arguments also identify the premises each law still requires.

#### Native Articulation Law {#native-articulation-law}

The Native Articulation Law (NAL) assigns an elementary presentation to each actual irreducible native transformation. Its motivation is immanence and structural fidelity: do not add a phenomenal fusion across independently operating transformations or a phenomenal division inside a native prime without further lawlike structure. It addresses organization rather than an ingredient list and needs no additional spectator. Its strongest objection is carrier-centered: an irreducible physical event can still have several carriers with separate qualitative states. Rejecting their experiential division already adopts the event-centered individuation criterion. Native elementary-event grain and qualitative duration also need independent specification. A content map could be constant, so one presentation per prime would not itself supply two qualitative counterparts and their relation.

#### Relational Manifestation {#relational-manifestation}

Relational Manifestation (RM) proposes that a concretely realized intrinsic relation has a manifest counterpart preserving its distinction, with some common presentation supported by its contributors. Ontological uniformity motivates the proposal: if intrinsic qualitative terms are real, wholly unmanifest connections between them should not simply be assumed. A serious rival nevertheless retains qualitative terms and real intrinsic relations without their being given together; it need not introduce another substance. Uniformity therefore supplies a reason to investigate the law, not a proof of its co-presentation clause.

This proposal also changes the object represented by the family. If $\mathcal R$ lists required realizing contributor supports, the least downward-closed family containing them is

$$
\mathcal F_{\min}=\{\varnothing\}\cup\{\{e\}:e\in E\}\cup\bigcup_{R\in\mathcal R}2^R.
$$

It records common realizing support. It is not the earlier family $K$ of qualitative occurrences jointly given. A subset of a realizing support need not be sufficient for the target. A compressed parity can preserve a distinction without presenting both full argument qualities. Identifying $\mathcal F$ with $K$ would need content fidelity and actual occurrence, neither supplied by set closure.

#### Intrinsic Qualitative Grounding {#intrinsic-qualitative-grounding}

Intrinsic Qualitative Grounding (IQG) adds: (i) actual concrete categorical character is qualitative; (ii) irreducible relational facts have categorical grounds; (iii) the nominated relation’s ground is a presentation in which that relation is manifest. The positive rationale integrates intrinsic actuality and accepted relational phenomenology without an added spectator. Clause (iii) is not a consequence of the first two: a relational ground can contain qualitative terms without one common presentation. The objection concerns assembly/ownership, not whether the relation exists. The broader categorical and whole-act alternatives are developed in [Module H](/consciousness/relational-awareness/module-h). None of these principles is assumed to establish the main paper’s native-human induction.

### 2.8 Continuous interaction and primitive support {#section-2-8}

A finite tick does not independently individuate a physical elementary event. Let independently fixed parts have connected open coordinate domains $X_i$, and let $f(t,x)$ be a $C^1$ vector field on their full product, with local uniqueness. At $t_0$, join parts $i,j$ when a cross derivative $D_jf_i$ or $D_if_j$ is nonzero somewhere in the intervention domain at times arbitrarily near $t_0$.

#### Proposition 2.5: Finest interaction-germ decomposition {#proposition-2-5}

The connected components of this symmetric germ graph give the unique finest fixed-part product decomposition of the vector-field germ.

**Proof.** Every missing cross-block edge vanishes on some time neighbourhood. Finitely many pairs supply a common neighbourhood. Connected Cartesian coordinate domains and vanishing cross derivatives imply that each block’s vector field depends only on that block, giving local product flows. Conversely, a product germ has zero cross-block derivatives, so every germ edge must have its ends in one block. Each graph component must consequently lie in any admissible block. $\square$

Separate smooth invertible part-coordinate changes, consistent part permutations and orientation-preserving smooth time diffeomorphisms preserve the zero/nonzero cross-dependence. Arbitrary mixing of native parts does not. For $\dot x=\sin(2\pi t)y$, $\dot y=\sin(2\pi t)x$ on $[0,1]$, the flow is $\exp(M\alpha(t))$ with off-diagonal $M$ and $\alpha(t)=(1-\cos2\pi t)/(2\pi)$. It returns to identity while its intermediate flow couples the inputs.

