How Much Can a Sophon Remember? A Bekenstein-Bound Argument on Memory, Mass and the "Proton" Description

1. Introduction

It is well known that the sophon is a proton (a single particle of ordinary matter) which Trisolaris unfolded from eleven dimensions into two, etched with circuitry and folded back. The survey literature of the Galaxy Era goes on to call the result a "supercomputer" of proton size (the term is the human rendering, and we say nothing of the Trisolaran original), as though size placed no limit on what the machine could hold. The description fails in the arithmetic of information (not in the engineering, about which we have nothing to add), since a body of given radius and energy can store only so many bits whatever is etched into it.

Let me set the figure down, because I have not seen it written in one place.

Our vantage is the Galaxy Era, from which the sophons are a closed chapter of the Crisis and of the eras that followed it. No Galaxy-Era year is printed for the present writing, and Section 2 states the vantage and the one point of usage which it raises.

The inputs are old and not in dispute. The limit on the information which a region of given radius and energy can hold has been derived many times (the survey on which we rely, for bodies with compact extra dimensions, is Ostrow, 150 GE), and the matching limit on the rate of elementary operations took its modern form later (Kavanagh, 230 GE). None of these works, nor the tables and statements used below, gives the number that matters here (how many bits a body of proton radius and proton mass can keep, and what each further bit costs).

We confine the paper to structure, memory and mass. How paired sophons communicate belongs to the companion paper by Nduka-Feld in this issue and the droplet is a separate technology that we leave aside. The argument is untestable by design (it reports no measurement of any sophon) and says only what the known limits force a body of this size to be.

In what follows, Section 2 sets out the system and its record, Section 3 obtains the capacity, the "bit price" and the rate limit, Section 4 checks the arithmetic against known bodies and the archived statements, Section 5 weighs the four ways out, and Section 6 gives our conclusion and what the records need.

Translator's note: this paper was rendered into English from the Chinese original by the Branch translation office, and the credit belongs to Aldous, T.

2. System Description

The system is a single particle, and in the account that Trisolaris gave (and that the survey literature has kept, Lindal, 90 GE) a proton was unfolded from eleven dimensions into two, a circuit was etched upon the unfolded surface, and the proton was folded back to its ordinary form, after which paired sophons gave Trisolaris contact with Earth.

Three quantities enter the argument, the radius R of the body after refolding (0.84 fm, the proton value of the nuclear tables, Brightwell, 120 GE), the energy E (the proton rest energy, 938.27 MeV, until Section 5 relaxes it) and the information I in bits, with ħc = 197.33 MeV fm throughout. A "proton" description, if true, fixes R and E at these values, and every departure considered below departs from one of the two.

As a methodological note on vantage and affiliation, we write from the Galaxy Era, once the Crisis, the Deterrence Era and the Bunker Era have closed, and treat the sophons, like cosmic sociology in its entirety, as a historical subject at this vantage. The Jupiter-shadow cities were destroyed in the dimensional strike at the end of the Bunker Era, and the Galaxy-Era institute that employs me kept the name of the destroyed Jupiter City 3 Branch. We follow the survey bodies in writing years in Galaxy-Era terms, and the collation of the statements in Bunker-Era terms.

Our source for the sophon's own description is the Sophon Statements Register (our term, and below simply the register), which is our own coding of the 23 items in the Bunker-Era collation (Venner, 14 BkE). The items are statements by Trisolaris, they reached Earth through the sophon channel, and humans had no independent means of checking them, so that the register is a record of testimony (and not of observation).

3. Analysis / Model

The Bekenstein bound limits the information I that a system of total energy E can hold inside a sphere of radius R, and it reads I ≤ 2πRE/(ħc ln 2). The derivation uses no property of the system except those two quantities (Ostrow, 150 GE), which is why it applies to a machine built by any engineering (even one that folds a particle through eleven dimensions), and why the number of bits that it gives is a ceiling and not an estimate.

For R = 0.84 fm and E = 938.27 MeV the bound is 36.2 bits, with a range of 34.5 to 37.9 bits for radii from 0.80 to 0.88 fm (the spread we adopt for the uncertainty of the tabulated radius), so that the ceiling is four and a half bytes (enough for a short register, and not for a program).

