PUM · Arithmetic audit

Scope: math-and-extraction audit only. I treated Bendall/PUM claims as formal arithmetic statements and recomputed them with exact rationals where possible.

Primary files checked:

Master ledger
Legend, verbatim
Corner panels, verbatim
sources/patent/PART_15_OF_20_-_DRAFT_518_400_B_KMV_-_MSAART_PATENT_APPLICATION_NOTES_-_THUR_22ND_SEPT_2022.txt
sources/patent/PART_17_TO_20_-_DRAFT_518_400_B_KMV_SOLOMON'S_MOLTEN_SEA_ARK_-_THUR_22ND_SEPT_2022.txt

1. Identity verification table

Notation: 5.555.. means the exact repeating value 50/9, not the finite decimal 5.555.

status	claim	computed (Fraction)	computed (decimal)	comment
OK	`SUN / 5.555.. = 155,520`	155520	155520	Ledger lines 46, 66-⁠75; patent PART 17 line 129.
OK	`SUN / 4.444.. = 194,400`	194400	194400	Ledger line 47; patent PART 17 line 130.
OK	`SUN / 3.333.. = 259,200`	259200	259200	Ledger line 48; patent PART 17 line 131.
OK	`SUN / 2.222.. = 388,800`	388800	388800	Ledger line 49; patent PART 17 line 133.
OK	`SUN / 1.111.. = 777,600`	777600	777600	Ledger line 50; patent PART 17 line 134.
OK	`SUN / 11.111.. = 77,760`	77760	77760	Ledger line 51.
OK	`MATTER / 11.111.. = 311,040`	311040	311040	Ledger line 52; patent PART 17 lines 411-⁠425.
OK	`AETHER / 5.555.. = 25,920`	25920	25920	Ledger line 53; corner panel line 16.
OK	`AETHER / 11.111.. = 12,960`	12960	12960	Ledger line 54.
OK	`MATTER / 6.666.. = 518,400`	518400	518400	Ledger line 55; corner panel line 39.
OK	`MATTER / 16 = 216,000`	216000	216000	Ledger line 56; corner panel line 37.
OK	`MATTER / 144,000 = 24`	24	24	Ledger line 57; patent PART 17 line 425.
OK	`MATTER / 266.666.. = 12,960`	12960	12960	Ledger line 58; patent PART 17 line 425.
OK	`SUN / 25,920 = 33.333..`	100/3	33.3333333333	Ledger line 59; corner panel line 41.
OK	`SUN / 518,400 = 1.666..`	5/3	1.66666666667	Ledger line 60; corner panel line 42.
OK	`SUN / 129,600 = 6.666..`	20/3	6.66666666667	Ledger line 61; corner panel line 43.
OK	`1D = 10/9`	10/9	1.11111111111	Ledger lines 66-⁠70.
OK	`2D = 20/9`	20/9	2.22222222222	Ledger lines 66-⁠71.
OK	`3D = 10/3`	10/3	3.33333333333	Ledger line 72.
OK	`4D = 40/9`	40/9	4.44444444444	Ledger line 73.
OK	`5D = 50/9`	50/9	5.55555555556	Ledger line 74.
OK	`6D = 20/3`	20/3	6.66666666667	Ledger line 75.
OK	`7D = 70/9`	70/9	7.77777777778	Ledger line 76.
OK	`8D = 80/9`	80/9	8.88888888889	Ledger line 77.
OK	`9D = 10`	10	10	This follows `9 * 10/9`; ledger line 78.
MISMATCH	`10D ladder value = 10` if ladder is `n * 10/9`	100/9	11.1111111111	Ledger line 79 says `10`; this is an exception or contradiction.
MISMATCH	`11D ladder value = 100/9` if ladder is `n * 10/9`	110/9	12.2222222222	Ledger line 80 uses `100/9`, i.e. the 10th ladder step, as an Aether-to-Matter factor.
