Overview
Q7 addresses a central structural question of the Cosmochrony programme: the coincidence between two independently derived three-dimensional structures.
On the one hand, Q5b extracts an effective co-metric on the Carnot structure of \[ \mathrm{Heis}_3(\mathbb{R}), \] whose spatial sector has three directions. That extraction is its Theorem 5.2, conditional on an unestablished spatial-limit hypothesis and on a lifting hypothesis; at principal-symbol order the central slot is degenerate; and the Lorentzian reading of the signature carries a further hyperbolicity hypothesis. The unconditional part is the homogeneous dimension four of the Carnot structure, which fixes the volume growth and the spectral dimension, not a non-degenerate metric.
On the other hand, O28 works with an admissible projection space \[ H_{\mathrm{eff}} = \mathbb{C}^3, \] supplied as a parameter of its protocol, whose per-pair covariance has threshold rank \( r_{\mathrm{eff}}^{1\%} = 3 \) — a finite-data value, which O28 itself declines to call an invariant. O23's adjoint-dimension theorem then gives three independent directions, conditionally on a supplied spinor carrier: it assumes the neutral sector is carried by an irreducible two-dimensional unitary representation, an input the framework does not derive. The value \( d_\rho = 2 \) is O26's minimality selection, and O29 states that no calculation there identifies these objects.
The objective of Q7 is: to state what an identification between these two “3”s would require, and to prove the requirements that follow from representation theory on a supplied carrier.
What the paper proves
- Algebraic obstruction: \( \mathfrak{su}(2) \) and \( \mathfrak{heis}_3 \) are non-isomorphic — semisimple against nilpotent — so no identification is available at the Lie algebra level. This is unconditional.
- Uniqueness of the spin-one module: up to isomorphism, \( \mathrm{Sym}^2(V_\rho) \) is the unique irreducible three-dimensional complex module of \( \mathfrak{su}(2) \), isomorphic to the complexified adjoint. Under the identification hypothesis, the measured space inherits that structure.
- Representation rigidity: once two isomorphic modules are supplied, nonzero equivariant maps exist and are unique up to a complex scalar. The remaining problem is their geometric origin and compatibility, not abstract existence. Choosing a positive scalar requires a phase convention.
- Additional geometric hypotheses: [ID] identifies the measured space, [ACT] supplies the target action, and [INV] requires compatible real horizontal rotations. Hermitian weight invariance alone does not equate the two horizontal coefficients. Their exact transport also requires an adapted unitary intertwiner [ADAPT], which Schur does not supply. The Casimir value on the supplied spin-one module is 2, independently of [ID]; interpreting it on the measured space requires [ID].
- Metaplectic symmetry: the chirped discrete Fourier transform commutes with the discrete Weil Laplacian, \[ [F_c, L_{\mathrm{Weil}}] = 0, \] for every odd prime and every character coprime to it. This is a statement about the discrete operator, and it does not transfer to the symbol of the continuum operator.
What the checkpoints measure
Version 2.0 re-types the paper's numerical section. The three stored basis rows of the O25 checkpoints are pure Fourier modes of indices \( \{0, -k, +k\} \), with spectral purity 1.000000 on all twelve tested pairs. Every diagonal entry of the compression in the stored basis is therefore a Rayleigh quotient in closed form, the off-diagonal entries following the index rule stated below:
\[ \langle e_m, L_{\mathrm{Weil}}\, e_m \rangle = 4 - 2\cos(2\pi m / q). \]
- The central value is \( 4 - 2 - 0 = 2 \) exactly, for every prime and every character: the additive \( 4I \) term of the Laplacian less the shift eigenvalue at the flat mode. It agrees numerically with the \( \mathfrak{su}(2) \) Casimir eigenvalue, but the agreement is not normalisation-invariant — halving the Laplacian gives 1 while the Casimir eigenvalue is unchanged — so it carries no evidential weight.
- The horizontal value is \( 4 - 2\cos(2\pi k/q) \) for the selected index, so the gap is the identity \[ A_H - 2 = 4\sin^2(\pi k / q). \] Its vanishing in the large-\( q \) limit is equivalent to \( \mathrm{dist}(k/q, \mathbb{Z}) \to 0 \), a property of the pipeline's mode-selection rule that no paper of the corpus derives. Here \( A_H \) and the central value name diagonal entries of this discrete compression; they are the quantities earlier readings identified with coefficients of the continuum symbol, and the paper keeps the two apart.
- An off-diagonal entry appears exactly when two stored indices differ by \( \pm c \bmod q \), which accounts for the two pairs previously reported as finite-size and sector anomalies.
What the pipeline reports is not that compression but its conjugate by the covariance eigenbasis. As matrices the two coincide at nine of the twelve pairs; at \( (61,3) \) they agree entry by entry in modulus but not in phase; and they differ outright at two. At \( (101,3) \) the flat mode is ordered last, which is the origin of the published values 2.096 and 2.048; at \( (151,5) \), the one pair where an undetermined rotation meets a block that is not scalar, the reported split and off-diagonal entry are solver-dependent. The invariant content is the spectrum of the compression.
A diagnostic shipped with the paper reproduces all of this from a 37 kB bundle of inputs with a published SHA-256 digest, and prints, per pair, the Fourier indices and purities, the measured and analytic values with their residuals, the cross-term check, the covariance spectrum with the gap inside the horizontal plane, the conjugated matrix, and the spectrum of the compression. The bundle is a derived extract: reproducing the extraction itself needs the O25 campaign checkpoints, which are not redistributed with the paper.
Conclusions withdrawn in version 2.0
Version 2.0 withdraws the following, as results and as supports. It takes no position on the editorial status of the papers concerned, whose claims are matters for those papers.
- the reading of the central value as a spectral signature of the Casimir;
- the power-law fit of the isotropy gap as evidence for asymptotic isotropy;
- the transfer of the block-diagonality of a discrete compression to cross terms of the continuum symbol;
- the attributions to O29 (the module identification and \( d_\rho = 2 \)), to O27 (an unconditional factorisation through \( \mathfrak{su}(2) \)), and to Q5b (a co-metric carrying a positive central coefficient, and a convergence rate in \( q \));
- the downstream supports previously invoked for the two coefficients.
The earlier claim that the ambiguity was removed, either by a unique bridge or by a definitive separation, is not retained for the broader construction: a new compatible target remains open, while the exact Q5b symbol is excluded by rank.
Relation to the Cosmochrony programme
Q7 sits between two branches: the O-series, which supplies the carrier and the admissible projection space, and the Q-series, which derives an emergent geometry from spectral limits. It types the edge between them rather than closing it.
A separate conditional obstruction applies to the Q5a route: under Q5a's depth and weight hypotheses, no common scalar normalisation of the published admissibility form yields a toric differential limit, so that route supplies no spatial second-order operator. It leaves open a bridge on a different filtration, under an anisotropic or non-scalar renormalisation, or on a non-toric target.
Open directions
- The mode-selection rule: derive \( q \mapsto k(q,c) \) from the breadth-first traversal and the Gram–Schmidt step, which is what would make the isotropy question testable.
- An extended spatial target: construct a new effective operator or form with a compatible action, then test the necessary positive graded form. Lower ordinary differential-order terms do not change the rank of Q5b's existing principal symbol.
- A source for the identification hypothesis: exhibit a construction supplying the module structure on the measured space, or an argument excluding it.
References
Jérôme Beau. Toward Three Spatial Directions from a Supplied Spinor Carrier: Requirements for an Equivariant Bridge, 2026.