At fixed radius each further bit needs a further ħc ln 2/(2πR) of energy, which at 0.84 fm is 25.9 MeV, (0.0276 proton masses). We call this the "bit price" (our term, and in quotation marks on every use), and a sophon of proton radius that holds N bits must weigh at least N times the "bit price" (the first 36 bits being paid for by the proton's own energy). The consequence is a steep one, with rising bit count, rising energy, rising mass, and a slope that no choice of encoding can change.

The rate limit is the other half of the argument. The Margolus-Levitin limit allows at most 2E/(πħ) elementary operations per second for a system of energy E (Kavanagh, 230 GE), which at proton energy is 9.07 × 10²³ per second, (one step in 1.1 × 10^-24 s). Divided across the 36-bit register, that is 2.5 × 10²² full sweeps of the register in every second, so speed is ample (for any task that fits in 36 bits) and memory is the bottleneck.

Table 1 gives the consequences for three round capacities (bytes of eight bits, decimal prefixes), with the energy needed at the radius of the proton, the resulting mass, the radius which the same capacity would demand at proton energy, and the scale ħc/R across that radius, which Section 5 uses.

4. Validation Against Field Data

A bound that is only arithmetic cannot be tested against data, and the checks below are checks of our own arithmetic (structural by construction) and of the register, not confirmations of the bound itself. The first is the normalisation. A body whose radius is its reduced Compton wavelength has RE = ħc and a bound of 2π/ln 2 = 9.06 bits whatever the particle, so that the electron, at 386.16 fm, returns 9.06 bits, and the proton, whose radius is 3.99 reduced Compton wavelengths, returns 9.06 × 3.99 = 36.2 bits (as in Section 3).

Our second check is that no known body is set above its bound by a count of its states (a count that cannot fail for a true bound, and shows only its slack). A nucleon has four spin–isospin states, so a nucleus of A nucleons holds at most 2A bits on that count (the same count is applied to the proton), and the bound exceeds it in every row of Table 2, by a factor of 18.1 for the proton and by 36.0 to 117.6 for the nuclei. Radii and nuclear (not atomic) energies are from the tables (Brightwell, 120 GE), and every bound uses unrounded inputs. The bound per nucleon ranges from 36.2 bits for the proton to 235.3 for lead-208, and the proton has the least slack of any body in the table on this count (the one body we are asked to treat as a computer has the smallest margin; with a single spin bit for a known proton it would be 36.2, level with the alpha particle's 36.0 and no longer the smallest).

Table 3 gives the coding of the register, which tests the other half of the description. Of the 23 items, 19 name a proton and 4 say only that the sophon is a particle (Table 3 gives the other codings), and none states a mass, a capacity or an operation rate. None, as we have coded the items, uses the word "supercomputer" (whatever the source of that characterisation, no item attaches a figure to it), and the silence on all three quantities is complete.

The behaviour that humans did observe points the same way from the other side. Over the Crisis, the Deterrence Era and the Bunker Era the sophons were seen to disturb accelerator experiments, to set a countdown before the eyes of a single observer, to follow the communications of the whole species, and in the end to operate the android "Sophon" (all of it history at our vantage, and none of it giving a mass or a capacity). The system did work that no 36-bit store could do, but the observations do not say where its bits were kept, and so the exits of Section 5 belong to the argument.

Against the whole exercise there is one main objection, that a bound derived for ordinary matter should not be applied to a machine made by a civilisation which could fold a proton through eleven dimensions. I shared that objection for years. What changed my mind was the observation that the derivation uses only the radius and the energy, and that a civilisation able to fold a proton has not thereby changed either of the two numbers which the "proton" description fixes (if it has changed them, that is one of the exits below and not a refutation of the bound).

5. Failure Modes

Four exits from the 36-bit ceiling are open, and nothing in this paper closes any of them. The first is that the sophon is far heavier than a proton. Table 1 gives the price as 221 proton masses for a kilobyte, 2.21 × 10⁵ for a megabyte (about 223 kilodaltons in the chemists' unit) and 2.21 × 10⁸ for a gigabyte, if the radius stays at 0.84 fm, and a heavier body does not contradict the bound while it does contradict the "proton" description, so the exit is open only to someone who gives that description up.

