MCCI: Unified Theory of Mental Holes--Causality--Choice Architecture

MCCI: Unified Theory of Mental Holes–Causality–Choice Architecture (With Definitions–Criteria–Theorems–Proofs–Verification Protocols, Compatible with WSIG / EBOC / RCA–CID) Auric (S-series / EBOC) Version: v1.7 (2025-11-05, Asia/Singapore) Keywords: Mental holes; Causal diagrams (SCM); Choice architecture (default/framing/order); Bias–noise decomposition; Loss aversion; Reference point; CATE; I-projection (KL/Bregman); WSIG; EBOC; RCA–CID MSC: 62Cxx; 62Pxx; 68Txx; 91Bxx; 94Axx Abstract We construct a theory of “mental holes” verifiable under the triple norm of probability– utility–causality: given a rational baseline strategy and an embedding of observable architecture variables, we define the total deviation functional and its four-dimensional decomposition (bias, noise, causal mismatch, architecture sensitivity), provide identification criteria via backdoor/frontdoor/instrumental variables/discontinuity/difference-in-differences, and specify minimal experimental designs. Under I-projection and Bregman geometry, we prove the “Pythagoras–decoupling” structure and derive realizable estimation–audit pipelines (DQC). In the WSIG dictionary, the I-projection of the rational constraint family is viewed as the “readout norm”, and deviations are written as KL/Bregman distances; in EBOC, the pipeline is implemented as “window selection leaf” rules; in RCA–CID, reversible logs guarantee intervention replayability and external audit. We also provide an in-model determination criterion for “loss aversion–love” via the indicator L=η(λ−1) (concern weight ×fracture coefficient). Core proofs follow Csisz´ar’s I-projection and Bregman–Pythagoras, Pearl’s causal criteria, and modern estimation theory. 1 Notation & Axioms / Conventions (WSIG–EBOC–RCA Unity) A1 (Measure–Strategy–Readout): The observation triple (H, w, D) induces windowed readouts; all strategies and distributions on the standard simplex are metrized by Bregman divergence Dϕ(·∥·) and KL; rational baseline given by I-projection on constraint families [?]. A2 (Calibration Identity, WSIG card): Under the unified calibration of scattering– information geometry, we adopt the mother scale φ′(E)/π =ρrel(E) = (2π)−1tr Q(E), where Q:= −iS†∂ESis the Wigner–Smith group delay matrix; as the measure coordinate connecting to this system [?]. A3 (Finite-order NPE discipline): All discrete–continuous transformations and windowed integrations uniformly adopt “finite-order Euler–Maclaurin + Poisson” three-term error closure, asserting non-increasing singularity and pole = primary scale. A4 (RCA–CID reversibility): Implementation and audit are uniformly mapped to Bennett reversible computation and Zeckendorf-encoded logs; guaranteeing reversible replay of interventions and estimation versions [?]. 1 2 Model and Baseline Norm Variables: Context X, action A∈ A, outcome Y, unobserved disturbance U;architecture variables C= (F, D, S) for presentation framing, default selection, presentation order. SCM: Directed acyclic graph Gand structural equations Vi:= fi(Pa(Vi), Ui). Rational baseline: Under identified intervention distribution P(Y|do(A=a), X) and utility u, the Bayes–decision optimal strategy π⋆(· | x)∈arg max π E[u(Y)|do(A∼π(· | x)), X =x]. Actual strategy: π(· | x, c) may explicitly depend on c. Divergence: Take KL or general Bregman divergence Dϕ. 3 Definitions: Deviation Functional and Four-Dimensional Decomposition of Mental Holes Definition 3.1 (Total Deviation–Repeated Review Unification).For each context X=x, fix abaseline presentation c0; let the r-th review’s strategy be π(r)(· | x, c0). Define