The MIND: Five Finite Constructions — P2 mathematical addendum

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The MIND: Five Finite Constructions
First research edition, with P1 and P2 mathematical addenda
Research prepared for Brian Theory
AI-assisted mathematical synthesis; not external peer review
5 October 2026
Abstract
We develop five bounded constructions concerning observation, resolution, factorization,
distributed consistency, and actuator calibration. Their common question is which distinctions
suffice for a specified task. Exact derivations and finite computations establish properties of the
models. They do not establish a marker of phenomenal consciousness. The constructions deliber-
ately retain counterexamples to attractive but invalid bridges. The accompanying source register
records reading depth, and the local baseline records manuscript versions and hypotheses. No
claim of mathematical priority is made. The P1 addendum gives an exact predictive-acquisition
criterion for the bounded RF poset class, with minimal and transcript-level obstructions. The
P2 addendum characterizes the exact mutual-information interval for binary supports with fixed
marginals and a positive cell floor, while preserving causal and recoding distinctions.
Contents
1 Contracts before interpretation 2
2 M1: distinctions, quotients and predictive closure 3
2.1 Model specification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 Minimal dynamic completion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.3 Symmetrization is not identification . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.4 Test and abandonment rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3 M2: equal group index, unequal information gain 4
3.1 Model specification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3.2 A decision-sensitive extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
4 M3: three obstructions to identifying structure with causal information 5
4.1 Model specification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
4.2 Support does not determine dependence . . . . . . . . . . . . . . . . . . . . . . . . . 5
4.3 Effective information and intervention redistribution . . . . . . . . . . . . . . . . . . 6
4.4 Information interactions depend on source coordinates . . . . . . . . . . . . . . . . . 6
4.5 Limits and programme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
5 M4: local consistency without a global assignment 6
5.1 Model specification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
5.2 Noise threshold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
5.3 Workspace interpretation and its failure test . . . . . . . . . . . . . . . . . . . . . . . 7
1

6 M5: calibration, memory and the limits of a self-model interpretation 7
6.1 States, observations and control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
6.2 Exact performance and ablations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
6.3 What is self-related, and what is not? . . . . . . . . . . . . . . . . . . . . . . . . . . 8
7 Additional negative results and review targets 9
7.1 A directed-order distance need not be a metric . . . . . . . . . . . . . . . . . . . . . 9
7.2 Plasticity is not automatically observable change . . . . . . . . . . . . . . . . . . . . 9
8 P1: predictive refinement under a repair-safe library 9
8.1 Fixed-state acquisition contract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
8.2 Compatibility with deterministic prediction . . . . . . . . . . . . . . . . . . . . . . . 10
8.3 P1-C1: a minimal supplied-but-unsafe pullback . . . . . . . . . . . . . . . . . . . . . 11
8.4 P1-C2: exact transcripts need not have a least refinement . . . . . . . . . . . . . . . 11
8.5 Checked scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
9 P2: attainable information over a binary logical support 12
9.1 Declared constraints and prior construction . . . . . . . . . . . . . . . . . . . . . . . 12
9.2 Sharp continuum bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
9.3 Finite grids and a quantitative rounding bound . . . . . . . . . . . . . . . . . . . . . 13
9.4 An identical observational law with different interventions . . . . . . . . . . . . . . . 14
9.5 Admissible recoding and verification limits . . . . . . . . . . . . . . . . . . . . . . . . 14
10 Open theorem and experiment targets 15
1 Contracts before interpretation
The local baseline identifiers B01–B09 refer to sections of 00-context/baseline.md; canonical
versions are in 00-context/local-versions.json. Source and idea identifiers are stable across
the library. M1–M5 are constructions, not additional idea counts. The executable companion is
experiments/finite models.py; it uses only the Python standard library. Its outputs are in
experiments/results.json. There is no Monte Carlo seed because all examples use finite enu-
meration or finite sums. Decimal entropies use floating point; counting and calibration probabilities
use exact integers or rational numbers where applicable.
Type Carrier and composition Required additional semantics
CM / LM 2 ×2 truth tables or formula en-
tries; native XOR–AND prod-
uct
Operand order, polarity and matched
frames; no probability law is implicit.
Support relation Subsets or 0–1 matrices with
OR–AND relational composi-
tion
This is a different semiring from native
CM composition.
Stochastic channel Nonnegative matrices with row
sums one; ordinary real product
Distributions, observation protocol
and interventions must be supplied.
Fuzzy relation Values in [0 , 1], selected t-norm
and aggregation
Membership is not automatically prob-
ability; thresholding need not preserve
composition.
2

