{"id":"ead11eb2-5d15-496e-8489-f10ea686d154","arxiv_id":"2608.09785","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"high","formal_verification":"none","parameter_count":5,"one_line_summary":"A relative-momentum-local construction from the t-J model treats cuprate superconductivity, pseudogap and strange metal as three organizations of one Cooper channel, and predicts zero-energy Cooper-surface ridges in 2e-ARPES.","lead":"A theory paper proposes that superconductivity, the pseudogap and the strange metal in copper-oxide superconductors are three different low-energy organizations of the same Cooper-pair channel. It predicts zero-energy pair ridges as a measurable two-electron photoemission signature.","discovery_kind":"unification","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Central claim is unsupported as stated: the paper postulates, rather than derives, the static and dynamic relative-momentum-local prescriptions, and Section 8.1 concedes that neither BKT disordering, the Cooper logarithm, nor Q→0 establishes this locality.","rationale":"The Reader's REJECT verdict is sound. I considered whether there was an even more decisive internal mismatch: H_PG is a zero-COM, static Hamiltonian, while Section 2 defines PG by finite widths ΔQ and ΔΩ; Section 8.1 explicitly calls this a 'matching ambiguity' and says the reduction to J_qδ_{qq'} remains to be derived. This strengthens the reader's concern rather than replacing it. The exact Cooper representation (Eq. 1), the exact block solution (Eq. 5) and the 2e-ARPES ridge prediction are genuine contributions, and I do not see them as fraudulent or careless; but they are not enough. The unification claim is precisely the statement that the same dressed pair kernel produces static SC, quasi-static PG and dynamic SM, and that statement fails unless the block-diagonal/dynamic forms are derived. The proposed test checks whether the off-diagonal terms omitted by H_PG are actually small; if they are, the paper should be reconsidered; if not, the central claim is unsupported. I therefore keep the reader's REJECT and recommend no verdict change.","tokens_in":16444,"tokens_out":12129,"duration_ms":114986,"concrete_test":"Compute the omitted off-diagonal d-wave interaction δH = -(2J/N_s)Σ_{q≠q'}φ_qφ_{q'}Δ†_qΔ_{q'} inherited from the exact Q=0 Cooper term, and evaluate its matrix elements between the exact block ground states of H_PG at x=0.10, T≈200K using Fig. 2 parameters. If max_{q≠q'} |⟨GS_{q'}|δH|GS_q⟩|/(J_q+J_{q'}) ≳ 0.1, the block-diagonal projection is uncontrolled and H_PG is not the low-energy sector of the t-J Cooper channel; if the ratio is an order of magnitude smaller, the static RML prescription has quantitative support. For the dynamic prescription, extract the retarded pair propagator from the particle-particle polarization at finite Q and check whether the low-frequency form is r_s+iℓ_sΩ+γ_s|Ω| over a window T,|ω|≪Ω_dyn.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's headline claim — that SC, PG and SM are three infrared organizations of one Cooper channel of the t-J model — requires that the dressed low-energy Cooper-pair kernel be block-diagonal in relative momentum, with static coupling J_q=√2Φ_dφ² (Eq. 4) and dynamic inverse propagator D^{-1}=r_s+iℓ_sΩ_m+γ_s|Ω_m| (Sec. 6). Both prescriptions are assumed, not derived. Section 8.1 states this explicitly: 'Neither BKT phase disordering, the Cooper logarithm nor the Q→0 limit establishes relative-momentum locality' and 'Deriving both RML prescriptions... remains an open problem.' The concern is not that RML locality is known false; it is an unproven projection from the exact interaction (Eq. 1) onto independent {q,−q} blocks. BKT disorder acts in real space, the Cooper logarithm governs the COM Q=0 channel, and Q→0 does not restrict relative momentum; none of these generates J_qδ_{qq'}. There is also a matching gap within the PG claim: Section 2 defines PG as a finite-width peak (ΔQ∼ξ_PG^{-1}, ΔΩ∼τ_PG^{-1}), but H_PG retains strictly zero-COM, static pair operators; Section 8.1 admits the reduction to J_qδ_{qq'} from the finite-width kernel 'remains to be derived.' The exact solvability and the one-loop MFL form are therefore results for a model with RML locality imposed, not consequences of the t-J model unless the missing derivation is supplied.