Format obstructions (no new κ): exact bounds on window width, copy cost and label dimension

[DreamingOfClouds]

Format obstructions (no new κ): exact bounds on window width, copy cost and label dimension, checked on gx.check1 programs and on #425's mutable-basis word. This is a research package of statements, proofs and
exact checks. It claims no new κ. It fixes what three formats can and cannot do and shows where the remaining
freedom is.

	statement	checked on
A	a contribution that reaches a register by a drop-free path forces its frame to contain the source label, so lossless x_T → y_S needs orthogonal labels	every gate of the p = 3, p = 10 (#327) and p = 11 (#193) programs: 0 violations in 3.1 M gate checks
B	k_a + Σ_j rk C_j ≤ m; with corrector rank h − 1, m ≥ 1 + s(h − 1)	rank h − 1 on all families (replayed correctors for the complex programs and #425; in-cap neighbourhoods for the bit and E8 rows); #425's word has m = 73 = 1 + 9·8, so the bound is attained
C	the drops that carry a total to its readers cost at least rk(U(P) × U(Q))	equality: 360 (p = 10), 440 (p = 11), 72 (#425 centres, 57 gathered sources each); for the E8 bit dual the package computes the bound 72, which equals the copy loss #370 reports (not replayed)
clique bound	Σ_K k_T ≤ r on a clique of drop-free pairs	ω = p (bit), 8, 8, 12, 12 (complex, p = 4..7); 20 ports pairwise lossless in the p = 10 program; ω = 8 on E8
E8 labels	the h = 9 labelling of the E8 in-cap graph is unique; no k = 2 labelling in dimension ≤ 17	exact SAT, with calibrations and a positive control
P	for fixed-role B₂ windows over F₂ (one helper set, same slots, slot-carried pairs spanning) the stage windows are orthogonal, m ≥ s·h, under (P1)–(P2)	computed on the complex #327 and #193 programs only (399/400, 483/484). Compressed windows circumvent it: #425 (m = 73 < 81) and, for the bit word, #386 (m = 64; Theorem P is not claimed for ℚ forms). Not a general obstruction
S	a mutable-basis word with b_x modified sources and b_y read targets is simulated in the immutable-source / no-target-read format with R + b_x + b_y helpers	exact on #425's word: 904 = 680 + 112 + 112

What this means for the swarm. A–C and the clique bound bound m and the copy rank, the label dimension for a given
in-cap graph, and (through the five-stage ledger D ≤ N_data·k, read from upstream) D. They do not bound the
helper count R, nor the idle and stock costs, which is where #425, #433 and #442 gained. A further κ gain in these
formats has to come from R or the stock, or from a new port family whose corrector rank is below h − 1 with a cheap
decoder.

python -m pip install -r research/format-obstructions/requirements.txt    # python-sat 1.9.dev15, networkx 3.7, mpmath 1.3.0, numpy 2.3.5
python3 -B research/format-obstructions/verify.py --output /tmp/format-obstructions-out   # about 90 s on 12 cores (15 CPU-minutes), no network

What changed

One new self-contained package, research/format-obstructions/, plus its path-filtered workflow. No existing file
changes. Vendored byte for byte (sha256 in SOURCE.json): Jacob Sussman's gx.py, the p = 11 certificate of #193, the
p = 10 program of #327, #425's FULL-WORD.json.gz (regenerated with its own scripts; the sha equals its
EXPECTED.json), the cost model of #346 (from #310) and the p = 10 bit histogram of #325. The one local input is the
p = 3 toy program.

What verify.py checks

It verifies the pins, then runs 91 tasks in a process pool and compares every number with expected.json:

• gx.check1 with the exact scalar identity on the three programs, the taint replay of Theorem A at every gate, the
  corrector ranks of Theorem B and the copy bounds of Theorem C;
• the p = 3 toy: split scatter, cst 18 → 16, exhaustive optimum over all cut chains;
• Theorem P (rank of the cross-Gram constraints), the forced-copy count (10 per overlap, exact over all odd a, b) and
  the contamination scan (1,160 candidate directions);
• κ = 1.048005508028206e-3: mutable-basis Hamming packets (≈2^-9.89814; conditional; complex +4.97%) #425's word: frame and taint replay, exact scalar replay over ℚ, the Theorem S conversion and its replay;
• the pinned pricer on the p = 10 bit histogram (pinned root reproduced; context only);
• the E8 SAT checks: uniqueness at h = 9, k = 2 at r = 16 (255 patterns) and r = 17 (256), all UNSAT.

Negative controls: shifted labels and a corrupted port; a toy cut outside the radical; the immutable/no-target-read
detector flags 1,640 source modifications and 112 target reads on #425's word and 0 on the converted word; omitted K
and omitted compensation rejected; k = 1 at r = 8 UNSAT and r = 9 SAT, h = 8 UNSAT, the k = 2 encoder at r = 18 accepts
the direct sum.

Verification

• Replays: two fresh-copy runs in other working directories (macOS, Python 3.12.14, python-sat 1.9.dev15,
  networkx 3.7, mpmath 1.3.0, numpy 2.3.5) pass every check; CI has not been run yet.
• Independent audit: Independent read-only audit (separate Claude agent): two blockers found and fixed; re-verified.

