{"id":"a2e6b765-aafb-454f-baea-aa69b3f64e32","arxiv_id":"2607.19681","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"The induced visual vertical is the half-argument of the first circular moment of stimulus orientation in the doubled-angle domain, reproducing the Li–Matin tracking, combination, and cancellation rules in closed form with no per-configuration angle parameters.","lead":"A single formula — half the argument of the first circular moment of stimulus orientation at doubled angles, φ = ½ arg Σ Aⱼ e^{i2γⱼ} — reproduces in closed form the three classic Li–Matin rules for how tilted lines bias the perceived vertical. If correct, it unifies perceived vertical and eye level as two readouts of one image-computable order parameter and hands the field two decisive untested predictions.","discovery_kind":"unification","skeptic_critique":{"model":"deepseek-v4-flash","headline":"VPEL dissociation rests on an unselected per-hemifield normalization assumption that the trial re-analysis cannot confirm because it was constructed to match it.","rationale":"The reader's weakest_assumption exactly matches my identified concern: the per-hemifield pooling and cross-hemifield combination is an explicit, structurally load-bearing assumption not derived from the order parameter. My independent reading of §5.5 confirms the author's own flag: \"This is an explicit structural assumption... nothing in §2 forces normalisation to stop at the midline.\" The confirmatory re-analysis's qualitative signs and nulls were constructed to match the sum/difference architecture, leaving only the k1 ≈ 0.84–0.90 sub-additivity as out-of-sample quantitative support. That single number, while suggestive, does not distinguish between the proposed two-level pooling and alternatives like a global c0 with a spatial weight kernel, or a per-hemifield renormalization with a scaling factor. The paper is admirably honest about this and about the VPV combination slope not being separable from averaging (p=0.29), but the title claim \"vertical and eye level as one orientation order parameter\" rests on this assumption. Thus the reader's CONDITIONAL verdict is appropriate: the VPV half is strong and well-supported, but the VPEL half requires either a derivation of the midline stop from first principles or a forward experimental test. My proposed concrete test—a same-hemifield unequal-salience pair—would discriminate between within-hemifield normalization and global pooling, which the current four-stimulus set cannot. I agree with the reader's assessment and recommend no change to the verdict.","tokens_in":26727,"tokens_out":1900,"duration_ms":19019,"concrete_test":"Run a new two-hemifield experiment with two lines of unequal salience placed in the same hemifield (e.g., both in the right field with 1:2 contrast ratio). Under the paper's two-level pooling, the within-hemifield c0 normalization predicts the pair's contribution to VPV follows averaging toward 1/n (k1 ≈ 0.5 for long lines), while the VPEL difference readout s_R−s_L should be blind to within-hemifield averaging and show a different combination coefficient. Alternatively, computationally compare the paper's per-hemifield pooling against global c0 pooling and per-hemifield renormalized pooling on the four stimulus configurations: if all three models can be tuned to match the 2013 data by adjusting kV and the cross-field discount, the concern lands. The decisive experiment is the unequal-salience same-hemifield pair, because it distinguishes within-hemifield normalization from global poolin","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central VPV claim, φ=½ arg c1, is mathematically sound and its combination-table fit is honest, but the VPEL half of the title, \"vertical and eye level as one order parameter,\" depends on a structural assumption the paper itself flags in §5.5: c0 pooling stops at the hemifield midline, and the two hemifield signals are then added/subtracted without further renormalization. The author states this is not forced by §2 and is justified by the projective-geometry argument and, ultimately, the fit to the 2013 data. That last justification is circular: the confirmatory re-analysis uses a stimulus set (1L, 1R, 2P, 2S) that was designed so that the sum/difference architecture predicts exactly the observed signs and nulls. The out-of-sample component claimed is only the sub-additive cross-field k1 ≈ 0.84–0.90, but that is a single quantitative point, not a test of the midline-stopping rule itself. If pooling were global (one c0 across both hemifields), or if each hemifield were internally renormalized before combination, the predicted VPV/VPEL dissociation would not follow. The paper's own acknowledgment that nothing in §2 forces the midline stop means the strongest load-bearing assumption is not derived from the order parameter but is an additional structure imposed on it. This does not invalidate the VPV analysis, but it means the claim that VPV and VPEL are readouts of the same order parameter is not established to the same standard as the VPV rules.