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a relative-momentum-local (RML) theory of cuprates in which the antiferromagnetic superexchange of the t-J model, rewritten as an exact spin-singlet Cooper-pair interaction (Eq. 1), organizes low-energy physics into three regimes: phase-coherent static d-wave superconductivity, a phase-incoherent quasi-static pseudogap, and a phase-incoherent dynamic strange metal. A static RML prescription (Eq. 4), coupling independent relative-momentum blocks with J_q(x,T) = √2 Φ_d φ²_{q,d}, yields an exactly solvable block Hamiltonian with nodal electron poles, antinodal gaps, resolution-controlled Fermi arcs, and gapless charge-2e Cooper surfaces; a Cooper-surface Lifshitz transition is predicted to produce logarithmic enhancements in c_V/T and the charge response without a spin singularity. A dynamic RML prescription, with a block-local pair-field inverse propagator D^{-1}_Ξ = r_s + iℓ_s Ω_m + γ_s |Ω_m|, yields a one-loop marginal-Fermi-liquid self-energy. The paper predicts zero-energy Cooper-surface ridges in two-electron ARPES as a decisive signature. Both RML prescriptions are labeled as postulates, and Section 8.1 states that deriving them from a common dressed Cooper kernel of the t-J model is an open problem.","tokens_in":16862,"tokens_out":25297,"duration_ms":199136,"significance":"The unification claim would be a substantial advance if the RML prescriptions could indeed be derived from the t-J model, since it would connect single-particle, pair-sensitive, and thermodynamic probes through a single Cooper channel. The manuscript has genuine strengths: the exact Cooper representation identity (Eq. 1) is a correct starting point; the 16-dimensional block solution of H_PG is genuinely exact and internally consistent (Eqs. 5-6); the Lifshitz-transition mechanism rests on a parameter-free symmetry argument at the van Hove points; the nodal scaling forms for arc width and length are explicit and falsifiable; and the paper names concrete experimental signatures (2e-ARPES ridges, charge-spin asymmetry at the Lifshitz doping, and ω/T collapse of −Im Σ/T). The paper is also unusually candid about its open problems. However, the central claim is not established: the PG and marginal-Fermi-liquid results follow from two assumed models with free parameters, not from the t-J model, and Section 8.1 identifies the derivation of the RML prescriptions as the 'central microscopic task' that remains open.","major_comments":[{"comment":"The static RML prescription is postulated, not derived. Equation (4) restricts the exact interaction (Eq. 1) to a block-diagonal Hamiltonian with J_q(x,T) = √2 Φ_d φ²_{q,d}, and the PG phenomenology that follows (gap ∝ φ²_{q,d} along the PG contour, Fermi arcs, Cooper surfaces) is a property of this assumed block structure. Section 8.1 concedes the point: 'Neither BKT phase disordering, the Cooper logarithm nor the Q→0 limit establishes relative-momentum locality,' and 'the reduction to J_q δ_{qq′} remains to be derived.' The three candidate mechanisms are indeed insufficient: BKT disordering acts in real space, the Cooper logarithm organizes the Q=0 channel, and Q→0 does not restrict the relative momentum q. A concrete test would be to evaluate the inter-block correlation ratio P_{qq′}(0,0)/[P_{qq}(0,0)P_{q′q′}(0,0)]^{1/2} in the Gaussian theory of Section 6; if this ratio is not parametrically small in the relevant window, the block-diagonal model (4) is not the infrared limit of the t-J Cooper channel. Until such a check is supplied, the exact solvability and the arc and Cooper-surface results are results for a model with locality imposed, not consequences of the t-J model.","section":"§4, Eq. (4); §8.1"},{"comment":"The dynamic RML prescription assumes the inverse propagator D^{-1}_{Ξ,0}(iΩ_m) = r_s + iℓ_s Ω_m + γ_s |Ω_m| with free coefficients r_s, ℓ_s, γ_s and a further free coupling constant J_s, and it is labeled 'postulated to have the same infrared form as D^{-1}_{Ξ̄}'. The