Mathematical and numeric facilities on CADR and Genera

System 46

System 46 declares package MATH with one load module, AI: LISPM2; MATRIX QFASL. The package declaration allocates 100 symbols, but package size is not an API count. The matrix manual and implementation agree on eight intended user functions. The core SYSTEM-INTERNALS numeric file is loaded independently of MATH; it supplies integer square root, square root, logarithm, exponential and trigonometric routines, and exponentiation support.

No RAT or INFIX module corresponding to the later System 303 files was found in the pinned System 46 load inventory. That is a statement about this source boundary, not about every MIT or LMI Lisp-machine release.

LM-3 System 303

System 303 still declares MATH as the single SYS: SYS2; MATRIX module. The base system separately loads:

This separation is historically useful: MATH is matrix algebra, while ratios, complex numbers, elementary functions, and infix reading are core Lisp facilities. The MATH package grows from 100 to 200 allocated symbols, but the exported matrix interface remains deliberately small.

Genera 8.5

Genera declares MATH as a subsystem of UTILITIES; its serial source component is still sys2/matrix. The base SYSTEM separately loads numeric definitions, transcendentals, floating-point support, bignums, double floats, complex arithmetic, and rational arithmetic. The infix reader belongs to Development-Utilities, not the MATH subsystem or the Common Lisp standard.

This load boundary explains three user-visible facts:

• ordinary Common Lisp numeric operations are available without loading MATH;
• matrix functions use the MATH: package and may be loaded as part of Utilities;
• the infix reader can be absent even when arithmetic and matrices are present.

Matrix representation and result types

Matrices are ordinary Lisp arrays. AREF, array element types, displacement, and the general storage system therefore govern their representation. There is no separate matrix object, sparse format, symbolic coefficient type, or matrix editor. Arithmetic dispatch is ordinary +, -, *, and division, so exact rationals and complex elements can participate when the surrounding numeric implementation and array element type permit them.

The later decomposition sources avoid allocating an integer-specialized LU array when division may produce non-integers: when the input element type is a subtype of integer, a general element type is chosen. Optional destinations give callers allocation control but also make shape and alias rules part of the contract.

Rank-one arrays changed meaning

The manuals' shorthand “matrix” obscures a real compatibility break:

System 303 patch 98.36 records failures on vectors and careless result-array types. Patch 98.42 then explicitly fixes vector × matrix. The final maintained source implements row-vector semantics but retains an old comment saying that a rank-one array is N×1. The code, patch history, manual, and runtime agree with one another; the comment is stale. Genera's later source comment and implementation agree on column-vector semantics. Portable code must not infer the orientation merely from the function name.

What the manuals and Help omit or contradict

The System 46 matrix manual describes both multiplication operands as two-dimensional, while its source already attempts vector support. The maintained System 303 array manual corrects this and explicitly allows one- or two-dimensional operands; its final implementation and the runtime agree, notwithstanding the stale N×1 source comment.

Genera's installed Help regresses to saying that multiplication matrices are two-dimensional. The licensed implementation accepts rank-one arrays, and the live matrix-by-vector result proves that this is reachable behavior in the inspected world. Help does accurately preserve a separate inversion limitation: conformally displaced arrays do not work. It also prints DECOMPOSE's fourth ignore argument without explaining it; direct source inspection establishes that the argument is genuinely unused compatibility surface. These are not editorial guesses—the first is a Help/source/runtime contradiction and the latter two are Help/source cross-checks.

Aliasing and displacement

System 303 and Genera protect the exact case where matrix-3 is the same forwarded array object as either input: multiplication uses a temporary, copies the result back, and returns the requested destination. The source explicitly checks identity, not overlapping displacement. Two distinct displaced arrays that share storage can therefore still overlap in ways the safeguard does not detect.

Genera's installed Help separately warns that inversion does not work on conformally displaced arrays. Its implementation uses DECODE-RASTER-ARRAY, spans, one-dimensional array registers, and direct storage access for speed; the restriction is not a general statement that displaced arrays are invalid Lisp arrays.

Algorithms and lineage

The matrix file credits the linear-equation routines to Forsythe and Moler's Computer Solution of Linear Algebraic Systems (1967). Across the releases:

• INVERT-MATRIX uses Gauss–Jordan inversion. For each row it chooses the largest absolute entry in an unused column, records the chosen columns, pivots, and finally permutes the result.
• DECOMPOSE forms a combined L-I+U array and a permutation vector. It scales rows by their largest-magnitude element, chooses a scaled pivot, and performs Gaussian elimination without physically moving rows.
• SOLVE interprets the permutation vector, performs forward substitution through the implicit unit-diagonal lower triangle, then back substitution through the upper triangle.
• System 46 DETERMINANT recursively expands cofactors and uses two bit arrays to track free rows and columns. System 303 and Genera reuse DECOMPOSE, multiply the diagonal of U, and correct the sign from the row permutation. A singular condition is converted to determinant zero.