Modal chain ring Nonzero vectors and explicitly
allowed effects over a specified
ring
Possible/impossible outcomes, primi-
tive effects and zero divisors matter;
no Born rule follows.
Hilbert model Complex vectors or density op-
erators and quantum channels
Physical realization, measurement and
decoherence are additional commit-
ments.
An interpretation associates a formal variable with a measured capacity. An exact map with-
in mathematics is not that interpretation. We distinguish computational tasks, representational
mechanisms, physical implementation, and claims about experience. All five models operate at the
first two levels.
2 M1: distinctions, quotients and predictive closure
2.1 Model specification
Let W be a finite latent carrier and Φ = ( ϕi)i∈I a family of Boolean tests. The observation is
q(x) = ( ϕi(x))i∈I, with realized carrier Q = im q. An available action may acquire one new test
ψ at a declared cost; latent deterministic dynamics is T : W → W . One transition is one time
unit; a test-acquisition step is a separate operation. The boundary is the analyst’s chosen W , and
the agent only receives requested test outcomes. The present model is noiseless; noisy observations
require a channel and do not inherit exact fiber equality as an empirical fact.
The positive topology is generated by ϕ−1
i (1). On realized signatures, the upward-open special-
ization order is coordinatewise comparison. If both positive and negative truth regions are included,
the result is the partition topology of signature fibers. Thus observational equality and positive
order retain different data (B02).
For W = {00, 01, 10, 11}, observing the first coordinate gives three positive opens. Observing
both coordinates gives six positive opens, whereas observing both signs of both coordinates gives
sixteen. The complete signature map is injective in the second case, although {00} is not positively
open. This example prohibits reading positive topology as a complete yes/no partition.
Proposition 1 (Exact deterministic descent) . For a surjection q : W → Q there exists a unique
¯T : Q → Q satisfying qT = ¯T q if and only if
q(x) = q(y) = ⇒ q(T x) = q(T y).
Proof. Necessity follows by applying the commuting equation to x, y. For sufficiency define
¯T (q(x)) = q(T x). The hypothesis makes this independent of the representative; surjectivity gives
existence on every element and uniqueness.
For q(a, b) = a and T (a, b) = ( b, a), states 00 , 01 are currently indistinguishable but their
successors have different first bits. The coarse state is therefore not a sufficient predictive state.
Enumeration of all 44 = 256 transitions finds exactly 64 that descend to this two-cell partition; this
verifies only the count for that carrier. The proof supplies the general claim.
2.2 Minimal dynamic completion
For a finite set of actions with transition mapsTa, successively refine the current equivalence relation
using the signatures
σk+1(x) =

q(x), (σk(Tax))a


, σ 0 = q.
3

Interpreting signatures by their equality classes, this process stabilizes after at most|W |−| Q| strict-
ly splitting rounds. At stabilization each action descends. Induction shows that two states remain
equivalent exactly when they give identical q observations along every finite action word. This is a
standard automata/bisimulation construction, proposed as a bridge to B03, not a new conscious-
ness theorem. If the supplied observation library cannot realize the required splits, the abstract
refinement is not implementable by the agent. An RF-IDENT-safe implementation additionally
requires the precise order and query contract in B06.
2.3 Symmetrization is not identification
Let R = {(0, 1), (1, 2)}. Adding converse pairs makes R symmetric but not transitive: (0 , 2)
is absent. Reflexive-transitive closure then identifies all three points in one equivalence class.
Symmetrization preserves undirected adjacency; equivalence closure forgets graph distance and all
within-component distinctions. This is a candidate reconstruction of one operation used in bi-logic,
not a derivation of Matte-Blanco’s psychology (see logic dossier). The separate stages must remain
visible.
2.4 Test and abandonment rule
Compare dynamic completion against the original partition, the fully resolved state and a finite-
history predictor. Measure predictive error and observation cost under held-out action sequences.
A new B-Theory contribution requires a useful constraint or invariant beyond standard partition
refinement. Abandon a claimed exact quotient when descent fails; retain it only as an explicitly
approximate or history-dependent model.
3 M2: equal group index, unequal information gain
3.1 Model specification
The four states are leaves of a rooted binary tree at depth two. Distances are d(x, x) = 0, 1 /2
for distinct siblings and 1 between root subtrees. The old observation is the first bit. Either
action resolves the left or right old cell into its two leaves. States are static during acquisition, the
boundary is the four-leaf tree, and one action costs one declared probe unit. A probability measure
µ describes sampled states; it is extra structure beyond the tree.
The full tree automorphism group has order eight. Its pointwise stabilizer of the two old cells
has order four; resolving either cell leaves a stabilizer of order two. Both refinements therefore have
index two and log index one. These are stabilizers inside the full combinatorial tree group. If one
instead restricts to measure-preserving automorphisms for nonuniform µ, the group changes and
the following group counts cannot simply be reused.
Proposition 2 (Local entropy increment) . For a partition refined by splitting a cell C into
C1, . . . , Cr,
Hµ(P ′) − Hµ(P) = µ(C)H
 µ(C1)
µ(C) , . . . , µ(Cr)
µ(C)

,
with zero contribution from a null cell.
Proof. Expand −P
A µ(A) log2 µ(A) in both partitions; unchanged cells cancel. Substitute µ(Ci) =
µ(C)pi in the remaining sum.
4