Second, the state may be kept in the compactified dimensions (where a four-dimensional statement of the bound need not apply). A store in the compact directions escapes the ceiling only if they are larger than the proton, at least 186 fm for a kilobyte, 0.186 nm for a megabyte and 186 nm for a gigabyte, at proton energy. Such an extent carries excitations at ħc/R (1.06 MeV, 1.06 keV and 1.06 eV in Table 1), and any field of the proton which reached into it would show them in nuclear or atomic spectra, whereas the structure of the proton confines the extra-dimensional extent of its fields to a small fraction of its radius (Pemberton, 260 GE). The store would therefore need degrees of freedom which do not couple to the proton's own fields, and would have to keep its state through the refolding (we know of no argument that excludes this, and none that supplies it).

Third, the sophon may hold almost nothing itself. Paired sophons, on the account of Trisolaris, kept it in real-time contact with Earth, and a body which relays its state across the link needs local memory only for what it must hold between exchanges (a buffer of four and a half bytes is small, but it is not nothing). The machine would then be the pair together with the Trisolaran installation behind it, with the proton as its terminal. This exit leaves the bound and the "proton" description intact, costs no mass and no extra dimension, and fits the work which the sophons were seen to do (Section 4), so it asks least of physics. Its price is paid in the link, which belongs to the companion paper by Nduka-Feld (the loop delay is bounded there at no more than 132 s, on an assumption about when each question was composed which that paper cannot test, and the limit leaves a far shorter delay open), and whether the link was fast enough for a terminal to work in real time is a question which this paper cannot answer.

Fourth, the "proton" description is Trisolaran testimony, and no human has ever verified it. Nineteen of the 23 items name a proton and none gives a mass, so that if the word was a convenience of the speaker (a name for the substrate that the listener would understand), the first exit is open at no cost to anything but the description.

The argument has weaknesses of its own. The bound counts information inside a sphere, and during the unfolding the body occupies a surface of large extent, so the unfolded phase has a capacity we leave unstated (the question is what survives the refolding). The radius is also a charge radius, and a strong-interaction radius which differed by 20% would move the ceiling from 36.2 bits to between 29.0 and 43.4 bits (it shifts the numbers, and does not remove the ceiling).

6. Conclusion

We find that a sophon of proton radius and proton mass can hold at most 36.2 bits, that each further bit costs 25.9 MeV at that radius, and that the Margolus-Levitin limit of 9.07 × 10²³ operations per second leaves speed unconstrained (so memory is the bottleneck). A proton-sized, proton-mass body is therefore likely excluded as the store of a "supercomputer", and the sophon, if it did the work it was seen to do, was likely not a self-contained machine. The finding concerns the body and not the system, and the one alternative that deserves weight is that the memory lay on Trisolaris with the sophon as a terminal, which remains open because it needs a link whose mechanism and speed we cannot test here. A later archive may supply the mass and the capacity which the register lacks, and the two figures together could settle two of the exits at once.

My school is the Lurker framework, in which the Lurker (also "Spirit", "Spirit Mother" in some translations) stands as the cosmogonic power behind the course of the dimensional-reduction war and of the ages after it, and I hold the Luo-Ji / Ye axioms to be valid in a local domain only, and from this vantage historical as well (a correct account of one epoch, and no more). Neither the Bekenstein bound nor the Margolus-Levitin limit depends on that view, although my interest in the compact dimensions does.

The records need correcting accordingly. The term "supercomputer" should leave the survey descriptions of a proton-sized sophon, or be marked as a human rendering whose Trisolaran original this paper does not claim to know, and the silence of the register on mass and capacity is a gap in the archive and not a detail (the 23 items fix the substrate and little else). Any later account of a sophon's memory, in the archives of the Bunker Era or in the survey literature, has to state which of the four exits it takes.

Let me close with an analogy that establishes nothing and has no standing in physics. A clerk who is told that a whole ledger was written on a single grain of rice does not doubt the informant, and does not doubt the rice, but counts the characters that the grain can hold (a few, and no more), weighs it, and asks whether the writing was on the grain or on the sack that came with it.