marginal-Fermi-liquid decay rate (Eq. 7), the Kramers–Kronig logarithm in Re Σ^R_s, and the logarithmic suppression of the quasiparticle weight Z_s(ω) all follow mechanically from this assumed form, so the MFL scaling is built into the model by construction rather than obtained from the t-J model. Section 8.1 acknowledges that 'reducing the finite-Q sector to a channel-independent Ξ_q requires microscopic justification' and that the MFL benchmark additionally needs r_s > 0, a causal ultraviolet completion, and a non-collapsing scaling window. The Gaussian particle-particle bubble of Section 6 provides a kinematic origin for Ohmic damping, but the step from γ_{αα′}(Q,T) to a channel-independent, approximately isotropic γ_s|Ω_m| form is not shown. I would regard the unification claim as supported only if the dressed inverse propagator obtained from the fermion determinant is shown to reduce to, or be bounded by, the assumed form in the transverse-intersection annulus T ≪ v̄_F |Q| ≪ E_F.","section":"§6, Eq. (7); §8.1"},{"comment":"There is a matching gap between the operational definition of the pseudogap and the model used to describe it. Section 2 defines the PG by a finite-width collective peak with ΔQ ∼ ξ^{-1}_PG and ΔΩ ∼ τ^{-1}_PG, whereas H_PG retains strictly zero-COM, static pair operators. Section 8.1 states that a normalized weighted average over |Q| ≲ ΔQ and |Ω| ≲ ΔΩ is only a 'possible matching prescription,' and that H_PG 'does not determine finite-Q pair propagation and lifetime' because the reduction to J_q δ_{qq′} 'remains to be derived.' Consequently, the abstract's central claim that the RML theory 'relates single-particle, pair-sensitive and thermodynamic probes to three infrared organizations of one Cooper channel' is not demonstrated: the three regimes are organizations of two separately postulated models, and the common dressed Cooper kernel that would unite them is exactly the missing step, as Section 7 also states ('deriving both RML prescriptions from a common dressed Cooper kernel of the t-J model remains an open problem').","section":"§2, §4, §7, §8.1"}],"minor_comments":[{"comment":"The manuscript contains numerous typesetting artifacts, including 'hostd-wave' in the abstract, 'thet-J model,' and inconsistent subscript rendering ('TΔd', 'T F', 'T (0) BKT'); a careful production pass is needed.","section":"Abstract, throughout"},{"comment":"The caption of Figure 1 reproduces the adjacent main-text paragraph almost verbatim and should be shortened to describe the panels.","section":"Figure 1"},{"comment":"The claim that the d-wave saddle has a lower Helmholtz free energy than the extended-s-wave saddle 'for the parameters used here' is asserted without a shown calculation; please display the free-energy comparison or provide a reference.","section":"§3"},{"comment":"The symbols D^{-1}_{Ξ,0} and D^{-1}_{Ξ̄} differ only by subscripts and are easy to confuse; distinct notation would help.","section":"§6"}],"recommendation":"reject","confidential_remarks":"The author is transparent about the postulates and open problems; the deficit is one of claim-to-evidence rather than diligence. If the journal publishes phenomenological model studies, a substantially reframed version — presenting H_PG and S_s as conjectured effective models of the t-J Cooper channel, with the unification claim downgraded to a conjecture — could be viable, but the current title and abstract promise a derivation that the manuscript itself says is missing. The predictive content is also diluted by the number of free inputs (band parameters, Φ_d, r_s, ℓ_s, γ_s, J_s, Γ, and the windows Q_0, Q_1, Q_2, Ω_c), so the falsifiability of the MFL and Fermi-arc results is weaker than that of the 2e-ARPES prediction. I found no evidence of misattribution or missing references."