L:= EXErDϕπ⋆(· | X)∥π(r)(· | X, c0). (Architecture sensitivity is separately measured by AS and its regularization term RAS; see Theorem ??.) Definition 3.2 (Same-Case Repetition and Four Components).For the same case x, repeat reviews A(r)∼π(r)(· | x, c). Here π(r)(· | x, c)denotes the action distribution of the r-th review (or reviewer); its Bregman centroid ¯πϕ(· | x, c) := (∇ϕ)−1Er[∇ϕ(π(r)(· | x, c))]. Define Bias(x) := Dϕπ⋆(· | x)∥¯πϕ(· | x, c), Noise(x) := ErDϕ¯πϕ(· | x, c)∥π(r)(· | x, c), CM(x) := E[u(Y)|A∼¯πϕ(· | x, c), X =x] −E[u(Y)|do(A∼¯πϕ(· | x, c)), X =x]2≥0, AS(x) := sup c,c′ Dϕπ(· | x, c)∥π(· | x, c′). Definition 3.3 (Strength Indicator).Given weights ω≻0, define Defect := EXhωbBias(X) + ωnNoise(X) + ωcCM(X) + ωaAS(X)i. Note: The four terms here correspond one-to-one with B,N,C,RAS in Section ??, where C= EX[CM(X)] and RAS is the penalty functional for AS. 4 Causal Embedding and Identification Criteria Architecture embedding: Incorporate Cas parent or co-parent of Ainto G:C→A→ Y; allow Cto alter information presentation and observation channels but not the structural equations of potential outcomes Y(a). 2 Backdoor criterion: If there exists Z⊂Xblocking all backdoor paths from Ato Y, then P(y|do(a)) = PzP(y|a, z)P(z) [?]. Frontdoor/IV/RD/DiD: For unobserved confounding, use frontdoor variables, qualified instruments (relevance, exclusion, monotonicity), regression discontinuity, and modern multiperiod DiD (including staggered treatment timing and continuous intensity) respectively [?,?]. 5 Three Core Theorems and Proofs Theorem 5.1 (Bregman–Pythagoras Dual Decomposition + Regularization).For each x, taking expectation over ryields ErhDϕπ⋆∥π(r)i=Dϕπ⋆∥¯πϕ+ErhDϕ¯πϕ∥π(r)i. Taking expectation over X, by Definition ?? we obtain L=EXDϕ(π⋆∥¯πϕ) | {z } B +EXErDϕ(¯πϕ∥π(r)) | {z } N . Introducing regularization to penalize causal mismatch and architecture sensitivity, define Laug := L+EX[CM(X)] | {z } C + ΨAS |{z} RAS ⇒ Laug =B+N+C+RAS, where C,RAS ≥0. Proof. The Bregman three-point identity Dϕ(x1∥x3) = Dϕ(x1∥x2)+Dϕ(x2∥x3)+⟨x1−x2,∇ϕ(x3)− ∇ϕ(x2)⟩, taking x1=π⋆, x2= ¯πϕ, x3=π(r)and conditional expectation over r, using ¯πϕ= (∇ϕ)−1E[∇ϕ(π(r))] to make the cross term 0 (Bregman centroid first-order condition), yields the first identity and baseline equality; CM(X) is defined as a nonnegative squared difference by Definition ??, ΨAS is the penalty functional for AS; incorporating both as regularization terms gives the augmented Laug [?]. Theorem 5.2 (Architecture Equivalence and Architecture Effect).If two presentations c, c′ only affect information channels without altering the structure of Y(a), then AS(x) = 0 ⇐⇒ π(· | x, c) = π(· | x, c′)almost surely. If AS(x)>0, there exists an architecture effect induced by pure presentation difference P(a|x, c)=P(a|x, c′). Proof. By positive definiteness of divergence and the definition, immediate. Theorem 5.3 (In-Model Determination of “Loss Aversion–Love”).Let s∈ {0,1}, reference point s∗= 1, other’s welfare weight η≥0, fracture loss coefficient λ > 1, U(x, y, s) = u(x) + η u(y) + v(s−s∗), v(z) = (α z, z ≥0, −λ β(−z), z < 0. where β(·)>0, β(0) = 0. Operationalize “love” as: WTP to reduce separation probability from ε↓0to 0exceeds the baseline implied solely by risk aversion of u. Then under the premise λ > 1, Love ⇐⇒ η > 0, L := η(λ−1) >0. Proof. At first-order approximation, WTP ∼εη·∆u+ (λ−1) ·β(1), where ∆urepresents the marginal difference in other’s welfare between s= 1 and s= 0; if η= 0, this term vanishes; if λ= 1, there is no loss aversion correction for separation. Both being positive yields positive WTP excess. 