The Game of Beliefs Protocol (GOBP): A Message-Passing Mechanism for CRSCE Decompression

Abstract

Cross-Sums Compression and Expansion (CRSCE) encodes arbitrary binary data as fixed-size bit-matrices and stores a small set of directional “cross-sum” projections (lateral, vertical, diagonal, anti-diagonal) together with a cryptographic hash chain. Compression is therefore structural, lossless, and content-independent, with a predictable output size. The corresponding decompression task is not “inflate-bytes” in the conventional sense; it is a constrained reconstruction problem closely related to discrete (binary) tomography, where a large binary matrix must be recovered from a few projections. This companion post defines the Game of Beliefs Protocol (GOBP): a protocol-layer abstraction for coordinating distributed probabilistic inference (especially loopy belief propagation) with incentive-compatible information sharing, producing an implementable path toward practical CRSCE decompression. The protocol treats decompression as inference on a factor graph, and it treats solver cooperation as a repeated game scored by strictly proper scoring rules, aligning individual incentives with truthful belief reporting.

Motivation and Context

CRSCE’s design goal—predictable, content-independent compression—moves “compression” away from statistical modeling of source redundancy and toward a deterministic structural representation. In CRSCE, a binary input stream is represented as a series of fixed-size $s \times s$ bit-matrices, and for each matrix, four directional projections (“cross-sums”) are computed (lateral, vertical, diagonal, anti-diagonal). A cryptographic hash chain (SHA-256) is used to ensure integrity and exact reconstruction.

This design has an immediate consequence: decompression is no longer a single-pass inversion of a local coding transform; it is the recovery of a high-dimensional discrete object from a small number of aggregate constraints. That is, given a binary matrix $X∈{0,1}^{s×s}$, CRSCE stores a set of line sums in several directions. Recovering $X$ from those sums is a well-studied theme in discrete tomography, where ambiguity (many solutions satisfy the same projections) is a central difficulty.

The hash chain addresses integrity, but it does not automatically make reconstruction computationally easy. SHA-256 is explicitly designed so that (in the absence of special structure) finding a preimage matching a digest is infeasible. CRSCE’s hope is that the projection constraints shrink the feasible set so aggressively that reconstruction becomes tractable in practice; nevertheless, implementing decompression “coming soon” is itself an admission that reconstruction is the hard part. It's taken thirty (30) years so far.

GOBP is proposed as the missing bridge between (i) the mathematics of message passing on graphical models and (ii) a systems reality in which decompression is performed by multiple cooperating (or partially adversarial) solver processes. It is a protocol because decompression will likely be iterative, distributed, and instrumented; and it is a “game” because in distributed settings, reliable outcomes often require incentive alignment (or, at minimum, robust aggregation mechanisms).

CRSCE Decompression as a Reconstruction Problem

What CRSCE Stores

CRSCE can be characterized as follows:

Arbitrary binary input is represented as fixed-size $s \times s$ bit matrices;
Four cross-sum projections are computed (lateral, vertical, diagonal, anti-diagonal);
A cryptographic hash chain (SHA-256) provides verifiability and exact reconstruction; and
The format is compact with a predictable compression ratio determined by its block size (s).

This description is enough to formalize the decompression challenge without depending on unpublished internal drafts. The key point is that the stored representation is dominated by aggregated constraints (projection sums) rather than local invertible transforms. That places decompression in the family of constrained reconstruction problems, not in the family of textbook entropy coding decoders.

Reconstruction from Projections and the Role of Ambiguity

In discrete tomography, a standard choice of directions is precisely the set CRSCE highlights: row sums, column sums, diagonal sums, and anti-diagonal sums. The reconstruction of a binary image/matrix from these projections is typically underdetermined; multiple binary matrices can share the same line sums, and the feasible set can be large.

This ambiguity is not a mere theoretical nuisance; it is the core systems issue. “Decompression” must pick the exact original matrix, not any matrix consistent with the projections. CRSCE’s cryptographic hash chain provides a definitive accept/reject oracle: a candidate reconstruction is correct if and only if it yields the required hash-chain consistency. But an accept/reject oracle is not a constructive inversion method; without a good search strategy, the oracle simply tells you that most candidates are wrong.

