15 KiB
Projectable Geometry Compressor Spec
Status: draft v0.1 Date: 2026-05-08 Scope: compression architecture, symbolic geometry, reversible accounting Primary claim boundary: this is not a physics theory, biological claim, financial claim, or proof of optimal compression. It is a receipt-gated compressor design for projectable geometry with explicit residual accounting.
1. Purpose
The compressor exists to edge the compression decimal by turning large structured objects into:
shared projectable geometry
+ compact carrier symbols
+ bounded residual sidecars
+ exact rehydration receipts
The core objective is not to discover true physics. The core objective is:
project -> preserve -> improve
Where:
project:
map a large object into a lower-dimensional control basis
preserve:
track every displaced coordinate in a residual lane
improve:
reduce per-instance cost once the shared model is amortized over a corpus
For a single object, the model cost may dominate. For a corpus, the test is:
K(model) + K(residuals | model) < K(raw objects)
No compression claim is promoted until that inequality is benchmarked against baselines.
2. Provenance
This spec is grounded in the receipt chain built from the Standard Model Lagrangian term-family probe:
equation wall
-> 12D term-family source plane
-> exact rational centroid
-> signed 16-axis envelope
-> 4D primitive keel
-> 12D residual lane
-> genus-3 residual boat
-> force-regime model
-> DNA carrier substitution
-> absurd genetics stress
-> extension failure boundary
Relevant receipt-bearing runners:
4-Infrastructure/hardware/standard_model_lagrangian_exact_average.py
4-Infrastructure/hardware/standard_model_12_to_4_reduction.py
4-Infrastructure/hardware/standard_model_genus3_residual_boat.py
4-Infrastructure/hardware/standard_model_force_regime_model.py
4-Infrastructure/hardware/standard_model_dna_substitution_alignment.py
4-Infrastructure/hardware/standard_model_absurd_genetics_stress.py
4-Infrastructure/hardware/standard_model_extension_failure_probe.py
The design rule extracted from those receipts is:
anything weird is allowed if it is reversible, conserved, and receipted
Reference Lean gate:
0-Core-Formalism/lean/Semantics/Semantics/ProjectableGeometryCanonical.lean
The Lean gate encodes the canonical dimensional representation and executable negative witnesses for broken residual and unresolved-shell cases.
3. Core Objects
3.0 Canonical Dimensional Representation
The current canonical representation is:
16D signed envelope
-> 12D source/residual plane
-> 4D primitive keel
-> genus-3 residual boat
-> 0D closure
Expanded accounting:
16D signed envelope:
12 exact source axes
+ 4 meta/control axes
12D source plane:
unreduced canonical source coordinates
4D primitive keel:
field / shear / packet / spectral
12D residual lane:
source_12D - lift(project(source_12D))
genus-3 residual boat:
three handle vectors carrying the residual lane
0D closure:
no unresolved mass debt
The representation law is:
source_12D =
lift(project(source_12D))
+ residual_12D
The genus-3 residual carrier law is:
packet_local
+ shear_torsion
+ spectral_field
= residual_12D
The dimensional shell closure prior prices the representation in twelfths:
visible_4d = 4/12
shadow_3d = 3/12
closure_0d = 1/12
lawbound = 4/12
unresolved = 0/12
total = 12/12
Promotion requires:
axis counts match 16 -> 12 -> 4 -> 3 -> 0
shell mass closes with unresolved = 0
three residual handles sum to residual_12D
lifted_4D + residual_12D reconstructs source_12D
source hash is present
receipt hash is present
Failure cases:
broken residual handle sum -> reject
unresolved shell mass debt -> reject
16D axes without typed semantics -> reject
genus-3 carrier without exact residual replay -> reject
Claim boundary:
16D is a typed routing/witness envelope.
12D is a source/residual accounting plane.
4D is a compact primitive control keel.
3D is a three-handle residual carrier.
0D is closure, not deletion.
3.1 Source Plane
The source plane is the unreduced coordinate system for the current object family.
Example:
12D source plane = twelve symbolic equation term-family axes
Requirements:
source axes must be named
source vector must be canonicalized
source vector must have a stable hash
source vector must support exact or bounded numeric representation
3.2 Primitive Keel
The primitive keel is the compact control vector.
Current four-primitives basis:
field
shear
packet
spectral
Interpretation:
field:
density/value surface coordinate
shear:
gradient, coupling, torsion, transformation coordinate
packet:
localized event, witness, claim, or receipt coordinate
spectral:
eigenmode, covariance, resonance, and residual-spectrum coordinate
The primitive keel must be canonical and normalized for the target family:
sum(primitive_keel) = 1
3.3 Projection Matrix
The projection matrix maps source axes into the primitive keel.
