The Third Verdict at Strong Coupling: Is the Interacting Entanglement Hamiltonian Irreducible to the Bisognano–Wichmann Local Energy Density? (Phase VII, and its certification Phase VII-b)
Scope label (load-bearing). Genuinely interacting quantum field theory in flat 1+1 Minkowski space. No gravity: not dynamical, not backreacting, not quantum. This is Exit (a) of Proposition 2 — the coupling axis, isolated from the gravity axis. Archived as Phase VII (with its certification Phase VII-b); not merged into the free-field Minkowski corpus of Phases IV–V, nor into the fixed-curved-background Phase VI. Phase VI closed the fixed-background branch with a structural obstruction (Proposition 2): for a free field on a fixed classical background the state is a functional of the background data, hence so is every modular observable, and grade (ii) is the ceiling — a genuine positive is structurally unreachable there. Exactly two exits survive: (a) genuine interaction, and (b) dynamical gravity. This paper attacks Exit (a), the minimal scope extension: still flat space, no gravity, only the coupling switched on. The object is the interval entanglement Hamiltonian K_A = −log ρ_A in the DMRG/MPS ground state of the mixed-field Ising chain — genuinely interacting, non-integrable, gapped, non-conformal — and the reference is the model's own interacting Bisognano–Wichmann family (identity plus a freely weighted local energy density plus first descendants), never the free stress tensor. The same structural logic caps the attainable outcome at a distinction inside grade (ii): REDUCE (the third verdict extends to coupling) versus IRREDUCIBLE (the first crack of the enclosure). Both were preregistered as publishable and hash-frozen before any production run. Phase VII returned a substantive but uncertifiable result: all four target residuals sit 5–30× below the controls, the residual anti-tracks the coupling, and the co-primary spectral reconstruction echoes a correlation-length ordering — yet the frozen multiplicative threshold (θ = 10 × floor = 1.62) exceeded the maximum response of any bounded foil (≈ 0.45), because the free-control floor was dominated by Jordan–Wigner representation mismatch, an instrument bias ~30× the target signals. The teeth precondition failed by arithmetic, not by physics, and the verdict was left with an asterisk. Phase VII-b removes it: an independently frozen rerun in the model's natural variables (the fermionic Jordan–Wigner-bilinear BW family, whose free-control residual is exactly zero at full range — the calibration the earlier instrument lacked), an additive threshold, and satisfiable contrast teeth. Teeth pass; all four targets lie 8–200× below the certified threshold and below every control, in both representations. Verdict: COLLAPSE (Exit (a)), certified. The interacting entanglement Hamiltonian reduces to the local Bisognano–Wichmann energy density; the reduction is representation-independent, and increasing the coupling makes K_A more BW-local, not less. The strong-coupling axis carries no grade-(ii)-irreducible modular orientation. No grade-(i) claim is made or implied (Proposition 2 forbids it for a fixed-Hamiltonian ground state). The genuine positive, if it exists, is pushed onto Exit (b) — dynamical geometry — named and out of scope here. Preregistration and reproducibility. Both investigations were governed by preregistrations frozen and hashed before any production run — Phase VII: SHA-256(§1–§6) = 5c53a0c0…fc09f (frozen 2026-06-30 22:45:34 UTC); Phase VII-b: SHA-256(§1–§6) = c63f8a10…39d46 — with append-only deviation logs (DEV-1…DEV-6 and DEV-1…DEV-4(VII-b)) below the hash boundary, and cautionary records of the two failed instruments preserved rather than discarded. Code, preregistrations, per-claim ledgers, and all emitted CSV/JSON data: https://github.com/PauDD/Simulador-Fase-VII Continues Phase IV (10.5281/zenodo.21261576), Phase V (10.5281/zenodo.21302338) and Phase VI (10.5281/zenodo.21321084) of the programme Rigidez de la Incidencia Geométrica (Geometric Incidence Rigidity).