For leaf masses ( .7, .1, .1, .1) the left increment is .8h2(.875) = .4348515546 bits; the right
increment is .2 bits. For the uniform measure both increments are .5 bits, still distinct from the
one-bit log index. B05’s group and entropy quantities answer different questions.
3.2 A decision-sensitive extension
For target variableY and loss L, choose an acquisition action to reduce Bayes risk, not automatically
to maximize H(X). If only the right-cell distinction changes the optimal response to Y , the smaller
entropy increment can have greater task value. The refinement preserves old outcomes; changing
the cost or the target can reverse its desirability. This is compatible with Blackwell comparison of
experiments, which assumes a specified decision problem and does not erase acquisition costs.
Exact labels are a noise-free idealization. Empirically estimate a channel P (z | x, a), compare
held-out risk to a constant or random probe, and check sensitivity to tree encoding, nonuniform
sampling and biased missing data. Abandon the symmetry-index-as-information proposal; retain
separate symmetry, entropy and utility coordinates. A resolution hierarchy alone supplies no latent
time, unconscious process, or spontaneous symmetry-breaking mechanism.
4 M3: three obstructions to identifying structure with causal in-
formation
4.1 Model specification
Here states are binary variable tuples or four microstates. Observations are support, joint fre-
quencies, or interventions, explicitly distinguished below. Actions either change an intervention
distribution or select a coarse-graining. Channels describe one-step transitions. There is no learn-
ing update. The boundary and variable factorization are analyst choices, and all distributions are
population-level specifications, not estimated data.
4.2 Support does not determine dependence
Both distributions
P0 =
1/4 1 /4
1/4 1 /4

, P 1 =
.4 .1
.1 .4

have the all-one Boolean support, which factors as a Cartesian product and has OR–AND Boolean
rank one. Yet IP0(X; Y ) = 0 and IP1(X; Y ) = 1 − h2(.2) = .2780719051 bits. Consequently no
support-only function can recover mutual information for all compatible positive distributions. In
particular a Boolean factorization defect is not a probability dependence measure. B07 may expose
useful support constraints, but a stochastic lift must be separately specified.
Even a complete observational law is insufficient for causal direction. With U ∼ Bernoulli(1/2),
models X = U, Y = X and X = U, Y = U have the same distribution X = Y . Under do( X = 1)
they give respectively P(Y = 1) = 1 and 1 /2. The distinction is between structural equations and
conditioning, not between two matrix notations.
5

4.3 Effective information and intervention redistribution
Use the four-state channel
K =


0 0 0 1
0 0 0 1
0 0 0 1
1/3 1 /3 1 /3 0

 .
The macro partition {0, 1, 2} | { 3} yields deterministic alternation and is strongly lumpable. Under
a uniform intervention over microstates, I(X; X ′) = h2(1/4) = .8112781245 bits. Under a new
uniform macro intervention, macro mutual information is one bit. Under the macro input distri-
bution induced by uniform micro interventions, it is again .8112781245 bits. The apparent gain
compares different intervention ensembles; it violates no data-processing inequality. This finite
example illustrates a known causal-emergence distinction [3]. It is not a calculation of IIT’s Φ.
4.4 Information interactions depend on source coordinates
Let A, B be independent fair bits and Y = A ⊕ B. Then I(Y ; A) = I(Y ; B) = 0 but I(Y ; A, B) = 1.
Under the invertible recoding ( C, D) = (A ⊕ B, B), I(Y ; C) = 1. A variable-wise synergy assertion
therefore requires a justified factorization into sources; global invertibility does not preserve it.
This does not select a unique partial-information decomposition. Physical locality or independently
intervenable variables could justify a factorization, but a convenient Boolean encoding alone cannot.
4.5 Limits and programme
A support-to-stochastic lift can be studied as a family of all compatible distributions, reporting a
range of information measures instead of a single score. With small samples, an unobserved event
is not a proven structural zero. Matched tests should vary sampling priors, causal models, coarse-
graining and invertible recoding. The theorem target is a characterization of what is identifiable
under the stated data; the abandonment rule rejects causal or phenomenal claims when equivalent
observational models disagree.
5 M4: local consistency without a global assignment
5.1 Model specification
Variables A, B, C take values in F2. The three observation contexts are AB, BC, CA. At one time
slice each context reports a relation
A + B = b1, B + C = b2, C + A = b3.
The event sheaf contains all assignments, with ordinary coordinate restriction. The allowed-support
subpresheaf imposes these relations. Each pair’s support projects surjectively onto the shared single
variable. Equipping each pair with the uniform distribution on its two allowed assignments also
gives matching overlap marginals. Actions may flip a constraint or transmit a selected assignment;
these are different interventions. No neural timing or workspace access mechanism is specified by
this static consistency test.
Proposition 3 (Parity obstruction). A global assignment exists if and only if b1 + b2 + b3 = 0 in
F2. When it exists there are exactly two assignments.
6