All three matrix sources still say that iterative improvement is not implemented. That is a concrete unfinished numerical path: SOLVE returns the direct LU result and does not perform residual-based refinement.

Genera 8 release notes report substantial performance improvements to multiplication, transpose, and inversion. Source inspection explains the direction of that work: the Genera file decodes raster arrays once, retains row spans, uses array-register declarations, and accesses storage through one-dimensional operations inside the inner loops.

System 46 boundary

The System 46 NUMER source implements ISQRT, SQRT, LOG, EXP, sine and cosine in radians and degrees, one- and two-argument arctangent, and hard or slow exponentiation paths. It handles fixnums, bignums, and short/full floating formats. No later-style RAT module was located at this boundary, so this article does not project System 303's ratio and complex representation backward onto System 46.

System 303 representation

System 303's numeric definitions add exact ratios and complex numbers as three-word extended-number objects. Constructors enforce canonical forms:

• a ratio never retains denominator zero, never retains denominator one, and is reduced to an integer when division is exact;
• a rational complex number with zero imaginary part becomes its real part;
• complex components are either both rational or use the same floating type;
• integer and rational arithmetic is exact until a floating operand requests a floating result.

The source defines a 17-bit short-float significand model and a 31-bit single-float significand with an 11-bit exponent field. These representation constants are implementation facts, not promises that every arithmetic path has the same intermediate precision.

The complete public rational/complex utility layer defined directly by System 303 RAT is:

RATIONALIZE uses continued-fraction reasoning to choose the simplest rational in the permitted interval. It is not merely a decimal printer or a fixed-denominator approximation.

The contemporary System 303 numeric manual and Genera Help preserve an important language boundary: ZetaLisp integer division truncates the quotient toward zero and leaves a remainder with the dividend's sign, whereas Common Lisp / returns an integer or exact ratio as appropriate for rational arguments. An infix expression's division meaning consequently depends on the readtable/file syntax selected by the later Genera reader.

Genera's Common Lisp numeric contract

Genera documents the Common Lisp numeric tower as:

number
├── real
│   ├── rational
│   │   ├── integer (fixnum or bignum internally)
│   │   └── ratio
│   └── float (single or double)
└── complex

Rationals are exact and limited by available memory rather than a fixed magnitude. The installed type documentation gives the Genera fixnum interval as -2^31 through 2^31-1; larger integers become bignums without changing their Common Lisp type relationship. Ratios are normalized, a positive denominator is used, and an integral quotient is represented as an integer. Complex numbers whose imaginary part is exact zero can canonicalize to a real.

For mixed arguments, the installed coercion table converts rational plus single-float to single-float, rational plus double-float to double-float, single-float plus double-float to double-float, and a complex/noncomplex pair to complex. MAX and MIN are documented exceptions that do not coerce their returned argument. A single-float computation is not silently widened merely because its result overflows while a double float could hold it; callers must request the wider format. These are language-level coercion rules, distinct from the implementation's intermediate arithmetic.

The Common Lisp and ZetaLisp names must not be conflated:

The licensed RAT source contains a commented-out complex branch in exact-rational conversion and says it cannot be right because the result must satisfy RATIONALP. This is an unfinished/dead path, not evidence that CL:RATIONAL accepts complex arguments.

Complete documented numeric interface grain

The Genera installed numeric dictionary and short-form category documentation were reconciled with the licensed implementation. At the promised public grain, the complete families are below. Names identified as legacy remain documented entry points; private % arithmetic dispatchers and compiler-only helpers are excluded.