MISMATCH	`12D ladder value = 12` if ladder is `n * 10/9`	40/3	13.3333333333	Ledger line 81 says `12`.
MISMATCH	`16-sector torus value = 16` if included in the same ladder	160/9	17.7777777778	Ledger line 82 is a separate sector count, not a ladder value.
OK	`LOS base 144; *1000 = AETHER`	144000	144000	Ledger lines 138, 31-⁠38.
OK	`MOS C = 24 * 11.111.. = 266.666..`	800/3	266.666666667	Ledger lines 139, 195; legend line 17.
OK	`POS base 864 = SUN/1000`	864	864	Ledger line 140.
OK	`RFEU = AETHER * 9/10`	129600	129600	Ledger lines 143, 221; legend line 30.
OK	`AOS base = 360`	360	360	Ledger line 144; legend line 27.
OK	`C octave: 266.666.. * 2 = 533.333..`	1600/3	533.333333333	Ledger line 197.
OK	`C octave: 266.666.. * 4 = 1,066.666..`	3200/3	1066.66666667	Ledger line 197.
OK	`C octave: 266.666.. * 8 = 2,133.333..`	6400/3	2133.33333333	Ledger line 197.
MISMATCH	`Bendall C / standard C ~= 1.0192`, taking standard C as `261.63`	80000/78489	1.01925110525	Relative error from `1.0192` is about `5.0e-5`, larger than the requested `1e-6`; this is ordinary rounded commentary, not a core identity.
OK*	`10/9 ~= 182 cents`	10/9	182.403712134 cents	Ledger line 208 is rounded.
OK*	`5/3 ~= 884 cents`	5/3	884.358712999 cents	Ledger line 210 is rounded.
OK*	`6/5 ~= 316 cents`	6/5	315.641287001 cents	Ledger line 211 is rounded.
OK	`SUN / AETHER = 6`	6	6	Ledger line 218; legend quadrant values lines 43-⁠44.
OK	`SUN * 6.666.. = AETHER * 40`	5760000	5760000	Ledger line 218.
OK	`TIME / AETHER = 3.6`	18/5	3.6	Ledger line 219.
OK	`TIME * 5/18 = AETHER`	144000	144000	Ledger line 219.
MISMATCH	`AETHER * 0.2 * 18 = AETHER`	518400	518400	Ledger line 219 parenthetical says `AETHER (= AETHER x 0.2 x 18)`, but `0.2 * 18 = 3.6`, so the RHS is `TIME`, not `AETHER`.
OK	`MATTER / AETHER = 24`	24	24	Ledger line 220.
OK	`MATTER * 1/24 = AETHER`	144000	144000	Ledger line 220.
OK	`RFEU / AETHER = 0.9`	9/10	0.9	Ledger line 221.
OK	`RFEU = 9 * AETHER / 10`	129600	129600	Ledger line 221.
OK	`GREAT_YEAR / AETHER = 0.18`	9/50	0.18	Ledger line 222.
OK	`GREAT_YEAR * 100 = 2,592,000`	2592000	2592000	Ledger line 223.
OK	`GREAT_YEAR / 12 = 2,160`	2160	2160	Ledger line 224.
OK	`GREAT_YEAR / 4 = 6,480`	6480	6480	Ledger line 225.
OK	`TIME / GREAT_YEAR = 20`	20	20	Ledger line 226.
OK	`1.666.. * 12 = 20`	20	20	Ledger line 226.
OK	`TIME * 60 / 100 = 311,040`	311040	311040	Ledger line 227.
MISMATCH	`SUN_DIAMETER^2 / 1000 = SUN_SQUARE = 3,456,000`	746496000	746496000	Ledger line 228 correctly self-flags this as false.
OK	`SUN_SQUARE = 4 * SUN_DIAMETER`	3456000	3456000	Ledger lines 228-⁠231; corner panel line 36; patent PART 17 line 299.