A two-sided germ can depend on later coupling. A coupling zero for $t\le0$ and $e^{-1/t^2}$ afterward is included at zero despite an idle past and identical instantaneous derivatives. Coupling 1–2 only before zero and 2–3 only afterward connects all three at zero without simultaneous edges. Product indecomposability of a germ is thus different from instantaneous exchange or co-presentation.

For independently fixed finite tensor factors, let $E_i$ replace factor $i$ by its normalized partial trace times identity. The $E_i$ commute and are Hilbert–Schmidt orthogonal projections. Define

$$
P_S=\prod_{i\in S}(\operatorname{Id}-E_i)\prod_{j\notin S}E_j,\qquad H_S=P_SH.
$$

Expanding $\prod_i[E_i+(\operatorname{Id}-E_i)]$ proves $H=\sum_SH_S$; mutually orthogonal projector ranges prove uniqueness. Each included factor is traceless and each omitted factor is identity. Local unitary frames preserve support, while their derivative terms add only singleton generators; a scalar shift changes only the empty support. Pointwise support needs regularity: a change on one time point in an integrable generator does not alter the propagator. Pair supports need not exhaust state/history structure: $[Z_1X_2,Y_2Z_3]=2iZ_1Z_2Z_3$, and the commuting chain $Z_1Z_2+Z_2Z_3$ acting on $|+++\rangle$ at angle $\pi/4$ produces a pure state with every one-qubit reduction maximally mixed. It is entangled across each bipartition although the primitive triple support is zero.

### 2.9 Action, articulation rates and quantitative support {#section-2-9}

On fixed factors $\mathcal H_L\otimes\mathcal H_R$, with dimensions $d_L,d_R$, remove scalar and local Hamiltonian terms by

$$
H_\times=H-\frac{\operatorname{Tr}_R H}{d_R}\otimes I_R-I_L\otimes\frac{\operatorname{Tr}_L H}{d_L}+\frac{\operatorname{Tr}H}{d_Ld_R}I.
$$

For locally integrable Hermitian $H$, define

$$
\mu(I)=\hbar^{-1}\int_I\|H_\times(t)\|_{\mathrm{op}}\,dt.
$$

It is additive under faithful subdivision. An absolutely continuous local frame adds only local derivative terms and conjugates $H_\times$; an increasing absolutely continuous time change rescales the generator and preserves the integral.

#### Proposition 2.6: Zero action and product subintervals {#proposition-2-6}

The action vanishes exactly when every subinterval propagator is a product unitary up to scalar phase. With $H_0=H-H_\times$ and its propagator $V_0$,

$$
\begin{aligned}
\|V(t,s)-V_0(t,s)\|_{\mathrm{op}}&\le\mu([s,t]),\\
\frac12\|V\rho V^\dagger-V_0\rho V_0^\dagger\|_1&\le\min\{1,\mu([s,t])\}.
\end{aligned}
$$

**Proof.** Zero integral makes $H_\times=0$ almost everywhere, so the local generators yield product propagation. Conversely, the tangent of a product-unitary propagator at almost every identity-time lies in the local Lie algebra, excluding a cross term. Duhamel’s integral expression for $V-V_0$, unitary invariance of operator norm and integration give the first bound. Expanding the density difference into two terms gives trace norm at most $2\|V-V_0\|_{\mathrm{op}}$; trace distance is also at most one. $\square$

For $H=g(t)Z_LZ_R$, $\mu=\int|g|/\hbar$ while the accumulated angle is $\theta=\int g/\hbar$. A forward angle $\pi/4$ followed by its inverse has identity endpoint and action $\pi/2$. Exchange SWAP generated by $g(I+XX+YY+ZZ)/2$ has cross norm $3|g|/2$ and action $3\pi/4$ at its SWAP endpoint. Endpoint permutation does not certify an interaction-free physical history.