Proof. Summing all equations cancels each variable twice, proving necessity. Given A, set B =
A + b1 and C = B + b2; the third equation holds exactly when the parity condition holds. The two
choices of A give the two solutions.
For (b1, b2, b3) = (1 , 1, 1) no global section exists, although pairwise projected supports and
marginals agree. This does not contradict sheaf gluing: we have not chosen mutually compatible
individual sections on the contexts. The full event sheaf glues such sections; the issue is extending
these allowed sets or marginal distributions [4].
The cover nerve is a triangle with no filled face. It has β0 = β1 = 1 for both even and odd parity.
Its ordinary homology cannot distinguish the two cases. One needs the restriction/constraint data,
not just nerve shape. Order-complex homology in B03–B04 is another construction again. A bridge
must provide a functor or an explicit comparison, not rename the invariant.
5.2 Noise threshold
Suppose each edge is anti-correlated with probability 1 − η, has equal single-variable marginals,
and the two satisfying and two violating outcomes are each equiprobable within their class. Every
deterministic assignment has at least one equal edge, so a joint distribution requires 3 η ≥ 1.
Conversely at η = 1 /3, the uniform distribution on the six nonconstant assignments has these
marginals. At η = 1, uniform 000 , 111 does too. Their convex mixture with weight (3 η − 1)/2 on
the latter realizes every η ∈ [1/3, 1]. Thus the threshold is exact for this symmetric family. Finite
samples require confidence bounds on all three edge errors before claiming violation; measured
context changes may instead break the common variable identification. This toy is not evidence of
physical quantum contextuality in a brain.
5.3 Workspace interpretation and its failure test
A proposed workspace may broadcast an assignment and ask modules to accept or reject it. Added
communication can reveal disagreement; it cannot make the odd parity constraints jointly true with-
out revising a constraint or changing the task. Compare a centralized constraint solver, distributed
message passing and naive majority voting at matched bandwidth and latency. Test solution accu-
racy and contradiction detection. Neither global consistency nor a nonzero obstruction is claimed
sufficient or necessary for experience.
6 M5: calibration, memory and the limits of a self-model inter-
pretation
6.1 States, observations and control
An actuator has an unknown fixed polarity θ ∈ { 0, 1}, initially fair. Calibration actions aj are
independent fair bits known to the full agent. The measured result is
yj = aj ⊕ θ ⊕ ϵj, ϵ j
iid∼ Bernoulli(η).
The agent stores zj = yj ⊕ aj, and estimates θ by majority vote with random fair tie-breaking.
This is a maximum-likelihood/MAP rule for known η < 1/2 and a fair prior. Calibration lasts
n fast observation steps; θ is constant over calibration and the following execution block. The
chosen boundary includes the actuator and its action/sensor loop; an external object with the
same input-output law would admit the same inference.
7

During execution, a target gt is supplied, the agent chooses at = gt ⊕ ˆθ, and the output is
ot = at ⊕ θ ⊕ νt, with independent execution noise νt ∼ Bernoulli(µ). The agent is not allowed
to learn from execution feedback in this experiment. Relaxing that restriction defines a different
controller. Forward-model motor control motivates this functional construction [5], but does not
establish its conscious interpretation.
6.2 Exact performance and ablations
The probability of correct calibration is
An(η) =
nX
k=0
n
k

ηk(1 − η)n−k
12k<n + 1
212k=n


, A 0 = 1/2.
At η = .2, the values for n = 0, 1, 3, 5 are .5, .8, .896, .94208. With utility Un = An − .03n, the best
member of this four-option menu is n = 3. Utility is accuracy minus probe cost, not survival or
thermodynamic free energy. No global optimal-stopping claim is made.
Probes An Un True survival IID shortcut
0 .50000 .50000 .49271000 .34375000
1 .80000 .77000 .78757400 .81443540
3 .89600 .80600 .88193048 .92066609
5 .94208 .79208 .92722159 .95564458
Survival here is an operationally defined event: at most two output failures during six executions,
with µ = .1. It is not demonstrated autopoiesis. Conditional on a correct estimate, failures are
Bin(6, µ); conditional on an incorrect estimate they are Bin(6 , 1 − µ). Hence
P(survive) = AnB6,µ(2) + (1 − An)B6,1−µ(2),
where BT,p(b) = Pb
k=0
T
k


pk(1 − p)T −k. Replacing failures by independent trials at the averaged
error rate loses their common dependence on ˆθ and gives the erroneous last column.
A no-memory decoder retaining only the final calibration sample has accuracy A1, even if it
paid for three probes. An action-blind decoder sees only y1, . . . , yn. Conditional on either θ, that
vector is uniform, so its best accuracy remains 1 /2. Replacing its action trace by independent
fair bits has the same consequence; arbitrary within-episode permutation is not asserted to remove
every information trace. A trivial fixed prior estimate has accuracy 1 /2; an oracle has calibration
accuracy one and still suffers execution noise. Comparisons require identical measurement and cost
budgets; a one-probe policy and a three-probe memory ablation have different costs despite equal
final calibration accuracy.
6.3 What is self-related, and what is not?
The estimated parameter controls the consequences of the agent’s own actions, which gives an
operational body-parameter relation. There is no second-order confidence report, introspection,
narrative identity, or phenomenal variable. The sufficient memory can be an integer vote count.
A compiled finite-state controller that stores that count and maps it directly to actions has ex-
actly the same performance without a separately named self-model module. Thus the ablations
establish contributions of memory and action-conditioned data, not the necessity of an explicit
self-representation architecture.
8