3 6 Identification and Estimation (DQC: Document–Counter–Causalize– Audit) D1 Document: Case file contains (X, C, A,objective,constraints). D2 Counter-framing: Apply two or more Cto the same case (gain/loss framing, default switching, order shuffling), compute c AS(x) = max c,c′Dϕˆπ(· | x, c),ˆπ(· | x, c′), flag as “architecture sensitive” if above threshold. D3 Causalization: Draw DAG and identify via backdoor/frontdoor/IV/RD/DiD criteria; prioritize small-scale randomization for randomizable cases. Estimate ATE = E[Y(1) −Y(0)], CATE(x) = E[Y(1) −Y(0) |X=x]. For observational data, use IPW/DR/TMLE and causal forests; perform Γ-sensitivity analysis for unobserved confounding [?]. D4 Audit (Noise audit): Same-case multi-evaluation estimates Noise and aggregates \ Defect = ωbb B+ωnb N+ωcb C+ωac AS. Distinguish “level noise/pattern noise/occasion noise” in reports and provide “decision hygiene” protocols (independent judgment, aggregation, multi-source evidence) [?]. 7 Identification Criteria and Minimal Experimental Design (Quick Reference) Backdoor: Select Zblocking all paths with arrows into A, use PzP(y|a, z)P(z) [?]. Frontdoor: When complete mediator Mexists and A→Mhas no backdoor, M→Yis backdoor-adjustable, P(y|do(a)) is identifiable [?]. Instrumental Variables (IV): Zrelevant to A, independent of Y(a), affects Yonly through A; under monotonicity identifies LATE [?]. Regression Discontinuity (RD): Continuity assumption at threshold guarantees local average causal effect identification [?]. Multi-period DiD: Under staggered treatment and heterogeneous effects, use Callaway– Sant’Anna / Sun–Abraham families and extensions to continuous treatment intensity [?]. 8 Estimators and Error Discipline (Non-Asymptotic Implementation) IPW / DR: Utilize double robustness of propensity score and outcome regression; report small-sample corrections and trimming robustness [?]. TMLE: Two-step substitution estimation respecting efficiency influence function of target functional, easy to integrate with ML; provide influence function standard errors [?]. Causal forests / Generalized random forests: Estimate CATE and uncertainty, handle cluster errors [?]. Sensitivity analysis: Rosenbaum Γ bounds, marginal sensitivity model and its sharper variants [?]. NPE error budget: For all discrete–continuous transformations, report three parts: aliasing, boundary layer (Bernoulli), and tail with total bounds. 4 9 Isomorphic Connection with WSIG / EBOC / RCA–CID WSIG (I-projection = Born readout): The I-projection q⋆= arg minq∈Q KL(p∥q) on rational constraint family Qis the “norm readout”; total deviation L= KL(q⋆∥pπ) is the readout–strategy relative deviation; Bregman–Pythagoras gives the additive “bias + noise” structure [?]. EBOC (static block): Case files and randomized designs are window selection rules on static block measures, not altering global measure; time is viewed as leaf reading of blocks, with order induced by selection rules. RCA–CID (reversible log): Embed DQC pipeline in reversible cellular automata; all intervention–estimation versions recorded in CID logs encoded in Zeckendorf normal form, and Bennett reversible embedding guarantees replayability and external audit [?]. Calibration alignment: In scenarios requiring convergence with energy spectrum calibration, cite φ′/π =ρrel = (2π)−1tr Qas universal coordinate; group delay–bandwidth resource constraints become global budget for DQC [?]. 10 Experimental Blueprint and Reproducibility Checklist A/B (default effect): Randomize D∈ {opt-in,opt-out}; test ∆ATE and c AS. Dual-framing review: Same notification presented in gain/loss versions; estimate CATE with TMLE [?]. Noise audit: Same-case multi-evaluation; distinguish level/occasion/pattern noise and report post-reduction magnitude and stability [?]