Computational Difficulty

Discrete tomography and related binary reconstruction problems are known to have deep computational complexity. Complexity analyses consider reconstruction when marginal sums are prescribed in multiple directions, including diagonal and anti-diagonal constraints. More broadly, binary tomography is frequently posed as a non-convex, combinatorial recovery problem; common formulations are NP-hard, motivating relaxations, heuristics, and specialized algorithms.

CRSCE decompression is therefore best approached as a solver problem: we must exploit the structure induced by the projections, plus the “ground truth” selection power of the hash chain, using algorithms that can navigate huge combinatorial spaces efficiently. Message passing (belief propagation and its loopy variants) is one of the few families of methods that scale to very large sparse constraint structures in practice, as evidenced by its role in decoding modern error-correcting codes on sparse graphs.

Graphical Models and Loopy Belief Propagation as a Decompression Engine

Factor Graph View of the CRSCE Block

Consider a single CRSCE block: an unknown binary matrix X with entries $x_{r,c}∈{0,1}$. The compressed representation supplies line sums. Abstractly:

Row constraints: $ \displaystyle {LSM}_{r} = \sum_{c = 0}^{s - 1}{CSM}_{rc} $ for each row r.
Column constraints: $ \displaystyle {VSM}_{c} = \sum_{r = 0}^{s - 1}{CSM}_{r,c}$ for each column c.
Diagonal constraints: sums over lines of constant $\displaystyle {DSM}_{r} = \sum_{c = 0}^{s - 1}{CSM_{r,(r + c)\ mod\ s}} $
Anti-diagonal constraints: sums over lines of constant $\displaystyle {XSM}_{r} = \sum_{c = 0}^{s - 1}{CSM_{r,(s - r + i - 1)mod\ s}} $

Each constraint is a high-arity relation among the participating bits. This is naturally represented as a factor graph: variable nodes for the bits, factor nodes for each line-sum constraint, and edges connecting a factor to the variables on that line. The factor-graph formalism exists exactly to capture such “global function factors into local functions” structure and to support message passing.

However, two obstacles appear immediately:

the factor graph is loopy (many short cycles on a grid-like structure), and
the constraints are “hard” (they are equalities, not soft likelihoods).

Both obstacles are addressable in practice, but they require careful engineering and a willingness to accept iterative approximate inference as an internal tool.

Belief Propagation and Its “Loopy” Extension

Belief propagation (BP) computes exact marginals on trees, and loopy belief propagation (LBP) applies essentially the same message updates on graphs with cycles, iterating until messages stabilize (if they do). The conceptual update is local: each message from node $i$ to neighbor $j$ summarizes what $i$ believes about $j$ based on $i’s local evidence and messages received from other neighbors.

LBP has a variational interpretation: fixed points correspond to stationary points of the Bethe free energy approximation. This helps explain both its effectiveness (good approximations in many sparse settings) and its pathologies (non-convergence, multiple fixed points).

For CRSCE decompression, the objective is not merely approximate marginals; it is an exact reconstruction. This is where LBP should be understood as a heuristic engine inside a larger exact search-and-verification loop. This “approximate inference guides exact decisions” pattern is common in sparse-graph decoding and constraint satisfaction: message passing concentrates probability mass; decimation or branching fixes confident variables; verification enforces correctness.

Why a Protocol Layer Is Needed

If decompression were a single-process local algorithm, the story could stop at “implement LBP plus verification.” But CRSCE’s reconstruction problem is large, iterative, and naturally parallelizable. Moreover, decompression for CRSCE is likely to be performed in environments where:

multiple solver instances explore different initializations, schedules, or heuristics;
hardware heterogeneity exists (CPU, GPU, clusters);
partial progress and intermediate beliefs are valuable artifacts (debugging, reproducibility, auditing);
there may be untrusted or partially trusted participants (e.g., crowd compute, federated solvers, third-party accelerators).

The Game of Beliefs Protocol (GOBP) is proposed to standardize how beliefs are represented, exchanged, aggregated, audited, and rewarded (or at least weighted), so that “distributed belief propagation” becomes an interoperable system rather than a bespoke research codebase.

Defining the Game of Beliefs Protocol (GOBP)

High-Level Definition

Game of Beliefs Protocol (GOBP) is a protocol for coordinating distributed probabilistic inference on a shared graphical model by:

standardizing message formats for beliefs and constraints,
defining aggregation rules for combining beliefs from multiple sources,
embedding an incentive mechanism (a repeated game) that rewards accurate, honest probabilistic reporting, and
providing cryptographic commitments and audit trails for solver outputs and intermediate claims.