Scope label (load-bearing). Genuinely interacting quantum field theory in flat 1+1 Minkowski space. No gravity: not dynamical, not backreacting, not quantum. This is Exit (a) of Proposition 2 — the coupling axis, isolated from the gravity axis. Archived as Phase VII (with its certification Phase VII-b); not merged into the free-field Minkowski corpus of Phases IV–V, nor into the fixed-curved-background Phase VI. Phase VI closed the fixed-background branch with a structural obstruction (Proposition 2): for a free field on a fixed classical background the state is a functional of the background data, hence so is every modular observable, and grade (ii) is the ceiling — a genuine positive is structurally unreachable there. Exactly two exits survive: (a) genuine interaction, and (b) dynamical gravity. This paper attacks Exit (a), the minimal scope extension: still flat space, no gravity, only the coupling switched on. The object is the interval entanglement Hamiltonian K_A = −log ρ_A in the DMRG/MPS ground state of the mixed-field Ising chain — genuinely interacting, non-integrable, gapped, non-conformal — and the reference is the model's own interacting Bisognano–Wichmann family (identity plus a freely weighted local energy density plus first descendants), never the free stress tensor. The same structural logic caps the attainable outcome at a distinction inside grade (ii): REDUCE (the third verdict extends to coupling) versus IRREDUCIBLE (the first crack of the enclosure). Both were preregistered as publishable and hash-frozen before any production run. Phase VII returned a substantive but uncertifiable result: all four target residuals sit 5–30× below the controls, the residual anti-tracks the coupling, and the co-primary spectral reconstruction echoes a correlation-length ordering — yet the frozen multiplicative threshold (θ = 10 × floor = 1.62) exceeded the maximum response of any bounded foil (≈ 0.45), because the free-control floor was dominated by Jordan–Wigner representation mismatch, an instrument bias ~30× the target signals. The teeth precondition failed by arithmetic, not by physics, and the verdict was left with an asterisk. Phase VII-b removes it: an independently frozen rerun in the model's natural variables (the fermionic Jordan–Wigner-bilinear BW family, whose free-control residual is exactly zero at full range — the calibration the earlier instrument lacked), an additive threshold, and satisfiable contrast teeth. Teeth pass; all four targets lie 8–200× below the certified threshold and below every control, in both representations. Verdict: COLLAPSE (Exit (a)), certified. The interacting entanglement Hamiltonian reduces to the local Bisognano–Wichmann energy density; the reduction is representation-independent, and increasing the coupling makes K_A more BW-local, not less. The strong-coupling axis carries no grade-(ii)-irreducible modular orientation. No grade-(i) claim is made or implied (Proposition 2 forbids it for a fixed-Hamiltonian ground state). The genuine positive, if it exists, is pushed onto Exit (b) — dynamical geometry — named and out of scope here. Preregistration and reproducibility. Both investigations were governed by preregistrations frozen and hashed before any production run — Phase VII: SHA-256(§1–§6) = 5c53a0c0…fc09f (frozen 2026-06-30 22:45:34 UTC); Phase VII-b: SHA-256(§1–§6) = c63f8a10…39d46 — with append-only deviation logs (DEV-1…DEV-6 and DEV-1…DEV-4(VII-b)) below the hash boundary, and cautionary records of the two failed instruments preserved rather than discarded. Code, preregistrations, per-claim ledgers, and all emitted CSV/JSON data: https://github.com/PauDD/Simulador-Fase-VII Continues Phase IV (10.5281/zenodo.21261576), Phase V (10.5281/zenodo.21302338) and Phase VI (10.5281/zenodo.21321084) of the programme Rigidez de la Incidencia Geométrica (Geometric Incidence Rigidity).
