Gated exciton fluids V: three dimensions - what the T-duality, holography and projection analogies do and do not give, measured

Fifth paper of the gated-exciton line (I: 10.5281/zenodo.23294615; II: 10.5281/zenodo.23297479; III: 10.5281/zenodo.23298292; IV: 10.5281/zenodo.23302787). It asks whether ideas borrowed from string theory (lifting to 8 or 24 dimensions, holography, projection, T-duality) help with the three-dimensional node-arrangement problem. Known fact first: in R^3 no configuration is universally optimal, because the Poisson dual of FCC is BCC and the two exchange roles at the self-dual Gaussian parameter alpha=pi. Two pre-registered rounds of experiments follow.

Round 1: among 3D Bravais lattices the minimiser is always FCC or BCC (29/29 kernel points; Gaussian crossover a*=pi to 3e-14); random-restart searches over non-Bravais cells with up to 16 points find nothing below min(FCC, BCC, HCP, A15) in 9,860 restarts; on S^3, where the 600-cell is a proved optimum, a local optimiser reaches its energy (to 1e-8) for long-range and soft kernels but not for stiffer Gaussians (version 5: for the soft Gaussian a=0.5 those energy-hits are not the 600-cell); along the Bain path the self-dual (isodual) lattice at alpha=pi is the energy maximum between the two degenerate minima, so the T-duality analogy explains why no 3D universal optimum exists rather than pointing to one.

Round 2 (negative throughout): F1, an alpha-homotopy from a soft kernel, reaches the proved 600-cell in 0, 0 and 6 of 50 seeds at a=1, 2, 4, while plain restarts at equal compute reach it in 50/50 at a=4; by the registered stop rule our 3D searches are declared unreliable. T2: Gaussian mixtures straddling the self-dual point give nothing below FCC/BCC/HCP in 5,100 restarts. T1: a search for formally self-dual configurations accepts nothing, but its objective cannot tell the known one (diamond) from configurations 1e-3 away, so the negative is uninformative. H1: the two-point linear-programming code failed its own one-dimensional sanity gate (under an admissibility tolerance fixed in code, not registered) and was not run in d=3. Deviations from the registrations are listed; two adversarial reviews and their resolutions are in the source archive.

Version 2 adds H1': a literature check found that H1's question at alpha=pi is already answered (Cohn and de Courcy-Ireland, arXiv:1603.09684, Table 1.1: two-point bound 0.2232 against a record of 0.2315 in d=3). An attempt at rigorous certificates was abandoned in development; a re-registered numerical, not certified run with the same LP code reproduces the published d=1, 2, 3 values at alpha=pi within the registered windows (d=1 and d=3 to the printed 8 digits) and gives gaps of 0.48%, 6.7% and 21.6% below the best lattice in d=3 at alpha=2, 4, 8: numerically, at polynomial degree up to 40, two-point bounds do not settle the 3D question at any alpha tested. A hostile review found no blocking item; its corrections are applied.

Version 3 adds H1'' and H1''': interval-certified two-point LP bounds (256-bit interior point, exact cutting planes, arb certificate, Cohn-Kumar bound as stated by Cohn and de Courcy-Ireland). At K=40 (polynomial degree 39) the d=3, alpha=pi validation gate failed (certified 0.2232177895, 3.05e-8 below the published 0.22321782, tolerance 1e-8), so the alpha=2, 4, 8 runs were gated. A second pre-registration changing only K passed at K=50 (degree 49; certified 0.2232178194). The V3 value rises monotonically with K (K=60: 0.2232178245): raising K is sufficient, though the cause of the K=40 shortfall is not identified. The registered results are certified lower bounds at alpha=2, 4, 8 lying 0.48%, 6.7% and 21.6% below the best lattice. The bound values are rigorous; that two-point bounds at this degree cannot settle the 3D Gaussian-core question there remained numerical in version 3 (it needed a dual certificate; see version 4). A hostile review found a defect in the certificate code (sub-ulp intervals where fhat < 0 between near-double roots were not checked); an independent re-check confirms all reported bounds, the K=60 value (not a gate) only after adding a small lift. One part of the registered prediction failed (alpha=8 certified value 8.3e-6 below the version-2 numerical value, against a predicted 1e-6); one evaluator deviation is disclosed.