The “belief” in GOBP is explicit: it is a probability distribution (or a parameterization of one) over a variable’s state, or over a factor’s local configuration. The “game” in GOBP is likewise explicit: participants choose what beliefs to publish, and their long-run influence (and optionally compensation) is determined by a scoring mechanism evaluated against realized ground truth when reconstruction completes.

Why “Game”: Incentives and Truthful Probabilities

GOBP’s incentive layer is anchored in a mature idea: strictly proper scoring rules. A scoring rule assigns a numeric score to a probabilistic forecast once the actual outcome is known. A scoring rule is strictly proper if a forecaster uniquely maximizes expected score by reporting their true belief distribution. Classic examples include the Brier (quadratic) score and the logarithmic score.

In a decompression setting, once the correct matrix is identified (and verified via the CRSCE hash chain), each bit’s realized value becomes ground truth. The protocol can then score prior probability reports about those bits (or about derived events), updating reputations, weights, or payouts accordingly. This is analogous in spirit to information aggregation in prediction markets, where scoring rules (including market scoring rules) are used to elicit and aggregate beliefs.

Game theory provides the language for analyzing such mechanisms: we want equilibria in which “truthful reporting and good-faith computation” are best responses. The foundational notion of equilibrium in strategic games is Nash equilibrium. GOBP does not require full rational-agent modeling to be useful, but it is designed so that—if participants are even approximately incentive-driven—honesty is aligned with reward.

Entities, Roles, and Trust Model

GOBP defines a small set of roles; a real system may collapse them or distribute them:

Problem Publisher: publishes the canonical CRSCE decompression instance (block projections + hash targets) and the derived factor graph schema.
Solver: runs inference (LBP, search, relaxations, hybrids) and publishes belief messages.
Aggregator: combines belief messages into consensus messages using a defined pooling rule.
Verifier: checks cryptographic commitments, message signatures, and (when a candidate solution is proposed) validates constraints and hash-chain consistency.
Ledger (optional): an append-only log of commitments, messages, scores, and final artifacts.

GOBP’s trust model is deliberately flexible: it can be run as a cooperative protocol among trusted solver processes (where the “game” reduces to structured telemetry and weighted ensembles), or it can be run among partially trusted parties in a future (as-yet-defined state), where cryptographic commitments and scoring are necessary to manage strategic behavior. The later is an ideal state where crowdsourced computation could make very large archives cost effective across organizations.

The Decompression Graph: From CRSCE Projections to a Factorization Suitable for Message Passing

The Core Challenge: High-Arity Constraints on a Loopy Grid

A line-sum constraint touches O(s) bits. Naively, a factor node connected to all bits in a row would have an exponential internal state space if represented as a full table. Practical message passing therefore relies on structured factors (e.g., “cardinality” or “sum” factors) whose messages can be computed efficiently using dynamic programming.

Discrete tomography literature frequently frames reconstruction from row/column/diagonal/anti-diagonal sums as solving a system of constraints that is often underdetermined. This underdetermination is not a reason to abandon message passing; it is a reason to expect posteriors to remain uncertain until additional information (priors, additional constraints, or hash-based verification feedback) breaks symmetries.

Converting Hard Equalities into Soft Potentials

BP is typically defined for strictly positive factors (to avoid degeneracies), whereas hard equalities create zero-probability configurations. There are multiple pragmatic approaches:

Soft penalty potentials: replace equality with a sharply peaked likelihood, e.g. $ψ(S) = exp(−λ (S − R)^2)$ where $S$ is the line sum and R is the required projection. As $λ→∞$, the distribution concentrates on satisfying assignments.
Constraint propagation + BP hybrid: maintain exact feasibility ranges for partial sums (a CSP view) while BP provides probabilistic guidance for branching.
Augmentation with auxiliary variables: represent a sum constraint by a chain of small factors that enforce incremental sums, enabling exact “local” constraint enforcement without exponential tables.

The third approach is especially compatible with factor graphs and message passing as presented in the sum-product literature. It is also conceptually aligned with the way sparse-graph codes are decoded: parity checks are constraints, yet message passing is still feasible because factors have exploitable algebraic structure.

