Documents the radix formulation from MathPunch-FiniteState
(coq/MathPunchFiniteStateAudit/Radix.v, MathPunchFiniteState/Radix.lean), which
is the correctness obligation underneath the encoder method.
The defect, proved rather than asserted: positional evaluation is not injective
on digit strings. Both collisions are machine-checked in Coq and Lean --
eval_digits 8 [0;1] = eval_digits 8 [1], and eval_digits 8 [] = eval_digits 8
[0]. Quantified: in base 8 over lengths 0-3, 585 strings collapse onto 512
values.
Two repairs, both BIJECTIONS onto correctly stated codomains:
framing : digit strings <-> U_n {n} x [0, b^n)
verified exhaustively base 8, n=0..4 (1,8,64,512,4096; no gaps)
DFA : canonical numerals <-> N+
verified base 8 to length 5 (32767 strings, values exactly [1,8^5))
Bijectivity is the operative property, not injectivity: injectivity says
encodings do not collide, bijectivity says decoding is TOTAL on the valid
codomain. Stating the codomain as N x N would make framing merely injective;
stating it correctly makes it bijective.
Distinctiveness assessed fairly: the content is classical numeration-system
material (regular numeral languages, canonical numeration, Cobham). The
distinctive move is making it an explicitly proved prerequisite of an encoding
pipeline, with the collisions exhibited as theorems in two proof assistants.
Also flags a maintenance hazard: the DFA and toDigits exist only in Radix.lean;
Radix.v has eval_digits and framed_value alone. Lean is regenerated from coq/*.v
in that repo, so regeneration would silently erase the automaton.
Corrects NOTATION_AND_BORROWED_TERMINOLOGY.md: the blanket 'no biology claim
anywhere' disclaimer was too strong and is now scoped to this submission, since
a Lean probe elsewhere deliberately models expanded genetic alphabets (4/8/12
letters) and proves an optimality statement about them.
Co-Authored-By: Claude Opus 5 <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_01WY6SfRYvm8zFKMX9GcjS8u
Documents the compact-form method: re-express opaque exact data in a
structurally-chosen fixed basis and read the coordinates, treating a successful
short exact encoding as a receipt of provenance rather than as compression.
Concrete instance, verified exactly: the challenge's eight large seed integers
decode in the Pascal basis to the Chudnovsky constants A=13591409,
B=545140134, S=426880, which in turn satisfy A = den(s2)-num(s2),
B = 6*den(s2), A/B = (1-s2)/6 for the CM invariant s2 = 77265280/90856689.
Chain: opaque integers -> Pascal coordinates -> Chudnovsky constants -> CM
invariant -> modular origin.
States the limits plainly:
- NOT a compressor. Measured 3.54x (588 -> 166 bits); unremarkable, and any
compression claim should be rejected. Consistent with the finding elsewhere
in this programme that char-poly encoding adds overhead vs an entropy-coded
baseline.
- Falsifiable, not numerology: basis fixed in advance, encoding exact with no
tolerance, and the recovered coordinates were pinned beforehand by an
unrelated classical formula.
- Distinctiveness assessed fairly: this is a disciplined exact-arithmetic
variant of established inverse-symbolic practice (PSLQ, ISC), unusual mainly
in targeting structured integer arrays and in carrying the encoding into the
proof. 'Unique' would overclaim.
Co-Authored-By: Claude Opus 5 <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_01WY6SfRYvm8zFKMX9GcjS8u
Several terms across this programme are borrowed from molecular biology and
genetics (hachimoji, scar/scar-map, strand/braid, carrier, chirality, mutation)
and could be misread as claims about biological systems. They are not. This
document states the intended mathematical meaning of each, names the source
field it collides with, and says explicitly what is not being claimed.
Part 1 covers terms actually used in the 2.8 submission (seed, jet, carrier,
mutation, deformation, compact form, certificate, red case, authority tag),
grounded in solution.tex and the certificate scripts. Part 2 covers the
biology-adjacent vocabulary of adjacent repositories and is RECONSTRUCTED FROM
WORKING NOTES -- the author should confirm each entry before circulation.
Also records the load-bearing EXACT / NUMERICAL WITNESS / LITERATURE
distinction: the main claim rests only on EXACT and LITERATURE items, and
numerical enclosures are never used as premises.
Co-Authored-By: Claude Opus 5 <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_01WY6SfRYvm8zFKMX9GcjS8u
Scored against the challenge's operative standard (section 4: a CAS-based
symbolic derivation is sufficient evidence) and its stated central risk
(section 1: retrieval vs reasoning), not a generic rigour bar.
Verdict: satisfies and exceeds the evidentiary standard. Findings:
F1 CRITICAL run_checks.sh exits 1 on any machine WITH SageMath installed;
set -euo pipefail makes the declared-optional Sage cross-checks
fatal, so the script passes without the optional tooling and
fails with it
F2 MEDIUM false assertion at p28_lattice_hypotheses_certificate.sage:107 --
component 1 is 216+154x as written, true value 216+262x
(b2=(1,2,1,0) contributes 216x, not 108x). Non-load-bearing:
the identity below it, Bcomb == 4x*Crow, is TRUE and Crow
reproduces the manuscript's C(x) exactly
F3 MEDIUM Sage 10.9 parent-coercion TypeError at line 43 masks F2
F4 MEDIUM novel-vs-imported content not stated plainly, though the
Chudnovsky import itself is cited precisely
F5 PASS proves the officially stated claim about the official object
F6 PASS evidence exceeds the section-4 standard, and is falsifiable
F7 PASS manuscript builds clean: 0 errors, 0 warnings, 17 pages
None of F1-F3 touches the mathematics; all are certificate plumbing.
Co-Authored-By: Claude Opus 5 <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_01WY6SfRYvm8zFKMX9GcjS8u
Reconstructs the Problem 2.8 data directly from the challenge statement
(R, u=2n+3, w, the 4x4 M(n), both integer seed rows) and verifies by exact
rational arithmetic that the manuscript's specialisation reproduces it:
M_N(x_0) == official M(N), all 16 entries, at N = 0,1,2,3,5,8,17,40
A_0 = A*C - (5/4)H_0 == official first seed row
A_1 = S*C == official second seed row
(14R-567)/9 == 236337691420383 (the deformed coefficient restores)
18 assertions, three with explicit negative controls. fractions.Fraction
throughout; no floating point, no CAS.
Closes the one gap no existing certificate covered: the other four verify
statements about the deformed family, none verified that the family is the
official object.
Co-Authored-By: Claude Opus 5 <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_01WY6SfRYvm8zFKMX9GcjS8u
