Cubrim — Global Addresser

The honest limit, stated first

A fixed short code can distinguish only a finite number of states, while the space of possible files grows exponentially with length — so a short address alone can never uniquely denote every long sequence. An address is meaningful only together with a catalog where the object already exists. And any honest result must charge the full cost: the addresses, the metadata, the residual data — and the shared matrix itself, which is only worth its size when amortized across many files and devices. Cubrim-2 therefore does not promise to “compress any file into 16 bytes”. It maps where global addressing genuinely beats local compression — and records, just as openly, where it structurally cannot.

How much space would ALL possible matrices take

We count in bits. An N-dimensional cube is a visual arrangement of a bit sequence: a 4×4×4 cube is 64 bits laid out in three dimensions. Dimensionality (1D/2D/3D/4D) is a way to arrange the same bits, not a different amount of them. Hence the number of all possible matrices of length B bits is 2^B and depends only on B: a 64-bit 1D string and a 4×4×4 cube share the same 2^64 ≈ 1.8×10^19 states.

B, bits	cube example	all possible = 2^B	store them all (bytes)	fits in 20,000×1 TB?
8	2×2×2	2.56×10²	2.56×10² B	yes
27	3×3×3	1.34×10⁸	4.53×10⁸ B	yes
51	—(порог/threshold)	2.25×10¹⁵	1.44×10¹⁶ B	yes
64	4×4×4	1.84×10¹⁹	1.48×10²⁰ B	no
125	5×5×5	4.25×10³⁷	6.65×10³⁸ B	no
256	4×4×4×4 (4D)	1.16×10⁷⁷	3.71×10⁷⁸ B	no
1000	10×10×10	1.07×10³⁰¹	1.34×10³⁰³ B	no
4096	8×8×8×8 (4D)	1.04×10¹²³³	5.35×10¹²³⁵ B	no

Capacity yardstick: 20,000 disks of 1 TB = 2×10^16 bytes = 1.6×10^17 bits. The full-enumeration threshold is B = 51 bits: the last length at which all 2^B matrices (with their contents, 2^B×B bits = 1.44×10^16 bytes) still fit; at B = 52 they no longer do (2.93×10^16 bytes). A 4×4×4 cube (64 bits): just the 64-bit addresses of all 2^64 matrices would take 1.48×10^20 bytes ≈ 147.6 EB — four orders of magnitude beyond the yardstick. At B = 256 there are 1.16×10^77 matrices — still slightly fewer than the atoms in the observable universe (~10^80); the “more than atoms” line is crossed at B ≈ 266. Beyond that, super-exponential growth with no physical storage prospect whatsoever.

The honest conclusion: “all possible matrices” cannot be enumerated — 2^B diverges super-exponentially at lengths below a single machine word. So the Addresser stores not all possible blocks but only the ones ACTUALLY ENCOUNTERED (CAS/deduplication): their number is bounded by the volume of real data and obeys the track’s measured laws — 89.9% of blocks in an uncurated matrix occur exactly once (AH-19), and an honest 16–64 bytes per object is achievable only on an exact match against the catalog (AH-05). This is the same fundamental limit as the block above: an address is meaningful only together with a catalog where the object already exists.

On dimensionality: 2^B does not depend on it, but dimensionality changes which bits land in one cube when REAL data is tiled — and therefore how many unique cubes occur in practice. That is measured by a scan (part B of the experiment), not by a formula.

Formulas: number of matrices = 2^B; storing the contents = 2^B × B bits; the ordinal address of one matrix = B bits (2^B states). All numbers are computed with exact integer arithmetic (python), not estimates.

How much space the ACTUALLY encountered matrices need (scan experiment)

A real CROSS-DEVICE corpus of 13.48 GB (a union of three hosts: arcana-devs 9.06 + arcana-www 3.38 + arcana-prod 1.04 GB; the hosts exchanged only cube hashes) was tiled into cubes of B = 4096 bits (512 bytes) in four layouts of the same length: a 1D string of 4096, 2D 64×64, 3D 16×16×16, 4D 8×8×8×8. A layout groups DIFFERENT bits of the file into one cube (strided tiling with a 4096-byte row), so the number of unique cubes on real data differs — even though the number of possible ones (2^4096) is identical.

layout	cubes scanned	unique	% unique	cross-host overlap (cubes)	store the unique
1D 4096	25 890 520	14 965 607	57.80%	1 847 617	7.14 GiB
2D 64×64	19 368 448	12 077 059	62.35%	1 260 924	5.76 GiB
3D 16×16×16	13 959 168	8 020 728	57.46%	943 354	3.82 GiB
4D 8×8×8×8	10 518 528	5 578 363	53.03%	768 085	2.66 GiB

There is NO saturation: the number of unique cubes grows almost linearly to the very end of the scan (devs-slice curve: 1D 10.9M unique at 7.3 GB → 12.3M at 8.3 GB) — real data at this cube length hardly “runs out”. On the devs slice, 72–82% of unique cubes were seen exactly once — the same measured law as AH-19 (89.9% on CDC chunks): an uncurated matrix is mostly dead weight. Cross-host overlap is real (0.77–1.85M cubes are shared between hosts), and it is what lowers the union corpus unique share to 53–62%.