Where the Hash Chain Fits

CRSCE uses a SHA-256 hash chain for integrity and exact reconstruction. In protocol terms, the hash chain is a global oracle: it certifies correctness, but it does not decompose into small local factors over bits in any obvious way.

GOBP therefore treats the hash chain primarily as:

a terminal verifier (accept/reject for a candidate block or sequence of blocks), and
a source of incremental pruning when combined with search strategies that propose partial reconstructions in a structured way.

Hash chains are a well-established cryptographic primitive (e.g., in one-time password systems and micropayment schemes). This matters for GOBP because the protocol leans on commitments and auditability: when a solver claims “I believed $bit(r,c)=1$ with probability 0.93,” that claim should be time-stamped, immutable, and attributable, so that scoring is meaningful and manipulation after the fact is infeasible.

The Protocol Proper: Messages, Rounds, Aggregation, and Scoring

Canonical Problem Instance Identifier

Each decompression task (typically: one CRSCE block, or one “work unit” derived from a block) is assigned a canonical identifier:

problem_id = H(
    version ||
    s ||
    projection_payload ||
    hash_chain_target ||
    normalization_rules ||
    factorization_schema_id
)

Here $H$ is SHA-256 (or another protocol-specified hash). The purpose is not cryptographic novelty; it is interoperability. Every participant must agree on exactly what problem is being solved and what “truth” means.

Belief Message Types

GOBP standardizes a small set of message types. A minimal core is:

VAR_BELIEF: marginal belief for a variable node, e.g. P(x=1) (or log-odds / LLR).
EDGE_MSG: directed message along an edge in the factor graph (LBP-style), either variable-to-factor or factor-to-variable.
FACTOR_STATE: optional summaries for structured factors (e.g., distributions over partial sums in a row/column chain).
COMMIT / REVEAL: cryptographic commitment to a batch of messages, enabling later auditing and scoring.
CANDIDATE: a proposed full reconstruction (or a compressed representation of one) intended for verifier checking.
CHALLENGE: a request to justify or reproduce messages (useful when solvers are untrusted or when debugging).

Messages can be encoded compactly. For binary variables, the most numerically stable representation is often the log-likelihood ratio (LLR), widely used in iterative decoding on sparse graphs. The protocol does not mandate LLR, but it should define a canonical conversion and numeric bounds.

Rounds and Schedules

GOBP is agnostic to synchronous vs asynchronous scheduling, but it standardizes round semantics:

Compute: each solver updates local messages using its current neighborhood state.
Publish: solvers publish messages and commitments for the round.
Aggregate: an aggregator computes consensus messages from the set of published messages.
Feedback: consensus messages become the next-round inputs; optional decimation events are announced.
Propose: when enough certainty exists, solvers (or the aggregator) propose a candidate reconstruction.
Verify: verifiers check projection constraints and the hash-chain target; accept or reject.

Loopy belief propagation’s behavior depends strongly on scheduling, damping, and normalization. Practical implementations emphasize these stabilization choices. GOBP therefore treats “message update rule” as an algorithm plug-in, but it requires that each message batch declare enough metadata to be interpretable: iteration index, damping used, normalization scheme, and factorization schema.

Aggregation as Opinion Pooling

When multiple solvers publish beliefs about the same variable or edge, the protocol must combine them. A natural family of aggregators are log opinion pools: $$ p_{consensus}(x) \propto \prod_k P_k(x)^{wk} $$ where $w_k ≥ 0$ are weights (e.g., reputations or inverse-variance estimates). This is a “product-of-experts” style merge: consistent strong beliefs reinforce; disagreement yields calibrated uncertainty. In LLR form for binary variables, this reduces to a weighted sum of logits—computationally conveniently.

In adversarial settings, the aggregator should be robust: e.g., trim extreme logits, cap influence, or require stake. GOBP does not prescribe a single robustness method, but it does require that the aggregation rule be declared and reproducible.

Scoring: Turning Beliefs into a Repeated Game

Once a candidate reconstruction is accepted, every variable’s realized value becomes ground truth for scoring earlier beliefs. GOBP proposes strictly proper scoring rules as the default incentive layer.