The second arc of the incidence–rigidity programme closed every tractableregime by one mechanism: a single effective background—a temperature, aboost, a generalized ensemble, a time-dependent profile—absorbs theregistered modular orientation, and the verdict reduces to the spectralcondition already in hand. The structural obstruction of Phase VI fixed theceiling for fixed backgrounds and left two exits; Phase VIII attackeddynamical geometry in its minimal lattice form, proved the blindnesstheorem—no single-time matter observable distinguishes quantum fromclassicalized geometry—and found that even the multi-time record of onematter sector collapses onto an effective background. This paper reportsthe continuation at the one locus that mechanism cannot reach byconstruction: a geometry that is itself in superposition, mediating betweentwo matter parties, where there is no single background to reduce to. Thetopology is the minimal Bose–Marletto–Vedral setup on a 1+1 lattice: twonon-interacting Ising chains coupled only through one quantum incidenceregister. Two layers are read off the same runs. Layer one is theentanglement witness as theorem and calibration: the mediatedlog-negativity grows to E_N = 0.124 while the fitted classical stochasticmediator of Kafri–Taylor–Milburn type, granted the quantum one-pointstatistics and full per-cell freedom, stays at the LOCC floor at every size(5.8e-11 to 6.4e-16)—and at the largest size it matches the quantum mutualinformation (0.042 vs 0.041) with negativity exactly zero. Layer two is theprogramme's question: is the joint multi-time record of the pair reducibleto any single shared effective background? Under a rule frozen by hashbefore the confirmatory tiers, two cells, (g, Ω) = (0.6, 1) and (0.6, 2),fire the record witness at Q = 13, 21 and 25 qubits—excesses +0.017 and+0.005 over a floor that saturates (0.0569 → 0.0666 → 0.0664), optimizerseed-scatter at 1e-16, Monte Carlo errors below every separation—while thefast-register row Ω = 4 collapses below the floor at all sizes: theadiabatic-elimination reduction of the earlier phases, recovered insidethis instrument as its null control. The verdict of the frozen rule isH1(record-discriminates): the joint record carries content irreducible tothe declared rival class, stably in size. The two witnesses dissociate inboth directions—more entanglement does not imply more record excess—so therecord defect is an information axis independent of the entanglementwitness, with its window where the mediator is a genuine dynamicalparticipant. The result is a graded lattice computation (grade (ii),minimal quantum-geometry sector); its deviations, calibrations and caveatsare reported in full, the field limit is left open, and the realinterferometric experiment is not modelled.
Phase IV closed the free/geometric-flow sector on the third verdict: the registered modular orientation is geometric and irreducible to any CPTP channel, yet it coincides with the Borchers–Wiesbrock spectral condition (energy positivity) — already-known QFT serving as the arrow. That verdict rests on a single structural fact: a KMS state has one modular temperature (Tomita–Takesaki). Any genuinely new arrow beyond the spectral condition therefore requires a state that is not KMS nor generalized-Gibbs. This paper attacks the one corner that is at once outside the third verdict, in scope (no gravity), and tractable: the current-carrying non-equilibrium steady state (NESS). Proceeding symmetrically, with preregistered decision rules frozen before any run: Island A (a Generalized Gibbs Ensemble after an integrable quench) closes null / horn b — the charge tower is absorbed by a single recalibration c(R) → 1, and a finite-size positive is diagnosed as a charge-basis conditioning artefact. The genuine NESS is decided in two stages. On the lattice, the preregistered instrument returns HORN A, grade (ii): the bulk-interval modular Hamiltonian is irreducible to the local-equilibrium-plus-current generator, with a clean law r∞ ≈ Δβ/2. A second, independently frozen preregistration takes the field limit a/β → 0 and returns global collapse: the decisive residual against the Bernard–Doyon boost-of-thermal reference vanishes as ~a² to the equilibrium floor across every drive, while the discriminator's teeth stay intact. The lattice HORN A is reclassified as a UV band-curvature (inflection-point) artefact; the growing distance-to-local-cone that produced it measures the intrinsic non-locality of the Cardy–Tonni boost-of-thermal entanglement Hamiltonian, a known-QFT feature, not new structure. The third verdict extends to the continuum non-KMS, positive-entropy-production sector. Any genuine new modular arrow is pushed to strongly-coupled / holographic NESS, named and left out of scope.
