Finding the structure behind identical behavior

A complete behavioral description tells us what a system does. A theory that locates experience in causal organization must also say which parts constitute the system, how they can change, and which operations belong to its realization. Those questions survive even when every decoded input–output trace is known.

The distinction becomes concrete in a pair of five-register systems. One contains two closed recurrent organizations. A nonlocal recoding yields another implementation whose essential-dependence graph is strongly connected. Decode its states appropriately and the two systems deliver the same complete service. Their native elementary changes, however, differ. Changing one original register can change several coordinates of the recoded implementation.

For Shadow Theory’s SPC-2 constitution, this reaches directly into the question of awareness localization. Its admission rule examines recurrent organizations with executable covering return and predictive nontriviality; its further rules assign endogenous predictive organization and provenance continuation. The realization that supplies these operations matters to the assignment. A public behavior table alone leaves part of that realization unspecified.

Paper 4 advances the question by replacing supplied coordinate labels with response evidence. Can a collection of interventions reveal a system’s binary coordinate structure even when the estimator receives no register names, labeled netlist, or complete elementary-overwrite algebra? Within a precise finite response model, the answer is yes. The identifying information lies in how the full distribution of possible responses varies across contexts.

This also clarifies what a fair comparison between theories requires. Renaming every part of one realization, building a different implementation with the same service, and choosing a grain for one fixed device are three distinct operations. Treating them separately gives the comparison its force.

Start with a completely distinguished set of $N=2^n$ states. Suppose each stationary response row is strictly positive and factors in an unknown bijective binary chart $c$:

$$Q_\alpha(s)=\prod_{i=1}^{n}q_i(\alpha)^{c_i(s)}[1-q_i(\alpha)]^{1-c_i(s)},\qquad 0<q_i(\alpha)<1.$$

The context $\alpha$ can index a preparation or intervention. Neither the binary chart nor the component probabilities are supplied to the reconstruction. The chart can be an arbitrary bijection of the observed states; it need not be affine in their public labels.

Take natural logarithms and subtract the uniform mean across output states:

$$L_{\alpha s}=\log Q_\alpha(s)-\frac1N\sum_{t\in S}\log Q_\alpha(t).$$

If this centered-log matrix has rank $n$, every compatible binary chart differs only by a permutation and independent complementation of coordinates. Those remaining choices change the names and orientations of the bits, while preserving the underlying cube.

The proof supplies a reconstruction. Write the coordinate signs as $b_i(s)=(-1)^{c_i(s)}$. The centered logarithms span these sign functions. On the complete cube, a linear combination that takes only the values minus one and one must reduce to a single signed coordinate when it is itself a row of another binary chart. Full rank therefore fixes the coordinate span and rules out more elaborate recodings.

Let $P$ project onto the row space of $L$. Its entries expose the geometry:

$$P_{st}=\frac{n-2d_H(c(s),c(t))}{N}.$$

Pairs at Hamming distance one are precisely cube neighbors. Recover that graph, choose a root and order its neighbors, and graph distances reconstruct a chart. The procedure checks that the graph really is a cube rather than forcing every response family into one.

The declared class is essential. Complete binary readout and product response structure exclude alternative carriers and hidden refinements from this identification problem. A single 32-valued variable would satisfy a different factorization condition. The theorem establishes uniqueness inside the specified binary class and gives that class an explicit evidential test.

Exact identifiability becomes useful when the same organization survives a quantified error. Let $\widehat L$ be an observed centered-log matrix, and suppose the nonempty class of compatible exact product models obeys $\|\widehat L-L\|_2\le e$. If its observed $n$th singular value satisfies

$$\widehat\sigma_n>Ne,$$

every model in that class has the same chart, up to signed permutation. The projector onto the leading $n$ right singular vectors recovers adjacency through the threshold

$$s\ne t,\qquad \widehat P_{st}>\frac{n-3}{N}.$$

The reason is geometric. Projector error is bounded by $e/\widehat\sigma_n$, which lies below $1/N$. That error is smaller than half the separation between a true neighbor and the closest non-neighbor. The entire admissible uncertainty class therefore shares one cube.

Nonemptiness and a justified error bound are part of the certificate. A spectral gap cannot turn an inconsistent response model into an identified realization. Nor is full rank claimed to be the weakest possible evidence. Its value is as a clear sufficient condition with a constructive stability guarantee.

The paper also exhibits a genuine ambiguity when a direction remains unexcited. Vary two Bernoulli biases on three bits while leaving the third fair. Both $(x_0,x_1,x_2)$ and $(x_0,x_1,x_2\oplus x_0x_1)$ make every response row product. The fair bit conceals a nonlinear recoding. This example shows why variation across contexts carries information that one undifferentiated response distribution can miss.

The finite benchmark uses the original state $x=(a_0,a_1,a_2,j_0,j_1)$ and update

$$F(x)=(a_1,a_2,a_0\oplus a_1,\neg j_1,j_0).$$

The recoding

$$E(x)=(a_0\oplus j_0,a_1,a_2,j_0\oplus a_1,j_1)$$

defines a second update $G=EFE^{-1}$. Reporting its state through $E^{-1}$ preserves every complete decoded service trace. A signed-coordinate control supplies a third implementation whose local structure should agree with the original.

During the synthetic diagnostic, the service update is paused. A pulse independently flips each modeled native bit with probability $p$, producing

$$Q_c(t\mid s)=p^{d_H(c(s),c(t))}(1-p)^{n-d_H(c(s),c(t))}.$$

The generator uses its native chart to produce the responses; the estimator sees only public state labels and the response table. At $p=1/8$, exact rational response laws recover each implementation’s chart. With the remaining realization fields held to their supplied contracts, the inherited extractor then returns:

Implementation	Log-score rank	Certified connected groupings	Assigned supports
Original	5	10	2
Nonlocal recoding	5	38	1
Signed control	5	10	2

The original supports contain three and two registers, with eight and four predictive classes. The recoded support contains all five registers, with 32 classes. The diagnostic laws reveal their difference: for every public preparation, the original and recoded response rows have total-variation distance $45/256$. The signed control preserves the original pulse metric.

