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: 2026-08-28T21:57:58Z · db:addressor_hypotheses

Hypotheses · 24

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AH-03 NO-GO W1 · 2026-07-13

Generated matrix (PRNG/pi/de Bruijn) — anti-hypothesis

If the matrix is made deterministically generatable (distribution cost ~0), then total_cost does not improve on any real data class, because an address into an exhaustive or random catalog costs at least as much as the content itself (information conservation).

Data class (Z)

any real class

Address target

n/a

Predicted lever

strict NO-GO (predicted)

Ceiling category

structurally cannot win (boundary)

Mechanism

M/2^512 coverage for a random pool; a k-gram's address inside a de Bruijn sequence occupies exactly the k-gram's own bits

Falsification test

exhibit a class where addressing into a generated pool beats the entropy cost — not expected

Full cost (total_cost)

address: by theorem >= the information content of the addressed fragment · metadata/residual: effectively the whole file · amortized: distribution ~0 — and it does not help.

Probe verdict

NO-GO — the arithmetic is now confirmed EMPIRICALLY: a real 64 MiB PRNG pool (seed=20260715, reproducible), 67.1M indexed windows, 100,000 real 64 B sequences from real files — 0 byte-exact hits (vectorized rolling hash + byte verification). The conservation-law expectation (~1e-141) agrees; any physical pool scales by the same law.

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

date: 2026-07-15
probe: W1-analytic-closure
verdict: NO-GO

empirics

date: 2026-07-15
probe: DR-prng-pool-v1 sha:1d8a05c33903
pool_windows: 67 108 801
real_sequences: 100 000
byte_exact_hits: 0

arithmetic

de_bruijn_address_cost: log2(256^k) = 8k бит = ровно информация самого k-грамма
de_bruijn_k8_length_bytes: 256^8 = 2^64 Б = 18.4 ЭБ (нехранимо)
random_pool_1TiB_expected_hits_of_64B_string: 2^40 позиций × 2^-512 = 2^-472 ≈ 0

verdict_note

NO-GO — the arithmetic is now confirmed EMPIRICALLY: a real 64 MiB PRNG pool (seed=20260715, reproducible), 67.1M indexed windows, 100,000 real 64 B sequences from real files — 0 byte-exact hits (vectorized rolling hash + byte verification). The conservation-law expectation (~1e-141) agrees; any physical pool scales by the same law.

AH-04 NO-GO W1 · 2026-07-13

Cube occupancy bitmap as a matrix section

If the operator's 'bits mark which bytes of the N-cube exist' is realized as a succinct rank/select (RRR) bitmap over the Cubrim-1 phi geometry, then the cost of transmitting the SET of occupied coordinates on sparse inputs (rho<0.3) approaches log2 C(L,k), because RRR encodes sets near-optimally.

Data class (Z)

sparse inputs (rho < 0.3)

Address target

n/a

Predicted lever

as a data carrier — NO-GO (predicted); as a search key — see AH-14

Ceiling category

shared context (dictionary / fragments)

Mechanism

succinct set encoding; but a set carries neither order nor values (Gotchas #2/#7)

Falsification test

share of the file's information held by the occupancy set; <5% on real classes -> dead as a carrier

Full cost (total_cost)

address: n/a · metadata: RRR bitmap ~ log2 C(L,k) bits · residual: permutation + values — 99%+ of the file's information (Gotcha #2) · amortized: the occupancy section's share of the matrix.

Probe verdict

NO-GO as a carrier — measured on 3,000 real files: the occupancy set ('which bytes exist') carries a median 3.07% of the file's information (p90 8.14%) — under the pre-registered 5% bar; order and values carry the rest (Gotchas #2/#7). The search-key role was tested separately in AH-14 and is also closed (recall gain 0.01 pp).

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

date: 2026-07-15
probe: W1-occupancy-perm-v1

sample

3000 реальных файлов 256Б..4МиБ (seed=42)

verdict: NO-GO

verdict_note

NO-GO as a carrier — measured on 3,000 real files: the occupancy set ('which bytes exist') carries a median 3.07% of the file's information (p90 8.14%) — under the pre-registered 5% bar; order and values carry the rest (Gotchas #2/#7). The search-key role was tested separately in AH-14 and is also closed (recall gain 0.01 pp).

occupancy_share_of_info_pct

max: 16.169%
p50: 3.067%
p90: 8.143%

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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.