An informative next test changes θ after calibration and compares recalibration, change-point
detection, a history-matched lookup controller, and an externally observing estimator. Report
target success, proper scoring rule loss, calibration cost and recovery time separately. The model
should be abandoned as a claim of privileged self-awareness if those matched alternatives explain
its entire advantage. It remains useful as an exact test of adaptive control and observational access.
7 Additional negative results and review targets
7.1 A directed-order distance need not be a metric
R003 [6] proposes, in its arXiv v1 printed p. 15, footnote 15, zero on the diagonal, shortest directed
order-path length between comparable points, and infinity otherwise. For the strict order x <
z, y < z, x ∥ y, this gives
d(x, y) = ∞ > d(x, z) + d(z, y) = 2.
Thus the formula fails the triangle inequality on general strict partial orders, and is not natural-
number-valued there. This is a counterexample to that formula under its stated generality, not a
rejection of the paper’s phenomenological programme. An undirected graph path metric on each
connected component repairs the triangle inequality but changes the semantics; it can connect
incomparable points. A total-order restriction is another, narrower route and requires explicit
assumptions about allowed edges.
7.2 Plasticity is not automatically observable change
R001 [7] motivates an objection to static substitutes. The main-text inference from nontrivial weight
change to necessarily changed input-output function requires qualifications: ft(x) = wtx − wtx = 0
has changing parameters and a constant function. Likewise, finite-horizon adaptive dynamics can
be written on an augmented state (ht, Wt) and composed for a fixed number of steps. This does not
imply cheap simulation, equal physical intervention responses, or a fixed-window solution for every
horizon. Compare input histories, memory, parameter access, resource budgets and admissible
perturbations explicitly. The article’s supplementary proofs were not successfully retrieved in this
edition, so no whole-paper refutation is claimed.
8 P1: predictive refinement under a repair-safe library
This addendum develops IL002/IL005 and M1. Its authoritative report, full proofs, provenance,
independent reviews and bounded checks are in p1/README.md. It extends the local RF target
from complete identification to reported classifications. The inherited RF and RFM sources are
the canonical records listed under B06, not a theorem for arbitrary height-two posets. Neither
novelty nor empirical relevance is established here.
8.1 Fixed-state acquisition contract
Let P = {a} + M + {z}, with distinct least and greatest elements and an antichain M. Fix an
explicitly supplied finite library Q of oriented, noiseless Boolean queries. The unknown x ∈ P
remains fixed during acquisition. Background signatures define the post- T0 order but are not freely
observed (A1). Complements are separate supplied queries. There is no cross-branch consumption,
changing library, extra history-dependent cost or state-changing acquisition.
9

At support S, a query q retains v ≤ w exactly when q(v) ≤ q(w). Legality requires the resulting
order-complex inclusion to be a homotopy equivalence, before conditioning. Branch cancellation
leaves the induced initial order. Set Qs = {q : q(a) ≤ q(z)}. The inherited local rule is
safe(S, q) ⇐ ⇒



