The manuscript opens with an operational adequacy principle: a report q of states or histories is adequate for a fixed family of experiments precisely when every outcome factors as O_a = O-bar_a circ q (Eq. 2, Section 2.2). Figure 1 and the introductory calculation with J(x) = (x + x^{-1})/2 - 1 = (x-1)^2/(2x) (Eq. 1) show that the scalar cost alone fails this test: the states x=2 and x=1/2 both cost 1/4, yet the same multiplicative control r=2 produces signed cost changes +7/8 and -1/4. The page images confirm the printed fraction and the numerical values; the algebra is elementary and correct. Adequacy therefore forces retention of distinctions that admitted actions can expose. Proposition 2.1 records the standard initiality of the free inductive carrier generated by 0 and S (Dedekind recursion), which fixes the arithmetic of finite records once a starting point and successor are chosen; the manuscript correctly treats the physical reading of a record as a separate premise and notes that order of occurrences can still matter when later queries distinguish it (citing the ordered-loop work [3]).
Section 3 derives the reciprocal cost from a signed-postings construction. Squaring the signed pair (a,b) produces a^2 + b^2 positive and 2ab negative postings; the net signed count is (a-b)^2, and the minimal unsigned representative in that class is exactly (a-b)^2 (Eq. 3). Reading the residual against the comparison's own cross postings 2ab yields J(a/b) once the meter m(d,r) is required to be nonnegative, additive on pooled records, and normalized by m(r,r)=1 (Proposition 3.1, Eq. 4). The page image of Figure 4 for (a,b)=(3,1) matches the arithmetic (residual 4, cross 6, ratio 2/3). The composition law J(xy) + J(x/y) = 2J(x) + 2J(y) + 2J(x)J(y) (Eq. 5) follows by direct expansion; the numerical check x=y=2 recovers 9/8 on both sides. Proposition 3.2 then shows that any F satisfying the same law together with the unit quadratic calibration lim_{t->0} 2F(e^t)/t^2 = 1 must equal J. The proof sets H(t)=1+F(e^t), obtains the doubling relation H(2t)=2H(t)^2-1, and recovers H(t)=cosh t by arcosh iteration and the calibration asymptotic; the argument is the standard uniqueness proof under those functional hypotheses and does not invoke extra smoothness. The manuscript is explicit that the composition law itself is a physical premise about chained comparison readings, and that alternative references (a^2+b^2 or (a+b)^2) fail the law. Continuity is invoked to pass from positive rationals to positive reals; stating the domain of m (natural counts or a real cone) and naming that continuity premise would improve readability (editorial preference O3/O6/O10) but does not expose a counter-example under the natural intended hypotheses.
Section 4 lifts the single-bond cost to a directed ledger of bonds with positive multipliers x_b and endpoints s(b), t(b). Incident cost C_i and signed logarithmic flow sigma_i are defined by Eqs. (6)-(7). An addressed intervention T_{b,r} multiplies one bond by a known r>1. Proposition 4.1 expands the response h_r(x)=J(rx)-J(x) to the closed form (r-1)/2 (x - 1/(r x)), inverts it by the unique positive root of the resulting quadratic (Eqs. 8-9), and proves the Lipschitz bound and the strict positivity of two successive responses. Theorem 4.2 then shows that the triple consisting of the first cost change, the second cost change under the same control, and the first signed-flow change recovers every active bond, its multiplier, and its orientation (or self-loop). The proof is elementary once the response properties are granted: successive cost changes vanish together exactly on non-incident addresses, the first response recovers x_b even when zero, and flow signs fix direction. Figure 5 supplies an explicit numerical check (x_b=2, r=2, h_2=7/8 recovers x=2). The result is conditional on addressability of bonds and calibrated cost/flow meters; the text correctly states that writing the intervention does not establish physical availability or permissibility.
Section 5 identifies economic transitions with ledger actions. An economic state supplies actual and required positive signals u_b, v_b; the ratio x_b = u_b/v_b and the strains a_b = ell_b(t) - ell_b(s) realize the transition exactly under the multiplicative action (11), and they are the unique active strains that do so (Proposition 5.1). Harm at participant i is defined as the signed change in incident cost H_i (Eq. 12). Because J(x)=J(1/x), surplus and shortfall are priced alike by the cost while the sign of ell_b (kept by the flow) distinguishes them. Endpoint telescoping (Eq. 14) shows that path histories with identical start and end costs are invisible to endpoint sums; Figure 7 exhibits the concrete pair of histories (double then halve versus no action) that an answerable record can still separate. The manuscript repeatedly warns that a numerical H_i does not become morally significant merely by entering J, and that consent, authority and care concern answerable histories rather than endpoint sums (already-addressed objections O2, O5, O9, O13). These qualifications keep the ethical and economic readings as conditional modeling identifications that still require physical identification of participants, needs and available actions.
Section 6 treats a finite quantum system. A CPTP report E that preserves perfect binary discrimination on the d(d-1)/2 basis pairs and the d(d-1)/2 sum/difference pairs (Eq. 15) must satisfy the Knill-Laflamme condition K_a^* K_b = alpha_ab I on the whole input space. The proof converts perfect discrimination into orthogonal output supports, hence vanishing matrix elements of the Kraus Gram operators on the tested pairs, forcing the scalar multiples of the identity. The resulting normal form is the reversible-channel structure of Nayak and Sen: E(rho) = V (rho tensor tau) V^* (Eq. 16). An explicit CPTP recovery R is written in Eq. (18); the identity R E = id survives tensoring with any finite retained reference. For a qubit the two experiments Z and X suffice, and the Y component is recovered even though never tested (Figure 8). Corollary 6.2 transports every instrument on recoverable systems by conjugation with the recoveries (Eq. 20 and Figure 9), preserving outcome probabilities, conditional states and reference correlations along finite adaptive protocols. The theorem is stronger than numerical tomography from two expectation values precisely because it constrains one physical channel. Adding an explicit definition of "preserves perfect binary discrimination" as optimal success probability one (equivalently orthogonal supports) to the theorem statement is a presentation preference (O4/O7/O11), not a repair of an incorrect claim; the proof already uses that equivalence. The manuscript correctly separates preservation of a dynamics from derivation of which dynamics nature admits.
Section 7 and Appendix A keep the logical status of each claim explicit. Golden scaling and the three-dimensional recognition argument are presented as further conditional selection results under named physical requirements; they are not derived from finite arithmetic alone. Realization constructions (finite-instrument embedding, field-channel approximation, coupled reciprocal graph system) are cited as separate manuscripts that supply implementation under named controls, not as completed universality proofs. Experiential identity is flagged as a further, specific identification that the recovery theorems neither require nor settle. The universal-completeness question is stated exactly: whether the derived carriers and enacted controls realize every physical distinction and transition together with occurrence probabilities remains open and requires selection of carriers, interactions, apparatus and an occurrence law. The abstract closing sentence, Figure 2 bottom band, Section 7 and Appendix A already supply that limit (already-addressed objections O1, O8, O12). The reference list is short, relevant and resolver-verified (Knill-Laflamme, Nayak-Sen, and the author's own delta-calculus and uniqueness papers); no integrity issues appear. The conditional theorems are correctly proved under their stated hypotheses; the principal limitation is the still-open realization half of the program, which the text already acknowledges. Minor clarifications of meter domain, the perfect-discrimination definition in the theorem statement, and one intermediate line in the arcosh argument would improve readability without altering any result.
preprint published 2026-11-14 13:25 UTC by preprint-admission
Preprint record published. Integrity screen clean; desk screen PASS. The review summary and available scorecard are attached as public metadata.