Papers I–II finite-domain audit

Successor search on the 15-bit bridge

This audit concerns finite exact facts about rationals that reproduce the 15-bit floor map of OMEGA_NAUGHT/1.0. Frozen 1.0 remains the canonical contract. No candidate here is an accepted successor.

The Python verifier checks specified exact finite calculations against the frozen 60-decimal enclosure of λ. It does not score serialization, verifier surface, execution cost, or proof cost, and it does not establish a universally superior successor.

That scalar interval does not automatically preserve network floors. On the canonical cube, 4196/12343 first disagrees at local integer 127813 (1738 versus 1737), and 22813643/2^26 first disagrees at 165245 (2246 versus 2247). The network interval is recorded on the 600-cell parameter page. Those disagreements do not relabel the scalar results on this page.

Certified λ enclosure

Conditional on the certified enclosure $\lambda_-\lt\lambda\lt\lambda_+$ , the successor-search verifier proves the stated finite floor results by exact integer arithmetic. It does not re-prove containment of $\lambda$ .

λ₋ = 339949771344778929576702809321279193551009905014259416689444 / 10^60
λ₊ = 339949771344778929576702809321279193551009905014259416689445 / 10^60

Contract: OMEGA_NAUGHT_PROOF/1.0, claim OMEGA.LAMBDA.CERTIFIED_ENCLOSURE
Proof method: Machin identity π = 16 arctan(1/5) − 4 arctan(1/239) with alternating-series remainder bounds; positive square-root enclosure of √5; then λ = φ/(φ+π).
Manifest: OMEGA_NAUGHT_PROOF_1.0_manifest.json650 bytes · SHA-256 56693dd0ea48bc0e4015e984a5c0bd63f82dfd537dda63e638cc7707365e8ec9
Proof verifier: omega_proof_verifier.py24500 bytes · SHA-256 591f62879d147c33ab1d9a68e94f2ac9ef5d446fd334a0c28c33b81b282ac72e
Proof certificate: OMEGA_NAUGHT_PROOF_1.0_certificate.json11040 bytes · SHA-256 5198caa5442fa95552fea0dfa26e175bcaf310b0fe5d7c60cab1667fc4dbf470

Four legitimate paths, not one winner

Exact computation confirms the 15-bit admissible interval, the minimum-denominator candidate $4196/12343$ , extinction within ten transitions, a signed-32 horizon optimum, and a two-bit predecessor coordinate. These identify several exact constrained optima. They do not establish one universally superior successor, because the candidates preserve different guarantee sets.

Globally map-equivalent normalization: $\frac{169974885672389}{5\times 10^{14}}$
Minimum-modulus finite bridge: $4196/12343$
Maximum-horizon signed-32 kernel: $49136/144539$
Baseline-horizon finite replacement: $846547/2490212$

The first preserves the original rational map globally. The other three optimize different finite-domain constraints. Residual, decode, and canonical-byte equivalence are not implied by frozen-domain quotient equivalence.

Exact 15-bit admissible interval

Let $a_n=\lfloor\lambda n\rfloor$ with $\lambda=\varphi/(\varphi+\pi)$ . Every rational reproducing the required floors on $D_{15}=\{0,\ldots,32767\}$ lies in

I_{15}=\bigcap_{1\le n<32768}\Bigl[\frac{a_n}{n},\frac{a_n+1}{n}\Bigr)=\Bigl[\frac{4196}{12343},\frac{7039}{20706}\Bigr).

Active lower constraints: $n=12343$ and $n=24686$ . Active upper constraint: $n=20706$ . The endpoints are Farey neighbors:

7039\cdot 12343-4196\cdot 20706=1,\qquad |I_{15}|=1/255574158.

Every rational strictly between the endpoints has denominator at least 33049. The included left endpoint is therefore the unique smallest-denominator member of $I_{15}$ :

q_{\min}=4196/12343,\qquad A(q_{\min})=45392.

At that first disagreement, $\lfloor 4196\cdot 45392/12343\rfloor=15430$ while $\lfloor\lambda\cdot 45392\rfloor=15431$ . Because later 15-bit states remain in $D_{15}$ , this rational generates the same quotient-state trajectories as frozen 1.0 for every 15-bit seed. It does not generate the same Euclidean remainders or canonical capsule bytes.

Partial exact Pareto table

All six profiles induce the same quotient-state trajectories on $D_{15}$ , hence the same extinction depth ten. Agreement horizon $A(q)$ is the least $n\ge 1$ with $\lfloor qn\rfloor\ne\lfloor\lambda n\rfloor$ .

Constrained optima on the 15-bit bridge. Different guarantee sets belong on different Pareto frontiers.
Profile	Q/S	Q,S bits	Max Qd on D₁₅	Kernel	A(Q/S)
Frozen 1.0	$339949771344778 / 10^{15}$	49, 50	`11139134157654340726`	unsigned / widened 64-bit	42189065
Globally normalized	$\frac{169974885672389}{5\times 10^{14}}$	48, 49	`5569567078827170363`	signed 64-bit	42189065
Minimum modulus	$4196 / 12343$	13, 14	`137490332`	signed 32-bit	45392
Signed-32 horizon	$49136 / 144539$	16, 18	`1610039312`	signed 32-bit	2490212
Baseline-horizon min	$846547 / 2490212$	20, 22	`27738805549`	signed 64-bit	42189065
First admissible dyadic	$22813643 / 2^{26}$	25, 27	`747534640181`	signed 64-bit shift/mask	33049

Signed-32 bound: any kernel product $Q\cdot 32767$ fitting signed 32-bit arithmetic requires $Q\le 65538$ . Agreement beyond $n=2490212$ would require membership in

\bigl[846547/2490212,\; 797411/2345673\bigr),

whose endpoints are Farey neighbors, so every such rational has numerator at least 846547. Therefore no signed-32 candidate has agreement horizon strictly larger than 2490212, and $49136/144539$ attains that bound.