For a binary variable $x∈{0,1}$ and a reported probability $p=P(x=1)$:

Log score: $S(p,x) = log(p)$ if $x=1$, else $log(1−p)$. (Strictly proper under standard conditions.)
Quadratic/Brier score: often expressed (up to affine transforms) as $S(p,x) = −(p−x)^2$.

Proposition (Incentive Compatibility; Proof Sketch). Let a participant’s subjective belief about $x$ be $q$, and suppose the protocol pays expected score $E[S(p,x)]$. If $S$ is strictly proper, then $p=q$ uniquely maximizes $E[S(p,x)]$. This is a central theorem in the scoring rule literature and is developed in modern generality by Gneiting & Raftery.

In GOBP, a participant’s long-run weight $w_k$ in aggregation can be an increasing function of cumulative score. This creates a feedback loop: accurate forecasters gain influence; inaccurate or deceptive forecasters lose influence. Conceptually, this resembles the way market scoring rules support information aggregation in prediction markets.

How GOBP Solves CRSCE Decompression in Practice

The Decompression Pipeline as “Inference → Decimation/Search → Verify”

A GOBP-based CRSCE decompressor is best understood as a pipeline:

Build a factor graph representation of the projection constraints for a CRSCE block.
Infer approximate marginals (LBP) under a soft constraint model.
Decimate / Branch: fix high-confidence bits or structured subsets; reduce the problem.
Propose a full candidate reconstruction when constraints are satisfiable and uncertainty is low enough.
Verify the candidate using the stored projections and the SHA-256 hash-chain target.
Backtrack / Restart if verification fails; incorporate failure as information (e.g., forbid assignments, adjust priors, perturb messages).

This pattern is not speculative in the abstract; it is the same conceptual architecture that makes message passing useful in other large combinatorial inference problems, such as decoding on sparse graphs (LDPC), where iterative message passing is combined with hard parity constraints and final codeword verification.

The critical difference is that CRSCE constraints are geometric (line sums in multiple directions), not algebraic parity checks. But discrete tomography shows that these constraints can be expressed as systems with large solution spaces; the question becomes whether iterative inference can drive the system toward the unique solution singled out by the hash chain.

Why Distributed Solvers Help

LBP on loopy graphs can converge to different fixed points depending on initialization, damping, and update schedule. In CRSCE decompression, this sensitivity is an opportunity: multiple solver instances can explore diverse trajectories through belief space, and an aggregator can combine them into a better-calibrated consensus. This is “ensemble inference” expressed as a network protocol.

Moreover, CRSCE blocks (as publicly described) are naturally partitioned by matrix, and thus decompression can be parallelized by block. Within a block, the factor graph can be partitioned spatially (sub-grids, diagonal bands) or by constraint families (rows vs columns vs diagonals), enabling partial-message specialization.

Integrating Verification Feedback without Breaking Cryptographic Assumptions

SHA-256’s security properties mean we should not expect to extract “local gradient information” from the digest. The hash is used as a binary oracle: correct/incorrect. Nonetheless, verification feedback is still useful because it confirms when the search has reached the unique satisfying reconstruction. In practice, solvers use hash mismatch as a signal to restart with perturbed beliefs or alternative decimation orders.

Hash chains and related constructions are widely used precisely because they are easy to verify and hard to invert. In GOBP, this translates into a clean separation of responsibilities: inference proposes; verification certifies.

Implementation Blueprint

Data Structures: Factor Graph Schema for CRSCE Projections

A practical decompressor should avoid explicit high-arity factor tables. Instead:

Represent each bit $x_{r,c}$ as a binary variable node.
Represent each line constraint as a structured factor implemented by DP over partial sums.
Cache factor-to-variable messages and variable-to-factor messages in compact numeric form (float32 LLRs are often enough, with care for saturation).
Use damping and normalization (as in practical LBP) to stabilize updates.

The discrete tomography literature explicitly highlights the common four-direction setting (row/column/diagonal/anti-diagonal) and emphasizes underdetermination. That should inform engineering: you should expect large regions of near-symmetry early in inference, and you should rely on decimation and verification to break ties.

Message Update Engine

LBP message updates can follow the standard sum-product semantics on factor graphs. The protocol does not force a single engine; it does, however, require that a solver declare:

update schedule (synchronous vs asynchronous; ordering);
damping parameter(s);
numerical stabilization (normalization, log-space computation);
stopping criteria (max delta threshold, max iterations).

