P2: Sharp information bounds for binary logical supports

Brian Theory · Result dated 5 October 2026 · Site section reading copy 1, 10 October 2026

Scope: this online reading copy reproduces Section 9 (P2), Propositions 7–9 and their proofs, from The MIND: Five Finite Constructions. It is a section reading copy, not a separate full manuscript. The unchanged 15-page PDF and its extracted text include M1–M5, P1 and P2.

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The result is an elementary specialization of established coupling mathematics. It establishes neither novelty, causal identification, an IIT/PID value nor a criterion for consciousness. Feasibility checks use exact rational arithmetic; displayed decimals use binary64 logarithms. Specialist screen-reader testing has not been performed; no accessibility conformance is claimed.

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 $X, Y \in {0, 1}$ have named intervention ports and a declared two-variable partition. Fix $a = ℙ (X = 1)$ , $b = ℙ (Y = 1)$ in $[0, 1]$ , exact support $S \subseteq {0, 1}^{2}$ , 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 $𝔽 2$ .

The Bernoulli coupling representation and Fréchet 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

P_{t} = (\begin{matrix} 1 - a - b + t & b - t \\ a - t & t \end{matrix}), J (t) = I_{P_{t}} (X; Y) = h_{2} (a) + h_{2} (b) - H (P_{t}) .

Here $t = ℙ (X = 1, Y = 1)$ , rows index $X$ , columns index $Y$ , and $0 \log_{2} 0 = 0$ . 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 $t_{0} = \min {U, \max {L, ab}}$ , its exact attainable-information set is

[J (t_{0}), \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 $L \leq t \leq U$ . On a feasible interval all cells are positive and

\begin{array}{l} J^{'} (t) & = \log_{2} \frac{t (1 - a - b + t)}{(a - t) (b - t)}, \\ J^{″} (t) & = \frac{1}{\ln 2} (\frac{1}{t} + \frac{1}{1 - a - b + t} + \frac{1}{a - t} + \frac{1}{b - t}) > 0 . \end{array}

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 $L = U$ . □

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 $0 < 𝜖 \leq 1 ∕ 4$ , the information interval is $[0, 1 - h_{2} (2 𝜖)]$ .

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 feasible 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 $P_{t}$ . □

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 $0 < 𝜖 \leq 1 ∕ 3$ , the unique table has three equal masses and $I = 2 h_{2} (1 ∕ 3) - \log_{2} 3 \approx 0.2516291674$ 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, suppose $Na, Nb, N𝜖$ are integers, and assume $L \leq U$ . The feasible grid points are $t = k ∕ N$ , $k = NL, \dots, NU$ . The grid maximum equals the continuous maximum. Its minimum is the smaller of $J (t_{-}), J (t_{+})$ , where

t_{-} = \min {U, \max {L, ⌊ Nab ⌋ ∕ N}}, t_{+} = \min {U, \max {L, ⌈ Nab ⌉ ∕ N}} .

Writing $J_{N, \min}$ for that minimum,

0 \leq J_{N, \min} - J (t_{0}) \leq \frac{1}{2 𝜖 \ln 2 N^{2}} .

If $t_{0}$ 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 $t_{0} = ab$ is interior, choose a nearest grid point, at distance at most $1 ∕ (2 N)$ . 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 $Na$ or $Nb$ 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.

$(a, b, 𝜖)$	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 $U \sim Bernoulli (1 ∕ 2)$ and $E \sim Bernoulli (1 ∕ 5)$ be mutually independent. Compare acyclic structural models

D : X = U, Y = X \oplus E; C : X = U, Y = U \oplus E .

Both have the full-support observational table $P = (\begin{matrix} 2 ∕ 5 & 1 ∕ 10 \\ 1 ∕ 10 & 2 ∕ 5 \end{matrix})$ , the same floor $1 ∕ 10$ , and $I (X; Y) = 1 - h_{2} (1 ∕ 5)$ . A surgical intervention replaces only the equation for $X$ and leaves the exogenous law and the equation for $Y$ unchanged. It gives

\begin{matrix} ℙ (Y = 1 ∣ do (X = 0)) & ℙ (Y = 1 ∣ do (X = 1)) \\ D & 1 ∕ 5 & 4 ∕ 5 \\ C & 1 ∕ 2 & 1 ∕ 2 \end{matrix}

For an independent fair intervention label $A$ that sets $X = A$ , $I (A; Y)$ is respectively $1 - h_{2} (1 ∕ 5)$ 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 $(X, Y) \mapsto (X \oplus r, Y \oplus s)$ , 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 $(C, D) = (X, X \oplus Y)$ gives $(\begin{matrix} 2 ∕ 5 & 1 ∕ 10 \\ 2 ∕ 5 & 1 ∕ 10 \end{matrix})$ , 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 denominators $4, 8, 12, 16, 20, 24$ , 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 $2 \cdot 1 0^{- 12}$ 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.

References cited in P2

[1] M. Kovačević, I. Stanojević, and V. Šenk (2015). On the entropy of couplings. Information and Computation 242, 369–382. doi:10.1016/j.ic.2015.04.003. Read copy: arXiv:1303.3235v2, Sections 2.1–3.1.

[2] A. Erdely (2026). A subcopula characterization of dependence for the multivariate Bernoulli distribution. Journal of Statistical Theory and Applications 25, article 4. doi:10.1007/s44199-025-00157-4, Section 3, equations (7)–(8).

Verification and provenance

The retained primary run passed 10,280 heterogeneous finite cases; the independent AI-assisted run passed 3,809. These are historical P2 runs, not new website computations or proofs of the continuum theorem. The analytic proofs above establish that interval. The full report, prior-art locators and exact manuscript source are in the source package. The verification package retains the checker, results, independent review, plans and validation receipt. Historical relative paths describe the original research archive.

Exact sources · Verification package · P2 results · Historical P2 validation · Site provenance and hashes