2. Algorithm sanity check

The extraction’s core geometric algorithm is correct: the right-triangle side labelled “3 side” is |mp|, not |mp|/3. This is directly supported by the patent’s Argon calculation: 189.4 (3 side) x 252.5333 (4 side) (PART 15 lines 240-⁠243), and by the hydrogen square/area calculation (PART 15 lines 151-⁠156).

One detail needs correction: master ledger line 16 describes the cascade as /24 /60 /60 /60 /60 /24 ..., but the worked hydrogen example uses /24 /60 /60 /60 /60 /60 /24 (ledger lines 119-⁠126). The patent hydrogen text uses /24 /60 /60 /60 /60 /60 /60 /24 after the area, not the shortened ledger algorithm (PART 15 lines 155-⁠158).

Cascade columns below are exact values after area / 24 / 60 / 60 / 60 / 60 / 60 / 24.

element	mp	area `(2/3)mp^2`	`/24`	`/60`	`/60`	`/60`	`/60`	`/60`	`/24`
H	259.2	1119744/25 = 44789.76	46656/25 = 1866.24	3888/125 = 31.104	324/625 = 0.5184	27/3125 = 0.00864	9/62500 = 0.000144	3/1250000 = 0.0000024	1/10000000 = 0.0000001
He	272.2	3704642/75 = 49395.2266667	1852321/900 = 2058.13444444	1852321/54000 = 34.3022407407	1852321/3240000 = 0.571704012346	1852321/194400000 = 0.00952840020576	1852321/11664000000 = 0.000158806670096	1852321/699840000000 = 0.00000264677783493	1852321/16796160000000 = 0.000000110282409789
Ne	248.6	3090098/75 = 41201.3066667	1545049/900 = 1716.72111111	1545049/54000 = 28.6120185185	1545049/3240000 = 0.476866975309	1545049/194400000 = 0.00794778292181	1545049/11664000000 = 0.000132463048697	1545049/699840000000 = 0.00000220771747828	1545049/16796160000000 = 0.0000000919882282617
Ar	189.4	1793618/75 = 23914.9066667	896809/900 = 996.454444444	896809/54000 = 16.6075740741	896809/3240000 = 0.276792901235	896809/194400000 = 0.00461321502058	896809/11664000000 = 0.0000768869170096	896809/699840000000 = 0.00000128144861683	896809/16796160000000 = 0.0000000533936923678
Kr	157.4	1238738/75 = 16516.5066667	619369/900 = 688.187777778	619369/54000 = 11.4697962963	619369/3240000 = 0.191163271605	619369/194400000 = 0.00318605452675	619369/11664000000 = 0.0000531009087791	619369/699840000000 = 0.000000885015146319	619369/16796160000000 = 0.0000000368756310966
Xe	111.8	624962/75 = 8332.82666667	312481/900 = 347.201111111	312481/54000 = 5.78668518519	312481/3240000 = 0.0964447530864	312481/194400000 = 0.00160741255144	312481/11664000000 = 0.0000267902091907	312481/699840000000 = 0.000000446503486511	312481/16796160000000 = 0.000000018604311938
Rn	71.15	2024929/600 = 3374.88166667	2024929/14400 = 140.620069444	2024929/864000 = 2.34366782407	2024929/51840000 = 0.0390611304012	2024929/3110400000 = 0.000651018840021	2024929/186624000000 = 0.0000108503140003	2024929/11197440000000 = 0.000000180838566672	2024929/268738560000000 = 0.00000000753494027802

Recognizable scalar hits:

H uniquely hits 0.5184; multiplied by 1,000,000, this is 518,400 = TIME (ledger lines 119-⁠123; patent PART 15 line 155).
H also hits 0.000144; multiplied by 1,000,000, this is 144, the LOS/light base, not 144,000 unless an additional 1000 scale is applied (ledger lines 123-⁠130; legend line 29 gives Light 144 / 0).
The other six noble-gas cascades do not land exactly on the named cardinal scalars in the same way. Helium’s fifth /60 value gives 158.806670096 * 1e6 scaling at that stage, Neon gives 132.463048697, Argon gives 76.8869170096, Krypton gives 53.1009087791, Xeon gives 26.7902091907, and Radon gives 10.8503140003. These are not exact PUM seed values.

Elemental value algorithm:

element	digits of `\|mp\|`	product `P`	`P^2`	ledger value	status
H	2,5,9,2	180	32400	32400	OK
He	2,7,2,2	56	3136	3136	OK
Ne	2,4,8,6	384	147456	147456	OK
Ar	1,8,9,4	288	82944	82944	OK
Kr	1,5,7,4	140	19600	19600	OK
Xe	1,1,1,8	8	64	64	OK
Rn	7,1,1,5	35	1225	1225	OK

This matches the ledger table (lines 102-⁠110) and patent examples for H and Ar (PART 15 lines 51-⁠54, 256-⁠258).

