The manuscript opens with the reciprocal cost J(x)=(x+x^{-1})/2-1=(x-1)^2/(2x) of equation (1). Direct expansion confirms the two algebraic forms agree. The states x=2 and x=1/2 both give J=1/4; multiplication of the ratio by 2 produces the signed cost changes +7/8 and -1/4 shown in Figure 1. The text correctly notes that any valuation obeying the three identity laws of the cited companion work collapses precisely the pair {x,1/x}, so a cost-only report loses only direction. Section 2 then defines adequacy by the factorization O_a=O_bar_a circ q for every admitted experiment a (equation (2)). Proposition 2.1 is the standard initiality statement for the free inductive carrier generated by 0 and S; its short recursion proof is correct and supplies the unique arithmetic operations used later. The domain of validity is finite records; completion to the reals is explicitly flagged as an extra step that the construction does not supply by itself. Order of occurrences is retained only when later queries can distinguish it, a point illustrated by the ordered-loop reference [3].
Section 3 derives J from signed postings. Squaring the pair (a,b) produces the residual (a-b)^2, whose minimal unsigned representative is given by equation (3). Reading that residual against the comparison's own cross-postings 2ab yields equation (4). Proposition 3.1 shows that any nonnegative, additive, normalized meter m(d,r) must be ordinary division; the monotone-squeezing argument on the nonnegative reals is standard and correct. The composition identity (5) follows by direct expansion of J. Proposition 3.2 then proves uniqueness under the fixed law (5) together with the local quadratic calibration lim 2F(e^t)/t^2=1. Setting H(t)=1+F(e^t) produces the doubling relation H(2t)=2H(t)^2-1; iteration with arcosh and the calibration asymptotics recovers H(t)=cosh t, hence F=J. The argument is self-contained and does not rely on global smoothness. The later rational-cost classification (cited from [7]) shows that every rational cost obeying balance, reciprocity and no-ceiling is of the form kappa(x^k+x^{-k}-2); the record reading selects the member k=1, kappa=1/2, whose forced combination rule is precisely (5). Smoothness alone does not force J, as the counter-family J+epsilon J^m (m>=3) demonstrates. All of these steps are conditional on the physical premises that the residual is formed by signed postings, that the cross-postings serve as reference, and that chained comparisons obey the fixed law.
Section 4 reconstructs the directed ledger. Incident cost C_i and signed flow sigma_i are defined by (6) and (7). The response function h_r(x)=J(rx)-J(x) expands to equation (8); the positive root of the resulting quadratic is (9), and the Lipschitz bound (10) follows from the identity h_r(x)-h_r(y)=((r-1)/2)(x-y)(1+1/(r x y)). Two successive responses cannot both vanish, separating absent from present bonds. Theorem 4.2 assembles three observations (first cost change, second cost change, first flow change) at every participant and every candidate address. The two successive cost increments vanish together exactly when the bond is non-incident; the first nonzero response recovers the multiplier via (9); flow signs orient the bond; a single incident participant identifies a self-loop. Adequacy (2) then supplies a unique decoder on the range of any report that preserves the observations. The reconstruction is therefore internal to the ledger model under the stated addressability and calibration premises; it does not by itself establish that the probes can be performed.
Section 5 treats ethics and economics as transformations of the same ledger. An action multiplies each bond by e^{a_b} (11); harm at participant i is the induced change of incident cost (12). Economic states are read as ledgers by the ratio of actual to required signal (13). Proposition 5.1 shows that the strains a_b=ell_b(t)-ell_b(s) realize any transition exactly and uniquely. Definition 5.2 of borne share splits each bipartite Delta J_b equally and assigns self-loops wholly to their participant; Proposition 5.3 then proves that the sum of borne shares equals the total ledger cost change while the sum of harms double-counts every bipartite bond. The numerical example (bond falling from 4 to 1, self-loops rising from 1 to 3) yields consistent figures H_i=-11/24 and beta_i=5/48. Endpoint telescoping (14) loses path information that a retained record can still answer (Figure 7); the text correctly notes that consent and authority concern histories, not endpoint sums. Moral significance is not automatic: the manuscript repeatedly requires identification of the relevant physical comparison.
Section 6 contains the strongest quantum result. Theorem 6.1 assumes a CPTP map E that preserves perfect binary discrimination for every basis pair and every real superposition pair (15). Kraus operators then satisfy the Knill-Laflamme condition K_a^* K_b=alpha_ab I on the whole input space. Diagonalization yields the reversible-channel normal form E(rho)=V(rho tensor tau)V^* of Nayak and Sen; the explicit recovery (18) is CPTP and satisfies R E=id, including after tensoring with any finite retained reference. For a qubit the two experiments Z and X already suffice; the untested Y component is recovered automatically once the channel form is forced. Corollary 6.2 transports every instrument by the sandwich E_C T_o R_A, preserving outcome probabilities, conditional states and reference correlations along finite adaptive protocols. The argument is a clean, finite-dimensional application of exact quantum error correction; it does not claim classical tomography from two numerical outcomes, a caveat the text already states.
Section 7 and Appendix A correctly separate what is proved from what remains open. Golden scaling and three-dimensionality are presented as conditional selection results under named physical substrates. Born weights are imported from the record-rule preprint [9] (register noncontextuality being the principal premise); the manuscript does not re-derive them here. Experiential identity and whole-subject composition are likewise flagged as further identifications whose physical realization is still required. The claim-to-evidence appendix lists each theorem together with the physical identification an application must still supply. No algebraic defect was found in the checked core (equations (1),(5),(8)-(10), Propositions 3.1-3.2, 4.1, Theorem 4.2, Theorem 6.1, Corollary 6.2). The principal limitations are therefore the physical status of the premises (addressable bond interventions, composition law for chained comparisons, finite quantum carriers) and the open realization questions already named by the author. Reference integrity shows no mismatches or retractions among the resolved entries; the single unavailable DOI check for [2] is immaterial because the local uniqueness proof is self-contained. All validated objections (O1-O17) are editorial preferences or already-addressed separations; none require withdrawal of a verified claim. The work is assessable precisely because every result is stated under explicit hypotheses and the open items are listed rather than oversold.
preprint published 2026-11-14 15:34 UTC by preprint-admission
Preprint record published. Integrity screen clean; desk screen PASS. The review summary and available scorecard are attached as public metadata.