In sparse-graph decoding contexts, efficient sum-product implementations often operate directly on LLRs, with well-studied approximations to reduce complexity. CRSCE structured sum-factors will require different DP internals, but the same numeric discipline applies.

Protocol Transport and Cryptographic Commitments

GOBP messages should be authenticated (signatures or MACs depending on trust model) and optionally committed in batches:

    commitment = SHA256( canonical_encode(message_batch) )

SHA-256 is standardized by NIST’s Secure Hash Standard. Batch commitments support auditability: after reconstruction, a solver reveals its batch, and others verify that the batch hash matches the prior commitment.

This “commit now, reveal later” logic is orthogonal to CRSCE’s internal hash chain, but it leverages the same cryptographic principle: one-way hashing produces cheap-to-check integrity with strong resistance to retroactive manipulation. Hash chains have long histories in security systems (e.g., S/KEY) and related schemes.

A Minimal Wire Format (Illustrative)

A compact binary encoding (CBOR, protobuf, flatbuffers) can represent:

    Message {
        uint32  version;
        bytes32 problem_id;
        uint32  round;
        enum    type;        // VAR_BELIEF, EDGE_MSG, COMMIT, REVEAL, CANDIDATE, ...
        uint64  author_id;
        bytes   payload;     // type-specific
        bytes   signature;   // optional, depends on trust model
    }

Payload examples:

VAR_BELIEF payload might be ($variable_id$,$p1$) with $p1$ encoded as float16/float32 or fixed-point logit.
EDGE_MSG payload might be (edge_id, direction, vector/params).
CANDIDATE payload might be a compressed representation of an entire reconstructed block (e.g., bit-packed matrix) intended for verification.

Termination and Correctness Guarantees

GOBP does not promise that inference converges (LBP does not guarantee convergence on loopy graphs). Instead, GOBP promises that any claimed reconstruction can be verified against:

the projection constraints (deterministically checkable), and
the CRSCE SHA-256 hash-chain target (deterministically checkable).

Correctness of the final output therefore reduces to correctness of the verifier checks, not to correctness of any participant’s inference method. This is an important separation: the protocol encourages good inference, but it does not require blind trust in it.

Security, Robustness, and Adversarial Considerations

Threat Model Sketch

A decompression protocol can face:

Noise / incompetence: some solvers are simply inaccurate.
Strategic misreporting: solvers publish beliefs designed to influence aggregation rather than to reflect genuine posterior beliefs.
Sybil behavior: an attacker spawns many identities to bias consensus.
Replay / equivocation: a solver changes its story after the fact.

GOBP’s defense posture is layered:

Commitments reduce equivocation and enable audit.
Strictly proper scoring rules make honest reporting a best response in expectation (when rewards matter).
Robust aggregation limits the influence of outliers.
Stake/reputation weighting makes influence expensive to acquire and easy to lose.

Why Proper Scoring Rules Are the Right Primitive

Proper scoring rules were developed specifically to assess and elicit probabilistic beliefs. The key is not merely “reward accuracy” but “reward calibrated probabilities.” In decompression, a participant that always reports extreme probabilities (0 or 1) might sometimes be right, but when wrong it should be punished sharply. The log score provides exactly that behavior.

Importantly, GOBP does not require monetary payments; “score” can control purely internal weights, rate limits, or acceptance thresholds. The game metaphor remains apt because the protocol defines payoffs (influence) and strategies (what beliefs to publish), and because equilibrium analysis is still the correct lens for understanding long-run behavior.

The author is not naive. In most cases, the GOBP solution will be implemented within a single organization's ecosystem, not crowdsourced across a diverse cloud of heterogeneous hardware. This is due partly to liability and risk as well as to regulatory constraints. Nonetheless, it is a prudent design consideration, especially where the decompressed data is ciphertext.

Limitations and Research Questions

The Fundamental Ambiguity of Four-Direction Projections

Reconstruction from row/column/diagonal/anti-diagonal sums can be ambiguous; projections may not uniquely determine the binary matrix. CRSCE resolves this at the data-format level by including cryptographic verification (hash chain), but computational ambiguity remains: a solver must still locate the uniquely correct matrix among potentially many projection-consistent matrices.

