Companion to "The Windowed Reading", sharpening its regrade of the two-modes proposal at the one point where that note stopped: the Wick identity - one spectral number appearing as Euclidean decay rate and as unitary oscillation frequency - was displayed there on the free spectrum only. This note displays it with interaction, exactly, and adds an equivalence. The positive-type property of the joint kernel - the engine of the link-reflection-positivity theorem - makes the transfer operator of the interacting quartic chain positive semidefinite; hence H = -ln T exists on the support of T as a real self-adjoint Hamiltonian, and the interacting excitation energies exist as theorems of the Euclidean data. They are then extracted twice, independently: as Euclidean decay rates (agreement with the spectrum to 9e-16 - machine precision) and as unitary oscillation frequencies (FFT, agreement at the 1e-3 resolution limit); a matrix-pencil analysis of the Euclidean correlator alone recovers two interacting levels with the correct parity selection (Delta E_1 to 1e-12, Delta E_3 to 5e-7); the identity holds across couplings g = 0, 0.5, 2.0, with the free control an exact benchmark (Delta E_1 = ln 2, derived, passed at 1e-6); the gap is grid-converged to nine digits (Delta E_1 = 1.303200265 at g = 0.5); the Euclidean reading passes into the unitary one as the boundary value of the tube (linear convergence); and the interacting poles sit at s = +-i Delta E_n, on-axis and gapped - the mechanism premise verified at the interacting level in this truncation. The control is the sharpest statement: with the antiferromagnetic joint, the transfer operator acquires a macroscopic negative eigenvalue (-0.391 against 1e-16 noise), -ln T turns complex, and no real Hamiltonian generating these data - no unitary reading - exists in this class. In this class, reflection positivity is thereby displayed as equivalent to the existence of the indefinite face: the second mode stands and falls, exactly and computably, with RP. No number is derived; every claim carries a kill criterion; all computations are supplied as self-contained scripts.
Companion to "The Windowed Reading", sharpening its regrade of the two-modes proposal at the one point where that note stopped: the Wick identity - one spectral number appearing as Euclidean decay rate and as unitary oscillation frequency - was displayed there on the free spectrum only. This note displays it with interaction, exactly, and adds an equivalence. The positive-type property of the joint kernel - the engine of the link-reflection-positivity theorem - makes the transfer operator of the interacting quartic chain positive semidefinite; hence H = -ln T exists on the support of T as a real self-adjoint Hamiltonian, and the interacting excitation energies exist as theorems of the Euclidean data. They are then extracted twice, independently: as Euclidean decay rates (agreement with the spectrum to 9e-16 - machine precision) and as unitary oscillation frequencies (FFT, agreement at the 1e-3 resolution limit); a matrix-pencil analysis of the Euclidean correlator alone recovers two interacting levels with the correct parity selection (Delta E_1 to 1e-12, Delta E_3 to 5e-7); the identity holds across couplings g = 0, 0.5, 2.0, with the free control an exact benchmark (Delta E_1 = ln 2, derived, passed at 1e-6); the gap is grid-converged to nine digits (Delta E_1 = 1.303200265 at g = 0.5); the Euclidean reading passes into the unitary one as the boundary value of the tube (linear convergence); and the interacting poles sit at s = +-i Delta E_n, on-axis and gapped - the mechanism premise verified at the interacting level in this truncation. The control is the sharpest statement: with the antiferromagnetic joint, the transfer operator acquires a macroscopic negative eigenvalue (-0.391 against 1e-16 noise), -ln T turns complex, and no real Hamiltonian generating these data - no unitary reading - exists in this class. In this class, reflection positivity is thereby displayed as equivalent to the existence of the indefinite face: the second mode stands and falls, exactly and computably, with RP. No number is derived; every claim carries a kill criterion; all computations are supplied as self-contained scripts.
October 25th, 2026: updates following the development of the Absolute Frame Theory. Why is the physical action a phase, exp(iS) (oscillatory, norm-preserving, interference-bearing), rather than a real weight, exp(−S) (diffusive, dissipative)? This is the difference between a quantum and a merely statistical world. In Absolute Frame Theory (AFT) it is the difference between the Lorentzian observable manifold ℳ and the Euclidean substratum 𝒜. The imaginary unit is introduced by the ℳ–𝒜 signature transition, the Wick rotation. We show that this transition is the single deepest posit beneath the AFT account of quantum theory. It is the same root from which the complex structure, the Born rule, the arrow of time, and the Tsirelson bound were separately shown to follow. We then reduce it as far as a physical theory can reduce its own ground. The signature transition factorizes into three layers: the existence of a time direction (the sliding of the embedding interface), the orientation (the arrow, fixed by informational irreversibility), and the complex/unitary character (the i and the phase). For the last and only open layer we prove three statements. First, the complex structure is not smuggled in: it acts on each real mode as the real, antisymmetric generator J₀ of the first-order phase-space closure of the mode's oscillation, so that the imaginary unit is an operator of the real phase plane, not an imported ℂ. Real time and the imaginary unit enter together, through the reconstruction. Second, we prove a theorem (for the free sector with uniform sliding) that the reconstructed evolution on ℳ is unitary (the phase exp(iS), with a single complex structure), and that no dissipative exp(−S) sector is admissible. The comoving frame is static, and Osterwalder–Schrader reconstruction delivers a positive Hamiltonian and unitary continuation. Third, that single complex structure acts compatibly across subsystems, J_S ⊗ 𝟙 = 𝟙 ⊗ J_S′ (an operator identity, not a shared scalar sign). Hence the reconstructed theory is locally tomographic, complex rather than real (rebit). The phase is therefore forced by the fact that 𝒜 is Euclidean. That fact is not an arbitrary extra assumption. It is entailed by 𝒜's role as the atemporal, stable, deterministic ground. Atemporality forbids a timelike direction, and a bounded-below action requires positive-definiteness. Hence the two-signature architecture reduces to one posit. Asking why 𝒜 is Euclidean is asking the theory to derive its own ground. That is a foundational question of self-referential, not identifiability, character. The reduction is sharp: it does not derive the posit from its origins, but it shows the posit to be single, non-arbitrary, and necessary given the role. The open residual is thereby narrowed to two sectors, the interacting and the curved/horizon, the latter addressed by modular reflection positivity in the companion dynamical reconstruction.