Disks for the scan itself: 2.7–7.1 GiB of unique matrices per 13.48 GB of cross-device data — a fraction of one 1 TB disk. Extrapolating to the world’s stored data (on the order of 10 ZB = 10^22 bytes; order-of-magnitude per IDC Global DataSphere reports): with the measured 53–62% unique share on the cross-device corpus and NO saturation, the matrices would take ~5.3–6.2 ZB — billions of 1 TB disks, i.e. the same order as the data itself. THE EXTRAPOLATION IS LINEAR AND MARKED AS AN ASSUMPTION: the world corpus differs in composition, and saturation at larger volumes is not excluded — it was simply not observed in the scan.

Conclusions: (1) even storing only the ENCOUNTERED matrices at world scale is the same order as the data itself — the Addresser’s win lives not in a “warehouse of all cubes” but in deduplicating the repeated share (38–47% on the union corpus) and in r≥2 curation (AH-19); (2) the optimal layout by the “fewer unique” criterion is 4D (53.0% unique vs 62.4% for 2D on the union corpus) — multidimensional grouping does gather repeating bits more often, with a moderate margin (~6–9 pp); (3) the comparison is honest with a caveat: layouts need different alignment (2D — 256 KB blocks, 4D — 2 MB), so corpus coverage differs — figures are per each layout’s covered share.

Scripts: probe_matrix_scan.py + matrix_scan_dump.py (strides in the header; the cross-device union exchanges hashes only, MTX-scan-crossdevice-v1); cube hash blake2b-96; saturation curve checkpointed every 500 MB; all figures measured, the extrapolation labeled.

Wave 1 — research complete

Wave-1 deep research is complete: each of the 24 hypotheses carries a real measured verdict (full-cost charged accounting, falsification test executed, script+SHA on the card). GO means the mechanism works and is measured; NO-GO means it is closed by measurement or strict arithmetic. Predicted levers remain predictions and are labeled; the measured numbers live in every card below.

NO-GO · 9 GO · 15

shared context (dictionary / fragments) · 4 identity dedup — reference ≪ payload on exact match · 3 structurally cannot win (boundary) · 4 infrastructure cost accounting · 10 near-match + delta · 3

Generated: 2026-08-28T23:44:47Z · db:addressor_hypotheses

Hypotheses · 24

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AH-17 NO-GO W1 · 2026-07-13

Inversion on small unique files

If chunk addressing is applied to small unique files (<4 KiB), then total_cost becomes worse than local compression, because fixed costs (manifest, address, catalog, sync) exceed any saving.

Data class (Z)

small unique files (<4 KiB)

Address target

n/a

Predicted lever

the measured inversion point X becomes the router rule for AH-18

Ceiling category

structurally cannot win (boundary)

Mechanism

constant costs do not amortize over a small size

Falsification test

net bytes vs zstd by size bin; absence of an inversion would refute (not expected)

Full cost (total_cost)

constant items (manifest + >=1 address + catalog record + sync) give cost/size -> infinity as size -> 0; the saving is bounded by the file size.

Probe verdict

NO-GO — the predicted inversion point X is ABSENT on the real mix: the charged Addresser wins in EVERY size bin including <=1 KiB (0.5225 vs 0.5937), because small files are the most duplicated in real data. For a strictly unique file the Addresser's loss is arithmetic (same compressed bytes + refs + manifest) and needs no threshold; the router rule (AH-18) keys on measured dup-ness, not size.

Measured result (raw research log; descriptors partly in the RU original)

bins

#1

bin: 1-1024
addr: 0.5225
files: 76 360
local: 0.5937
addr_wins: true

#2

bin: 1025-4096
addr: 0.2764
local: 0.3986
addr_wins: true

#3

bin: 4097-16384
addr: 0.2313
local: 0.3311
addr_wins: true

#4

bin: 16385-65536
addr: 0.197
local: 0.2922
addr_wins: true

#5

bin: 65537-262144
addr: 0.1955
local: 0.3037
addr_wins: true

#6

bin: 262145-1048576
addr: 0.7053
local: 0.7621
addr_wins: true

#7

bin: 1M-4M
addr: 0.6237
local: 0.705
addr_wins: true

#8

bin: >4M
addr: 0.2538
local: 0.3942
addr_wins: true

date: 2026-07-14
probe: W1-inversion-v1
corpus: real 3-tree mix, 8.5 GiB, per-file transmission model, charged refs+manifest
verdict: NO-GO

verdict_note

NO-GO — the predicted inversion point X is ABSENT on the real mix: the charged Addresser wins in EVERY size bin including <=1 KiB (0.5225 vs 0.5937), because small files are the most duplicated in real data. For a strictly unique file the Addresser's loss is arithmetic (same compressed bytes + refs + manifest) and needs no threshold; the router rule (AH-18) keys on measured dup-ness, not size.