#### Finite-Rate Articulation {#finite-rate-articulation}

Finite-Rate Articulation (FRA) proposes joint-presentation occurrences with survival $S(\mu)$. If $S(0)=1$, continuity at zero and action-homogeneous memoryless survival imply $S(x+y)=S(x)S(y)$, then $S(\mu)=e^{-c\mu}$ for $c\ge0$. To prove this, continuity excludes a zero survival at finite action by repeatedly halving it; $-\log S$ is continuous and additive, hence linear. Probability of at least one occurrence is $1-e^{-c\mu}$. Finite response, composability and invariance under descriptive subdivision motivate the proposal, but no such property earns $c>0$, selects the norm, or makes survival memoryless.

For an external equal-probability lottery over actions $0,2x$, tokenwise conditioning gives $\tfrac12(1-e^{-2cx})$, not $1-e^{-cx}$; their gap is $\tfrac12(1-e^{-cx})^2$. A state-sensitive completion is also needed if the proposal is to distinguish $|00\rangle$ from $|++\rangle$ under $ZZ$: their action agrees despite different state evolution. Finally, alternating $g=\pm1$ on $2n$ equal bins gives fixed action one and maximum signed phase $1/(2n)$. Its propagator path tends uniformly to idle while the proposed occurrence probability stays $1-e^{-c}$. This is a dependence on actual control history, not a failure of invariance under redescribing one unchanged history.

#### Reciprocal Articulation Field {#reciprocal-articulation-field}

An alternative, the Reciprocal Articulation Field (RAF), introduces actual amplitudes $a_1,a_2$ with potential

$$
V=\frac r2(a_1^2+a_2^2)+\frac u4(a_1^4+a_2^4)-\kappa a_1a_2-h(a_1+a_2),\qquad r,u>0.
$$

At $h=0$, $\kappa\ge0$, the zero state’s Hessian eigenvalues are $r\mp\kappa$. For $\kappa>r$ the aligned minima have $a_1=a_2=\pm\sqrt{(\kappa-r)/u}$ and Hessian eigenvalues $2(\kappa-r),4\kappa-2r$. Small forcing satisfies $um^3+(r-\kappa)m=h$, giving susceptibility $[3um^2+r-\kappa]^{-1}$ along a stable aligned branch. Relaxation additionally requires a kinetic law: under $\dot a_i=-\gamma\partial_{a_i}V$, $\gamma>0$, the linear relaxation rates are $\gamma$ times the Hessian eigenvalues. If $\kappa$ depends on a physical coordinate, differentiating $V$ produces a corresponding backreaction force; the field is therefore a physical hypothesis, not a cost-free phenomenal label.

Those predictions do not validate the field or its qualitative interpretation. At every strictly positive finite temperature its Gibbs density is positive throughout the plane, giving positive $\mathbb E a_i^2$ even below threshold and zero $\mathbb E a_i$ in the symmetric equilibrium even above it. Quartic domination makes the finite partition function locally analytic in $\kappa$; the zero-temperature bifurcation is not a universal biological-temperature cutoff. Nonzero forcing already gives a nonzero response below threshold. If each uncoupled core is already ordered, replace $r$ by $-\alpha$, $\alpha>0$: aligned and antialigned energies are $-(\alpha+\kappa)^2/(2u)$ and $-(\alpha-\kappa)^2/(2u)$. Every positive $\kappa$ favours aligned global minima, while antialigned states remain locally stable only for $\kappa<\alpha/2$. Thus ordering does not settle a one-subject fusion boundary. FRA and RAF concern different physical/constitutive commitments and cannot be combined by sharing an articulation label.

