# The distinctions a presented form requires

Jeremy Rodgers · Independent Researcher

Version 1.4 · 16 October 2026

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

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

## Presented form, content signatures and current sensory basing {#section-9}

A single presented form can already give us a relation. A bent contour presents differently directed arms; an extended patch presents differently situated aspects. This chapter develops that minimum, calculates the exact distinctions a content law would need, and separates the current sensory basis of an act from the question of its physical bearer.

These contributions have distinct warrants: phenomenological and constitutive argument, a conditional mathematical theorem, and a defeasible epistemic principle. They sharpen what is presented, how its content is constrained, and what makes it the basis of a particular act. They do not replace the positive performance-identification argument in [Module H](/consciousness/relational-awareness/module-h#section-8).

### 9.1 Relational aspects of one presented form {#section-9-1}

The minimum target requires qualitative counterparts and their relation together in an actual presentation. It does not require two independently existing objects, two original subjects, two input channels, or an overt comparison answer. One visibly extended patch can already have differently situated aspects; a visibly bent contour can have differently directed arms. Those aspects need not be numerically separate experiences or ontological parts of the bearer.

#### Principle 9.1: Finite spatial aspects {#principle-9-1}

If a current presentation genuinely has spatially extended or differentiated qualitative character, with sufficient presented resolution for at least two apparent localities or directions, then their spatial arrangement is an aspect of that presentation’s character.

The support for this principle is the distinction between presenting extent and merely signifying it. A nonspatial signal can mean “extended”, support a shape name, or control an accurate drawing. Those achievements do not make the signal itself visibly extended. What distinguishes the proposed antecedent is an apparent spatial organisation, rather than a label that refers to an organisation elsewhere. At the stated finite grain, differently located or directed features and their arrangement belong to how this one form appears. This is an account of presented character, not an inference from the geometry of a stimulus or an encoding.

The presented-resolution restriction does real work. An unresolved point, an undifferentiated shape-associated quality, or a sequence of separately experienced positions followed by an inferred shape need not satisfy it. Accurate recognition alone does not establish resolved aspects. A critic can also deny that apparent form has the proposed aspect structure; behavioral success does not decide that phenomenological disagreement. The principle does not infer a phenomenal token for every mathematical point, an infinite-resolution continuum, or the simultaneous appearance of every detail later accessible by shifting attention.

Once the principle is accepted for an independently warranted form token $a$, the target extraction is elementary. Let

$$
W=(a,F,i,j,q_i,q_j,r_{ij})
$$

record its actual form character $F$, two genuinely given aspects $q_i,q_j$ and their given arrangement $r_{ij}$. Then

$$
W\longmapsto(a;q_i,q_j,r_{ij})
$$

retains a witness to minimum relational appearance in the very same presentation. No inference combines separately anchored experiences into a new common field. This record already contains the relevant aspects; the projection is a certificate of their common witness, not a reconstruction algorithm that discovers jointness. The argumentative advance is the spatial-aspect justification for obtaining that witness from a form anchor.

This removes an evidential requirement for a two-object comparison answer. It does not establish that comparison, attention, drawing or covert inspection can be physically abolished while the same appearance survives. That causal preservation claim would require further evidence. A new form first constructed at current uptake $J$ can supply the witness: earlier sensory tokens need neither persist nor fuse. The spatial content also does not require a spatially congruent physical bearer. An exclusive qualitative bearer $\mathbb{B}$ could instantiate precisely this form. Aspect structure therefore clarifies jointness while leaving the distinct bearer argument intact.

### 9.2 The exact covariance contract for a content law {#section-9-2}

Let a finite group $G$ act on a set $X$ of physical descriptions and a set $C$ of candidate complete contents. Write $gx$ and $gc$ for the actions. An anchor supplies a particular $x\in X$ and $c\in C$. In a phenomenal application, the warrant that $c$ actually occurs, its subject, its phase and its relation to an activity must be supplied independently. Merely writing $c$ beside $x$ supplies no such warrant.

A deterministic law $F$ is equivariant on the physical orbit $Gx$ when

$$
F(gy)=gF(y)\qquad(g\in G,\ y\in Gx).
$$

The action has two importantly different interpretations. Under a **passive change of coordinates**, it redescribes one token; covariance is consistency between descriptions once token identity and the content chart are fixed. Under **active replacement, permutation of components, or transition to another token**, covariance is a substantive transport premise. An isomorphism of controlled physical kernels does not independently establish transport of actual experience. Only compositions of the specified $G$-actions are covered here. An additional intervention outside that family requires a new compatibility check.

