Binary Information Bounds

Logical possibility and statistical weight

A logical support lists allowed pairs of values. A probability law assigns weights to those pairs. Keeping the same support can leave many different laws and information values available; the variables and their partition must also be declared.

P2 concerns two named binary variables, X and Y. It fixes their probabilities of being 1, an exact support, and a positive floor on each allowed pair. The floor is an additional weight constraint, not a consequence of logical possibility. The result concerns population probabilities in bits, with no estimation or dynamical claim.

The exact result

With all four pairs allowed, every law with the fixed marginals has one free parameter: the probability of (1,1). The floor confines that parameter to a closed interval. When the interval is nonempty, mutual information has a unique minimum and reaches its maximum at an endpoint; continuity supplies every intermediate value. Independence is allowed only when all four product-law cells meet the floor. Otherwise a nonempty family has a strictly positive minimum, even though its support is rectangular.

With a proper support, a forbidden pair fixes the free parameter. The fixed marginals and floor then yield either an infeasible family or one law. This uniqueness does not follow from support alone with unspecified marginals.

For equal marginals of one half and a floor of one tenth, the continuous information range is approximately 0 to 0.2780719051 bits. With marginals of one quarter and the same floor, independence is excluded and the range is approximately 0.0270944047 to 0.1436583459 bits. The reader supplies the exact entropy expressions and proofs.

Finite checks and continuous statements

Aligned rational grids have the same maximum as the continuous family, but their minimum can be larger. P2 identifies the two candidate grid points for the minimum and bounds the gap. The grid values form a finite set, not the entire interval between their extrema. Without a common positive floor, exact full support instead has an open parameter interval: its largest information value is a supremum outside that support.

The retained primary run passed 10,280 heterogeneous cases and the independent AI-assisted review passed 3,809. Feasibility uses exact integers and fractions; logarithms are floating-point evaluations, not interval-certified values. These historical checks corroborate the analytic proofs. Website integration did not rerun or extend the research computations.

Statistical information does not identify causation

P2 gives two structural models with the same full joint law and observational information, but different responses when X is set by intervention. A fair intervention label yields positive information in the direct model and zero in the common-cause model. Those causal assumptions and the intervention design must be stated separately.

Local bit flips preserve information when support, marginals, probabilities and intervention labels are transported together. A global XOR recoding can preserve whole-state entropy while changing mutual information, because it changes the variable partition and mixes the named intervention ports. It lies outside P2’s declared admissible recoding group.

Place in The MIND programme

This result develops M3’s distinction between support, weights and causal information in The MIND. It supplies a small statistical baseline for Information and boundaries, whose system/environment update model remains proposed.

Structured Boolean Factorization studies logical decomposition over disjoint variable blocks. P2’s rectangular-support discussion does not identify that decomposition with independence under arbitrary weights or a cell floor. Its support indicators use relation semantics, distinct from native CM/LM arithmetic over the two-element field.

Sources and limits

The online Section 9 reading copy retains the exact source statements, proofs, citations and proposition numbers. The combined PDF includes the earlier constructions and P1 as well as P2. The source package, verification package and provenance hashes preserve the source/evidence distinction.

Brian Theory leads the programme; AI tools assisted with synthesis, computation and internal review. The underlying binary-coupling and entropy ingredients are established mathematics. The narrow prior-art check does not establish priority. P2 supplies no larger-alphabet algorithm, causal identification, IIT or PID calculation, biological test, or evidence of consciousness.