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Mathematical note

Burden-qualified contestable decisions

math/contestable-decision-record.md

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Direct repository links only; no document-level evidence status is implied.

Scope

This note formalizes the narrow systems residue from the legal evidence/procedure audit: evidence state, authority, burden, procedure, decision, review, remedy, and finality remain distinct. The legal rules are jurisdiction- and date-bound; the transfer is an experiment contract, not a claim that adjudication is an optimizer or a truth oracle. Its evidence boundaries are C-679C-704.

Decision record

For decision version vv, retain

Dv=(J,A,P,H,B,S,R,O,Q,D,G,V,F),\mathcal D_v=(J,A,P,H,B,S,R,O,Q,D,G,V,F),

where JJ is jurisdiction/rule authority, AA authorized decision-maker, PP parties or affected interests, HH claim and requested action, BB burden bearer, SS governing standard/decision rule, RR admitted record, OO objections and preserved issues, QQ disclosure/access/response state, DD disposition, GG stated grounds and reasons, VV review/remedy state, and FF finality/reopening conditions. These are typed fields, not commensurable scores.

For evidence item ii,

ei=(ai,pi,ui,ci,αi,δi,wi,hi,oi),e_i=(a_i,p_i,u_i,c_i,\alpha_i,\delta_i,w_i,h_i,o_i),

where aia_i is artifact identity, pip_i proponent, uiu_i permitted purpose, cic_i custody/provenance, αi\alpha_i authentication ruling, δi\delta_i admissibility ruling, wiw_i weight or likelihood contribution, hih_i supported claim/scope, and oio_i objections/contrary evidence. Authentication does not imply content truth; admissibility does not imply weight; weight does not imply sufficiency; sufficiency does not grant authority.

Statistical and authority boundaries

For formal hypotheses H1,H0H_1,H_0, the likelihood ratio

Λ(e)=p(eH1)p(eH0)\Lambda(e)=\frac{p(e\mid H_1)}{p(e\mid H_0)}

is dimensionless and depends on the evidence model. It is neither p(H1e)p(H_1\mid e) nor a universal translation of a verbal proof standard. A decision threshold follows only after hypotheses, priors/evidence distributions, consequence vector, protected constraints, authority, and the decision rule are declared.

Keep the outcome vector visible:

L=(N10,N01,Twrong,Nprocess,Nprotected,C,TH,TD,E),\mathbf L=(N_{10},N_{01},T_{\mathrm{wrong}},N_{\mathrm{process}}, N_{\mathrm{protected}},C,T_H,T_D,E),

where false positive/negative counts N10,N01N_{10},N_{01} are decisions, TwrongT_{\mathrm{wrong}} is time under an incorrect or unauthorized effect in person-seconds or effect-seconds, NprocessN_{\mathrm{process}} and NprotectedN_{\mathrm{protected}} are violation counts, CC is currency at a stated price year, THT_H human effort in person-seconds, TDT_D elapsed delay in seconds, and EE lifecycle energy in joules. Protected procedure cannot be silently traded for mean accuracy through an unstated scalarization.

Review is selected and scoped

For N0N_0 first decisions, NAN_A reviewed, and NRN_R reversed/remanded,

rA=NAN0,rR=NRNA.r_A=\frac{N_A}{N_0}, \qquad r_R=\frac{N_R}{N_A}.

Both are dimensionless. rRr_R is not the first-stage error rate because issue preservation, selection into review, review standard, harmless/prejudice rule, remedy, settlement, and reviewer error intervene. A review record therefore binds the issue, record version, standard, permitted scope, identified error, prejudice rule, disposition, remedy, successor version, and later validity.

Falsification boundary

The held composition loses if typed workflow plus provenance, access control, calibrated selective prediction, rule/citation graphs, independent review, red-team challenge, conflict-of-interest controls, and full recomputation match its protected outcome–cost frontier. Reason text is not credited as faithful causal introspection; finality is not truth; reopening is not free rollback; and similarity to a prior decision does not establish applicable authority.