q(a) ≤ q(z), a, z ∈ S,
q(a) ≤ q(v) for all v ∈ S, a ∈ S, z /∈ S,
q(v) ≤ q(z) for all v ∈ S, z ∈ S, a /∈ S,
true, a, z / ∈ S.
With both endpoints, 00 and 11 permit elementary edge–triangle collapses, 01 deletes nothing, and
10 has nonzero relative H1. On an endpoint star a descending edge disconnects the complex. A
useful legal query on an endpoint-containing support therefore belongs to Qs.
Proposition 4 (P1-T1: target classification) . For a finite-valued target c : P → C, a legal tree
reports c(x) exactly if and only if Q separates differently labelled middle pairs and Qs separates
every differently labelled pair involving an endpoint.
Proof. For necessity, follow each pair’s shared history to its first separator. An endpoint pair re-
quires a useful legal separator in Qs. For sufficiency, split Qs into equal-endpoint tests E and
ascending-endpoint tests A. Peel informative E tests on the branch retaining both endpoints. De-
tached branches are antichains, classifiable using Q. The remaining core agrees with the endpoints
on every E test. If A is empty, the endpoint separation condition makes the core homogeneous.
Otherwise stop if homogeneous, or ask any A test to produce a lower and an upper star. A remain-
ing leaf with a different label from its endpoint has an A separator: an E separator is excluded
by core membership. This is legal throughout the respective star. Peel such leaves into antichain
branches, classify those, and stop when each endpoint branch is homogeneous.
Direct scans give O(n2 + mn2) decision and construction time, including order validation, and
at most n − 1 internal nodes. Depth optimality does not follow.
Proposition 5 (P1-T2: finest attainable transcript) . Partition P by Qs signatures. Keep each
class containing an endpoint intact and refine every other class by full Q signatures. Let R be the
resulting equivalence relation. Then c is legally reportable exactly when R ⊆ ker c. Some legal tree
realizes exactly R as its leaf partition.
Proof. R is the equivalence closure of the unavoidable pairs in P1-T1. Apply that proposition to
the R-class label. Correct classification separates distinct R classes; no legal tree separates states
within one. Its leaves are therefore precisely those classes.
8.2 Compatibility with deterministic prediction
Separately supply an output b : P → O and known deterministic maps Tu : P → P . These define
counterfactual predictions; they are not executed or freely observed during acquisition. M1 gives
E0 = ker b, E k+1 = Ek ∩
\
u
(Tu × Tu)−1(Ek).
The stable E∗ is the largest transition congruence inside ker b, with at most n − | b(P )| strict
refinements.
Proposition 6 (P1-T3: predictive acquisition) . A legal tree reports the E∗ class if and only if
R ⊆ E∗. This is equivalent to the existence of any legally reportable transition congruence F ⊆ ker b.
10

Proof. P1-T2 proves the first equivalence. Any such F satisfies R ⊆ F ⊆ E∗ by reportability and
maximality. Conversely E∗ itself supplies F when the inclusion holds.
If the coarsest predictive partition is blocked, every finer predictive partition is blocked. A
witness is an R-equivalent pair separated by E∗, with a finite action word giving unequal b outputs.
If compatible, acquisition may retain extra transcript information and discard it in the reported
class. No quotient-order or topological property of this compression is asserted.
There is a dual closure:
G0 = R, G k+1 = Eq
 
Gk ∪
[
u
(Tu × Tu)(Gk)
!
.
Its limit G is the least transition congruence containing R, since every containing congruence
contains every iterate. Compatibility is equivalently G ⊆ ker b. Reportable predictive congruences
are then exactly the transition congruences F with G ⊆ F ⊆ E∗: G is the finest and E∗ the coarsest
reported partition.
8.3 P1-C1: a minimal supplied-but-unsafe pullback
On a < x < z , take b = 010, Q = {010, 110} and T (a) = x, T (x) = x, T (z) = z. This T is
order preserving. The base partition is {a, z} | { x}, but bT = 110 separates a, z, so E∗ is discrete
after one round. The pullback is already supplied. Nevertheless R = {a, z} | { x}: 110 is unsafe
initially and remains the descending test 10 on the zero branch {a, z} of 010. Ordinary acquisition
has depth two; RF acquisition is impossible. Replacing 110 with the actually supplied complement
001 repairs it.
Three states are minimal for strict deterministic predictive refinement: on at most two states a
nonconstant base is injective, while a constant base stays constant after every word. Two queries
are minimal when the nonconstant base is safely obtainable and ordinary acquisition obtains a
strict refinement: one query cannot split its own fibres. These are conditional minimality claims.
Passive histories (b(x), b(T x)) are 01, 11 and 00 and identify the initial state after one transition.
This is not RF acquisition: the zero branch becomes {x, z}, where the second query is unsafe. For
the timing control T = ( a, z, x) and Q = {010, 100}, locally legal two-time sampling is possible
under an explicitly added image-support update rule. Such state-changing protocols lie outside the
fixed-state theorem and do not forecast before the later sample.
8.4 P1-C2: exact transcripts need not have a least refinement
Let P = {a} + {x, y} + {z}, with identity dynamics and target B = {a, x, z} | { y}, and supply
0001, 0011, 0111, 0101. A legal tree using 0001 then 0011 has leaves P1 = {a, x} | { y} | { z}. One
using 0111 then 0101 has leaves P2 = {a} | { y} | { x, z}. Both report B. Every root query splits
{a, x, z}, so no exact leaf partition is B. The only common coarsening of P1, P2 that still refines
B is B, which is unavailable. Thus no coarsest attainable exact-leaf refinement exists, although
the coarsest predictive reported partition does. This distinguishes two meanings of implementable
refinement.
8.5 Checked scope
The coordinator checks 964 cases across nine named libraries: 75 target-classification cases, 888
predictive cases, and one exact-leaf audit. A separate verifier adds 188 checks, including simplicial
11

