Memory by Subtraction A class-encoded memory architecture, its algebra, its read head, and the experiments that would kill it

In three words: Holonomy-Formatted Memory. A record whose bit is a class rather than a magnitude — a count of holes cut into a solid, deposited on a loop, and read back as an integer. If you only read one line of this paper, read that one; everything else is what it costs, what it buys, and what would prove it wrong.

BOTTOM LINE UP FRONT Every bit in every deployed memory is a magnitude. A retardance, a resistance, a charge, a reflectivity. Magnitudes decay, and every archival lifetime figure in existence is an extrapolation of how slowly one particular magnitude slides below a readable threshold. This paper asks what the other column costs and buys. Store an integer instead — a count of holes. A count does not slide. It is or it isn’t.

Four things follow, and three of them are somebody else’s:
1. The design gate. Name your structure group before you fabricate. Abelian and discrete → exactly computable, no convention to argue about. Non-abelian, or reduced modulo a symmetry → defined only up to a choice you must publish. This sorts memory proposals on paper, before a fab budget.

2. The algebra of the cut, which is the contribution. A sealed cavity adds a 2-sphere and gives you an abelian ℤⁿ readout — an address. A through-tunnel adds a handle and gives you a free non-abelian readout — a sentence. The map between them is the abelianisation. You choose your readout algebra at write time, which no storage scheme currently does.

3. The write head already exists. Nanovoids in fused silica, characterised twenty years ago, with period, count and connectivity already on dials.

4. The read head is a sixty-year-old theorem. For a planar domain with h holes the heat-trace constant term is a₀ = (1 − h)/6. You can hear the number of holes. You cannot hear the shape — and that turns out to be a security property rather than a limitation.

What we are not claiming. Not the fabrication. Not the read mathematics. Not density — we lose on density and say so early. Not a comparison against anyone’s product. And nothing here is measured by us: this is theory and synthesis with a specified experimental programme, and Section 9 lists the five experiments that would end it.

Keywords: topological memory · class-encoded storage · Betti numbers · Euler characteristic · Hodge Laplacian · fused silica · femtosecond laser writing · nanovoid · microexplosion · write throughput · parallel femtosecond writing · spatial light modulator · electron beam · acoustic vortex · ultrasound · homotopy · abelianisation · structure group · heat trace · Kac · isospectral domains · persistent homology · physically unclonable function · retention tiers · archival storage · Chladni · hydrogel