The compatibility surface hidden by the phrase “legacy aliases” is finite. The numeric dictionary additionally names GLOBAL:+$, GLOBAL:PLUS, GLOBAL:-$, GLOBAL:DIFFERENCE, GLOBAL:MINUS, GLOBAL:*$, GLOBAL:TIMES, GLOBAL:/, GLOBAL:/$, GLOBAL:QUOTIENT, GLOBAL:1+$, GLOBAL:ADD1, GLOBAL:1-$, GLOBAL:SUB1, GLOBAL:GCD, GLOBAL:\, GLOBAL:REMAINDER, GLOBAL:EXPT, GLOBAL:^, GLOBAL:^$, GLOBAL:SQRT, GLOBAL:LOG, GLOBAL:ATAN, GLOBAL:ATAN2, GLOBAL:DFLOAT, GLOBAL:FLOAT, GLOBAL:FIX, GLOBAL:FIXR, and GLOBAL:RANDOM. The corresponding predicate/comparison compatibility names are FIXP, FIXNUMP, FLONUMP, BIGP, SIGNP, LESSP, and GREATERP. The integer-bit compatibility group adds GLOBAL:LOGAND, GLOBAL:LOGIOR, GLOBAL:LOGXOR, GLOBAL:HAIPART, GLOBAL:HAULONG, and GLOBAL:BIT-TEST.

BOOLE's documented operation constants are BOOLE-CLR, BOOLE-SET, BOOLE-1, BOOLE-2, BOOLE-C1, BOOLE-C2, BOOLE-AND, BOOLE-IOR, BOOLE-XOR, BOOLE-EQV, BOOLE-NAND, BOOLE-NOR, BOOLE-ANDC1, BOOLE-ANDC2, BOOLE-ORC1, and BOOLE-ORC2. These select Boolean truth tables; they are not sixteen additional arithmetic algorithms.

The same documentation exposes the usual numeric constants and controls: PI, MOST-POSITIVE-FIXNUM, MOST-NEGATIVE-FIXNUM, and the MOST-POSITIVE-*, MOST-NEGATIVE-*, LEAST-POSITIVE-*, LEAST-NEGATIVE-*, *-EPSILON, and *-NEGATIVE-EPSILON families for the Common Lisp short, single, double, and long float names. The SCL:LEAST-{POSITIVE,NEGATIVE}-NORMALIZED-* family distinguishes normalized extrema for those four names. Genera implements two underlying float precisions; the four Common Lisp format names include aliases rather than proving four distinct representations. SYS:SINGLE-FLOAT-P and SYS:DOUBLE-FLOAT-P are also indexed low-level predicates.

Reader/printer controls are *PRINT-BASE*, *PRINT-RADIX*, *READ-BASE*, *READ-DEFAULT-FLOAT-FORMAT*, SCL:*PRINT-EXACT-FLOAT-VALUE*, legacy BASE and IBASE, and the internal documented extended-ibase signed/unsigned reader variables. The special form WITHOUT-FLOATING-UNDERFLOW-TRAPS establishes a dynamic region in which underflow does not enter the normal trap path.

The implementation goes deeper than the public summary. It handles normalized and denormalized single and double floats, trap/exception state, integer promotion, and machine arithmetic. It also contains NEXTAFTER, although that name was not found in the installed short-form public dictionary and is therefore recorded here as source-visible rather than promoted into the documented interface.

Genera 8 release notes report fixes for denormalized square roots, double-float square-root rounding, logarithms of large bignums and ratios, logarithm coercion, and a FLOOR underflow path. These are evidence that the numeric substrate was an actively corrected implementation, not only a static Common Lisp compatibility table.

Lexical rules

Full System 303 grammar

The table is ordered from strongest to weakest binding. Decimal numbers are the source's binding powers.

The old implementation's DEFINFIX, DEFPREFIX, DEFDELIMITER, and DEFREPINFIX macros make additional operators mechanically definable. The contemporary reader manual gives only a short example and directs readers to the source for more detail; the table above is therefore intentionally source-derived.

Full Genera 8.5 grammar

Genera keeps the central arithmetic and conditional language but removes several System 303 forms:

The Genera source explicitly rejects prefix [; the old list literal is not merely undocumented. Dot/LIST*, at-sign/APPEND, the code-006 membership operator, strings, and percent comments are also absent from the inspected tokenizer/parser; the comparison glyphs remain. The source uses internal setup functions and a table of infix-key structures, so extension is technically possible, but installed Help explicitly says the notation is not intended as an extensible or full language.

The Genera source includes a substantial TEST-INFIX-READER fixture. Its automatic invocation is block-commented with an initialization/loading-order concern. In the live Listener the function ran, but many comparisons differed only because expected forms were compiled in SI/ZetaLisp package context while actual forms were read in CL-USER; rebinding *PACKAGE* removed some but not all differences. The test is therefore useful source evidence, not a context-independent installed conformance command.