LBP Stability and Fixed Points

LBP is approximate and may not converge; convergence depends on coupling strength, loop structure, damping, and schedule. A GOBP implementation should treat “non-convergence” as normal and incorporate restart/perturbation strategies, plus decimation heuristics that convert soft beliefs into hard commitments.

The Hash Factor Remains Global

SHA-256 is not decomposable into small independent local constraints in any straightforward way. Any attempt to treat the hash as a factor node in the same graph as projection sums will lead to an enormous factor with no efficient message computation (unless one introduces additional structure not present in standard hashes). GOBP’s stance is therefore pragmatic: use hashes for verification and audit; do not pretend they are inference-friendly.

Conclusion

CRSCE compresses by replacing raw bit matrices with a small set of directional projections plus a SHA-256 hash chain, achieving predictable, content-independent compression while preserving losslessness. Decompression is therefore a discrete reconstruction problem aligned with discrete tomography: large binary objects must be recovered from a few projections, a task known to be underdetermined and computationally challenging in general.

The Game of Beliefs Protocol (GOBP) is proposed as a systems-level answer: treat decompression as inference on a factor graph; coordinate many solver processes via standardized belief messages; combine beliefs via principled aggregation; and align incentives for truthful probabilistic reporting with strictly proper scoring rules evaluated after cryptographic verification provides ground truth.

In short: GOBP is not “a new inference algorithm.” It is the protocol scaffolding that makes large-scale, distributed, and auditable CRSCE decompression feasible, using well-established foundations from factor graphs and message passing, discrete reconstruction theory, and incentive-compatible probability elicitation.

References

Balázs, P. (2008). On the Ambiguity of Reconstructing hv-Convex Binary Matrices with Decomposable Configurations. Acta Cybernetica, 18, 367–377.

Brier, G. W. (1950). Verification of Forecasts Expressed in Terms of Probability. Monthly Weather Review, 78(1), 1–3.

Caldwell, S. (2025). Loopy Belief Propagation: Theory and Practical Implementation in Probabilistic Graphical Models. samcaldwell.net.

Gallager, R. G. (1962). Low-Density Parity-Check Codes. IRE Transactions on Information Theory.

Gneiting, T., & Raftery, A. E. (2007). Strictly Proper Scoring Rules, Prediction, and Estimation. Journal of the American Statistical Association, 102(477), 359–378.

Haller, N. M. (1994). The S/KEY One-Time Password System.

Hanson, R. (2002). Logarithmic Market Scoring Rules for Modular Combinatorial Information Aggregation.

Herman, G. T., & Kuba, A. (Eds.). (1999). Discrete Tomography: Foundations, Algorithms, and Applications. Birkhäuser.

Kadu, A., & van Leeuwen, T. (2018). A Convex Formulation for Binary Tomography. arXiv.

Kschischang, F. R., Frey, B. J., & Loeliger, H.-A. (2001). Factor Graphs and the Sum-Product Algorithm. IEEE Transactions on Information Theory, 47(2), 498–519.

MacKay, D. J. C. (1999). Good Error-Correcting Codes Based on Very Sparse Matrices. IEEE Transactions on Information Theory, 45(2), 399–431.

Nash, J. F. (1950). Equilibrium Points in n-Person Games. Proceedings of the National Academy of Sciences, 36(1), 48–49.

National Institute of Standards and Technology (NIST). (2015). FIPS 180-4: Secure Hash Standard (SHS).

Richardson, T. J., & Urbanke, R. (2001). The Capacity of Low-Density Parity-Check Codes Under Message-Passing Decoding. IEEE Transactions on Information Theory.

Rivest, R. L., & Shamir, A. (1996). PayWord and MicroMint: Two Simple Micropayment Schemes.

Weiss, Y. (2000). Correctness of Local Probability Propagation in Graphical Models with Loops. Neural Computation, 12(1), 1–41.

Wainwright, M. J., & Jordan, M. I. (2008). Graphical Models, Exponential Families, and Variational Inference. Foundations and Trends in Machine Learning, 1(1–2), 1–305.

Yedidia, J. S., Freeman, W. T., & Weiss, Y. (2005). Constructing Free-Energy Approximations and Generalized Belief Propagation Algorithms. IEEE Transactions on Information Theory, 51(7), 2282–2312.