Calculations, experiments and open topics

Date: 24 October 2026 Priority rule: do the cheapest calculation that can close a route before adding ontology.

Tranche 0: normalization and category firewall

The genus-one normalization and first central-charge budget are now executed in the preflight evidence. They establish \(Z_\Lambda=J+24\), track removal of the 24 weight-one currents, and show that a bare chiral \(c=24\) sector does not fit unchanged as the internal sector of a four-dimensional critical bosonic, heterotic or type-II string. The remaining tasks are:

1. Plot \(J(\tau)\), \(|J(\tau)|\) and a normalized measure separately on a fundamental domain. Mark the cusp divergence. This prevents the absolute value from being called a probability by appearance alone.
2. Write a candidate Bloch model only if an explicit \(H(k)\) is supplied. Test whether any map \(k\mapsto\tau\) respects periodicity, degeneracies and symmetry. If not, close the Bloch reading.
3. Maintain a carrier table distinguishing: Leech vector, VOA state, worldsheet modulus, response matrix, target-space field, spacetime point and detector record.

Decision rule: any proposed identification that lacks a map, invariant and inverse/fibre description remains analogy.

Tranche 1: genus-two relational test

This is the closest extension of the existing corpus.

The characteristic-orbit and separating-order parts are now complete in the seam-closure result. They show that the two sheets close only under an order-16 finite stabilizer, while full modular covariance requires all ten even characteristics. They also show that the seam changes separating zero-mode order, not only bookkeeping. The remaining tasks are:

1. Compute the Leech and Monster genus-two partition data in the same sewing normalization.
2. Match Fourier coefficients to pair-Gram strata \(0,12,15,16\) where the comparison is mathematically defined.
3. The reduced Majorana--Weyl (T1/S1) spin action is now complete, and its robust data are relative fixed/cycle ratios rather than a lone absolute phase. Construct or locate the missing genus-two fermionic Baby-Monster spin amplitudes, derive their anomaly line, combine the two factors, and retain only common-phase-invariant ratios. Then lift the established finite determinant sign to the analytic \(\chi_5\) line and construct or rule out a junction to the distinct \(2B\)/Leech Rule-60 seam. Do not substitute the additive bosonization sum or a chosen anomaly trivialization for this calculation.
4. If the analytic line closes, construct a derived physical response map taking the separating divisor to the response-root rank-one boundary \(\det\Psi=0\). The canonical map \(Q=\operatorname{Im}\Omega\) has now failed this test: separation leaves \(Q\) positive definite. Shared sign parity without a different zero-locus/transport map does not identify the modular and response Pfaffian lines.
5. Implement the response map as a two-quadrature coherent observable, or derive the physical phase reference selecting one real seam normal. A power-only scalar necessarily measures the squared, orientation-blind seam.
6. Keep the binary character and continuous loop phase distinct. The \(\chi_5\) sign maps to \(0/\pi\) holonomy, where the three-mode directional odd channel vanishes; use referenced interference for the sign unless the full sewn multiplier supplies a canonically transportable phase beyond \(\mathbb Z_2\).
7. Test factorization, positivity and whether the seam produces a distinct correlation/sewing observable.

Success: either a modularly consistent ten-sector completion or a precisely marked subgroup defect with a new computable invariant. Failure: the orientation bit cannot be included without breaking factorization, positivity or the declared subgroup/full-group covariance. Value either way: closes the strongest present Leech--Jordan join.

Tranche 2A: worldsheet gravity route

1. Perform an exact central-charge and chirality budget for bosonic, heterotic and type-II placements of the Leech/Monster sector.
2. For every surviving budget, enumerate massless spin-2, spin-1, scalar and fermion states after all projections.
3. Test modular invariance, level matching, anomaly cancellation and tachyon absence.
4. Couple target backgrounds and compute the leading beta functions.
5. Compactify to four dimensions and derive the Einstein-frame effective action.

Kill conditions: no consistent left/right completion; no four-dimensional graviton; unavoidable anomaly/tachyon; or moduli/vacuum incompatible with a controlled low-energy regime.

Tranche 2B: induced-action route

Start modestly: do not claim emergent spacetime.