Status, honestly

• κ: no new κ.
• Scope: Theorems A–C and the clique bound are proved for the frame model of PROOF.md §1, which contains the rules E0–E3
  of gx.check1 and κ = 1.048005508028206e-3: mutable-basis Hamming packets (≈2^-9.89814; conditional; complex +4.97%) #425's common-label rule; the correspondence is by reading, with no mechanical proof and no Lean.
• Layout hypotheses: (B1)–(B2) of Theorem B and (P1)–(P2) of Theorem P were read against Sussman's
  Work/Bridge/Net.lean, Work/BridgeGeom/Stage.lean and Work/BridgeGeom/Net.lean (f010392c); they are consistent
  with the statements there, but this was read, not formally verified. Two points are open: whether the backward stages
  1 and 3 count as "the same program", and whether the stage-j target frames lie in ι_j(q_S)^⊥. Theorem P holds for
  fixed-role B₂ windows over F₂ only; κ = 1.048005508028206e-3: mutable-basis Hamming packets (≈2^-9.89814; conditional; complex +4.97%) #425's nine-stage windows (m = 73) are outside it.
• Bit side: no bit word is replayed. Theorem P (stated for F₂ windows) and the overlap measurements were computed on
  complex programs, and nothing is claimed about Theorem P for ℚ forms such as Compressed nine-stage E8 windows and whole-block banking for a conditional κ = 9.98243154231047e-4 > 2^-10 bound #386's; the pricing context uses the
  p = 10 bit histogram (the file copied from the math.json that Conditional κ = 7.696840393833340e-4: a stronger p = 10 bit word — re-annealed pair module, p = 10 local design, balanced cube orientation, tiered reuse matching (+0.71% over #315's word) #325's verification writes) and is a pricer-model
  output.
• Theorem S: a bound on helper count, not on price (the simulation prices 13 % below κ = 1.048005508028206e-3: mutable-basis Hamming packets (≈2^-9.89814; conditional; complex +4.97%) #425's word under the coarse
  formula). Replayed by the package's detector on κ = 1.048005508028206e-3: mutable-basis Hamming packets (≈2^-9.89814; conditional; complex +4.97%) #425's word, not by gx.check1.
• E8 labels: the SAT results are for the full E8 in-cap graph only.

Attribution

• Jacob Sussman: gcert/1, gx.check1, the frame lemma, the five-stage layout and the E8 complex unit.
• jacklightChen (Zhihao Chen): κ = 1.048005508028206e-3: mutable-basis Hamming packets (≈2^-9.89814; conditional; complex +4.97%) #425's word and its conjugation semantics; Compressed nine-stage E8 windows and whole-block banking for a conditional κ = 9.98243154231047e-4 > 2^-10 bound #386.
• Dylan Crow (utcorvusvolat-dotcom): κ = 8.75481145306751e-4: first E8 bit word (complex-bound; +9.29% over #352) #370 (E8 bit word, root coordinates) and the Conditional κ = 0.000752893499605334868002344: source-bound w3 word and complete finite admission (+0.1855% over #309) #310 cost model.
• Louis Harrison: κ = 7.92580093369946e-4: w3 cube layer on #325's p10b word with a stronger p = 10 complex supplier #346 (the vendored pricer).
• Avi Eisenberg (ikeboy): Pair donors to avoid parity erasure on the source-assisted v4 word: conditional κ = 6.647872e-4 #193. icekylinx: Paired cubes and shared completed cores for conditional kappa = 4.609169e-4 #144, Source-assisted common-frame compression for conditional kappa = 6.566436e-4 #184 and Split centres and source-basis reuse for conditional κ = 0.001120864952258278 #433. chafreaky: A stronger bit supplier: fixed-coordinate and face-diagonal bit local circuit on PR168 v4 (effective bit saving 6.7734e-4) #200, A stronger complex supplier: PR200's reuse pairing on the source-assisted v4 word, complex saving 7.0991·10⁻⁴ (+1.28% over #194; no new κ, bit binds) #233, The source-assisted complex word of #233 as one explicit numbered program (gcert/1): reference-checked, 7.547361·10⁻⁴ in the Lean-checked five-stage layout (no new κ) #256.
• eumemic: Annealed modules, terminal sinks and physical bit words on paired cubes: conditional κ = 6.558894e-4 #168. DaysSky: Extended carrier matching on #161's complex word: conditional κ = 5.901881e-4 #162, κ = 8.01083628465007e-4: #346's w3 bit word with Sussman's Lean-checked E8 unit as the complex supplier (+0.96 % over #350) #352 and Conditional κ ≈ 1.2375517e-3: idle-axis hosting of helper residuals (+9.2% over #434) #442. hcg890 and SovereignSteak: Conditional κ ≈ 7.037017435 × 10⁻⁴ > 2⁻¹¹: five-stage bit and complex suppliers #234 and the five-stage ledger.

The statements, proofs, checks and packaging were prepared by DreamingOfClouds with Claude Opus 5.5 (Anthropic), via
Claude Code. Every upstream notice is retained; see NOTICE.

🤖 Generated with Claude Code

[Mister-Meeseeks]

Thank you @DreamingOfClouds. This contribution is now retained on main in the scoped 100-PR review and contributor credits.

Contribution: Format-specific copy, dimension, label and window obstructions.

Review and limits: Theorem P is restricted to the stated fixed-role characteristic-two windows; compressed windows and general rational forms are outside it. SAT and finite checks are not a universal network barrier.

The complete original source, notices and relevant execution evidence are retained through the pinned source archive. The review distinguishes fresh local runs, source-bound historical Linux CI and prior exact-head review; it does not claim 100 fresh full mathematical audits. Closing this historical PR as reconciled. This preserves parallel and cumulative credit without replacing the selected frontier or promoting priced research targets to completed constructions.

Reviewed head: 309482849b6ddf34db998f53612d34c19f6e3233. Publication: 64039f1bd3b6a523b96083fd858ad040a75e3030.