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper claims that the three Li–Matin rules for induced visual vertical (VPV) are closed-form consequences of a single order parameter: φ = ½ arg c1, c1 = ∫ E(θ)e^{i2θ}dθ = κ̂1ΣA_j e^{i2γ_j} (Eq. 1), the first circular moment of the orientation distribution in the doubled-angle domain. R1 (unit-slope tracking), R2 (symmetric cancellation), and R3 (phasor-additive combination) follow immediately; R4 gives a |cos 2θ| strength law. The combination coefficients for 2-, 3-, and 4-line stimuli are matched with mean |Δ|=0.030 using a single length-saturation function taken from Li–Matin's own fit. The paper then extends the same order parameter to perceived eye level (VPEL) by reading the two hemifield signals as a sum and a difference, and reports a confirmatory re-analysis of 30-observer trial data. The paper contains explicit caveats that the VPEL step requires an additional structural assumption, that the long-line data cannot separate the model from simple averaging, and that the square-frame magnitude envelope is supplied by an auxiliary cardinal-anisotropy term.","tokens_in":27140,"tokens_out":10116,"duration_ms":127551,"significance":"If the VPV half is correct, the paper provides a clean unification: the induced vertical is the principal axis of the orientation structure tensor / population vector, and the Li–Matin rules are signatures of any doubled-angle circular mean. The derivations are exact and the paper is exemplary in stating what it cannot decide (long-line averaging, square-frame envelope, VPEL architecture). However, the title's stronger claim — that VPV and VPEL are one order parameter — is not established to the same standard: the VPEL sum/difference architecture is an additional postulate, and the confirmatory re-analysis was constructed to match it. The VPV quantitative fit also depends on an external length-saturation function rather than on Eq. (1) alone. The paper would be a solid theoretical contribution if the scope claims were aligned with the evidence and the VPEL architecture given a discriminating test.","major_comments":[{"comment":"The VPV/VPEL extension in §5.5 rests on an explicit structural assumption: c0 pooling is per hemifield, and the two hemifield signals are combined as s_L+s_R and s_R−s_L without further renormalization. The paper correctly flags that 'nothing in §2 forces normalisation to stop at the midline,' but the confirmatory re-analysis cannot establish this assumption. The four stimuli (1L, 1R, 2P, 2S) were selected so that the sum/difference architecture predicts exactly the observed signs and nulls; the only out-of-sample quantity is the cross-pool discount k1≈0.84–0.90, a single scalar that does not discriminate hemifield-stopped pooling from global pooling or per-hemifield renormalization, and for which no uncertainty is reported. Because the title/abstract claim that VPV and VPEL are 'one order parameter' depends on this assumption, the paper should either add a test that discriminates the po","section":"§5.5 (Table 2; confirmatory re-analysis)"},{"comment":"The text attributes the predicted averaging slope k1 = 1/n to the c0 normalization of the first moment, saying 'The c0 denominator is “the whole is less than the sum of its parts.”' But φ = ½ arg c1 is unchanged by dividing c1 by c0, and for n aligned inducers arg c1 = 2γ for every n. The slope k1 = S(nℓ)/nS(ℓ) in §5.2 is therefore not a consequence of the normalized first moment; it is an additional length-saturation assumption taken from Li and Matin's own Fig. 5. As written, the derivation suggests that Eq. (1) plus c0 normalization predicts the combination table, whereas the table's magnitudes are actually fixed by an external scalar law. Please correct this attribution and state explicitly that the 1/n law in the saturated limit is a property of the length-saturation model, not of Eq. (1) alone.","section":"§5.1–§5.2 (Eq. 1; k1 = S(nℓ)/nS(ℓ))"},{"comment":"For the square frame, c2 has constant magnitude |c2| = 2, so the non-monotonic rod-and-frame magnitude envelope is supplied by an unterived cardinal-anisotropy term peaking near 24.6° against the observed 15–20° (a 5–9° overshoot). The paper acknowledges this, but it means the square-frame account is not a closed-form prediction of the order parameter; it is the order parameter plus a separate cardinal-reference model. Because the square is the classic rod-and-frame stimulus and §5.4 presents c2 as the resolution of where the square's energy resides, the scope of the 'closed-form' claim should be stated even more prominently. Together with the hemifield assumption and the length-saturation function, this means the full 'one order parameter' story is actually the core c1 plus at least three auxiliary structures. This does not affect the 1- and 2-line c1 results, but it is central to how t","section":"§5.4 and §5.6 (c2 and cardinal anisotropy)"}],"minor_comments":[{"comment":"The phrase 'the 30-observer trial data confirm the predicted sub-additive cross-field combination' overstates the evidential value of a re-analysis whose signs and nulls were constructed to match the sum/difference assumption. Consider 'are consistent with' or explicitly noting that only the quantitative discount is out-of-sample.","section":"Abstract and §5.5"},{"comment":"The acknowledged 5–9° overshoot of the rod-and-frame peak should appear in the abstract or conclusions, not only in the body, to avoid the impression that the square-frame prediction is exact.","section":"§5.4"},{"comment":"The closing poetic stanza is unusual for a journal article. Consider moving or removing it; it does not affect the scientific content.","section":"§7 / closing verse"},{"comment":"The statement 'we are not aware of a prior account that explicitly formulates the induced visual vertical as this doubled-angle first moment' is well hedged, but given the extensive related work already cited (Dakin & Watt 1997; Tzvetanov 2012; Kass & Witkin 1987), the novelty paragraph could be tightened to emphasize that the contribution is the transfer to Li–Matin's specific rules, not the operator itself.","section":"§4 / novelty discussion"}],"recommendation":"major_revision","confidential_remarks":"The paper is mathematically careful and unusually candid about its own limitations, but the title/abstract overreach the evidence for the VPEL half of the claim. The core VPV analysis is sound and the issues are fixable by reframing and, ideally, a targeted experiment that discriminates the hemifield-stopped pooling from global pooling or per-hemifield renormalization. I recommend major revision rather than rejection. The disclosure of AI use is appropriate; the reliance on a preprint (Shavit et al., 2026) for part of the empirical support should be clearly marked as such in any revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This paper divides cleanly into two claims, and you should keep them separate when you read it. The first — that Li and Matin's three induced-vertical regularities are the closed-form behavior of one doubled-angle first-moment readout, φ = ½ arg c₁ — is exact. R1–R4 follow by algebra once you grant the definition; the paper is upfront that the operator itself is standard (Dakin & Watt's centroid, the structure tensor, a population-vector decode); and the novelty is honestly scoped to the transfer: the explicit formulation of induced VPV as this moment, the derived combination slope k₁ = S(nℓ)/nS(ℓ), and the VPV/VPEL sum-difference. The fit to the published 2-, 3-, 4-line table with no per-configuration free parameters (mean |Δ| = 0.030) is real, and the paper deserves credit for stating the p = 0.29 non-separability from simple averaging and the open |cos 2θ| and triplet tests rather than burying them.\n\nThe second claim — eye level as the antisymmetric readout of the same order parameter — is where the soft spot is, and the stress-test note locates it correctly. VPV ∝ s_L + s_R and VPEL ∝ s_R − s_L require c₀ pooling to stop at the hemifield midline, and nothing in §2 forces that. The paper flags it explicitly in §5.5, which is honest, but the consequence stands: the qualitative signs and nulls of the 2013 re-analysis were constructed to match that assumption, and the only out-of-sample number is the cross-field discount k₁ ≈ 0.84–0.90. The projective-geometry argument (roll maps to same-sign tilts in both fields, pitch to opposite) is a genuine independent motivation, so I'd call it conditional, not circular — but it is an added structure, and the title's 'one order parameter' overstates what is established.\n\nSmaller items: the effective-salience front end Aⱼ and the cardinal-reference term carry unspecified degrees of freedom, and the trial-level re-analysis needs the 30-observer dataset and code deposited under a commit hash — a referee should be able to check the 0.84/0.90 numbers themselves. The self-citations are mostly to the author's own published 2013 study; that's legitimate here, and the paper discloses the re-analysis as secondary.\n\nWorth a serious referee: yes. The VPV account is a clean, falsifiable unification with unusually candid reporting; the VPEL half is the part a referee should push on — ideally by demanding either a derivation of the midline stop or a direct hemifield manipulation that separates midline-stopped from global pooling. I'd take it as conditional acceptance, with the two claims kept at different confidence levels.","headline":"VPV half is a genuine closed-form unification with honest quantitative contact; the VPEL half is an explicitly assumed extension, and the decisive experiments are still open.","tokens_in":27666,"tokens_out":7841,"would_cite":true,"duration_ms":77728,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper claims that the visually perceived vertical is the half-angle of the circular mean of the scene's orientation votes, and that the same vote-sum read as sum and difference across hemifields also explains perceived eye level.","keywords":["visually perceived vertical","perceived eye level","orientation order parameter","doubled-angle circular mean","Li-Matin rules","structure tensor","population vector","sub-additive combination"],"falsifier":"Measure the induced vertical for a symmetric pair of lines held at equal tilt about vertical while sweeping the half-separation θ from 0 to 90 degrees: the model predicts strength proportional to |cos 2θ| with a null at exactly 45 degrees and a reversal beyond; a plain averaging model predicts a flat, never-nulling effect.","tokens_in":26516,"feed_emoji":"🧭","tokens_out":3532,"duration_ms":37896,"temperature":0.7,"pith_summary":"This paper tries to show that three empirical regularities of the visually perceived vertical — a single tilted line shifts it roughly one-for-one, several lines combine sub-additively, and symmetric tilts cancel — are not separate findings but closed-form consequences of one quantity: the first circular moment of the stimulus orientation distribution in the doubled-angle domain. Because orientation is an axis with no arrowhead, a