For $x$ and $c$, define their stabilizers

$$
H=\operatorname{Stab}_G(x),\qquad G_c=\operatorname{Stab}_G(c).
$$

Thus $H$ consists of the transformations that leave the nominated physical description unchanged. The relevant obstruction is a member of $H$ that changes the particular anchored content; physical symmetry by itself is not an obstruction to every content law.

#### Theorem 9.2: Anchored orbit law {#theorem-9-2}

An equivariant map $F:Gx\to C$ with $F(x)=c$ exists if and only if $H\subseteq G_c$. When it exists, it is uniquely determined on $Gx$ by

$$
F(gx)=gc.
$$

**Proof.** If $h\in H$, equivariance gives $hc=F(hx)=F(x)=c$, proving necessity. Conversely, $gx=g'x$ implies $g'^{-1}g\in H\subseteq G_c$, so $gc=g'c$. The formula is well defined. Associativity proves equivariance, and the anchor and equivariance determine every orbit value. No assertion follows about a different physical orbit. $\square$

This elementary group-action result is useful because it makes a proposed content law answerable to a precise contract. It does not assume that equal observable likelihoods decide phenomenal identity, and it supplies neither the anchor nor the active transport premise.

### 9.3 An attained minimum content signature {#section-9-3}

Suppose the physical description omits distinctions needed for the anchored target. At the fixed base $x$, add a finite signature $z$ with an $H$-action and an $H$-equivariant recovery map taking $z$ to $c$. Put

$$
K=\operatorname{Stab}_H(c)=H\cap G_c.
$$

#### Theorem 9.3: Target-specific signature {#theorem-9-3}

Every such signature satisfies

$$
\operatorname{Stab}_H(z)\subseteq K,\qquad |Hz|\ge[H:K]=|Hc|.
$$

The bound is attained by $H/K$. A compatible augmentation over the entire physical orbit is

$$
G/K\longrightarrow G/H,\qquad gK\longmapsto gH,
$$

with recovery $gK\mapsto gc$. Each physical base has exactly $[H:K]$ signature values.

**Proof.** A transformation fixing $z$ must fix its recovered value $c$, giving the stabilizer inclusion. Orbit–stabilizer then gives the inequality. On $H/K$, the map $hK\mapsto hc$ is well defined and equivariant, and attains the bound. Since $K\subseteq H$, the displayed map from $G/K$ to $G/H$ is well defined; since $K\subseteq G_c$, so is recovery. Left multiplication commutes with both maps. The fibre above $gH$ is $\{ghK:h\in H\}$ and has $[H:K]$ members. Neither subgroup needs to be normal: these are coset sets with group actions, not quotient groups. $\square$

The count concerns distinguishable admissible signature states. It is not a count of neurons, material parts or measured memory bits. A fixed-width binary code would need at least $\lceil\log_2[H:K]\rceil$ bits, but actual geometry, history, boundary conditions or existing omitted variables might already supply the distinctions. The abstract construction does not show that a particular device can physically realize all its signature states. A proposed law that instead uses additional nonphysical inputs owes an account of those inputs; the theorem does not rule it out by definition.

If the target is coarsened by an equivariant map $\rho:C\to T$, use $t=\rho(c)$. Every transformation fixing $c$ fixes $t$, so $K\subseteq K_t=H\cap\operatorname{Stab}_G(t)$. Consequently

$$
[H:K_t]\le[H:K].
$$

Recovery of the coarser target can require fewer distinctions than recovery of complete content. It retains the group action, the anchor and the appropriate coset signature, rather than reconstructing a physical microstate or a finer phenomenal description.

#### Four interchangeable labels {#four-label-signatures}

Consider an abstract physical base fixed by all permutations of four labels, so $G=H=S_4$. Candidate contents are labeled relations on those four individuals. These are mathematical candidates, not claims that a four-port machine has experience.

| Target | $\lvert K\rvert$ | $[H:K]$ |
| --- | ---: | ---: |
| Strict line betweenness, unchanged by reversal | 2 | 12 |
| Unoriented four-cycle adjacency | 8 | 3 |
| Equal-distance complete four-point relation | 24 | 1 |
| Field occurrence alone, without geometry | 24 | 1 |

A strict order has 24 labelings, but reversing it preserves betweenness; there are $24/2=12$ distinct betweenness relations. A four-cycle has four rotations and four reflections, leaving $24/8=3$ adjacency relations. The complete equal-distance relation is unchanged by every permutation. Thus a line target needs twelve residual distinctions under this contract, whereas the square target needs three. Insisting on a complete oriented order would retain distinctions unnecessary for betweenness.