### 2.10 Collective capacity and a conditional content-grounding bound {#section-2-10}

Pairwise generators can encode globally irreducible physical distinctions. For $n$ qubits with control $c$, take the bounded autonomous Hamiltonian

$$
H=\frac{\pi\hbar}{2T}\sum_{j=2}^n|1\rangle\langle1|_c\otimes(I-X_j).
$$

Its commuting pair terms implement controlled-$X$ at $T$, mapping $|\pm\rangle_c|0\rangle^{\otimes(n-1)}$ to $|\mathrm{GHZ}_\pm\rangle=(|0^n\rangle\pm|1^n\rangle)/\sqrt2$. The states are orthogonal and $X^{\otimes n}$ distinguishes them; tracing out any factor removes their differing coherence, so every proper reduction agrees. Primitive pair arity therefore supplies no upper bound on collective physical distinction order. This is not an observation of a GHZ presentation.

If, additionally, a reduced density operator is a sufficient target aperture and its finite content probabilities are affine on all density operators, they have POVM form $F(\rho)_k=\operatorname{Tr}(E_k\rho)$ with $E_k\ge0$, $\sum_kE_k=I$. An affine extension to Hermitian operators gives the trace representation; positivity on all pure states and normalization give the effects. Consequently $\operatorname{TV}(F(\rho_0),F(\rho_1))\le\tfrac12\|\rho_0-\rho_1\|_1$. A binary target distinguishing equiprobable preparations with average error at most $\epsilon$ thus requires

$$
\frac12\|\rho_0-\rho_1\|_1\ge1-2\epsilon. \tag{9}
$$

The lower bound follows because the two correct-decision errors sum to at most $2\epsilon$. Sufficient grounding and affinity are extra correspondence premises; a physical ensemble representation alone does not supply them.

For an explicit changing aperture, let $H=\hbar\omega|1\rangle\langle1|_L\otimes(I-X_R)$ act on $|\Phi_\pm\rangle=(|00\rangle\pm|11\rangle)/\sqrt2$. With $\theta=\omega t$,

$$
|\psi_\pm\rangle=\frac{|00\rangle\pm ie^{-i\theta}\sin\theta|10\rangle\pm e^{-i\theta}\cos\theta|11\rangle}{\sqrt2}.
$$

Direct partial traces give $D_{LR}=1$, $D_L=|\sin\theta|$, $D_R=0$. If the phase distinction were actually presented with error at most $.1$, then at $\theta=\pi/4$ neither local density aperture meets the required $.8$ contrast. At $\theta=\pi/2$ the states are $|\pm\rangle_L|0\rangle_R$, so the same target becomes locally realizable. No subject migration, preserved original token, or actual qubit appearance follows. Generator, realizing support, encoded distinction and qualitative presentation are separate objects.

### 2.11 Descriptive gluing versus actual common occurrence {#section-2-11}

The global-structure proposal of Lee-Youngzie et al. (2026) motivates testing whether independently measured local qualitative structure determines a broad structure. A presheaf $\mathcal F$ has identity/composition restrictions; a sheaf adds existence and uniqueness of a glue for every compatible local family. A category-valued dissimilarity object needs a specified section/underlying-set construction before these set-valued conditions apply.

Finiteness alone does not turn existence into uniqueness. On a discrete two-point space, let $\mathcal F(X)=\{0,1\}$ and let every proper-open set have a singleton section set, with both globals restricting to it. Every matching family glues: a cover containing $X$ fixes its global member, and a proper cover admits either global. The cover by the two singleton opens has two glues. Injective restrictions or a jointly separating restriction family can instead supply uniqueness, but their independent role, not finiteness, does the work.

Nor does an incompatible family disprove the sheaf condition. Binary functions on three positions form a sheaf; left/middle and middle/right assignments that disagree on the middle have no glue because they are not matching. An impossible three-dimensional interpretation likewise need not mean the actual two-dimensional scene lacks unity.