Version 4 adds H1d: rigorous upper bounds on the two-point (linear-programming) bound for Gaussian energy in R^3 at density 1, by an energy adaptation of R. Li's discrete-reduction dual (arXiv:2206.09876, sphere packing): fold samples of an admissible function modulo a grid, solve a finite dual LP, verify an exact rational dual point in interval arithmetic. Validation: in d=1, where the bound is sharp, the certified upper bound is within 7.3e-8 (relative) of the exact value. In d=3 the certified upper bounds lie below min(E_FCC, E_BCC) by 0.414%, 5.82% and 18.3% at alpha=2, 4, 8, so the two-point bound (as stated in Cohn-de Courcy-Ireland Prop. 2.2) cannot prove FCC or BCC optimal there (the full Prop. 2.2 class is reached by a lemma added after registration); at alpha=pi the upper bound 0.2242866642 is below the coexistence record 0.23153532. With version 3's lower bounds the two-point bound is bracketed (alpha=pi: [0.2232178193, 0.2242866642]). The gaps were visible numerically before; the certificates are new. 29 of 31 registered grids certified (2 solver failures disclosed); an evaluator flaw found in a pre-run review was fixed during the runs (disclosed; it only tightens the criterion); all certificates were re-verified with independent code. A literature search found no published rigorous upper bound on this LP bound in d=3 (limited search).

Version 5 adds two things. H1d2/H1d3: the version-4 certified upper bounds extended to 13 values of alpha in [0.5, 8] (H1d2's d=1 sanity gate failed by exactly the a-priori cost of the LP's epsilon buffer, a design error disclosed before evaluation; H1d3 re-registered an epsilon-aware gate; 52 of 53 grids certified, all re-verified independently). At all 13 alpha the certified upper bound lies below min(E_FCC, E_BCC): the two-point bound as stated in Cohn-de Courcy-Ireland Prop. 2.2 is nowhere sharp for FCC/BCC on this grid. Normalised by the Siegel (expectation) bound, the certified gap is largest at the self-dual point alpha=pi (0.94%; true gap pinned by the version-3 lower bounds to [0.94, 1.09]%) and falls to 0.04% at alpha=8 and 1e-9 at alpha=0.5; a rigorous lower bound on a gap captures at most the stated fraction. F1s/F1s2: the round-2 F1 stall re-tested with a better-conditioned energy (mean-subtracted, long double, relative-reduction stop at one ulp), as an external reviewer of version 3 asked. From a0=0.25 the states reach the 600-cell energy to 1e-12 but its geometry in 0 of 150 starts, stopping on the new, 256-times-lower floating-point floor in a landscape degenerate above degree 7 (the states are 5-designs; Cohn-Kumar 2007); a floor-free continuation keeps descending without reaching it. The floor explanation of the stall height is confirmed, the expectation that a lower floor reaches the 600-cell is withdrawn, the reviewer's question (is a converged soft-kernel state the 600-cell?) stays open, and the round-2 verdict stands. A registered control from a0=0.5 (F1s2) fails too (0/50 at a0, 0/0/7 at the ladder ends), limited by its iteration cap rather than the floor, and corrects a premise: round 1's '50/50 at a=0.5' was an energy-only count, and a re-run of those starts with coordinates kept finds none at the 600-cell (C1's verdict is unchanged). Four hostile reviews (H1d3, F1s, F1s2, whole draft) are applied; the F1s review rejected our first reading by experiment, the F1s2 review three statements of our first write-up; all corrections and the reviewers' scripts are in the source archive.

Everything is small-scale, classical and single-engine; nothing uses the unrefereed planar lattice theorem; no claim about excitons. Written by an AI system; checked by independent instances; not refereed.