The scalar induced-action preflight now rules out using \(196884\) as an untyped light-species count. It also shows that the unencoded choice \(\xi=0\) versus \(\xi=1/6\) changes the quadratically induced Einstein coefficient from nonzero to zero. The remaining tasks are:

1. Choose the smallest source-derived finite spectrum with declared spins, masses and statistics on a background \(g_{\mu\nu}\).
2. Define its covariant kinetic operators and curvature couplings.
3. Compute Seeley--DeWitt coefficients \(a_0,a_1,a_2\).
4. Read off induced vacuum energy, Newton coefficient and curvature-squared terms.
5. Test signs, regulator dependence, species scaling and radiative stability.
6. Compare the induced connection/compensator content with the existing (Spin(2,3)) gravity checkpoint.

Success: a positive, parametrically controlled Einstein term and healthy correction scale tied to a non-arbitrary lattice invariant. Failure: wrong sign, uncontrolled cosmological term, regulator-only relation, ghosts below the cutoff, or no unique mapping from lattice states to 4D fields.

Tranche 2C: emergent-continuum route

1. Specify a local microscopic complex. Leech vectors may label states or links, but must not substitute for incidence.
2. Define local amplitudes and reflection-positive Euclidean or unitary Lorentzian evolution.
3. Find a tunable critical point and measure correlation length.
4. Infer spectral/Hausdorff dimension under coarse graining.
5. Test for an emergent symmetric rank-two correlator with two transverse polarizations.
6. Test approximate diffeomorphism Ward identities and an Einstein-dominated derivative expansion.

Kill conditions: no local continuum limit; dimension not near four; scalar/vector contamination cannot decouple; Lorentz violation remains relevant; or no universal spin-2 pole.

Tranche 2D: AdS₃ benchmark

1. Treat Monster CFT/AdS₃ as a lower-dimensional control, not the target theory.
2. Reproduce the vacuum character and modular images.
3. Test positivity, discreteness and factorization of the proposed bulk partition data.
4. Identify precisely which step fails in known pure-gravity saddle sums.
5. Ask whether the corpus's seam supplies an allowed additional sector rather than assuming that it does.

Success: a sharper lower-dimensional bulk dictionary or a proved exclusion. No promotion: even complete success here does not yield AdS₄ or four-dimensional EFE.

Tranche 2E: direct Cartan-action route

The Cartan seam gate now proves that the seam cannot itself be the compensator or connection, and that \(V\leftrightarrow -V\) is a connected two-time-plane rotation rather than a physical binary sector. The remaining route is:

1. Treat the existing independent \(Spin(2,3)\) connection and constrained compensator as the target interface.
2. Seek a microscopic model that produces the seam and compensator jointly, with a gauge-invariant defect/orientation coupling.
3. Require an explicit term-by-term map into a MacDowell--Mansouri/Stelle--West action.
4. Re-run the existing quadratic spectrum, BRST/measure and radiative-closure gates with the new microscopic input.

Kill conditions: the lattice datum is gauge-removable, supplies only a discrete label with no action term, or leaves the connection/couplings arbitrary. The seam-alone version now meets this kill condition. Success: a new microscopic vacuum or complete action, the precise reopen condition of the globally exhausted finite gravity search.

Tranche 2F: entanglement/thermodynamic route

The thermodynamic preflight now proves that Monster/Cardy entropy is a global grade entropy and that imposing \(S=A/(4G)\) fixes only \(A/G\), not area or Newton's constant. The remaining tasks are:

1. Derive subregions and reduced states from the microscopic carrier; a global graded trace is insufficient.
2. Calculate entanglement entropy and its regulator dependence.
3. Test an area law and identify the geometrical area variable.
4. Establish a local first law or entanglement-equilibrium identity.
5. Derive the field equation and check conservation, Newton normalization and higher-curvature corrections.

Kill conditions: no local factorization/subregion algebra, volume rather than area scaling, state-dependent Newton coefficient, or no causal-diamond interpretation. The \(J\)-alone version already fails the first gate.

Physical and engineering experiments

The gravity question is presently theoretical, but nearby structures are testable.