An expansive restriction postulated to obey

$$
d_X(x,x')\le d_V(r(x),r(x'))
$$

cannot forget a positively distinguishable feature: equal restricted values force zero broad dissimilarity. Restricting the left coordinate of binary pairs with normalized Hamming distance violates this for $(0,0),(0,1)$. An attentional manipulation may instead create a new context or token and sharpen discriminability. The arrow between attention contexts must then be distinguished from passive part restriction of one unchanged presentation; unchanged retinal input does not establish the latter.

Attach independently supported token sets $T_i(d_i)$ to local descriptions $d_i$: these are the nominated actual presentations of which the datum is warranted as an aspect. A common occurrence witness exists exactly when

$$
\bigcap_iT_i(d_i)\ne\varnothing.
$$

Compatible descriptions separately witnessed by $\tau_1,\tau_2$ need no common token. A new $\tau_3$ genuinely presenting both can supply the common witness without fusing $\tau_1,\tau_2$. Descriptive gluing does not establish that lifting. Even a common token needs independent evidence that the relation, rather than just separate counterparts, belongs to it. Equal binary local marginals conceal this: equal-pair and opposite-pair distributions both have uniform margins, while their joint-equality probabilities are one and zero. Independently measured broad relational data, with fixed readers and token/context indices, can discriminate them.

### 2.12 Native replacement and the corrected acquaintance condition {#section-2-12}

The fading/dancing-qualia argument supplies a cognitive-coherence motivation for organizational transfer (Chalmers, 1995). Its exact physical premise is stronger than boundary-output agreement. Fix components, local states, native independent preparations, controlled kernels, output ports, clock, initial law and environmental coupling. A product of local bijections and a component permutation, with corresponding interventions, is a native isomorphism only if every controlled kernel and those boundary/clock conditions commute. Induction proves equality of all finite trajectory laws, including adaptive policies that choose corresponding actions from matched prior outputs. The induction concerns physical trajectories; it does not identify a detector with direct acquaintance.

For example, $F(a,b)=(b,a)$ and $G(r,s)=(r\oplus s,s)$ obey $hF=Gh$ for $h(a,b)=(a,a\oplus b)$. Yet across the sixteen state/single-port/value preparations, eight corresponding same-port clamps fail: four first-port failures have clamped value different from $a$, and four second-port failures have $a=1$. No product bit-bijection with permutation conjugates them: under such maps both swap outputs remain cross-only dependencies, whereas $G$ has an isolated self-dependent second component. Global predictive recoding is not a physical-copy theorem.

Likewise, let a current/remembered comparator update $(x,m)'=(u,x)$, and suppose a material chart sends both presented values to $q_z(v)=v\oplus z$. Switching $z$ changes absolute labels in all four states while preserving their presented equality. A change between tokens does not itself produce a presentation comparing the old and new qualities. Occurrence transfer is weaker than exact qualitative-chart transfer, but still needs its own warrant.

The corrected no-silent-loss schema is especially instructive. Let $r_v$ mean occurrence, $k_v$ genuine current acquaintance, $B_v$ independently specified checking background, and $D_{uv}$ a possible physical discrepancy on a replacement edge. The proposed condition is

$$
B_u\wedge k_u\wedge r_u\wedge\neg r_v\quad\Longrightarrow\quad D_{uv}.
$$

Native matching supplies $\neg D_{uv}$. One anchored edge can yield $r_1$, but not $k_1$. On a three-vertex path, take $r=(1,1,0)$, $k=(1,0,0)$, all backgrounds true and no discrepancy. The schema holds in both directions on every edge, while endpoint occurrence fails. An occurrence-edge principle $r_u\wedge\neg r_v\Rightarrow D_{uv}$ repairs the first-loss proof; alternatively add $B_v\wedge r_v\Rightarrow k_v$ at every vertex. Each is a new premise, not an implication of transported occurrence. Interval occurrence also does not automatically mean an attended presentation at the checking instant. These are the exact obligations that a transfer account must earn, and they are distinct from the main paper’s narrow continuing-human capacity induction.