Maintained System 303

The fresh CADR session d27-math-20260718, generation 1, booted load band System 303-0 in the Xvfb harness. Exact listener inputs established:

The session ran from 2026-07-18 08:02:20 to 08:11:22 EDT. The base and private disk hash was bb16e46ad81decfe1efe691d36b6aa4ce3fd4ffb82474365de3520989d397cb5 at start and stop. The System source copy was Fossil check-in 4df393c68d7f083ce42d5c377039d26043cc18a9031ace28258dc97f4137eb91 with tree hash 21f5215de973aa6ccbddb817f2d64edd95ee1014c3028a9b0711ea7c741b807e; it did not change. Both usim_sha256_at_start and usim_sha256_at_exec were 707a77d23e28ea1c45ae0eb0145dc181fa7ba649b9defc30044d4f847ac2c5be. The simulator exited zero without a forced stop, and state_may_be_incomplete is false.

The selected window was LOCAL-CADR [running], X window 2097202, at 768×963 on display :90. Current public revisions at start were system 4df393c68d7f083ce42d5c377039d26043cc18a9031ace28258dc97f4137eb91, usim 330d8248ec2e12af071e287920e681600f75df9ffd854aada5f8a64c9adad64d, l d1250f90044f09b6c92014a9aef65f9574e1bcbf8a7163004e53cc6dbed0f2d6, chaos db2953fde68d726a605d1d1699bab6c926ef252bd4991f692bae6ee5a634764e, and usite 8f717978b458b40adf1e238aaf177f5bc54ef46881268e03b787ba57b0d30a0e. The private copy-time revisions for system, chaos, and usite matched those values. Their tree hashes were respectively 21f5215de973aa6ccbddb817f2d64edd95ee1014c3028a9b0711ea7c741b807e, 34ab197641aae909e9a224edc307020fddec263e732207a74573d51dac0daa87, and adbb720339db225e6635977a869cf3f3d50b507e614b37a976f4a6548d212a81; none changed after copying.

Private machine-artifact hashes at start were promh.mcr 2c667f99f014a7130a55b255d31df02588d9396beace78abfe9325269e4ff3e6, promh.sym e9e3dd6a541511dd9541ae96b99dae19cb185d8b79fa09959f21fa52224f233d, and ucadr.sym 9071decf16fa8f11d7970c4662db0d6e95600fe43ec86ac41c77b37dbd7caa2a. The reproducible toolchain used Guix channel commit 230aa373f315f247852ee07dff34146e9b480aec, manifest SHA-256 3adae999bbe420182f22adc2499fcc82449a46eaf580a362de9c0e718fa6b37d, Python 3.11.14, Xorg Server 21.1.21, ImageMagick 6.9.13-5, and xdotool 3.20211022.1. Final run.json SHA-256 was b191e52b8b730e80f73fcb383fcf79d2c34f904cdefcb122f66b90874bd281b8; the cleanly stopped simulator state was f7ae340093e34cf6258842238fc3a25dd98bf672b4a358f34bd3e7bfa51e9a84.

Genera 8.5

The fresh session d27-math-genera-20260718, generation 1, ran the licensed Genera-8-5.vlod through the repository's isolated Genera harness. It confirmed matrix multiplication, Common Lisp rational conversion, Genera's column-vector orientation, and presence of SI:INFIX-READ. It also exposed a serious conflict:

The last three results are not documented Genera semantics. The licensed source links CL:/ to rational quotient, implements Common Lisp / through exact rational division, and uses that operation in decomposition and inversion; the manuals define ratios as exact. CL:RATIONAL and CL:RATIONALIZE also behaved correctly in the same run. The evidence therefore establishes a reproducible defect or incompatibility in this exact world/VLM execution path, not that Genera normally truncates Common Lisp division. It remains open whether the cause lies in the world build, public VLM arithmetic-op implementation, invocation context, or another compatibility boundary.

The action log preserves all 18 input intents and their linked dispatch outcomes in order. The research sequence was: the combined product/determinant/conversion form; the determinant alone; decomposition with both returned values; Control-Z from the resulting debugger; simple rational-quotient arities; the quotient/division/inverse form; vector-first multiplication; Control-Z from its incompatibility error; matrix-first multiplication plus FBOUNDP; the direct string/Altmode parser attempt; Control-Z; the self-test in the Listener package and then in SI; a function-link identity probe that included the problematic literal ZL:// reader spelling; Control-Z; two narrower identity-probe retries separated by the literal KP_Subtract key. The last three function-identity probes never yielded a trustworthy Lisp value and support no claim in this article. This distinction is why the source link declaration, rather than a runtime identity result, grounds the CL:/ linkage statement.