. “Love” indicator: Construct insurance-type choice with small-probability separation on voluntary sample; estimate b L= ˆη(ˆ λ−1) and link with satisfaction/reciprocity secondary endpoints. Governance and fairness: Report CATE, c AS for key subgroups; set “architecture fairness” thresholds and notification norms. 11 Further Properties and Corollaries Corollary 11.1 (Backdoor Adjustment ⇒Causal Mismatch Term Vanishes).If there exists Z satisfying the backdoor criterion, and when computing CM full adjustment is performed on Z, then C= 0 [?]. Corollary 11.2 (KL Special Case Centroid).When Dϕ= KL and the first argument is on the simplex, ¯πϕis a geometric-mean-type centroid, ensuring the cross term in Theorem ?? vanishes [?]. Corollary 11.3 (Sufficiency of Decision Hygiene).Independent judgment and de-echo-chamber aggregation on the Bregman platform are equivalent to minimizing Er[Dϕ(¯πϕ∥π(r))], thus directly reducing N[?]. Corollary 11.4 (Group Delay Budget).In systems calibrated with tr Q, total complexity of windowed evaluation is constrained by group delay–bandwidth product upper bound, serving as resource budget for DQC [?]. 12 Proof Details (Selected) (I) Bregman–Pythagoras: Banerjee et al.’s general treatment of Bregman three-point identity and clustering centroid, combined with Csisz´ar I-projection geometry, gives the first-order condition ¯πϕ= (∇ϕ)−1E[∇ϕ(π(r))], hence cross term is 0 [?]. 5 (II) Causal identification: Pearl’s backdoor/frontdoor; Angrist–Imbens–Rubin IV and LATE; Hahn–Todd–van der Klaauw RD; Callaway–Sant’Anna (and subsequent extensions) multi-period and continuous treatment DiD [?]. (III) Estimation theory: Bang–Robins DR; van der Laan–Rubin TMLE; Athey–Wager causal and generalized random forests; Rosenbaum and recent sensitivity reviews [?]. (IV) WSIG calibration: Wigner–Smith group delay and Birman–Kre˘ın formula provide equivalent coordinates of calibration–phase–spectrum, used as measure coordinate converging with this theory [?]. (V) RCA–CID reversibility: Bennett’s logical reversibility and Zeckendorf theorem guarantee reversible replay and unique factorization of logs, enabling external audit of intervention– estimation versions [?]. 13 Implementation Blueprint (Engineering Minimal Set) 1. Diagramming and criteria: Each online decision flow first draws DAG and marks backdoor sets/available instruments/possible thresholds and temporal staggering. 2. Online DQC: Case file template + dual-framing questionnaire + small-scale randomization; automated IPW/DR/TMLE/causal forests; accompanied by Rosenbaum Γ report [?]. 3. Audit and governance: Report CATE, c AS and \ Defect (including b B,b N,b C,c AS) for key subgroups; set “architecture fairness” thresholds and review frequency. 4. RCA–CID: Use Zeckendorf-log to carry versions; declare reversible replay interface and audit API. One-Sentence Summary “Mental holes” are decomposable deviations of strategy relative to rational baseline; via causal criteria and I-projection, they are operationalized into measurable indicators; DQC converts doubt into institutionalized improvement and guarantees auditability and portability within the unified language of WSIG / EBOC / RCA–CID. References [1] Csisz´ar, I. I-Divergence Geometry of Probability Distributions and Minimization Problems. Ann. Probab. 1975. https://projecteuclid.org/journals/annals-of-probability/ volume-3/issue-1 [2] Smith, F.T. Lifetime Matrix in Collision Theory. Phys. Rev. 1960. https://link.aps. org/doi/10.1103/PhysRev.118.349 [3] Bennett, C.H. Logical Reversibility of Computation. 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