The invariant last row prevents a mistaken occurrence inference. Both the constant-present law and the constant-absent law are equivariant. A symmetry calculation therefore selects neither. An actual anchor together with independently warranted active transport would constrain occurrence on that transport orbit; the group calculation does not supply those premises. Nor does a signature’s physical carrier automatically become the bearer of the recovered appearance.

#### A nontrivial physical orbit {#nontrivial-physical-orbit}

For a separate check, let $S_3$ act on the unordered physical pair $x=\{0,1\}$, while the anchored target is the ordered pair $c=(0,1)$. Here $|H|=2$ and $K$ is trivial. There are three physical bases, six augmented ordered pairs, and two signature values above each base. Coarsening back to the unordered pair needs one. This example checks the construction when the physical base itself changes, rather than only at an $S_4$-fixed base.

### 9.4 Stochastic content and limited observations {#section-9-4}

A stochastic equivariant law at an $H$-fixed base has an $H$-invariant distribution. On a finite transitive content orbit $Hc$ of size $m$, every member has equal probability, conditional on that orbit. This follows because any two orbit members are related by an $H$-action, which preserves their probabilities. The conditional entropy is $\log_2m$. Different orbits may receive different total masses; uniformity across them does not follow. An intrinsically stochastic law can therefore have asymmetric realized outcomes without choosing a fixed preferred orientation. A deterministic description of the realized outcome would have to include its distinguishing draw, context or history.

For an unaugmented deterministic choice $c$ at an $H$-fixed base, covariance holds for exactly $|K|$ of the $|H|$ transformations. Its mismatch fraction under uniform enumeration is therefore $1-1/m$. For the line and square examples this is $11/12$ and $2/3$, respectively. These are symmetry-test fractions, not behavioral error rates.

#### Proposition 9.4: Limited observation of a uniform content orbit {#proposition-9-4}

Let a random content $Q$ be uniform on an $m$-member orbit, and let an observation $Z$ take at most $k$ values. For any observation channel and any decoder $d$,

$$
\Pr\{d(Z)=Q\}\le\min(1,k/m),\qquad H(Q\mid Z)\ge\max(0,\log_2m-\log_2k).
$$

The success bound is attainable.

**Proof.** The best deterministic decoder has success $\sum_z\max_q\Pr(Q=q,Z=z)$. Each summand is at most $1/m$, and there are at most $k$ summands. Randomized decoding cannot improve this linear objective. For $k\le m$, partition the contents into $k$ nonempty cells, reveal the cell, and choose one representative per cell: exactly $k$ contents are correctly decoded, giving $k/m$. For $k\ge m$, reveal the content itself. Finally, mutual information is at most $H(Z)\le\log_2k$, so $H(Q\mid Z)=\log_2m-I(Q;Z)$ gives the entropy bound, together with nonnegativity. Attainability of the success bound does not assert that the entropy bound is attained for every $m,k$. $\square$

Thus a two-valued observation cannot decode a uniform twelve-way line target with success above $1/6$, or a uniform three-way square target above $2/3$. These limits concern the stipulated information architecture. An uncounted analogue variable or history violates its observation-cardinality premise. The proposition does not identify experienced content with a decoded symbol, measure human memory, or select a phenomenal bearer.

### 9.5 Current basing without physical-port constitution {#section-9-5}

Current basing asks whether a particular actual relational presentation is the operative sensory basis of a nominated comparison, not merely an appearance accompanying a correct output. Let $B_{\mathrm{cur}}(Q,A)$ express that $Q$’s relational aspect is such a basis of the current activity $A$. This differs from identifying $A$ with an independently nominated physical performance $P$, or making a physical content vehicle a literal constituent of $Q$. Those latter questions concern constitution and bearer placement.

Four inputs keep the basing inference assessable:

1. An independently credible phenomenal anchor warrants a current $Q_J$ and its presented relational aspect $\rho$. Source geometry, a relation bit and answer correctness do not supply this alone.
2. Genuine current acquaintance $K_J$ addresses $\rho$ of this presentation. An equal-valued memory or another field is not the same source merely because its content agrees.
3. A nominated reason vehicle $R$ receives that comparative content nondeviantly, preserving its source and epistemic role. A start signal from $K_J$ followed by a cached answer is not such conveyance.
4. The nominated activity is the agent’s current exercise of that apprehension, with $R$ serving its authored comparative role. Ordinary source-specific practical knowledge, source selection, instruction-dependent retasking and checks for bypasses can support this application. Causal efficacy or acquaintance considered separately does not establish it.

These inputs separate phenomenal evidence, addressed epistemic contact, transmission history and practical use. Their conjunction is not established by defining the symbols. In particular the fourth input retains a substantive current-activity commitment; it cannot be inferred merely from a downstream controller’s success.

