{"id":"9d5fc4dc-e176-402b-a03b-7ba046ccff8e","arxiv_id":"2607.07950","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":5.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"CI* (normalized −c3 of the characteristic polynomial) is the unique natural scaling of that coefficient that preserves average consistency under leave-one-out submatrices, hence is size-independent.","lead":"The paper shows that a consistency index for pairwise comparison matrices is size-independent if it equals the average of the same index on all leave-one-out submatrices, and that the known CI* index satisfies this exactly. This gives a clean criterion for when a fixed numerical threshold can be compared across different numbers of alternatives in AHP.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified","rationale":"The Reader correctly isolates Claim 2 as a modelling choice rather than a theorem, and correctly notes that this is the sole reason for a CONDITIONAL rather than ACCEPT verdict. No deeper technical flaw exists: the combinatorial counting argument of Lemma 7 is elementary and correct, the scaling that produces the average-preserving property is forced once that identity is granted, and the coincidence with the already-published CI* is immediate. Because the paper never claims uniqueness of the axiom, only that CI* satisfies a natural size-independence property, the normative status of Claim 2 does not invalidate the mathematical result. The Reader’s assessment therefore needs no adjustment.","tokens_in":17345,"tokens_out":443,"duration_ms":5200,"concrete_test":"Independently expand both sides of the identity in Lemma 7 for a generic 5\times5 reciprocal matrix (or any fixed n≥4) by enumerating the triples in each C3(k); verify that every triple appears in exactly n−3 of the sub-sums. If the count holds, the subsequent scaling argument of Theorem 8 is forced and the central claim stands.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper’s central mathematical claim is correct and self-contained: Lemma 7 gives the exact combinatorial identity ∑ c3(A(k)) = (n−3)c3(A), and the scaling φ(A) = −6/(n(n−1)(n−2))c3(A) then yields the average-preserving equality of Theorem 8 by elementary algebra. The identity with CI* follows at once from the already-known relation (8). Claim 2 is openly presented as a modelling choice that motivates the average-preserving axiom; it is not asserted as a derived necessity. Consequently the only potential soft spot is normative rather than technical, and it does not undermine the correctness or the internal logic of the strongest claim.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The paper studies size-independence of consistency indices for pairwise comparison matrices (PCMs) in AHP. It introduces sub-PCMs (leave-one-out principal submatrices) and super-PCMs, motivates an average-preserving property φ(A)=(1/n)∑ φ(A(k)), and proves that the scaled coefficient φ(A)=−6/(n(n−1)(n−2))c3(A) of the characteristic polynomial is a consistency index satisfying that property (Lemma 7, Theorem 8). By the known identity of Brunelli et al., this index coincides with Peláez–Lamata CI*. Bounds relating Saaty’s CI of a super-PCM to its sub-PCMs are derived from Horn–Johnson (Theorem 10). Numerical experiments with random Saaty-scale PCMs for n=4…7 visualise the super/sub relationship for CI*, CI and CR.","tokens_in":17540,"tokens_out":1056,"duration_ms":10392,"significance":"If accepted, the average-preserving property supplies a clean, size-independent axiom that singles out CI* among common indices and gives a rigorous reason to prefer it (or an equivalent scaling of −c3) when thresholds must be comparable across matrix orders. The combinatorial identity of Lemma 7 is elementary, self-contained and correct; the reduction to CI* is immediate. The sub-/super-PCM viewpoint also opens a practical diagnostic route (identify items whose removal most improves consistency). Strengths include an explicit, parameter-free derivation, transparent counting argument, and reproducible random-matrix experiments. The contribution is incremental rather than transformative, but it is a solid, usable clarification of a long-standing practical question in AHP.","major_comments":[{"comment":"Claim 2 / Definition 8 (Section 3.2) is presented as the normative foundation of size-independence, yet it is only one modelling choice among several (e.g., worst-case sub-PCM, median, or max-min). The paper correctly notes that individual sub-PCMs can differ substantially from the super-PCM, but never tests whether average-preservation is the property practitioners actually need when they apply a fixed threshold such as 0.1. A short discussion or counter-example showing when