Reflection positivity (RP) is the single positivity statement through which a Euclidean path integral is promoted to a unitary real-time quantum theory (Osterwalder–Schrader reconstruction). Its classical proofs consult a geometric reflection — an involutive automorphism fixing a hyperplane — which a randomly grown geometry generically lacks. This preprint shows that on ensembles of sequentially assembled geometries the positivity structure survives, and that it is carried entirely by the measure: RP holds if and only if the growth dynamics is memoryless on isomorphism classes. Characterization. Class-level Markovianity, class-local product form with terminal normalization, and history-independent Gram realizability of the conditional chain forms are equivalent; the multiplicative cocycle identity is derived as the Markov property in multiplicative form, not hypothesized separately. Explicit spectrum. Under these conditions the chain form is block-diagonal over terminal classes with rank-one blocks; the positive spectrum equals the arrival measure of the class process, independent of the phase rule — phases act only on the eigenbasis, identifying the chain form with the decoherent diagonal. Sharpness. Outside the class, failure is not gradual: a history-dependent weight family violates the class Markov property with exact rational witnesses (1/60 ≠ 9/260, 1/30 ≠ 3/130), and no Gram realization exists. Physical anchoring. The covariant Rideout–Sorkin family of classical sequential growth dynamics for causal sets satisfies the hypotheses, so all theorems apply verbatim: its chain forms are positive semidefinite at every depth. A dictionary instantiates the framework in causal dynamical triangulations, group field theory and spin foams, tensor/matrix models, and causal sets. Verification. All statements verified by exhaustive exact enumeration in rational arithmetic through graph size eight (isomorphism classes 1–3–14–86–709–8534; full depth-two assembly over all 85,393 paths), across six weight families and three degree bounds.
In the context of many-body quantum mechanics, one often encounters the need to evaluate expectation values of operators or their matrix elements. For instance, given a Hamiltonian https://www.w3.org/1998/Math/MathML" display="inline"> H ^ , we typically want to evaluate its expectation value, which is the energy of the system, https://www.w3.org/1998/Math/MathML" display="inline"> E = ⟨ H ^ ⟩ . In second quantization, the matrix element of an operator can always be expressed as a vacuum expectation value. This generally yields an expression which is hard to evaluate. Fortunately, a powerful tool to simplify these calculations exists: Wick's theorem. This theorem provides a systematic way to break down expressions involving products of creation and annihilation operators into sums of simpler terms.
A theoretical and numerical study of when the anti-holomorphic (conjugate-variable) dependence of a two-point transfer kernel is forced by the underlying open-system dynamics rather than removable by gauge, basis, or preparation. For two counter-winding mode ladders coupled unidirectionally (reservoir-engineered, Lindblad-type, per-pair Jordan double-pole structure), the transfer kernel sums to a closed form involving the Lerch transcendent in the product of conjugated coordinates. Reciprocity is shown to be equivalent to the existence of an exchange-Hermitian mirror partner of the kernel; unidirectionality removes that partner. All symbolic claims are certified by exact Wirtinger-derivative computation. A recirculating-fiber-loop implementation and a time-domain protocol are proposed, with noise/imperfection analysis and explicit falsifiability criteria. Version 3, additive over v2 — the boundary arc (Sections 3–5): the kernel's intrinsic boundary at |x| = 1 with its generic-existence statement (the wall belongs to the class; its location is set by the tail of the weights); the chiral phase law of the branch-cut jump (machine-exact ratio, linear drift at rate kappa/2delta, family-independent); the log-periodic interference calendar on the log-scale coordinate, with prior-art positioning against discrete scale invariance (Sornette; Derrida–Itzykson–Luck); the Fourier floor theorem with full proof (the residual relief of large kernel sums equals the window-contrast of the Fourier transform of the weighted winding measure; finite atomic ladders keep the calendar forever by Bohr almost-periodicity, continua forget by Riemann–Lebesgue); closed-form floors (Ci/Si integrals, cardinal sine); and an inverse spectroscopy reading the hidden weighted distribution of winding speeds from the measured relief, with proven limits (exact mirror blindness of modulus-only data; window = aperture). Non-Hermitian point-gap topology references added (Wanjura et al. 2020; Zirnstein et al. 2021). Title-date correction (July 10, 2026). No physical claim is added; all results are machine-verified structural characterizations. Note added at deposit time: subsequent to the sealing of this version, the mirror-blindness limit of the inverse spectroscopy has been lifted by a reference-needle heterodyne protocol (demodulation law, asymmetry law, quadrature recovery of the full complex transform), machine-verified and judge-certified; traces and harnesses are in the public repository below. This upgrade will be the subject of the next linked deposit. All FINDINGS and machine harnesses backing every number in Sections 3–5, plus an independent external from-scratch reproduction audit, are public: https://github.com/antparis/oneway-kernel (MIT).
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.