AH-18 GO W1 · 2026-07-13

Router: Addresser + Cubrim-1 (competitive selection)

If a two-phase codec is built — phase 1: exact dedup against the matrix (whole-file AH-05, then CDC AH-02); phase 2: residual into the local Cubrim-1 codec — then total_cost on real mixed user data is <= min(either method alone) + epsilon and strictly better on mixes with non-zero dup mass, because the levers are orthogonal.

Data class (Z)

real mixed user data

Address target

n/a

Predicted lever

strict no-worse guarantee + additive gain on dup mass

Ceiling category

identity dedup — reference ≪ payload on exact match

Mechanism

global repetition and local statistics do not compete for the same bytes; regression-proof per the Gotcha #4 pattern

Falsification test

the dup-mass threshold below which integration overhead eats the gain

Full cost (total_cost)

sum of phases: dedup line items (AH-02/05) on covered mass + the full Cubrim-1 total_cost on the residual + scheme bytes and phase boundaries (integration overhead).

Probe verdict

GO — deepened with the REAL Cubrim-1 backend (the zstd stand-in replaced, directive executed): router 0.7435 vs cubrim-only 0.9879 (−24.4 pp) on a device-external matrix; 55% of files are whole-file hits. Operational finding: per-file Cubrim CLI archives are costly on small files (0.988; Cubrim-1's world-bench strength is on large corpus files) — the router should pick the local backend by size (zstd-22 reference on this sample: 0.7484). The wave-1 phase-1 enable threshold (~10% dup fraction) stands.

Measured result (raw research log; descriptors partly in the RU original)

date: 2026-07-14
probe: W1-router-v1

ratios

router: 0.4504
local_only: 0.5535
addresser_only: 0.4528
router_with_catalog: 0.4533

routing

files_to_local: 107 567
whole_file_hits: 95 557
files_to_addresser: 96 278

verdict: GO
corpus_bytes: 8.19 GiB (8 788 989 359 B)
deciles_note: 0-9% покрытия -> addr выбран у 1.0% файлов; >=10% -> 96.6-100%

real_backend

date: 2026-07-15
probe: DR-router-cubrim-v1 sha:4c480631606a
matrix: prod∪www (device-external, без само-надувания)

ratios

cubrim_local_only: 0.9879
router_cubrim_charged: 0.7435
zstd_ultra22_reference: 0.7484

sample

bytes: 41.42 MiB (43 430 770 B)
files: 900

backend: РЕАЛЬНЫЙ Cubrim-1 CLI v0.2.2 (commit 3528bd1), round-trip верифицирован

routing

to_addresser: 508
whole_file_hits: 493

verdict_note

GO — deepened with the REAL Cubrim-1 backend (the zstd stand-in replaced, directive executed): router 0.7435 vs cubrim-only 0.9879 (−24.4 pp) on a device-external matrix; 55% of files are whole-file hits. Operational finding: per-file Cubrim CLI archives are costly on small files (0.988; Cubrim-1's world-bench strength is on large corpus files) — the router should pick the local backend by size (zstd-22 reference on this sample: 0.7484). The wave-1 phase-1 enable threshold (~10% dup fraction) stands.

catalog_bytes: 24.04 MiB (25 204 723 B)
router_beats_both: true
epsilon_scheme_bytes_pct: 0.0023%
dup_fraction_enable_threshold_pct: 10%

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Where the Addresser cannot beat local compression

Six boundaries are fixed by wave 1 as explicit anti-hypotheses and no-win zones: unique high-entropy data (personal media, encrypted streams — nothing repeats globally); tiny unique files below the inversion point (fixed catalog costs exceed any saving); mathematically generated matrices (an address into an exhaustive or random pool costs at least as much as the content itself); a byte histogram or hash used as the data carrier (order is lost and buying it back costs the file’s entropy); long-tail content fetched roughly once (the first transfer is never repaid); and fragment schemes whose gain collapses into what a shared dictionary already provides. A tiny fixed-size reference per object (tens of bytes) is honestly achievable only on an exact match against a catalog that already stores the object; everywhere else the win criterion is simply that the reference plus all charged costs stay well below the payload it replaces — the reference size is a metric to minimize, not a hard gate.