The proposed transmission warrant is ordinary and defeasible: when an agent genuinely apprehends a present relation, its content is conveyed without source substitution, and the agent knowingly exercises that apprehension in this current comparison, the relation remains an operative qualitative basis of the activity. The positive reason is source-guided execution, rather than accidental agreement between an output and a concurrent field. Failed source selection, misunderstood content, substitution or loss of the current exercise can defeat the inference. It is neither a theorem about physical bits nor a claim that every truth-maker must efficiently cause the action.

This warrant does not require $R$ to constitute $Q$. An explicit consistent countercase has one agent, an actual $Q_J$ borne exclusively by $\mathbb{B}$, genuine $K_J$, faithful delivery to physical $R$, and a current psychophysical comparison $A$ that uses this very apprehension. Then $B_{\mathrm{cur}}(Q_J,A)$ can hold even though $Q_J$ is not a mode of $R$ or of physical $P$. The rival expressly denies the transfer of the qualitative mode to $P$; merely assigning the name $\mathbb{B}$ to a property instance of $P$ would not make it exclusive. Identification of $A$ with $P$ remains the separate [CMP argument](/consciousness/relational-awareness/module-h#section-8-10). This countercase establishes nonnecessity of port constitution for current basing. It does not show that every complete contract using port constitution is logically stronger than every contract using addressed transmission: their other requirements can differ.

#### Two substitutions that matter {#basing-substitutions}

A cached-answer route lets genuine $K_J$ trigger an action while a different record supplies its comparative value. This defeats faithful comparative conveyance. A different countermodel preserves genuine ancestry:

$$
Q_a/K_a\longrightarrow M\longrightarrow Q_b/K_b\longrightarrow R_b\longrightarrow\mathrm{action}.
$$

Here $M$ faithfully conveys the relation apprehended in $Q_a$, but a newly constructed $Q_b$ supplies the action’s current sensory basis. Both fields may genuinely occur and agree in content. Content fidelity and causal ancestry do not establish $B_{\mathrm{cur}}(Q_a,A)$. If $Q_b$ first appears at current uptake $J$, it may itself satisfy the target. The correct repair is to anchor and nominate that current field, not to demand persistence of an earlier one.

### 9.6 What a selective physical route contrast establishes {#section-9-6}

A physical contrast can strengthen content-provenance evidence while leaving the epistemic application explicit. Suppose a pre-gate code $r\in\{-1,+1\}$ has been nominated for a relational source. An outgoing gate either transmits $r$ or replaces it by a fair independent bit $k$. Let $\xi$ include all relevant pre-gate context and retained alternative routes. Conditional on $\xi$, the output has a stable mean $f_\xi(k)$. Gate assignment must average over the same conditional law of $\xi$ at fixed $r$; replacement must be independent of that context; and no gate flag, collateral timing change or feedback may reach the output except through $k$.

Under precisely this exclusion contract,

$$
\mu_{\mathrm{live}}-\mu_{\mathrm{repl}}=r\,\mathbb{E}\!\left[\left.\frac{f_\xi(+1)-f_\xi(-1)}{2}\right|r\right].\tag{36}
$$

For each $\xi$, replacement gives $[f_\xi(+1)+f_\xi(-1)]/2$; subtracting this from $f_\xi(r)$ proves the identity, after averaging over the common context law. Thus a direct-route controller with mean $\eta k$, $0<\eta<1$, has contrast $\eta r$. A controller that ignores $k$ and recomputes the same relation from a retained source has zero contrast, despite identical natural outputs. All output probabilities can remain positive.

The identity identifies a signed physical route contribution. It requires no reconstruction of the whole context. It does not identify a qualitative source, prove authorship, establish current basing, or label every individual episode in a heterogeneous ensemble. A gate inserted between $M$ and $Q_b$ in the preceding countermodel can affect the action even though $Q_a$ is not its current basis. Conversely, if $K_a$ is already independently warranted as the current comparison’s acquaintance, a gate after that uptake tests downstream expression; it does not discover the earlier basing. If the gate precedes field formation, separate anchors in the two arms can support separate ordinary applications, but not transport of one fixed pre-gate field.

The archival comparison in [Module G](/consciousness/relational-awareness/module-g#section-7) contains no such selective outgoing gate. Its initial sensory calibration and appearance-directed instructions support the ordinary current-scene inference at their stated strength; jointness and basing were not measured by dedicated probes. Separating basing from port constitution therefore repairs an unnecessary premise without inventing additional validation. Epistemic ancestry, operative current basis and downstream expression remain distinct, as do ordinary warrant, physical intervention algebra and constitutive bearer attribution.