collapse and relative-homology certificates. All assertions pass. Caps are eight states, eight queries,
twelve libraries and 4096 combined cases; exhaustive transition enumeration stops at four states.
No all-library search, general-poset classification, transition learning, noise model or empirical test
is claimed. The general statements depend on proofs, not extrapolation from these checks.
9 P2: attainable information over a binary logical support
9.1 Declared constraints and prior construction
This addendum develops IF001, IR002 and M3. The variables X, Y ∈ {0, 1} have named interven-
tion ports and a declared two-variable partition. Fix a = P(X = 1), b = P(Y = 1) in [0 , 1], exact
support S ⊆ {0, 1}2, and ϵ > 0. Feasible laws have zero mass outside S and mass at least ϵ on each
allowed cell. The floor is an additional weight constraint, not a consequence of logical support. All
information is in bits, at the population level; no estimation or temporal mechanism is claimed.
Real probability arithmetic remains distinct from the local CM/LM algebra over F2.
The Bernoulli coupling representation and Fr´ echet bounds are established results [2]. At fixed
marginals, maximizing mutual information is minimizing joint entropy on a transportation polytope
[1]. The floor-clipped binary characterization below is an elementary derived consequence, not a
priority claim. The narrow search and inspected passages are recorded under P2S001–P2S002 in
the source register and in p2/prior-art.md.
Every joint law with these marginals has the form
Pt =
1 − a − b + t b − t
a − t t

, J (t) = IPt(X; Y ) = h2(a) + h2(b) − H(Pt).
Here t = P(X = 1, Y = 1), rows index X, columns index Y , and 0 log2 0 = 0. The entropy formula
also handles degenerate marginals.
9.2 Sharp continuum bounds
Proposition 7 (Full-support characterization). For S = {0, 1}2, put
L = max{ϵ, a + b − 1 + ϵ}, U = min{a − ϵ, b − ϵ}.
The family is empty if and only if L > U . Otherwise, with t0 = min {U, max{L, ab}}, its exact
attainable-information set is
[J(t0), max{J(L), J(U)}].
Both extrema and every value between them are attained. The minimum law is unique. If L < U ,
all maximum laws are endpoints; both endpoints can tie.
Proof. The four cell inequalities give exactly L ≤ t ≤ U. On a feasible interval all cells are positive
and
J ′(t) = log2
t(1 − a − b + t)
(a − t)(b − t) ,
J ′′(t) = 1
ln 2
1
t + 1
1 − a − b + t + 1
a − t + 1
b − t

> 0.
The numerator minus denominator in the odds ratio is t − ab. Thus J decreases below ab and
increases above it on the positive-cell domain, giving the clamped minimizer. Strict convexity gives
the endpoint maximum. Continuity and the intermediate-value theorem give the entire closed
image interval, including the case L = U.
12

Independence is feasible exactly when the four product cells ab, a(1 − b), (1 − a)b, (1 − a)(1 − b)
are all at least ϵ. For a nonempty feasible family, otherwise the minimum is strictly positive, despite
rectangular support. For a = b = 1/2 and 0 < ϵ ≤ 1/4, the information interval is [0 , 1 − h2(2ϵ)].
Proposition 8 (Proper-support characterization). For a proper support, every forbidden cell fixes
t: the cells (1, 1), (1, 0), (0, 1), (0, 0) respectively require t = 0 , a, b, a + b − 1. The family is feasi-
ble if and only if all these requirements agree and the resulting table satisfies every support/floor
constraint. When feasible, its attainable-information set is the singleton {J(t)}.
Proof. There is at least one forbidden cell; its affine coefficient of t is 1 or −1. It fixes the only free
parameter. Checking the remaining constraints is necessary and sufficient. Normalization is built
into Pt.
This includes degenerate marginals; empty support is infeasible. It does not identify weights
from support alone when marginals are unknown. For S = {(0, 0), (0, 1), (1, 0)}, a = b = 1/3 and
0 < ϵ ≤ 1/3, the unique table has three equal masses and I = 2h2(1/3) − log2 3 ≈ 0.2516291674
bits.
If full support is constrained only by strict positivity and fixed 0 < a, b < 1, the parameter
interval is open, with endpoints max {0, a + b − 1} and min{a, b}. Its information image is [0 , M),
where M is the larger endpoint information on the closed nonnegative-cell completion. Strict
convexity excludes an interior maximizer. Thus a supremum must not be reported as attained
within exact full support. The positive floor changes this family and makes it compact.
9.3 Finite grids and a quantitative rounding bound
Proposition 9 (Aligned rational grids) . For full support, let N be a positive integer, sup-
pose N a, N b, N ϵ are integers, and assume L ≤ U. The feasible grid points are t = k/N,
k = N L, . . . , N U. The grid maximum equals the continuous maximum. Its minimum is the
smaller of J(t−), J(t+), where
t− = min{U, max{L, ⌊N ab⌋/N}}, t + = min{U, max{L, ⌈N ab⌉/N}}.
Writing JN,min for that minimum,
0 ≤ JN,min − J(t0) ≤ 1
2ϵ ln 2N 2 .
If t0 is an endpoint, the gap is zero.
Proof. Alignment places both endpoints on the grid. Monotonicity on either side of ab gives the
two-candidate minimum rule. When t0 = ab is interior, choose a nearest grid point, at distance at
most 1/(2N). The second derivative is at most 4 /(ϵ ln 2) throughout the segment, and J ′(ab) = 0,
so Taylor’s theorem gives the bound. The minimum over the grid can only be smaller than this
candidate value. Endpoint minimizers are already on the grid.
The nearest parameter need not minimize asymmetric J; evaluate both candidates. The finite
set of grid values is not the whole interval between its extrema. If N a or N b is nonintegral the grid
family is empty. If only the floor is unaligned, the feasible grid endpoints round inward and the
continuous maximum need not be attained on the grid.
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(a, b, ϵ) Feasible t Continuous MI, bits Grid N = 20 min
(1/2, 1/2, 1/10) [1 /10, 2/5] [0 , 0.2780719051] 0
(3/10, 2/5, 1/20) [1 /20, 1/4] [0 , 0.2512271879] 0 .0058021490
(1/4, 1/4, 1/10) [1 /10, 3/20] [0 .0270944047, 0.1436583459] 0 .0270944047
(1/2, 1/2, 1/4) {1/4} { 0} 0
Displayed decimals approximate exact entropy expressions. In the second row independence has
t = 3/25, absent from the N = 20 grid; the N = 40 minimum is about 0 .0003569247. In the third
row independence t = 1/16 violates the floor, and the continuous minimum is positive.
9.4 An identical observational law with different interventions
Let U ∼ Bernoulli(1/2) and E ∼ Bernoulli(1/5) be mutually independent. Compare acyclic struc-
tural models
D : X = U, Y = X ⊕ E; C : X = U, Y = U ⊕ E.
Both have the full-support observational tableP =