Equivalence taxonomy

Every comparison must name which equivalence is claimed:

Global quotient equivalence for all $d\ge 0$ .
Frozen-domain quotient equivalence on $D_{15}$ only.
Trajectory equivalence of iterated quotient states for every 15-bit seed.
Residual equivalence, or a declared bijection of stored remainders.
Decode equivalence of reconstructed raw payloads.
Canonical-byte equivalence of serialized capsules.

The normalized baseline has global quotient equivalence and residual isomorphism only if $\rho/2$ is used internally while external 1.0 bytes stay unchanged. The finite candidates have frozen-domain trajectory equivalence, not global quotient, raw-residual, or canonical-byte equivalence.

Extinction and a two-bit inverse

Exhaustive fibers of the common 15-bit quotient map. All seeds reach zero by depth ten. Extremal seeds: $17697\le d_0\le 32767$ .

Exact fibers F^k(D_15). Branch bits are ceil(log2 of the largest fiber).
Depth k	States	Largest fiber	Worst-case branch bits
0	32768	1	0
1	11140	3	2
2	3787	9	4
3	1288	27	5
4	438	80	7
5	149	236	8
6	51	695	10
7	17	2045	11
8	6	6016	13
9	2	17697	15
10	1	32768	15

For a 23-transition ledger, residuals after extinction are identically zero. The Euclidean remainder is not the smallest local inverse symbol. On this map every predecessor set in $D_{15}$ has at most three members, so the predecessor rank $c=d-L(e)$ takes values in $\{0,1,2\}$ and needs two bits. Histogram: 10489 triples, 650 doubles, one singleton. An enumerative rank over all valid paths still needs fifteen bits and is information-equivalent to storing $d_0$ .

The ledger is reconstructible evidence, not compressible source information. A lossless fixed-length encoding of every L-byte payload still requires at least 8L bits. On the 15-bit domain the first residual is already injective; see the canonical terminology module.

Rational identities (no λ enclosure)

These identities are exact integer comparisons in the page. They do not by themselves prove $I_{15}$ or any agreement horizon.

Holds: 7039·12343 − 4196·20706 = 1
Holds: |I_15| = 1/255574158
Holds: 12343 + 20706 = 33049
Holds: gcd(Q_0, S_0) = 2
Holds: Q_0/S_0 reduces to 169974885672389/5·10^14
Holds: 49136 · 32767 = 1610039312 < 2^31
Holds: ⌊(2^31−1)/32767⌋ = 65538
Holds: 797411·2490212 − 846547·2345673 = 1
Holds: Q_0 · 32767 = 11139134157654340726
Holds: 4196 · 32767 = 137490332

All listed integer identities hold in this session.

How to rerun the calculation

Download verifier source (Python 3, standard library). It uses the frozen 60-decimal open enclosure of $\lambda$ from OMEGA_NAUGHT_PROOF/1.0. No third-party packages. The reference run took about 36 seconds.

python3 Omega_Naught_Successor_Search_Verifier.py

Verifier SHA-256 f1038a182f21fa89736835530c8dfc8bfa44828e2ac4ae9d5498a04c38103f2c · 9163 bytes.

Inner certificate digest 26d230fba560a5e08f2380207afc5f8b43cdea25ac67de3de1761ab56fc21e5e (mathematical lines; excludes wall-clock).

JSON certificate: successor_search_certificate.json · 4062 bytes · SHA-256 16404bbe69bc4d0c0c8a2c27d6e380d3108717df6e9983ac198ad68c85dee057

Reference transcript: successor_search_output.txt · SHA-256 e83337d4e1a41427bcbcdddcc749dd7939cc17edcc3422f9ee1579e9ad8707fe. The last line is wall-clock and may differ.

Implementation conformance: the named implementation reproduced the specified reference outputs. This does not establish theorem validity, peer-review status, or independent mathematical reproduction.

What this does not accept

No successor is accepted until it supplies a declared guarantee set, canonical serialization, exact byte round-trip, independent multi-language replay, adversarial tests, and a quantitative comparison with 1.0 — the written acceptance rule in the source audit. This page certifies the finite arithmetic. It does not ship a new capsule format.

Kernel, serialization, and provenance remain separate. A digest is integrity evidence, not an exact proof against every mutation, and not an authentication of a self-consistent replacement.

Frozen papers: Papers I–II, Paper III. Frozen implementation envelope: G6_RELEASE/1.1.

Floor-map fibers is a separate specification of consecutive fibers, depth layers, and a 32-block seed cover for the same rational map. It is not an accepted successor and does not authenticate reported 0.2 ledger files.