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First thing to know: this paper has a real exact result, and it is honest about its own main gap. The exact spin-singlet Cooper representation of superexchange, Eq. (1), is a useful formal identity. The block-factorized PG Hamiltonian is exactly solvable and reproduces nodal-antinodal dichotomy, Fermi arcs as resolution-broadened features, and gapless charge-2e Cooper surfaces. The Cooper-surface Lifshitz transition with log anomalies in c_V/T and charge response, but not spin, is a concrete and falsifiable prediction. The 2e-ARPES zero-energy ridge signature is a clear target. The paper clearly separates what is derived from what is assumed, and Section 8.1 lists the open problems with unusual candor.\n\nThat said, the abstract's central claim — that SC, PG and SM are three infrared organizations of one Cooper channel of the t-J model — is not supported as stated. The static and dynamic RML prescriptions are postulates. The paper explicitly concedes that neither BKT disordering, the Cooper logarithm, nor the Q→0 limit establishes relative-momentum locality, and that the reduction of a finite-width PG kernel to J_q δ_{qq'} remains to be derived. The MFL self-energy follows from assuming D^{-1}=r_s+i l_s Ω_m+γ_s|Ω_m| with free coefficients; it is a consequence of the dynamic RML action, not of the t-J model. So the unification is a phenomenological framework, not a derivation. That is a load-bearing gap, not a minor one.\n\nThe free parameters are numerous — band parameters, static J_q shape, dynamic r_s, l_s, γ_s, J_s, resolution broadening, integration windows — so agreement with the broad features of cuprate phenomenology is not surprising. The paper does not claim quantitative fit. The value is in the explicit solvable model and the sharp experimental signature.\n\nThe author is a serious thinker and the paper deserves a serious referee. I'd send it out, asking the referee to weigh whether the RML prescriptions can be derived from the dressed Cooper kernel, and to keep the author honest about the distinction between model and derivation. The right outcome might be acceptance after major revision that reframes the title and abstract as a phenomenological RML theory with a solvable PG sector, rather than a derivation from t-J. I would not desk-reject this; there is enough new content and the community should see it.","headline":"Clever and transparent, but the central unification claim rests on postulated relative-momentum locality that the paper itself concedes is underived.","tokens_in":17386,"tokens_out":2554,"would_cite":true,"duration_ms":22119,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper argues that d-wave superconductivity, the pseudogap with Fermi arcs, and the strange metal are three low-energy organizations of one Cooper-pair channel of the t-J model, unified by a relative-momentum-local theory.","keywords":["cuprate superconductors","pseudogap","strange metal","marginal Fermi liquid","Cooper channel","t-J model","Cooper surface","2e-ARPES"],"falsifier":"In the pseudogap regime, a momentum- and energy-resolved two-electron photoemission experiment should see zero-energy charge-$2e$ ridges along $S_{2e}:\\xi_q=0$ and $S_{2h}:\\xi_q=J_q$ near the nodal points, with the two ridges meeting at the pseudogap nodes; observing no such ridges when the pair-removal matrix elements are symmetry-allowed and the resolution is sufficient would falsify the Cooper-surface part of the theory. In the strange-metal regime, measuring the dressed finite-centre-of-mass pair continuum and the single-particle self-energy in the same doping and temperature window should reveal $\\omega/T$ collapse of $-\\operatorname{Im}\\Sigma^R/T$ and a $T$-linear decay rate; their absence would falsify the dynamic RML and marginal-Fermi-liquid claims.","tokens_in":16177,"feed_emoji":"⚛️","tokens_out":11703,"duration_ms":92059,"temperature":0.7,"pith_summary":"The paper tries to establish that cuprate superconductivity, the pseudogap, and the strange metal are not separate phenomena with separate mechanisms but three low-energy organizations of the same Cooper-pair channel that arises from antiferromagnetic superexchange in the t-J model. It builds a relative-momentum-local (RML) theory in which low-energy pair correlations factorize into independent relative-momentum blocks, and shows that a static version of this factorization yields an exactly solvable pseudogap Hamiltonian that produces nodal electron