the average and the worst sub-PCM diverge would strengthen the claim that CI* is the privileged size-independent index.","section":null},{"comment":"Section 4 and Figure 3: the numerical comparison of CR with the average-preserving line is suggestive but incomplete. For n=4 the black dots deviate markedly; for n≥5 they appear closer. Because RI itself is an empirical average of CI over random matrices, the apparent improvement may be an artefact of that normalisation rather than evidence that CR inherits average-preservation. A quantitative summary (mean absolute deviation of the black points from the diagonal, or a formal test) is needed before concluding that “normalisation by the RI has achieved its intended effect.”","section":null}],"minor_comments":[{"comment":"Abstract and Introduction: the phrase “refine our previously proposed index … demonstrating that it coincides with the existing consistency index” is accurate but under-states that the coincidence is already known (Brunelli et al., Prop. 5). A single clarifying sentence would avoid any impression of rediscovery.","section":null},{"comment":"Table 2 caption and footnote: the RI values for n=12 are flagged as possibly erroneous in Saaty; the source (Tone) should be cited more prominently in the table itself.","section":null},{"comment":"Figure 3: grey and black dots are hard to distinguish in monochrome print; a different marker or transparency would improve readability.","section":null},{"comment":"Theorem 10: the second inequality is simply the algebraic rearrangement of the first; stating both is redundant and can be condensed.","section":null},{"comment":"Notation: the symbol φ is used for the new index while c3 (or a3) is used for the characteristic-polynomial coefficient; a short glossary or consistent Greek/Latin choice would help readers.","section":null}],"recommendation":"minor_revision","confidential_remarks":"The mathematical core is correct and the paper is short and readable; the main risk is that the normative status of the average-preserving axiom is under-argued. I would not reject on that ground, but the authors should be asked to acknowledge alternative size-independence notions. Fit for a methods / decision-analysis journal is good; novelty relative to Peláez–Lamata and Brunelli et al. is modest but the sub-PCM framing is useful."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"The one thing worth knowing is that they give a short, correct combinatorial argument for size-independence of consistency indices and show that the natural scaling of −c3 (i.e., Peláez–Lamata’s CI*) is the unique index among the usual suspects that satisfies it. That is new, even though the index itself is not.\n\nWhat they do well: they introduce sub-PCM / super-PCM language, prove the exact triple-counting identity (Lemma 7: sum of c3 over the n principal submatrices equals (n−3)c3), and then scale so that the average of the sub-indices recovers the super-index (Theorem 8). The algebra is elementary and checks out; the reduction to CI* is immediate from the Brunelli et al. identity they cite. The eigenvalue bounds for Saaty’s CI (Theorem 10) are standard Horn–Johnson consequences and are stated cleanly. The random-matrix scatter plots simply confirm what the theorems already say. Self-citations are limited to their earlier algebraic work on c3, which is re-derived here, so no circularity burden.\n\nSoft spot, kept in proportion: the average-preserving property rests on Claim 2 (“average consistency of the sub-PCMs is comparable to that of the super-PCM”). That is a modelling choice, not a derived necessity. If one preferred a max- or worst-submatrix axiom, CI* would lose its privileged status. The paper is open about this, but a referee will want a short discussion of why the average is the right normative target. Everything else is solid.\n\nThis is for people who write AHP software, textbooks, or consistency-index surveys. The math is light enough for a methods journal; the result is genuine if modest. I would send it to peer review without hesitation. Worth a look if you care about axiomatic foundations of consistency measures; skip if you only need a new numerical index.","headline":"Clean combinatorial axiomatization of size-independence that correctly selects the already-known CI*; modest but solid AHP contribution.","tokens_in":18120,"tokens_out":486,"would_cite":false,"duration_ms":6295,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["90B50","15A18","62C99"],"pacs":[],"model":"grok-4.5","headline":"A consistency index for pairwise comparison matrices is size-independent exactly when it equals the average of the indices of all its one-item-smaller submatrices.","keywords":["analytic hierarchy