A registered result in its source context
Original ledger classification: Identity [I] (bijection + exact multiset gradings + symbolic binomial identity) + Physical [P] (residue re-type)
Recorded statement: SELF-COUNTING CHANNEL (the Pascal closure is an IDENTITY of the certified channel, not an extra requirement; closes the state half of the QBL analytic core at the structural level): (1) CANONICAL BIJECTION [I] - for odd g negation w -> -w…
Statement, assumptions and sources
Paper context: Architecture, E8 and the electromagnetic fixed point
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CAR.COUNT.01
Original ledger classification: Identity [I] (bijection + exact multiset gradings + symbolic binomial identity) + Physical [P] (residue re-type)
Original claim text
SELF-COUNTING CHANNEL (the Pascal closure is an IDENTITY of the certified channel, not an extra requirement; closes the state half of the QBL analytic core at the structural level): (1) CANONICAL BIJECTION [I] - for odd g negation w -> -w maps the S+ weights bijectively into the opposite sheet (the CAR.PAIR.01 parity stated as a bijection), so the Cartan-neutral kernels of the certified channel S+xS- are exactly the pairs (w,-w), ONE PER CODE STATE: # neutral certified kernels = dim S+ = 2^{g-1} exactly (4, 16, 64 for g = 3, 5, 7) - the channel counts the code. (2) PASCAL PARTITION [I]: the same neutral set graded by pair-degree m (zero modes of Lambda^{2m}(C^{2g})) has sizes C(g,m) for m = 0..K = (g-1)/2; g=5 -> (1,5,10). (3) THE CLOSURE IS AN IDENTITY [I]: two countings of one set give 2^{g-1} = sum_{m<=K} C(g,m), symbolically verified for all odd g = 3..13 - the closure CONDITION of CAR.LADDER.01 is the channel counting itself. (4) HODGE FOLD [I]: the code's minus-sign grading C(g,2j) equals the folded binomial C(g, min(2j, g-2j)) with min(2j,g-2j) <= K - every code sector appears as a pair-degree <= K sector (g=5: code (1,10,5) vs channel (1,5,10), same multiset, folded order). (5) LADDER GENERICITY: the identity holds for every odd g and carries NO g-selection - the selection stays with rank-8 (g + N_fam = 8, CAR.RANK/v14) and ladder integrality (CAR.LADDER.01). (6) RESIDUE RE-TYPE [P] (recorded, not claimed): QBL clause (b') loses its closure burden - given the sheet-odd kernel (CAR.PAIR.02a) the channel automatically contains exactly one neutral kernel per code state at pair-degree <= K, so (b') is the definition of the channel; the remaining physical input is 'the seam certifies through a single scalar 2-point kernel' (for the established free c=8 seam net Wick extends the single kernel to the entire even tower by determinants); the carrier selection itself is carried by rank-8 + integrality - QBL is the structural consistency frame, no longer a missing selector
Original classification detail
Identity: Pascal closure = two countings of the channel's neutral set; one kernel per code state; QBL residue = single 2-point-kernel input
Dependencies as recorded
CAR.PAIR.01; CAR.PAIR.02; CAR.QTRANS.01; CAR.LADDER.01; CAR.RANK; ARCH.QUAD.01 (QBL)
Open the original ledger source, row 196Location recorded in the ledger
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- Architecture, E8 and the electromagnetic fixed point
Navigation through a shared verification script
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- How to read this documenttfpt 1 architecture e8
- Honest statusinput 4 origin theory
- Front matter / supportE8 audit and bootstrap
- Honest statusorigin theory
- The hard reduction: the live residual as typed interfacesTFPT — Introduction and reading guide
- Group-theoretic audits (optional)input 5 tfpt 3 e8 audit bootstrap
- How to read this documenttfpt 1 architecture e8
- Honest statusOrigin Theory
Registered verification modules
- v112_selfcounting_channel.pyRegistered verification module · registry
Read the description recorded in the module registry
SELF-COUNTING CHANNEL (the Pascal closure is an IDENTITY of the certified channel, not an extra requirement): CANONICAL BIJECTION — for odd g negation w ↦ −w maps S⁺ bijectively into the opposite sheet, so the Cartan-neutral kernels of the certified channel are exactly the pairs (w,−w), ONE PER CODE STATE: #neutral kernels = dim S⁺ = 2^{g−1} exactly (4, 16, 64 for g = 3,5,7) — the channel counts the code; PASCAL PARTITION: the same neutral set graded by pair-degree m has sizes C(g,m) — g=5: (1,5,10); THE CLOSURE IS AN IDENTITY: two countings of one set give 2^{g−1} = Σ_{m≤K} C(g,m) (symbolic, all odd g=3..13) — v108's closure condition is the channel counting itself; HODGE FOLD: code minus-grading C(g,2j) = C(g,min(2j,g−2j)) with min ≤ K — every code sector appears at pair-degree ≤ K; carries NO g-selection (stays with rank-8 g+N_fam=8 + integrality); RESIDUE RE-TYPE [P]: QBL clause (b') is now the channel's definition — the remaining physical input is 'the seam certifies through a single scalar 2-point kernel' (free c=8 seam net: Wick extends it to the whole even tower); QBL = structural consistency frame, no longer a missing selector
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