poles, antinodal gaps, and resolution-broadened Fermi arcs. A dynamic version of the same factorization yields a one-loop self-energy with marginal-Fermi-liquid scaling, and the underlying pair spectrum contains gapless charge-2e Cooper surfaces whose Lifshitz transition produces logarithmic heat-capacity enhancements. If the argument is right, a single pair-physics thread connects single-particle, pair-sensitive, and thermodynamic measurements, and zero-energy Cooper-surface ridges become a decisive 2e-ARPES signature.","feed_headline":"Three cuprate regimes collapse into one Cooper channel","feed_subtitle":"Fermi arcs, T-linear scattering, and d-wave pairing emerge from one pair channel, with 2e-ARPES as the test.","key_machinery":"The workhorse is the exact operator identity that rewrites the superexchange interaction as a sum over Cooper-pair operators: $\\hat H_J = -\\frac{2J}{N_s}\\sum_{Q,\\alpha=s,d}\\sum_{q,q'}\\phi_{q,\\alpha}\\phi_{q',\\alpha}\\hat\\Delta^\\dagger_{Q,q}\\hat\\Delta_{Q,q'}$, with $\\phi_{q,d}=(\\cos q_x-\\cos q_y)/2$ and $\\hat\\Delta_{Q,q}$ the spin-singlet pair operator at centre-of-mass momentum $Q$ and relative momentum $q$. On top of this, the paper postulates relative-momentum locality: at low energy the pair kernel factorizes into independent $\\{q,-q\\}$ blocks. The static prescription sets the intra-block pairing coupling to $J_q(x,T)=\\sqrt{2}\\Phi_d(x,T)\\phi^2_{q,d}$, making $\\hat H_{PG}=\\sum_q'[\\xi_q(\\hat n_q+\\hat n_{-q})-2J_q\\hat\\Delta^\\dagger_q\\hat\\Delta_q]$ exactly solvable in a 16-dimensional Fock space per block; the dynamic prescription uses a block-local pair field with inverse propagator $D^{-1}_{\\Xi,0}=r_s+i\\ell_s\\Omega_m+\\gamma_s|\\Omega_m|$ to generate Ohmic damping and a marginal-Fermi-liquid self-energy. The two prescriptions are the mechanism that translates one Cooper channel into the three observed regimes.","core_discovery":"The central claim is that the exact spin-singlet Cooper representation of antiferromagnetic superexchange in the t-J model, in which the interaction is written as attractive scattering of Cooper pairs conserving centre-of-mass momentum, is the common thread behind superconductivity, the pseudogap, and the strange metal. The paper derives, from a static relative-momentum-local prescription with coupling $J_q(x,T)=\\sqrt{2}\\Phi_d(x,T)\\phi^2_{q,d}$, an exactly solvable block Hamiltonian whose ground states are empty, singly paired, or doubly paired blocks; this produces nodal poles, antinodal gaps, Fermi arcs controlled by the energy resolution, and two gapless Cooper surfaces $S_{2e}:\\xi_q=0$ and $S_{2h}:\\xi_q=J_q$. It then shows that a dynamic RML prescription with pair-field inverse propagator $D^{-1}=r_s+i\\ell_s\\Omega_m+\\gamma_s|\\Omega_m|$ gives a one-loop fermion self-energy with marginal-Fermi-liquid scaling $-\\operatorname{Im}\\Sigma^R_s(q_F,\\omega)\\simeq\\lambda_s\\,\\omega\\coth(\\omega/2T)$. The same pair spectrum supports a Cooper-surface Lifshitz transition at the doping $x_{L,2e}=0.1934$ where a Cooper surface passes through the band-structure saddle point, yielding logarithmic enhancements of $c_V/T$ and the charge response with no spin singularity. The paper concludes that superconductivity is marked by phase stiffness and global coherence, the pseudogap by phase-incoherent quasi-static pair weight and gapless Cooper surfaces, and the strange metal by phase-incoherent dynamic pair fluctuations, with zero-energy Cooper-surface ridges as the decisive experimental signature.","pith_inferences":["Extension: the same static and dynamic RML split could be tested in controlled toy models, such as attractive Hubbard or BCS-BEC crossover models, by computing the block-resolved pair correlator $P_{qq'}$ and checking whether it is nearly diagonal in relative momentum before relying on exact-solvability shortcuts.","Extension: the Cooper-surface Lifshitz transition predicts a rounded logarithmic peak in $c_V/T$ at the reconnection doping, but superconductivity preempts the zero-temperature limit; a field-tuned normal-state measurement in the underdoped-to-optimally-doped range could look for the charge-spin