Minimum law:
P : source_n -> primitive_4
rows(P) sum to 1
Projection is allowed to be lossy. Loss is lawful only if the residual lane is emitted.
3.4 Lift
The lift is the deterministic low-cost reconstruction from the primitive keel back into source coordinates.
The lift is not expected to recover the original source by itself.
lift(project(source)) != source
Instead, the compressor must emit:
residual = source - lift(project(source))
3.5 Residual Lane
The residual lane carries all displaced information required for exact rehydration.
Law:
lift(project(source)) + residual = source
Acceptance:
rehydration_l1_error = 0
For lossy modes, the receipt must explicitly mark the loss budget and the irreversible boundary.
3.6 Genus-3 Residual Boat
The genus-3 residual boat is the current structured residual bucket. It has three handle vectors:
packet_local
shear_torsion
spectral_field
Law:
packet_local + shear_torsion + spectral_field = residual
Metrics:
hull_capacity_l1 = ||residual||_1
handle_l1
dominant_handle
zero_drift
closure_l1_error
Acceptance:
three_handles_sum_to_residual = true
closure_l1_error = 0
zero_drift = true
The genus-3 boat is a residual carrier, not a cosmological topology claim.
3.7 Signed Envelope
The signed envelope records positive, negative, and origin/control coordinates.
Current test envelope:
12 exact mirror axes
+ 4 meta/control axes
= 16 signed axes
Purpose:
make projection distance navigable
record mirror closure
separate exact axes from measurement or residual axes
4. Carrier Alphabets
The compressor must separate the mathematical basis from the carrier alphabet.
Current compatible carrier:
A -> field
T -> shear
G -> packet
C -> spectral
The carrier may be:
binary tags
packed structs
glyphs
DNA-like bases
hachimoji-style extensions
Typst/logogram symbols
virtual baud symbols
Carrier substitution is lawful only when it preserves accounting.
5. Carrier Laws
5.1 Primitive Conservation
Decoded carrier vector must equal the primitive keel.
decode(encode(primitive_keel)) = primitive_keel
Failure:
primitive_roundtrip_error
5.2 Normalized Keel
Carrier mass must preserve the normalized primitive total.
sum(carrier_keel) = 1
Failure:
keel_total_not_one
5.3 Decode Completeness
Every active carrier symbol must have a decode rule.
Failure:
nonzero_unmapped_extension_mass
missing_core_base_decode
5.4 Primitive Coverage
Every primitive must remain representable.
Failure:
primitive_coverage_failure
primitive_loss
5.5 Residual Closure
Residual sidebands must close.
sum(residual_sideband) = 0
unless explicitly carried as rehydration payload.
Failure:
residual_drift
5.6 Fail Closed
Ambiguity must reject instead of silently decoding.
Failure:
primitive collision
ambiguous aliasing
unknown primitive target
checksum/hash mismatch
6. Extension Boundary
Extra carrier bases or symbols are permissible only in three cases:
inert:
extra symbol carries zero mass
decoded split:
extra symbol carries mass but maps back to an existing primitive exactly
balanced sideband:
extra residual sideband carries signed mass but sums to zero
Forbidden extension behavior:
untracked extension mass
primitive collisions
missing decode rules
residual drift
primitive loss
ambiguous aliasing
This boundary is receipt-backed by:
4-Infrastructure/hardware/standard_model_extension_failure_probe.py
7. Virtual Baud Reconstruction Layer
The decompressor should be treated as a signal reconstruction path.
Pipeline:
compressed archive
-> framing decoder
-> virtual baud reconstruction layer
-> control-bit interpreter
-> glyph/kernel dispatch
-> primitive keel reconstruction
-> residual boat replay
-> exact output
Lanes:
DATA:
carrier symbols, literals, glyph/eigen descriptors
CTRL:
mode switches, kernel dispatch, page/domain framing
CLOCK:
frame/tick boundaries, phase buckets, resync points
REPAIR:
residual bytes, correction vectors, patch ops
WITNESS:
hashes, type witnesses, closure checks, receipts
One virtual baud tick is:
one admissible reconstruction event
Examples:
emit literal
switch carrier alphabet
apply primitive projection
replay residual handle
verify frame hash
resync stream
The baud layer is architectural only if it constrains parsing and recovery. If it merely names a metaphor, it is not part of the codec.