process","pairwise comparison matrix","consistency index","size independence","average-preserving property","characteristic polynomial","sub-PCM","CI*"],"falsifier":"Construct or sample a family of pairwise comparison matrices for which the average of the submatrix indices systematically differs from the super-matrix index under CI*, or exhibit another natural axiom of size-independence that CI* fails while another index succeeds.","tokens_in":18237,"feed_emoji":"📐","tokens_out":715,"duration_ms":7429,"temperature":0.7,"pith_summary":"In the analytic hierarchy process, people judge how much more important one item is than another and assemble those judgments into a matrix. Because human judgments are never perfectly consistent, every method needs a number that says how far the matrix is from perfect consistency. Saaty’s classic numbers use a fixed threshold of 0.1 no matter how many items are being compared; it has never been clear whether that threshold means the same thing for three items as for ten. This paper answers the question by relating a full matrix (the super-matrix) to the smaller matrices obtained by dropping one item at a time (the sub-matrices). It proposes that a size-independent index must equal the average of the indices of its sub-matrices. The authors then show that a simple rescaling of the coefficient of the characteristic polynomial satisfies this average-preserving property and is identical to an already-known index called CI*. Random-matrix experiments confirm that CI* sits exactly on the equality line while Saaty’s indices only approach it for larger sizes. The result gives a clear mathematical criterion for deciding whether any future consistency number can be used across different numbers of alternatives.","feed_headline":"One average makes a consistency index size-independent","feed_subtitle":"A matrix’s score must equal the mean of the scores of all matrices that drop one item","key_machinery":"The average-preserving property: for every n ≥ 4, φ(A) equals (1/n) times the sum of φ over the n submatrices obtained by deleting one row and column. The property is proved by counting how many times each triple of entries appears when the characteristic-polynomial coefficients of all submatrices are summed.","core_discovery":"The consistency index defined by φ(A) = −6/(n(n−1)(n−2)) times the degree-(n−3) coefficient of the characteristic polynomial of a pairwise comparison matrix A equals the average of the same index on every principal submatrix of size n−1. That average-preserving identity characterises size-independence and, by a known algebraic relation, shows that φ coincides with Peláez and Lamata’s CI*.","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["Consistency index equals average on all leave-one-out submatrices","φ preserved as mean of values on every principal submatrix of size n-1","Size-independent index via averaging principal submatrices coincides with CI*","Average of smaller matrices yields size-independent consistency score","Submatrix-average property makes consistency index independent of size"],"cache_read_input_tokens":128,"weakest_assumption_plain":"The claim that the right way to compare consistency across sizes is to require that a matrix’s index equals the average of the indices of its one-item-smaller submatrices, rather than matching every submatrix or the worst one.","fun_headline_variants_meta":{"raw":{"variants":["Consistency index equals average on all leave-one-out submatrices","φ preserved as mean of values on every principal submatrix of size n-1","Size-independent index via averaging principal submatrices coincides with CI*","Average of smaller matrices yields size-independent consistency score","Submatrix-average property makes consistency index independent of size"]},"model":"grok-4.5","effort":"low","cost_usd":0.003022,"raw_usage":{"total_tokens":1077,"prompt_tokens":812,"num_sources_used":0,"completion_tokens":72,"cost_in_usd_ticks":30220000,"prompt_tokens_details":{"text_tokens":812,"audio_tokens":0,"image_tokens":0,"cached_tokens":128},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":193,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":812,"tokens_out":72,"duration_ms":2551,"temperature":1.0,"reasoning_tokens":193,"cache_read_input_tokens":128,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-10T14:51:01.682888+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Construct or sample a family of pairwise comparison matrices for which the average of the submatrix indices systematically differs from the super-matrix index under CI*, or exhibit another natural axiom of size-independence that CI* fails while another index succeeds.","supporting_citations":[],"review_version":1}