### 2.13 Awareness, access and the construction of a personal event {#section-2-13}

The awareness-first account distinguishes the proposed knowing aspect of reality, denoted $U$ in the monograph, from the differentiated contents available through a particular apparatus and history. This ground is not an additional localized subject. The monograph’s starting commitment and the development paper’s operational distinctions have different roles (Rodgers, 2026e,c). The former proposes an ontological basis; the latter distinguishes a source, the access available to an observer, and the representation constructed through that access. An operational model can characterize dependence on apparatus or acquired constraints without deriving the fundamental commitment. Conversely, granting that commitment does not determine the extension of a particular experiential predicate.

This distinction matters even if one accepts that all differentiated terms belong to one underlying reality. Two differentiated processes may have separate perspectives; they may contribute to a new shared presentation; or their interaction may introduce no additional joint perspective. A premise about the underlying basis alone does not decide among those local possibilities. The present paper’s current-act argument supplies a scoped positive attribution rather than treating fundamental awareness as a substitute for its activity and bearer premises.

Perceptual presentation is formed through bodily and sensory processing and acquired organisation, including classifications, memories, expectations, feelings and beliefs. These conditions shape the sensory presentation itself; they are not merely descriptions added after a consciously viewed raw image. Ongoing seeing is perceptual processing, which can continue when reflective naming, commentary or personal appropriation is suspended. Perceptual construction, recognition and the sense of an event happening to me are different functional roles; they can operate concurrently within one activity. A practical history can therefore alter the organisation of current access itself. In the development account, a relational product can be retained and become available to subsequent interactions; it need not be a third substance (Rodgers, 2026c, §3.1). Retained constraints then shape activity, and activity can revise them. The reflective-alignment paper develops that reciprocity (Rodgers, 2026b, §4.1); its operational description does not by itself establish one phenomenal subject.

The personal observer is also constructed within this process. A central self-image organizes ownership, relevance and narrative continuity. The proposed reflective recognition of its constructed character need not erase memory, practical identity or the usable self-model. It changes the model’s claim to be the exhaustive centre (Rodgers, 2026c, §5.4). Whether an artificial or biological implementation has such recognition as experience requires its own warrant; recursive updating or an articulate self-report does not settle it (Rodgers, 2026j,b).

### 2.14 A conditional connection to the full constitution {#section-2-14}

The current-act argument is developed independently of assuming the full A1–A3 constitution. A defensible application of SPC-2 requires a justified physical realisation and native operating doctrine (Rodgers, 2026e, ch. 17). **A1** assigns a perspective to a qualifying maximal internal strongly connected component. Qualification requires an executable covering return and at least two distinct native predictive classes in the actual operating/protocol type and resource context. Incompatible control modes cannot be combined into a fictitious return. These conditions are not strengthened here by a new requirement for language, autobiographical reflection or a rich human scene.

**A2** assigns phenomenal relational organisation as a structural copy of the core’s complete endogenous predictive object over all physically admitted finite continuations. The actual phenomenal point corresponds to the predictive class of the realised intrinsic state. Native operation and outcome labels, finite test laws, compatible horizon restrictions and well-defined class transition instruments are preserved. This is the full structural assignment, not a selected finite report profile or an investigator’s preferred time horizon. **A3** then determines episode continuation through nonbranching realised process provenance; matching memories or outputs alone do not supply that continuity.