asymmetry the paper predicts.","Extension: because the paper leaves the derivation of both RML prescriptions from the dressed Cooper kernel open, direct numerical diagonalization of small t-J clusters could test whether block-diagonal pair autocorrelation dominates and whether the effective $J_q$ tracks $\\sqrt{2}\\Phi_d\\phi^2_{q,d}$.","Extension: if the marginal-Fermi-liquid self-energy is real, the same pair continuum should contribute to optical conductivity; computing the current vertex consistent with the self-energy via the Ward identity would connect the single-particle $\\omega/T$ scaling to the measured $A_1 T$ resistivity, which the paper flags as an open problem."],"forward_implications":["In the pseudogap regime, the visible Fermi arc is a thermally redistributed and resolution-broadened spectral feature, not a continuous locus of gapless quasiparticles; its width is set by the energy resolution $\\Gamma$ and its length scales as $\\sqrt{\\Gamma/\\kappa_n}$.","The pseudogap hosts two gapless charge-$2e$ Cooper surfaces, $S_{2e}:\\xi_q=0$ and $S_{2h}:\\xi_q=J_q$; when a Cooper surface crosses the band-structure saddle point at $x_{L,2e}=0.1934$, a neck-reconnection Lifshitz transition produces logarithmic enhancements in $c_V/T$ and the charge response but not in the spin susceptibility.","In the strange-metal regime the single-particle decay rate is $-\\operatorname{Im}\\Sigma^R\\simeq 2\\lambda_s T$ for $|\\omega|\\ll T$ and $\\lambda_s|\\omega|$ for $T\\ll|\\omega|$, so the quasiparticle weight vanishes logarithmically and the spectral function is incoherent; whether this produces $T$-linear dc resistivity requires a separate treatment of momentum relaxation and current vertices.","The pseudogap-to-strange-metal crossover can be tracked by a spectral-weight-transfer fraction $f_{SWT}(x,T)$ in the physical pair spectrum, while the superconducting transition is identified by nonzero phase stiffness and global phase coherence rather than by pair correlations alone.","Zero-energy Cooper-surface ridges in two-electron ARPES would be a decisive signature of the theory; their absence at the predicted nodal geometry, with allowed matrix elements and sufficient resolution, would falsify the Cooper-surface prediction."],"supporting_citations":[{"why":"Provides the Gutzwiller-renormalized factors $g_t(x)=2x/(1+x)$ and $g_J(x)=4/(1+x)^2$ used to build the renormalized t-J Hamiltonian from which the Cooper representation is drawn.","marker":"[33]"},{"why":"Establishes that a weak Cooper attraction is marginally relevant and that the $Q=0$ sector carries the leading infrared divergence, motivating the superconducting organization.","marker":"[56]"},{"why":"Supplies the phenomenological pseudogap Green's function with pole pockets and Fermi arcs, which the static RML result is compared against as the pseudogap benchmark.","marker":"[51]"},{"why":"Supplies the marginal-Fermi-liquid self-energy ansatz whose $\\lambda[\\omega\\ln(X/\\Omega_c)-i(\\pi/2)X]$ form the dynamic RML one-loop result reproduces.","marker":"[52]"},{"why":"Provides the universal-jump criterion used to estimate the bare BKT phase-ordering scale $T^{(0)}_{BKT}=\\frac{\\pi}{2}\\rho_d^{(0)}$ separating the pseudogap from superconductivity.","marker":"[57]"},{"why":"Defines the Luttinger surface of zeros in non-Fermi liquids, used to identify the punctured zero contour $S_L:\\xi_q=J_q/2$ in the exact pseudogap Green's function.","marker":"[58]"},{"why":"Demonstrates two-electron photoemission in solids, the experimental probe proposed to resolve zero-energy Cooper-surface ridges.","marker":"[61]"},{"why":"Advances momentum-resolved mapping of electron correlation in two-electron photoemission, supporting the 2e-ARPES proposal.","marker":"[62]"}],"fun_headline_variants":["Cooper channel unifies SC, pseudogap, strange metal","Exact pair channel explains three cuprate regimes","Zero-energy Cooper ridges: test for unified theory","One Cooper representation rules cuprate phases"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The framework rests on the assumption that, at low energies, pairing interactions in the t-J model act independently in