The manuscript supplies a conditional uniqueness theorem for a yes/no predicate on three-number bilateral records. With address (o, w, d), sources x = o - d and y = w + d, Theorem 5.3 states that a candidate law L meets the six reading requirements R1, U, R5, RS, R4 and H if and only if L equals A: yes exactly when 0 < x and 0 < x + y. The proof technique is elementary orbit reduction. Lemma 5.1 uses R1 (whole posting of m) to collapse every address to the slice d = 0, so L becomes a function ell(x, y). Lemma 5.2 then shows that U (positive scaling), R5 (acknowledged-part diagonal moves that preserve x + y) and RS (vertical moves that preserve the overlaps s+ and s- of the two departures) force ell constant on the three regions P = {x > 0, x + y > 0}, N = {x < 0, x + y < 0} and Z = {x(x + y) <= 0}. The remaining two requirements fix the three bits: H separates the balanced record (0, 0, 0) from the unit half-posting (1, 0, 0), while R4 (raising the other's record never turns a yes into a no) forces the unique assignment yes on P and no elsewhere. The converse is immediate by direct substitution. No step invokes the cost J. Theorem 5.4 supplies an explicit counter-model for each dropped axiom, confirming necessity; the table on page 14 lists the six witnesses (A restricted to d = 0, the half-unit threshold, B, W, negation of A, and always-no). Figure 1 makes the geometry of the regions and the anchor line visible. These internal checks hold under the stated hypotheses; the domain is every real triple, with no regularity restriction beyond the definitions of the moves.
The paper is explicit that Theorem 5.3 is a characterization, not an independent derivation. Section 1 and the status table of Section 12 state that the six requirements were found by working backwards from A, that nothing grounds them independently of the law they select, and that no observation tests the identification H1 ("wronged" is precisely a reading of the pair record that meets the six). Rival requirement sets are constructed rather than dismissed. Theorem 7.7 shows that R1, R2, R3, R4 and R5 force the netted wedge W (|y| < x); Theorem 7.9 proves that RS and R2 are mutually exclusive already under R1 and H alone; Theorem 7.10 shows that the authorship sentence AU yields only the one-sided law B (0 < x); Theorem 8.5 shows that undirected averaged harm yields E (0 < x(x + y)). Lemma 3.4 isolates the exact record sets on which each rival parts from A, and Theorems 6.3-6.4 locate which of the six carry the contested answers (R4 + H fix only the anchor line; RS decides the two-sided departure; R5 decides the acknowledged excess). The under-determination result is therefore internal and cleanly stated: uniqueness holds inside the chosen six, not among all natural readings of the same three numbers.
Two distinct ledgers appear. The amount reading (Definition 2.4, Lemma 2.6) posts the gaps f_s - 1 and -(f_t - 1) and is used for the one-ledger premise H3, the standing-breach accrual of Section 9, and the books realization of Section 11. The ratio reading posts log(f_s / f_t) and carries the cost J; it is used only for the harm check of Section 8. Lemma 2.6 records that the two agree on whether the own records disagree and on which is higher, yet differ on records that agree. Consequently the averaged-harm identity of Lemma 8.2 (verified algebraically from J(u/v) = (u - v)^2 / (2uv)) and Theorem 8.3 recover A only on positive figures, with equal weights fixed by the exchange symmetry (Lemma 8.1) and with the entry taken in its writer's favour. Theorem 8.4 shows that, inside the four symmetries, the sign of the averaged harm carries only the single bit that the balanced record is no wrong. The two routes are therefore not interchangeable; the manuscript notes the distinction but does not join them by any theorem.
Extensions rest on further identifications. Section 9 proves that a pair's net source moves only under an end's own entry (Theorem 9.1) and that, under the amount-ledger premise H3, the remedy due equals the standing amount of an open undertaking (Theorem 9.3). "Bound" is characterized by the four posting requirements K1-K4 (Theorem 10.1), leaving two candidate rules that differ only at the extorted address (1, 1, 0); the open premise H7 selects K_code. "Permitted" is defined (H8) as wronging nobody and therefore coincides with meeting every non-extorted undertaking once A is in force (Theorem 10.3). Section 11 shows that the same law A, read through the amount translation, is the unique law compatible with the books' own basis (Theorem 11.3); Theorem 11.4 records that the books forget both authorship and the third record. All of these results are conditional on the listed hypotheses; the final assembly Theorem 12.1 simply packages them.
Relative to the author's own criterion paper [3], the second demand (requirements that fix every answer without writing any answer in) is met by Theorem 5.3 together with the polarity and anchor analysis of Section 6. The third demand (preservation across every faithful realization) is met only for one re-encoding of the same three numbers; the manuscript itself notes that the books are not independent evidence. The first demand is the identification H1, which remains an assumption. The cost uniqueness Theorem 2.2 is imported from [2] via d'Alembert's equation and is not used in the main characterization. Self-citations to institute manuscripts and prior editions dominate the reference list; three of eight items are unresolved by standard resolvers, though none appears fabricated. No engagement with external moral, legal or economic literatures is attempted.
Validated-objection reconciliation: every listed objection (O1-O30) is either already addressed on the page (H1 as identification, open grounds of RS/R4, two-ledger separation, preservation limit, underdetermination, conditional extensions, imported framework) or is an editorial preference about further tightening of title/abstract framing. No confirmed mathematical defect remains. The mathematical core is sound under its hypotheses, the status language (proved / identified / open) is unusually precise, and the rival-law analysis is a genuine contribution. The title, abstract and definite-article claim nevertheless present a stronger physical-moral derivation than the conditional characterization that is actually proved. Independent motivation for H1, for the open grounds of RS and R4, and for preservation beyond a single re-encoding would be required before the stronger reading could be sustained; absent that motivation the work is best read, and should be framed, as an exact if-then uniqueness theorem inside a stipulated identification.
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