A finite-sample route uses the gap between distance-one and distance-two responses, $g=p(1-2p)(1-p)^{n-2}$. For five bits at $p=1/8$, 24,804 trials per preparation—793,728 trials per synthetic data set—give a conservative failure bound of 0.00999898. All 32 fixed-seed simulations recovered their anonymous charts. Exact graph checks also covered all 40,320 labelings of the three-cube.

This demonstrates the inference chain from responses to chart to conditional assignment. A physical implementation must additionally establish that its perturbations probe the relevant storage organization, with characterized preparation, timing and readout. The present data are synthetic; the assignment also retains its separate assumptions about resources, records and provenance.

The comparison becomes more demanding when stochastic responses are introduced. Exactly transporting one process through a recoding and adding independent noise separately in two native charts are different constructions. A legitimate common process must satisfy

$$K^G_\epsilon(Ey\mid Ex)=K^F_\epsilon(y\mid x).$$

If both native descriptions require product response rows, they must also preserve conditional independence. Replacing a correlated transported row by the product of its marginals would alter the process being compared.

The identification theorem immediately constrains this possibility. A strictly positive common family with full centered-log rank can be product in both charts only when their correspondence is a signed permutation. Paper 4 then classifies the exceptional affine cases, including deterministic coordinates and fair bits, where common product laws remain possible.

For $Z=AX\oplus b$, let $S$ index the genuinely random input coordinates and let $J$ collect the output coordinates touched by them. Independence requires $|J|=|S|$. Within that surviving face, each biased input must correspond to a weight-one row of the inverse transformation. Fair coordinates are the special directions that can absorb otherwise visible mixing.

For the benchmark’s recoding, near-deterministic independent errors remain product after transport precisely when $p_1=p_3=0$. Full-support compatibility instead requires $p_0=p_3=1/2$, preventing convergence to the deterministic endpoint. Consequently, no fully supported common native product-kernel family approaches the frozen deterministic pair. Sparse common families do exist—for example $(p,0,p,0,p)$.

Exact tests cover 3,125 parameter choices, of which 725 are compatible. With equal independent error rates $p=1/8$, two transported outputs have covariance $21/256$; the official PyPhi constructor rejects that exact joint law as conditionally dependent. It accepts the compatible sparse and full-support examples. The distinction is therefore expressed both as an algebraic result and as an executable admission test.

Recovering a binary microchart and selecting an intrinsic macrograin solve different problems. The response method can supply candidate microstructure for SPC-2, IIT or another constitutive theory. IIT’s intrinsic-unit rules then evaluate admissible organizations on a supplied substrate. Paper 4 asks whether the inherited formal comparison survives an explicit extension beyond single native units.

The official PyPhi search admits macro units with at most two micro constituents, one update per unit and one hierarchy level. It examines every nonconstant binary pair map modulo output complementation, with the official assembly, admissibility and exclusion rules. Five native units plus seven maps on each pair footprint yield 75 raw unit candidates before validity filtering.

All 32 matched states are evaluated for the original, recoded and signed-control systems under both complete presets: 192 primary cells. Under the 2023 preset, 19 units survive validity per cell. Under the 2026 preset, the constituent-positivity gate leaves the five native units. Across this entire declared search domain, no positive macro selection replaces the inherited native selections:

Preset	Original implementation	Recoded implementation
IIT 2023	Supports {a₁, a₂} and {j₀, j₁}; each has system integration 2	All five native units; system integration 2
IIT 2026	No positive complexes	No positive complexes

The completed run contains 17,800 evaluation records. Every matched state–preset comparison with the signed control agrees on the retained unit domain, evaluations, positive selections and ties. The search also preserves zero-valued objects returned by the software separately from positive complexes; counting the raw collection would misstate the scientific result.

The original SPC-2 three-register support and IIT 2023’s selected two-register support show that the comparison concerns the composition of organizations as well as their number. Its reach is the specified pair-macro domain. Larger spatial grains, temporal grains and changed background treatment define further comparison problems. The 2026 deterministic gate follows its own integration rules and provides no universal conclusion about every deterministic description at every grain.

The consciousness monograph states the philosophical and constitutive position. The research sequence makes the operational questions increasingly explicit. Paper 2 investigates robustness, composition and what observational equivalence can identify. Paper 3 tests the production and selection of learned effective interfaces. Paper 4 reaches beneath that interface to ask what intervention evidence can establish about a realization’s coordinate structure.

The resulting architecture keeps three achievements distinct and connected: identifying service behavior, identifying a decomposition within a response class, and applying a constitution to an established realization. Each has its own evidence. This makes the framework more concrete because the unresolved physical questions acquire explicit contracts and measurable conditions.

The mathematics builds on identifiable-component methods, finite-alphabet independent-component analysis and causal abstraction. Its contribution here is the finite cube reconstruction, the quantitative projector certificate, the recoding-specific stochastic classification, and the executed bounded comparison. The proofs and reported computations remain available in the complete chapters below, together with the attribution, verification record and research disclosures.

The next physical step is clear: characterize a real diagnostic intervention independently, estimate its context-dependent response laws, justify the model and error bound, and reconstruct the chart with reference labels withheld. Establishing phenomenal attribution is a further question requiring experience-facing evidence. The present result provides a constructive account of the structural evidence a constitutive framework can use.

Read Paper 4 on Zenodo, or continue through the complete technical account on this site.