each momentum pair $\\{q,-q\\}$, so that the simple block-local coupling $J_q=\\sqrt{2}\\Phi_d\\phi^2_{q,d}$ and the dynamic inverse propagator $D^{-1}=r_s+i\\ell_s\\Omega_m+\\gamma_s|\\Omega_m|$ capture the full pair physics—an assumption the paper itself says is not yet derived from BKT disordering, the Cooper logarithm, or the $Q\\to 0$ limit.","fun_headline_variants_meta":{"raw":{"variants":["Cooper channel unifies SC, pseudogap, strange metal","Exact pair channel explains three cuprate regimes","Zero-energy Cooper ridges: test for unified theory","One Cooper representation rules cuprate phases"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000294,"raw_usage":{"total_tokens":1836,"prompt_tokens":1199,"completion_tokens":637,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":815,"completion_tokens_details":{"reasoning_tokens":575}},"tokens_in":815,"tokens_out":637,"duration_ms":6474,"temperature":1.0,"reasoning_tokens":575,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T10:47:44.688043+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"In the pseudogap regime, a momentum- and energy-resolved two-electron photoemission experiment should see zero-energy charge-$2e$ ridges along $S_{2e}:\\xi_q=0$ and $S_{2h}:\\xi_q=J_q$ near the nodal points, with the two ridges meeting at the pseudogap nodes; observing no such ridges when the pair-removal matrix elements are symmetry-allowed and the resolution is sufficient would falsify the Cooper-surface part of the theory. In the strange-metal regime, measuring the dressed finite-centre-of-mass pair continuum and the single-particle self-energy in the same doping and temperature window should reveal $\\omega/T$ collapse of $-\\operatorname{Im}\\Sigma^R/T$ and a $T$-linear decay rate; their absence would falsify the dynamic RML and marginal-Fermi-liquid claims.","supporting_citations":[{"cited_title":"A renormalised Hamiltonian approach to a resonant valence bond wavefunction,","cited_arxiv_id":null,"evidence_quote":"Provides the Gutzwiller-renormalized factors $g_t(x)=2x/(1+x)$ and $g_J(x)=4/(1+x)^2$ used to build the renormalized t-J Hamiltonian from which the Cooper representation is drawn."},{"cited_title":"Renormalization-group approach to interacting fermions,","cited_arxiv_id":null,"evidence_quote":"Establishes that a weak Cooper attraction is marginally relevant and that the $Q=0$ sector carries the leading infrared divergence, motivating the superconducting organization."},{"cited_title":"Phenomenological theory of the pseudogap state,","cited_arxiv_id":null,"evidence_quote":"Supplies the phenomenological pseudogap Green's function with pole pockets and Fermi arcs, which the static RML result is compared against as the pseudogap benchmark."},{"cited_title":"Phenomenology of the normal state of Cu-O high-temperature superconductors,","cited_arxiv_id":null,"evidence_quote":"Supplies the marginal-Fermi-liquid self-energy ansatz whose $\\lambda[\\omega\\ln(X/\\Omega_c)-i(\\pi/2)X]$ form the dynamic RML one-loop result reproduces."},{"cited_title":"Universaljumpinthesuperfluiddensityoftwo-dimensional superfluids,","cited_arxiv_id":null,"evidence_quote":"Provides the universal-jump criterion used to estimate the bare BKT phase-ordering scale $T^{(0)}_{BKT}=\\frac{\\pi}{2}\\rho_d^{(0)}$ separating the pseudogap from superconductivity."},{"cited_title":"Some consequences of the Luttinger theorem: the Luttinger surfaces in non-Fermi liquids and Mott insulators,","cited_arxiv_id":null,"evidence_quote":"Defines the Luttinger surface of zeros in non-Fermi liquids, used to identify the punctured zero contour $S_L:\\xi_q=J_q/2$ in the exact pseudogap Green's function."},{"cited_title":"Two electron photoemission in solids,","cited_arxiv_id":null,"evidence_quote":"Demonstrates two-electron photoemission in solids, the experimental probe proposed to resolve zero-energy Cooper-surface ridges."},{"cited_title":"Mapping the electron correlation in two-electron photoemission,","cited_arxiv_id":null,"evidence_quote":"Advances momentum-resolved mapping of electron correlation in two-electron photoemission, supporting the 2e-ARPES proposal."}],"review_version":1}