For a region of a free-fermion ground state the entanglement Hamiltonian K_A = log[(1 - C_A) C_A^-1] splits into a part proportional to the identity and a trace-free remainder. The identity part shifts every level rigidly, so every observable built from level differences is blind to it. This paper is about that part. It is not a bulk quantity. The block carries one ladder per entangling cut, the trace is the difference of the two cut-phase charges, and on the exactly dimerised chain with a single cut carrying the whole charge it has the closed form 2 artanh(v / sqrt(v^2 + t^2)), derived in one line and measured to 10^-11. In exact arithmetic the trace is a definite number: zero to thirty-three digits when the two cut phases agree, and 0.786153199019314523 when they do not. The apparent divergence reported in earlier work was double-precision truncation of a finite sum, and is retired here. Along the two lines through the Rice-Mele plane where the half-line entanglement Hamiltonian is known in closed form, it is proportional to the Bisognano-Wichmann deformation T = (1/2){X, H} with no additive constant, and the ladder spacing and phase are fixed by the literature. Between those lines the commuting operator survives, measured at 10^-13 against controls that fail at 1.5 and 29, but the proportionality does not: the level map acquires an intercept b that is zero to 3 x 10^-9 on both lines and of order 10^-1 in the interior, and is identical whether the region has one entangling cut or two. That intercept is the measured failure of exact Bisognano-Wichmann proportionality. At first order in the staggering v the intercept is derived rather than fitted. The symbol's determinant is det(z - P) = z(z - 1) independently of momentum, so the factorisation is canonical everywhere; its logarithm is affine in an involution R = H/E with R^2 = I; the abelian Szego-type boundary functional is identically even in v and therefore cannot carry an odd intercept; and the first-order piece follows from the chiral zero mode of the dimerised chain, whose amplitudes are (-r)^j C(2j,j) 4^-j in closed form. The result is b/v = 2(E' - K')/(1 - r^2) + K', K' = K(1 - r^2), E' = E(1 - r^2), r = t1/t2, whose expansion in the modulus u is, in exact rationals, h(u)/(pi/8) = 1 - (5/8) u^2 - (9/64) u^4 - (65/1024) u^6 - (595/16384) u^8 - ... So pi/8, -5/8 and -9/64 are consequences rather than recognitions, and the natural elliptic candidate [9E - K]/16 - pi/8 is excluded by 7.7 percent on the first coefficient it does not get to choose. The closed form agrees with direct measurement extrapolated to v -> 0 by two to three orders better than the fitted series it replaces; a previously reported u^6 coefficient of -0.0648 is corrected here to -65/1024 and identified as truncation. The trace-free half is treated in parallel. A Clausius relation with a single temperature holds exactly when that half is proportional to the region's own Hamiltonian. At a ground state it never is, and the measured temperature is zero, which is correct rather than a failure. In a thermal state the same instrument returns 1/beta to five significant figures, and the boundary of the region where it does so collapses onto two dimensionless ratios. Two further results are included. The offset vanishes when the two cuts sit at the same phase in the unit cell, without requiring chirality, time reversal, inversion or cleanliness, and a mirror-symmetrised disordered chain gives a larger offset than its unsymmetrised control, which rules reflection out as the mechanism. And in two dimensions the offset is exactly additive along the boundary for a transverse coupling proportional to the identity in sublattice space, for a reason short enough to state, with a derived crossing past which additivity fails and a sigma_z coupling that breaks it smoothly. Every claim carries one of four grades: [T] proven here, [V] computationally verified with stated controls and a stated failure mode, [C] imported from the cited literature, [H] interpretation. The two exact lines, the commuting deformation, the boundary-locality of the entanglement spectrum, and the first law of entanglement entropy are not this work's and are used as given. Sixteen retired claims, including two withdrawn from earlier versions of this manuscript, are listed with the measurement that killed each. Files in this deposit: the_offset_belongs_to_the_boundary_v8_submission.pdf — 35 pages, gate output verbatim, script listings supplied separately. the_offset_belongs_to_the_boundary_v8_complete.pdf — 59 pages, identical body with all scripts printed inline. offset_v8_scripts.zip — 23 files. Every number quoted in the paper is printed by one of them; the two gate files reproduce their stored output byte for byte. KEYWORDS entanglement Hamiltonian; entanglement spectrum; modular Hamiltonian; Bisognano-Wichmann; Rice-Mele model; SSH model; free fermions; boundary charge; cut-phase charge; corner transfer matrix; Toeplitz operators; Wiener-Hopf factorisation; Szego-Widom asymptotics; algebraic Heun operator; complete elliptic integrals; entanglement first law; area law; Landau-Widom counting function ADDITIONAL NOTES Supersedes earlier versions of this manuscript. Two claims present in versions 1 to 3 are withdrawn and recorded in the kill list: that the incommensurate offset has no value, and that the offset is one quantum. Both were double-precision truncation of a finite sum. The u^6 coefficient of the shape function reported in the companion results ledger is likewise superseded by the derived value. Computational verification, gating and drafting assistance by an AI collaborator (Claude, Anthropic). Every gate is independently re-runnable from the ancillary scripts.