2/5 1 /10
1/10 2 /5

, the same floor 1/10, and I(X; Y ) =
1 − h2(1/5). A surgical intervention replaces only the equation for X and leaves the exogenous law
and the equation for Y unchanged. It gives
P(Y = 1 | do(X = 0)) P(Y = 1 | do(X = 1))
D 1/5 4 /5
C 1/2 1 /2
For an independent fair intervention label A that sets X = A, I(A; Y ) is respectively 1 − h2(1/5)
and 0. The common cause U may be hidden; causal sufficiency and a known DAG are not assumed.
The observational interval therefore identifies no unique causal channel. This is neither an IIT Φ
nor a PID calculation.
9.5 Admissible recoding and verification limits
Take the admissible group to be the four local bit flips (X, Y ) 7→ (X ⊕r, Y ⊕s), transporting support,
marginals, law and intervention labels. They preserve individual and joint entropies, hence each
MI value and the entire interval. An asymmetric marginal constraint must be transported: these
maps need not preserve a single numerical marginal slice.
For the preceding law the global invertible map ( C, D) = ( X, X ⊕ Y ) gives

2/5 1 /10
2/5 1 /10

, whose
coordinates are independent. Whole-state entropy is preserved, but MI drops to zero. This source-
mixing map is outside the declared port-preserving group; it is an intentional negative control, not
a counterexample to the stated invariance.
The primary suite checks 6320 integer tables and 3930 support/marginal groups for denomina-
tors 4, 8, 12, 16, 20, 24, plus 30 named grid, boundary, recoding and causal cases: 10280 total, one
run. A separate reviewer checks 3809 cases in one run and reconstructs the proofs. All pass. The
respective per-run caps are 25000 and 5000 cases, 60 seconds each, with at most three runs and
denominators at most 240. Feasibility uses exact integers/rationals; logarithms use binary64 with
absolute tolerance 2 · 10−12 in the primary suite. No grid enumeration proves the continuum result
or certifies the displayed logarithms by interval arithmetic. Search used two primary anchors; exact
retrieval counts and locators are in the P2 report. This is a bounded statistical characterization,
not a larger-alphabet algorithm, novelty certificate, causal identification or evidence of phenomenal
consciousness.
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10 Open theorem and experiment targets
1. Section 8 resolves reported predictive acquisition for B06’s fixed bounded-poset contract and
supplies an exact-transcript obstruction. A state-changing acquisition model remains sepa-
rate; do not extrapolate to arbitrary height two.
2. Optimize task risk plus observation cost over a hierarchy, keeping group index, entropy and
action value as separate outputs. Compare to ordinary decision trees before introducing
additional topology.
3. Section 9 resolves fixed-marginal binary support lifts with a positive cell floor. Larger alpha-
bets and uncertain marginals require a separately motivated extension and ordinary coupling
baselines.
4. Relate constraint-presheaf obstruction data to the particular order-complex relative homology
used in B03, with an explicit map and counterexamples.
5. Study drifting actuator parameters with calibrated confidence and cost-matched compiled
controllers. A cognitive interpretation requires independent behavioral evidence; phenomenal
claims need more than this task.
These targets are feasible research programmes, not promises of positive results. Experiments here
inspect bounded cases and do not certify scaling. RF-IDENT search, all partitions, and rich sheaf
calculations can grow rapidly; the initial next runs should keep declared finite limits.
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