circular-mean readout must work at twice the angle, so the proposed formula is the natural way to average orientations. If the account is right, the same order parameter, read as a sum and a difference across the two visual hemifields, yields perceived vertical and perceived eye level, unifying two research lines under one image-computable descriptor. The paper matches published 2-, 3-, and 4-line combination slopes using a single length-saturation function, with no per-configuration free parameters in the angles.","feed_headline":"One orientation vote-sum rules perceived vertical and eye level","feed_subtitle":"Same first-moment readout, read as sum and difference across hemifields, reproduces both integration rules with no fitted angles.","key_machinery":"The order parameter φ = ½ arg c₁ with c₁ = ∫ E(θ) e^{i2θ} dθ — equivalently the principal axis of the orientation structure tensor or the population vector of cosine-tuned orientation units. Orientation is a director, so the readout must use doubled angles; the first circular moment is the unique circular mean respecting that symmetry. The normalization by c₀ = ∫ E dθ produces the sub-additive combination law, and the length-saturation function S(ℓ) from the original data sets all magnitudes.","core_discovery":"The central claim is that the induced vertical is φ = ½ arg c₁, where c₁ is the first circular moment of the orientation-energy distribution taken at doubled angles: each line at orientation γ contributes a phasor at 2γ, weighted by its salience, and the readout is the direction of the sum, halved back. This is also the principal axis of the image structure tensor and an orientation population vector. The angle doubling is forced by the director symmetry of orientation (θ ≡ θ+π). From this single definition four rules follow: unit-slope tracking for one line, exact cancellation of a symmetric pair, phasor-additive combination that reduces to linear averaging at small tilts, and a |cos 2θ| st","pith_inferences":["If the order parameter is the readout, natural scenes with a coherent oblique bias should produce larger induced vertical than balanced cardinal-dominated scenes; the paper notes the extension but does not test natural images.","The 45-degree null experiment would also distinguish pooled averaging from winner-take-all nonlinear readouts, a discriminator that generalizes beyond this paradigm.","Because the first-order readout is position-blind, adding an eccentricity weighting kernel would yield quantitative predictions for peripheral inducers; the paper flags this as future work.","The sum/difference structure suggests individual differences in frame dependence are a single gain applied to an unchanged order parameter, testable by measuring both tasks in the same observers."],"forward_implications":["The Li–Matin rules are signatures of any doubled-angle circular-mean readout, not independent empirical laws.","The same order parameter, read as hemifield sum and difference, explains the reversed integration rules for perceived eye level.","A symmetric pair about vertical should show a |cos 2θ| strength curve with a null at 45 degrees and a reversal beyond — a decisive, not-yet-run test.","Complete summation is refuted; combination slopes approach 1/n as inducers saturate, making 'the whole is less than the sum of its parts' quantitative.","A square frame has zero first-moment vertical; its residual rod-and-frame effect lives in the second harmonic, explaining the 90-degree periodicity and the 45-degree null."],"fun_headline_variants":["One circular moment sets perceived vertical and eye level","Angle-doubling order parameter explains Li-Matin rules","Perceived vertical and eye level from a single orientation vote","Closed-form law: doubled-angle sum sets vertical and eye level","A single phasor sum predicts vertical and eye level"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The eye-level half of the account rests on the assumption that orientation signals are pooled within each hemifield and then combined across the midline without further renormalization — if pooling were global, or each hemifield normalized before combination, the VPV/VPEL dissociation would not follow.","fun_headline_variants_meta":{"raw":{"variants":["One circular moment sets perceived vertical and eye level","Angle-doubling order parameter explains Li-Matin rules","Perceived vertical and eye level from a single orientation vote","Closed-form law: doubled-angle sum sets vertical and eye level","A single phasor sum predicts vertical and eye level"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000795,"raw_usage":{"total_tokens":3434,"prompt_tokens":939,"completion_tokens":2495,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":683,"completion_tokens_details":{"reasoning_tokens":2429}},"tokens_in":683,"tokens_out":2495,"duration_ms":17304,"temperature":1.0,"reasoning_tokens":2429,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T12:01:55.412781+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Measure the induced vertical for a symmetric pair of lines held at equal tilt about vertical while sweeping the half-separation θ from 0 to 90 degrees: the model predicts strength proportional to |cos 2θ| with a null at exactly 45 degrees and a reversal beyond; a plain averaging model predicts a flat, never-nulling effect.","supporting_citations":[],"review_version":1}