This synthesis condenses a multi-phase research programme on the geometric origin of the measurement arrow: whether the registered temporal orientation of a quantum measurement is irreducibly geometric or reducible to an effective (CPTP) channel. The question is converted into a theorem of empirical non-equivalence and computed where it can be decided — on a genuine type III₁ continuum object, outside the type I shadow in which the two readings provably coincide. The instrument returns a third verdict: the registered multi-time orientation is geometric and irreducible to any effective channel — in type III₁ the restricted state admits no density-matrix representation to dilate, so no Stinespring dilation can readmit it as "merely an ancilla," which excludes, within the stated framework, its reduction to einselection — and it simultaneously coincides with the Borchers–Wiesbrock spectral condition: the positivity of the energy rather than a new arrow, neither new physics nor an imported thermodynamic gradient. The frontier this verdict localized was then pursued with the same instrument across every in-scope regime it could reach — non-geometric flow, weak non-Gaussianity, integrable and genuine non-equilibrium, a fixed curved background, strong coupling, and a minimal quantum promotion of the geometry itself — and in every one the third verdict held: the registered orientation reduced to already-known structure (a single spectral temperature, a boost of one, an evolving Gibbons–Hawking temperature, or a functional of an effective dynamical background), with the apparatus repeatedly returning apparent positives that each dissolved under continuum and operator-level controls. A structural proposition accounts for the pattern and places any genuinely new positive outside every background-functional setting. At the one in-scope locus with no single background to reduce to — a geometry register that itself superposes and mediates between two matter parties — a first lattice measurement finds the joint multi-time record irreducible to any single shared effective classical background in its declared rival class, stably across three sizes: a record witness carrying information beyond the entanglement witness of the interferometric proposals, bounded to a figure of merit at lattice scope and grade (ii), with its field limit open. The aim of the document is documentary: every verdict is conditional on a declared chain of assumptions, every theorem and dataset lives in the companion phase records (DOIs below), and the vocabulary of decision names outcomes of preregistered, hash-frozen decision rules.
Parts VIII–XI of this series reduced the program to a single missing object: a quantum measure μ(C) over causal histories. This Part XII specifies that object, realizes it in prototype, and tests whether it selects the observed effective spacetime dimension. The fundamental object is a Hermitian, positive, bi-additive Sorkin decoherence functional D(A,B) with physical measure μ(A)=D(A,A); classical measures obey the two-event sum rule I2=0, while quantum measures have I2≠0 but obey the Sorkin triple sum rule I3=0. An explicit such measure is constructed on the growth histories of small causal sets and verified numerically to satisfy I3=0 to machine precision while I2≠0. The fourth gate is sharpened beyond the standard Lorentzian-continuum requirement: the measure must concentrate on causal orders carrying coherent local proper time (a stable interval volume–proper-time law), the distinctive addition of this program. A systematic test on controlled ensembles then asks whether any non-circular intrinsic ingredient selects the 3+1 sector, with a uniformly negative result. The free measure is dominated by the entropic Kleitman–Rothschild phase (d_MM ≈ 2.37); proper-time coherence with propagation escapes that phase but settles on the effective 2+1 sector; the Benincasa–Dowker quantum phase produces only seed-unstable windows that vanish at the physically natural phase scale; and the Pauli–Jordan operator and the Sorkin–Johnston state, though intrinsic and phase-based, are monotone in dimension. A circularity audit shows that every selector whose optimum lies at four dimensions either inserts the dimension (local gravitational degrees of freedom, bound-matter stability, a (d−4)² term) or was tuned to it. The honest conclusion is that proper-time coherence explains the exit from the entropic phase but not the final 3+1 selection: the bare measure with coherence predicts an effective 2+1 phase, and the open problem is localized as the emergence of the third spatial direction (2+1 → 3+1). This yields a falsifiable prediction and identifies the missing object as a covariant, Bell-causal growth law whose continuum limit would fix the gravitational phase normalization. Supplementary Python scripts reproducing all numerical tests are included.