8. Packet Shape
The general packet shape is:
PGC1 packet =
magic/version
family id
source basis id
projection id
carrier alphabet id
primitive keel payload
residual boat payload
extension sideband payload
witness/checksum/hash trailer
The current finance-claim harness uses FCL1/FCS1 for a narrower payload family. PGC1 is the proposed general projectable geometry compressor envelope. It should not replace FCL1/FCS1 until it can reproduce or improve those receipts.
9. Compression Gain Test
A candidate packet is accepted only if its expected gain is positive after decoder cost.
gain =
baseline_size
- (
model_reference_cost
+ projection_payload_size
+ carrier_payload_size
+ residual_payload_size
+ witness_payload_size
+ amortized_decoder_cost
)
Accept:
gain > 0
Keep but mark exploratory:
gain <= 0
and structural receipts pass
Reject:
rehydration fails
carrier laws fail
baseline comparison missing
claim boundary missing
10. Codec Baselines
Every benchmark receipt should compare:
canonical JSON or canonical source bytes
zlib
CBOR when available
MessagePack when available
Protobuf/Nanopb-style schema when available
FlatBuffers-style schema when available
packed-struct/custom bitpack
projectable geometry packet
Missing optional libraries are skipped, not failures.
No competitive claim is allowed until the corpus is larger than a toy sample set.
11. Receipts
Every compressor run emits a JSON audit envelope even if the wire format is binary.
Required receipt fields:
schema
generated_utc
surface_id
source hashes
basis ids
projection ids
carrier alphabet ids
primitive keel
residual lane
residual boat
extension sidebands
closure checks
roundtrip checks
benchmark table
claim boundary
stable hash
timestamped receipt hash
lawful
Stable hashes exclude timestamp-only fields. Timestamped receipt hashes include the generated timestamp.
12. Failure Codes
Minimum failure vocabulary:
bad_magic
unsupported_version
bad_checksum
unknown_basis
unknown_projection
unknown_carrier
missing_decode_rule
primitive_roundtrip_error
keel_total_not_one
nonzero_unmapped_extension_mass
primitive_coverage_failure
primitive_loss
residual_drift
residual_closure_error
rehydration_l1_error
baseline_missing
gain_not_positive
claim_boundary_missing
Failures that affect exactness must fail closed.
13. Implementation Phases
Phase 0: Freeze Laws
Deliverables:
this spec
wiki tiddler
failure vocabulary
receipt schema draft
Phase 1: General Harness
Build a projectable geometry compressor harness that can read a source vector, projection matrix, carrier map, and residual policy.
Commands:
encode
decode
verify
bench
stress-carrier
stress-extension
Phase 2: Binary Envelope
Define PGC1 as a compact binary envelope.
Required tests:
bit flip rejects
unknown carrier rejects
missing residual rejects
extension mass leak rejects
exact rehydration passes
Phase 3: Corpus Benchmarks
Run over multiple object families:
FinancialClaimPacket
symbolic equation graphs
DNA/base sequence surfaces
Typst/logogram render packets
Phase 4: Optimization
Optimize projection matrices and carrier alphabets:
local search
H200 burst optimizer dry-run, then optional rented run
noisy recovery simulator
virtual baud decoder profiling
Phase 5: Committee Evidence
Export:
Jupyter Book chapter
receipt bundle
benchmark tables
failure-mode appendix
claim-boundary appendix
14. Acceptance Gates
A compressor candidate is lawful only if:
source canonical hash is recorded
projection rows satisfy stated laws
primitive keel roundtrips
residual lane rehydrates exactly
genus/residual carrier closes
carrier alphabet is bijective or explicitly extended lawfully
extension sidebands are inert, decoded, or balanced
bad mappings fail closed
benchmark baselines are present or explicitly skipped
claim boundary is present
The current known positive evidence:
12D -> 4D reduction closes with exact residual rehydration
genus-3 residual boat closes with zero drift
DNA carrier substitution aligns exactly
absurd genetics lawful cases survive
broken non-bijective carrier fails closed
extension failure boundary is identified
The current known limitation:
This has not yet demonstrated competitive compression over a large corpus.
It has demonstrated structural stability and exact accounting under carrier recoding.
15. Non-Claims
This spec does not claim:
new physics
genomic physics
biological implementation
financial correctness
legal/audit validity
compression superiority
Kolmogorov optimality
Hutter Prize competitiveness
It claims only a design:
projectable geometry with explicit residual accounting can be made carrier-stable
and fail-closed under known extension failures.