To connect the present case to those laws, an appropriate phase of its performance would have to be shown to belong to a justified qualifying core, with the relational mode $Q$ identified as an aspect of the core’s current scene. A performance’s executing contributors and a subject’s boundary are different nominations requiring a bridge. An environmental source or instrument can contribute causally without thereby becoming part of that subject. With the realisation supplied and SPC-2 accepted, phenomenal states, their timing and continuation are assigned at its declared native temporal resolution. No such complete application is demonstrated here. The local current-act argument neither validates A2 nor independently selects the first onset of a scene, and it does not turn the subject boundary into the set of everything causally involved in the performance. First presentation at current uptake, $J$, remains an admitted option.

### 2.15 The October draft and the scope of the recollective thesis {#section-2-15}

The unpublished 12 October 2026 draft of *Experience as Recollective Interpretation: Awareness, Scaffolding and the Question of Machine Experience* (Rodgers, 2026d) defines experience as the interpretation of a retained occurrence as having occurred. Its Section 5 explicitly allows minimal retention without language, explicit temporal concepts or autobiographical memory. An ongoing act can interpret its recent history: the original process need not be complete before reflective uptake starts. The relevant temporal ordering concerns the particular registered material and its uptake, rather than a requirement that seeing stop before reflection can begin. It does not place a consciously viewed, unprocessed image before all interpretation: perceptual construction already shapes what is seen.

Table 1 makes the uses explicit. It records the roles of the terms rather than treating the companion draft’s vocabulary as a proof of its constitutive thesis.

#### Table 1: Terminology across the present argument and the unpublished October draft {#table-1}

| Term | Present paper | October recollective draft |
| --- | --- | --- |
| Occurrence | Actual taking place; a qualitative occurrence requires its own warrant. Retention is a separate question. | Something happening in a system and registered or retained by it; seeing can occur without being classified as experience. |
| Appearance | Actual qualitative character of a constructed presentation, including the joint relation targeted here. | No independent admission rule for this target is supplied; seeing can continue without reflective personal classification, while presence is identified with construction. |
| Interpretation | Perceptual construction and reflective uptake have distinguishable, possibly concurrent roles. | Uptake of retained material as prior to its interpretation; explicit concepts of time are unnecessary. |
| Experience | Used for a qualitative episode or mode unless a narrower use is specified. | Recollective interpretation of a retained occurrence as having occurred. |
| Personal experience | Appropriation and interpretation as happening to a person. | The further subjective or personalised layer organised through concepts, feeling and a constructed self. |
| Awareness | An ontological starting commitment, distinct from evidence for a specified local appearance. | The proposed undivided ground remaining when interpretative divisions are removed. |

The last column records that draft’s definitions and proposals, especially Sections 1, 5 and 7; it does not promote them to independently established results.

Part of the apparent tension is therefore terminological. The companion draft can withhold its defined word *experience* from seeing without reflective self-classification while leaving perceptual processing in place. This does not remove the acquired organisation shaping that processing. Part is a difference in scope. The present paper defends a local sensory-capacity projection and a contestable bearer attribution; the companion proposes an account of all experience and discusses artificial systems. Its Section 10 expressly acknowledges that the universal recollective thesis is not established by the relational-awareness results. In particular, allowing first appearance at current uptake, $J$, does not make every appearance a recollection or make every retained descriptor appear.

A substantive question remains after those distinctions. Section 7 of the companion draft argues that presence is no additional ingredient beyond the constructed process, and Section 8 declines a functional account that would leave presence as a further open fact. Those claims are not merely a preference for *personalised* over *subjective*. Their weakest reading denies a separate inner spectator; that is compatible with the present act/mode argument. A stronger reading identifies actual qualitative appearance with suitably organised retentive interpretation under the awareness premise. That identification needs substantive warrant. The present paper neither derives it for every interpreting system nor establishes an irreducible remainder that would refute it. The companion’s Section 7 itself records that the move from reflective classification to experiential constitution has not been proved, and that a defender of seeing without reflective classification as experience has not thereby been refuted. The precise unresolved choice is which constitutive identification is intended and what supports its extension beyond the locally warranted case.