3. Extraction error scan

Likely source/poster transcription or poster-label issues:

Aether/Light quadrant: 144,000 / 4.444 = 32,400 [SUN R] is arithmetically correct, but the label is suspect. 32,400 is the Hydrogen digit-product value in the patent (PART 15 lines 51-⁠55), not a Sun radius unless using a hidden /13.333.. scale. See corner panel line 17.
Aether/Light quadrant: 144,000 / 3.333 = 43,200 [SUN R] is arithmetically correct at 43,200; as a Sun-radius label it is scale-shifted from 432,000, so it should be annotated as scaled if retained. See corner panel line 18.
Legend item 8 says 1,620 x 16 = 25,920 light frequency of -259.2 C (legend line 23). The arithmetic is OK, but the label “light frequency of -259.2 C” is not reconciled elsewhere; the hydrogen cascade reaches 0.5184 and 0.000144, not 25,920 directly (ledger lines 119-⁠126; PART 15 lines 155-⁠160).
Matter panel MOON row 9 x 96 = 864 [SUN D] is probably a poster/source label slip if the row is meant to stay in the Moon family; your note already flags this (corner panel lines 59-⁠63).
Centre cluster says MOON SQUARE = 8,640 ; 108 x 2,160 (corner panel line 88). But 108 x 2,160 = 233,280; 8,640 is 4 x 2,160, matching the Sun panel line 35.
Centre cluster says EARTH SQUARE = 31,680 ; 108 x 864,000 miles (corner panel lines 89-⁠91). 31,680 is 4 x 7,920; 108 x 864,000 = 93,312,000, an Earth-Sun distance relation. This looks like two different facts collapsed into one label.
Time panel labels 25,920 seconds = 432 minutes [SUN R] and 360 degrees of arc = 21,600 mins of arc [MOON R] are arithmetically correct but scale-shifted labels (432 vs 432,000, 21,600 vs 2,160). See corner panel lines 73 and 80.
Patent uses inconsistent spellings/signs: Xeon for Xenon and Krypton sometimes without the negative sign (PART 15 lines 136-⁠140, 288-⁠289). The ledger normalizes these as Xe and negative noble-gas melting points, which is sensible.

4. Internal contradictions

Dimension ladder contradiction: ledger line 66 says the ladder is n * 10/9 for n = 1..12+, but lines 79-⁠82 use 10, 100/9, 12, and 16 for entries that do not follow that formula. Treat 10D, 11.111.., 12D, and 16 sectors as named constants or exceptions, not ladder outputs.
Aether/Light role overlap: the ledger’s cardinal table defines AETHER (Light) = 144,000 (lines 31-⁠38), while the dimension table separates 5D Aether = 50/9 and 6D Force/Light = 20/3 (lines 74-⁠75). The poster quadrant also labels AETHER / LIGHT / ENERGY - 144,000 - 6D (legend line 43). This is not a numeric contradiction, but it is a symbol-role ambiguity.
The hydrogen cascade algorithm and worked example disagree about the divisor sequence. Ledger line 16 has /24 /60 /60 /60 /60 /24 ...; ledger lines 119-⁠126 have /24 /60 /60 /60 /60 /60 /24. Patent PART 15 lines 155-⁠158 continues with another /60 before /24.
Ledger line 219’s parenthetical AETHER (= AETHER x 0.2 x 18) is wrong as written. AETHER x 0.2 x 18 = TIME, while TIME x 5/18 = AETHER.
RFEU has two displayed scales: 1,296 and 129,600 in the legend (line 30), and 129,600 in the ledger (line 41). This is probably intended octave/scaling notation, but pum-core should store them as separate aliases or scaled forms.
LIGHT has scale ambiguity: legend line 29 gives Light 144 / 0, ledger line 35 uses AETHER (Light) = 144,000, and the hydrogen cascade reaches 0.000144, i.e. 144 after multiplying by 1e6 (ledger lines 123-⁠130). These should not share a single unqualified symbol.

5. Suggested ledger improvements

Split constants into typed namespaces: Seed.AETHER_144000, LOS.LIGHT_144, Scaled.RFEU_1296, Seed.RFEU_129600, etc. Avoid one symbol with multiple powers-of-ten meanings.
Encode the dimension ladder as a generator plus explicit exceptions. dim(n) = 10n/9 works through n=9; 10D, 11.111, 12D, and 16 sectors should be separate named entries with source citations.
Make every generated value carry provenance: value, exact_fraction, scale, operation_chain, and source_lines. This would make poster labels like [SUN R] vs scaled 432 auditable rather than implicit.
Implement element(symbol) as two independent projections: digit_product_value(mp) and triangle_cascade(mp, divisor_sequence). Keep the patent cascade sequence configurable because the ledger summary and patent text do not use exactly the same tail.