Modular spectral emulator

Build a programmable photonic or microwave network whose mode multiplicities realize the first finite coefficients of a selected theta/character function. Perturb boundary conditions through modularly related configurations and test spectral invariance.

• observable: resonance frequencies and degeneracies;
• nuisances: fabrication disorder, loss, mode mixing and finite-size splitting;
• decision: whether modularly related controls preserve the declared spectrum within a preregistered tolerance;
• payoff: theoretical engineering and a detector for modular-state design, not a test of EFE.

Pair-Gram response device

Use two coupled response channels and engineer four discrete correlation strata corresponding to the Leech minimal-pair Gram determinants.

• observable: calibrated (2\times2) response matrix and determinant;
• decision: whether the same physical device realizes positive-cone transport and the predicted stratum multiplicities;
• payoff: tests the physicality of the corpus's strongest lattice--Jordan carrier map.

Seam/defect experiment

Implement a sign-reversed double and affine boundary twist in a topological, acoustic or photonic network.

• observable: defect-localized phase, spectral flow or holonomy;
• decision: distinguish a true orientation-sensitive response from relabelling;
• payoff: tests whether the extra seam data has operational residue.

The fully specified first implementation uses a complex programmable resonator coupling and IQ/homodyne detection. Its preregistered discriminator is linear odd interference versus quadratic even power under coupling-phase reversal.

The detector reassignment result now gives this referenced protocol priority for a direct \(\chi_5\)-character test. The three-mode ring remains the correct detector for a genuinely continuous intrinsic flux, but the binary modular sign alone reaches only reciprocal \(0/\pi\) sectors and cannot predict its directional odd channel.

None of these experiments tests quantum gravity directly. They determine whether the proposed interface can exist as controlled physics before it is asked to carry gravity.

Open-topic register

Mathematical

• full genus-two Monster partition/sewing data in the chosen normalization;
• relation of affine seam defects to known VOA modules and topological defect categories;
• explicit intertwiner, if any, between the Siegel modular carrier and the response-cone carrier;
• higher-genus consistency and factorization;
• whether pair-Gram multiplicities determine more than the genus-one character.

Dynamical

• Hamiltonian or transfer operator behind the proposed “subroutine”;
• origin of time and distinction between worldsheet, target and laboratory time;
• selector that turns an orbit of admissible states into a realized history;
• locality and causal propagation;
• metastability, dissipation, measurement and records.

Gravitational

• complete action and variational principle;
• four-dimensional Lorentzian signature and dimension selection;
• conserved stress tensor;
• independent connection/coframe and compensator provenance;
• healthy massless spin-2 spectrum;
• cosmological constant and higher-curvature control;
• equivalence principle and universal coupling;
• quantum measure, anomalies and radiative closure.

Empirical

• a distinctive observable not absorbed by conventional EFT parameters;
• detector bandwidth and quotient losses;
• finite-size and disorder scaling in analogue devices;
• astrophysical/cosmological channel only after a complete EFT exists.

Interpretive

• whether the lattice is ontology, internal state space, code, boundary data or calculational representation;
• whether admissibility is fundamental or a disciplined constraint on models;
• whether “computation” means an update rule, a quotient/classifier, or merely effective predictability.

Exhaustion statement

Given the presently named ingredients--Golay, Leech, Monster, modular (j/J), the pair-Gram (H_2(\mathbb R)) map, (Sp(4)≅(Spin(2,3)), and EFE--the logical routes above exhaust the distinct ways the lattice object can reach gravity without changing the question:

1. it is a worldsheet/internal sector whose consistency yields target equations;
2. it is matter/spectral data integrated into a 4D effective action;
3. it is microscopic spacetime data with a continuum limit;
4. it is boundary data with a holographic bulk.
5. it selects or derives a direct Cartan gauge-gravity action.
6. it supplies local entropy/entanglement whose equilibrium law is Einstein's equation.

A future proposal should be placed in one route or shown to add a genuinely fifth output type. The next high-value work is the genus-two seam calculation and the central-charge budget, followed by the finite induced-action toy. Those calculations are capable of producing closure, not just more resemblance.