This is the demonstration layer of the Phase III central proposition of the research programme on the rigidity of geometric incidence. The predecessor framework supplies the quantum-measurement collapse arrow as a purely geometric datum — the causal structure and temporal orientability of Minkowski space — while keeping the global evolution unitary, and records the irreversibility of a measurement event as a causal monotone tied to the four-volume of the causal diamond between event and observer. The same data admit two readings: a geometric reading, in which the functional form of the irreversibility is rigidly determined by the modular geometry of the diamond (fixed up to an overall normalization) and is invariant under a change of geometry without recalibration; and an effective reading, in which the law is merely carried by a thermodynamic registration channel whose coefficient is free and must be recalibrated for each geometry. The paper poses the decision between them as a single theorem of empirical non-equivalence, built symmetrically so that either verdict — rigidity (new physics) or equivalence (reduction to einselection with an imported thermodynamic arrow) — would be a valid, publishable outcome. Working in algebraic quantum field theory, where the local algebras are type III₁, it establishes (on a reference state with geometric modular action — the vacuum, via Bisognano–Wichmann) that the irreversibility functional is determined by the diamond's modular data up to one normalization constant (Tomita–Takesaki, Casini–Huerta–Myers, Araki relative entropy), with the entire residual freedom localized in the modular status of the record-carrying state. Whether that normalization is itself geometric (the value π/24) is graded as a conjecture and is not load-bearing; the decisive content is invariance under a change of geometry, and the value is adjustable. In the genuine type III₁ continuum, both halves of the discriminating apparatus return a third verdict: the registered antisymmetric orientation is geometric and irreducible to any CPTP channel (in type III₁ there is no reduced density matrix to dilate, so no Stinespring dilation can readmit it "as only an ancilla"), yet it coincides with the Borchers–Wiesbrock spectral condition (energy positivity) — already-known QFT serving as the arrow, rather than a new irreducible mechanism. Any genuinely new positive is localized to two named sites: the interacting / non-geometric-flow sector (β ≠ 2π/a, where the Connes–Rovelli thermal-time reading must be rebutted, not assumed) and the orientation of the modular inclusion. A preregistered finite-N Gaussian discriminator returns equivalence in the simulable (type I) class — reported honestly as inconclusive for the fundamental reading, since the decisive type III₁ regime is not reached at any finite N. Scope: special-relativistic (Minkowski); gravity and quantum gravity are deferred. All results are explicitly graded (theorem / proposition / conjecture / computation / inherited), and the type III₁ instruments and verdict data are included with the deposit. Changes in v2 (2026-07-08). This version adds Phase IV, step 3 — the two-axis closure of the region left open by the free-sector verdict. Route (b), non-geometric Gaussian flow (multi-interval): the multi-interval vacuum modular flow is genuinely non-geometric (bilocal strength ρbi ∈ [0.004, 0.064]) yet carries a single modular temperature on every channel (D ≡ 0 at βmod = 1 to 10-12, continuum-stable), with the independent-scales einselection foil detected at D = 0.148 — NULL. Route (a), perturbative non-Gaussianity (tree-level λφ3): the T-odd connected cumulant onsets at first connected order (|A3| ∝ λ) but its separating content is zero (D = 0, dD/dλ = 0), while the multi-scale and FDT-breaking foils fire at 0.138 and 0.177 — NULL / recalibrable; the third verdict extends to the perturbative interacting sector. Both axes were pre-registered before running (frozen 2026-06-26 and 2026-06-25 respectively); no v1 result is altered. New verified references: Peschel (2003), Casini–Huerta (2009), Peschel–Eisler (2009), Arias–Casini–Huerta–Pontello (2018).