The Genera session ran from 08:12:04 to 08:21:43 EDT in separate user, mount, network, PID, IPC, and hostname namespaces. It had no external or default route and no guest-visible host file service. The archive was 206,213,430 bytes with SHA-256 89fb3e76b91d612834f565834dea950b603acf8f9dbacacdd0b1c3c284a2d36e; the 54,804,480-byte base and private world hash was a8ee5e86cc7e322f7385af3e0cd579d7650d4dcfc3ce328acbf8b25515dd0672 at start and stop. The VLM hash was 9f5e18d5770f973879716182b6856ef5a8ee9d3b2bb907476ea0cf35986aa4c7, and the action log contains 36 linked intent/outcome records with SHA-256 c7fc6bad3bb717c0b745375a9b838f190c0ca2a7b208f68885af5bbe06639547. Both exact X-protocol substitutions and the supervised RFC 868 request/reply were observed before the harness reported the system running.

The selected main window was Genera on DIS-LOCAL-HOST, X window 4194310, at 1200×900 and host-display position (72,55) on display :200. In addition to the world and VLM hashes above, the debugger hash was 2db918cfe8f35f52c7ff4b7695b0ecd3bb85e41a3327ea5a94874edf05edb54a, the exact-command ifconfig preload was f45f45461622975996ab41138f64bb84a4b17c51fba0dbb649208914898c26b7, the X compatibility preload was acd71dbcb948f05b7fd2730b2b4706c08f16f46d792bd9aa6aa64370e855e4b1, the RFC 868 responder was cc3a2274149c5593b52e6608d732d4048518c766134df5e0f018746ad5cf98bb, and configuration was 5ce6509f5adf2cf2d054d34eb4ba777ce462285b8cd9b01bc071bf819139e086. Xvfb did not advertise MIT-SHM. The responder evidence hash was 290773368004b41628d3cede011ad5066e850f10eae3df2c19857855f59c36bc, its completion record was 4318628ff9f26622e10dacc316aa1b4c418418408b9a4a3c1e48274820395da8, and the responder exited zero.

Tracked harness-source hashes were: entry point e10d07a1c745d37044f1a97903455d334d6dcdb0c1d0e6854598e10fab24fa05, Python harness bc9276ac766913bc15018dd334a2a2704ae5a926e1fcbc30ccfcff08af8cb48a, namespace helper 17a3e297930eef45a6f59a349f92ec1f6dc99b2c4d5caa2392dc0521636af01c, VLM helper cbf9ee0520b4892325266ed17afba8f1b663e7d266fea6d80de9cf98de17d2f8, ifconfig interposer source a4d126dbb6fd6f4903835bbb41c39652cfc53c91e942267dc9166c1c938c36e7, and X interposer source 4db1dee8e71d5ddc5cfd8289ecc3607738370ac97f856853786cfe713e94e392. The toolchain used the same Guix channel and manifest recorded for CADR, with Python 3.11.14, Bubblewrap 0.11.0, Xorg Server 21.1.21, GCC 15.2.0, ImageMagick 6.9.13-5, and xdotool 3.20211022.1. Final run.json SHA-256 was 4d04c2d06bc4c701a2782e366b4bfaa517da7bf7c1f04a677dbdc9ac8ced5975.

The private world did not change. The harness did not invoke Save World or create a host process checkpoint; save_world_performed and guest_checkpoint_created remain unknown rather than inferred. On shutdown the VLM accepted confirmation and began cleanup, then reached the already documented Cold Load mutex stall and required bounded SIGKILL. Accordingly forced_stop and state_may_be_incomplete are true, orderly_vlm_host_shutdown is false, and only unsaved Lisp state was discarded.

Public source files

Licensed Genera evidence

The following hashes identify local inputs without reproducing them:

The installed Help source artifacts used for cross-checking were the array chapter (76a25fea7a5dabe863128518b1b4d263fc2618fe3294fe9b2ea8f1d19a007154), condition chapter (0df549ca1d2c3518c62111e5620c7dbc12da610424b6192ce4d1df4f11730550), numeric short forms (77ee6d5e1e378ce873b84acddacd09d29221b0c37e0086f7206da72dee7a59b5), numeric types (4e41f0abcc75396c485a4dc909c3f81559109515cdeb361aa99cbf4c2ee38dcc), and the no-prefix numeric dictionary index (eeaebd5a682eb37c1b95e4e2bcfa366e64c3b2dfec96df72b27d5d425f67d68d). These are source-payload hashes from the inert extractor metadata, not hashes of tracked recovered prose.