The comparison also limits inferences about artificial systems. Section 8’s narrow verdict concerns interpretation and recollection without the human personal layer. It does not supply the present paper with an independent appearance anchor, jointness test or physical bearer identification for an artificial interpreter. Conversely, failure to replicate human autobiographical or emotional organisation alone would not exclude every possible appearance. Both claims concern the scope of an inference, not a verdict about a particular system’s consciousness.

### 2.16 Occurrence without recognition and the limiting memory question {#section-2-16}

Something occurring, its currently appearing, its recognition as a particular object or event, its appropriation as happening to a person, and its subsequent interpretation are distinct questions. They can overlap in ordinary life. The locally supported current presentation need not await a later autobiographical account, and no such account is required by the concrete-performance principle. This does not decide the companion draft’s universal identification of experience with recollective interpretation.

The limiting condition that removes all acquired scaffolding and the ability to form memory must be distinguished from suspending naming or personal recognition. Its proposed underlying basis does not itself specify an impression carrying a discriminating record. Something occurring and its recognition as an event are different claims; absence of recognition does not establish absence of every process or retained dependence. Neither the absence of a named object nor the inability to describe a condition entails that no change took place. Equally, the stronger limiting condition should not be replaced with an ordinary recognised scene and a failed report.

The positive recollective model in the October draft explicitly includes retention. Its Section 5 distinguishes such minimal retention from conceptual or autobiographical memory; its limiting seeing case withholds interpretative experience while preserving an occurrence. Those claims do not establish an unrecognised retained trace in the separate limiting condition that removes the ability to form memory. No retention mechanism for that stronger condition is supplied by this comparison.

The remaining explanatory question is specific. If later interpretations differ with a particular antecedent, what makes their dependence on that antecedent possible and their historical reference warranted? The claim that memory categories are supplied later does not by itself answer that question. The following elementary result identifies a stronger condition than absence of conscious, conceptual or autobiographical memory. It is a conditional diagnostic; it does not establish that the limiting condition actually satisfies that stronger condition.

#### Proposition 2.7: No antecedent discrimination without a dependence route {#proposition-2-7}

Let $A$ denote a nominated preceding condition. At the start of a later interpretation let $S$ include every variable available to that interpretation, including restored scaffolding, apparatus state and relevant continuing environment. Suppose $\operatorname{Law}(S\mid A=a)=\nu$ is the same for every admitted $a$, and a common stochastic kernel $T(\cdot\mid S)$ generates a later account $Y$ without a further $A$-dependent input. Then $\operatorname{Law}(Y\mid A=a)$ is the same for every admitted $a$.

**Proof.** For every account event $E$,

$$
\Pr(Y\in E\mid A=a)=\int T(E\mid s)\,\nu(ds),
$$

which is independent of $a$. An external input whose conditional law varies with $a$ would violate the stated completeness or common-kernel premise. $\square$

No conscious recollection, named object, faithful replay or material trace is assumed in this proof. The converse claim that every actual dependence must be a memory trace is not proved. A continuing external source, nonconceptual state or separately proposed nonphysical relation would constitute a different dependence route and would need to be specified and justified. Mere persistence of one undifferentiated basis does not yet explain discrimination among antecedents. Absence of an ability to form memory also does not by itself establish the independence premise for all these variables. The stronger no-dependence stipulation and the no-memory limiting condition must therefore not be silently identified.

If the apparent historical reference is instead entirely a present construction, the account must explain what warrants relating it to one particular antecedent. This is an open explanatory obligation, not an established retained-information mechanism. The current-act conclusion does not depend on choosing a resolution: ordinary current sensory evidence and the constitutive argument have their own stated domain.

The further proposal that fundamental awareness may be known as conceptual scaffolding falls away is philosophical and motivated by first-person experience. It does not follow from the impossibility of reproducing an occurrence in a description. The distinction preserves that proposal while keeping particular claims of occurrence, content, basing and bearer assessable.